IN MEMORIAM
FLORIAN CAJOR1
MxL/L^J
PLANE AND SPHERICAL
TRIGONOMETRY
BY
LEVI L. CONANT, Pii.D.
?
PROFESSOR OF MATHEMATICS IN THE WORCESTER
POLYTECHNIC INSTITUTE
NEW YORK :. CINCINNATI : CHICAGO
AMERICAN BOOK COMPANY
COPYRIGHT, 1909, BY
LEVI L. CONANT.
ENTERED AT STATIONERS' HALL, LONDON.
CAJOR1
PREFACE
IN this work the author has attempted to produce a text-
book which should present in a concise and yet thorough
manner an adequate treatment of both the theoretical and the
practical sides of elementary trigonometry. The material here
presented has been gathered and tested during the course of
many years of experience in the class room, and the arrange-
ment and method of presentation are the result of numerous
experiments made for the purpose of ascertaining what could
be done most effectively in the limited time usually devoted
to this subject.
The problems given in connection with the different cases
under the solution of triangles are nearly all new, and are well
graded and sufficiently numerous to give the student ample
preparation for the various problems that arise in plane sur-
veying and in elementary astronomical and geodetic work.
That portion of the book which treats of theoretical trigo-
nometry has been written in the attempt to present this aspect
of the subject in the simplest and clearest manner, and at the
same time with the design of equipping the student for the
more advanced work in pure and applied mathematics which
is pursued in the later years of his college course.
The best English, French, and Italian text-books have been
consulted, as well as those published in this country. For
assistance in the preparation of the work thanks are due to my
colleague, Professor Arthur D. Butterfield, to Professor W. B.
Fite of Cornell University, to Professor O. S. Stetson of Syra-
cuse University, to Mr. C. G. Brown, head of the department
of mathematics in the Englewood, New Jersey, High School,
and to Mr. J. A. Bollard, instructor in mathematics in the
Worcester Polytechnic Institute.
LEVI L. CONANT.
WORCESTER POLYTECHNIC INSTITUTE,
WORCESTER, MASS.
918256
ENGINEER'S TRANSIT, WITH GRADIENTEK
4
CONTENTS
PLANE TRIGONOMETRY
CHAPTER PAGES
I. THE MEASUREMENT OF ANGULAR MAGNITUDE . . 7-19
II. TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE . 20-30
III. VALUES OF THE FUNCTIONS OF CERTAIN USEFUL ANGLES 31-35
IV. THE RIGHT TRIANGLE 36-50
V. THE APPLICATION OF ALGEBRAIC SIGNS TO TRIGO-
NOMETRY . . . . . . . . . 51-73
VI. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE . . 74-84
VII. GENERAL EXPRESSION FOR ALL ANGLES HAVING A
GIVEN TRIGONOMETRIC FUNCTION .... 85-91
VIII. RELATIONS BETWEEN THE TRIGONOMETRIC FUNCTIONS
OF Two OR MORE ANGLES ...... 92-105
IX. FUNCTIONS OF MULTIPLE AND SUBMULTIPLE ANGLES . 106-113
X. INVERSE TRIGONOMETRIC FUNCTIONS .... 114-121
XI. THE GENERAL SOLUTION OF TRIGONOMETRIC EQUATIONS 122-130
XII. THE OBLIQUE TRIANGLE 131-155
XIII. MISCELLANEOUS PROBLEMS IN HEIGHTS AND DISTANCES 156-165
XIV. FUNCTIONS OF VERY SMALL ANGLES HYPERBOLIC
FUNCTIONS TRIGONOMETRIC ELIMINATION . . 166-175
SPHERICAL TRIGONOMETRY
XV. GENERAL THEOREMS AND FORMULAS .... 177-193
XVI. SOLUTION OF SPHERICAL TRIANGLES 194-213
GREEK ALPHABET
Greek is written with the following twenty-four letters :
FORM
NAME
LATIN
EQUIVALENT
A
a
alpha
a
B
13
beta
b
r
7
gamma
g
A
8
delta
d
E
epsilon
e
Z
r
zeta
z
H
g
eta
e
@
></
theta
th
I
t
iota
i
K
/c
kappa
c, k
A
\
lambda
1
M
p
mu
m
N
ZV
nu
n
5
f
xi
X
O
o micron
6
n
7T
pi
P
p
P
rho
r
T
CT ?
T
sijma
tau
s
\
t
T
V
upsilon
y
4>
<#>
phi
P h
X
%
chi
ch
i/r
psi
ps
n
CO
omega
o
PLANE TRIGONOMETRY
CHAPTER I
THE MEASUREMENT OF ANGULAR MAGNITUDE
1. The size and shape of a plane triangle can be completely
determined when any three of its six parts are known, provided
at least one of the known parts is a side.
By means of certain ratios called trigonometric functions,
which will be defined later, trigonometry enables us to investi-
gate and to determine the unknown parts and the area of a tri-
angle when any three of the parts are known, provided at least
one of the known parts is a side. Hence, in its most elemen-
tary sense,
Trigonometry is that branch of mathematics which treats of
the solution of triangles. During the past two centuries the
sense in which the word " trigonometry " is used has been greatly
extended, and it is now understood to include the general sub-
ject of mathematical investigation by means of trigonometric
functions.
Plane trigonometry treats of plane triangles, and of plane
angles and their functions.
2. Angles. In its geometric sense the word " angle " is
defined as the difference in direction of two intersecting lines.
In trigonometry, however, this word receives an extension of
meaning, which must be fully understood at the outset.
Suppose two straight lines, OA and OB, are drawn from the
point in such a manner that they very nearly coincide. Let
one of the lines, OA, remain fixed in position, while the other,
OB, revolves on the point as a pivot. We are now free to
revolve OB, either back into actual coincidence with OA, or
7
8 PLANE TRIGONOMETRY
forward, so as to enlarge the opening between the lines. At
any point of the revolution the angle AOB may be said to
have been formed, or generated, by the revolution of the
line OB.
In plane geometry angles greater than 180 are seldom em-
ployed, but in trigonometry the freest possible use is made
of such angles. Trigonometry even considers angles greater
than 360, meaning by an angle of that magnitude merely the
amount of revolution that has been performed by the moving
or generating line.
As an illustration of the meaning of the word " angle " used
in this sense, consider the movement of one of the hands of the
clock. Let the minute hand start from the position it occupies
at noon. In fifteen minutes it will move over or generate an
angle of 90 ; in thirty minutes an angle of 180 ; in forty-five
minutes an angle of 270 ; and in one hour an angle of 360.
Continuing, we may say that in two hours the minute hand
will move over an angle of 720, in three hours an angle of
1080, in four hours an angle of 1440, in n hours an angle of
rax 360, etc.
Again, suppose a runner to be competing in a two-mile race
on a circular track a quarter of a mile in length. If we sup-
pose a line to be drawn connecting the position of the runner
with the center of the circle formed by the track, the position of
the runner both on the track and in the race can be described
at any instant with perfect accuracy by giving the magnitude
of the angle through which this line has revolved since the
beginning of the race.
Thus, when the line has revolved through an angle, and hence
the runner has traversed an arc, of 180, he has completed one
eighth of a mile ; when he has traversed an arc of 360, he has
THE MEASUREMENT OF ANGULAR MAGNITUDE !>
completed one fourth of a mile ; and when he has finished
the race, he has run around the track eight times. In other
words, when he has finished the race the line that connects him
with the center of the track has revolved through an angle of
8 x 360, or 2880. During this time the runner has traversed
an arc of the same magnitude, i.e. of 2880.
It is at once seen that an idea is here introduced which is an
extension of the idea of the angle as it is ordinarily used in
geometry. This idea, which is fundamental in all work in
trigonometry involving angles, is the idea of formation or
generation in connection with the angle. Evidently a defini-
tion of this word is required which differs from that to which
the student has become accustomed in geometry ; and in the
extended sense here used, the term "angle " may be defined as
follows :
An angle is that relation of two lines which is measured by
the amount of revolution necessary to make one coincide with
the other.
3. The point about which the generating line revolves is
called the origin. The generating line is called the radius
vector. The line with which the radius vector coincides when
in its original position is called the initial line ; and the line
with which it coincides when in its final position is called the
terminal line.
4. Positive and negative angles. It is convenient, and often
necessary, to know not only the size of an angle, but also the
direction in which the radius vector has moved while generating
the angle. For this reason it is customary to speak of angles
as being either positive or negative.
If the radius vector moves in a direction opposite to that of
the hands of a watch when the face of the watch is toward the
observer, the angle it generates is said to be positive. The
motion of the radius vector as it generates the angle is then
said to be counter- clockivise.
If the radius vector moves in the same direction as the hands
of a watch when the face of the watch is toward the observer,
the angle it generates is said to be negative. The motion of
the radius vector is then called clockwise.
10
PLANE TRIGONOMETRY
The angles AOB 1 and AOB 2 are positive angles, and the
angles AOB 3 and AOB 4 are negative angles. The initial line
in each case is OA, and the terminal lines are OB V OB^ OB S ,
OB^ respectively. The direction of rotation for each angle
is indicated by the arrowhead. .
5. Angles are often described by referring them to some
position with reference to two intersecting lines, at right angles
to each other, of which one is horizontal and the other vertical.
It is customary to regard the horizontal line extending toward
the right as the initial line for all angles, when nothing is said
to the contrary.
If the radius vector, as shown in the figure, occupies any po-
sition between OX and OY, then the angle XOB l is said to be
in the first quadrant. If the
radius vector is between OY
and OX', the angle XOB 2 is
said to be in the second quad-
rant. Similarly, XOB 3 is said
to be in the third quadrant, and
XOB in the fourth quadrant.
These expressions only mean,
of course, that the terminal lines
lie in the first, second, third, and
fourth quadrants respectively.
6. In practical work the unit of measure that is always em-
ployed in dealing with angular magnitudes is the right angle
or some fraction of the right angle. This unit is chosen because:
(i) The right angle is a constant angle.
(ii) It is easy to draw or to construct in a practical manner,
(iii) It is the most familiar of all angles, entering as it does most
frequently into the practical uses of life.
THE MEASUREMENT OF ANGULAR MAGNITUDE 11
In geometry the right angle is the unit universally used.
In trigonometry two systems of measurement, involving the
use of two different units, are in common use.
7. The sexagesimal system. In this system the unit of
measure is the right angle. The right angle is divided into 90
equal parts, called degrees; each degree is divided into 60 equal
parts, called minutes; and each minute is divided into 60 equal
parts, called seconds. The symbols 1, 1', 1", are employed to
denote one degree, one minute, and one second respectively.
60 seconds (60") = one minute.
60 minutes (60') = one degree.
90 degrees (90) = one right angle.
This system is almost universally employed where numerical
measurements are to be made. It is, however, inconvenient
because of the multipliers, 60 and 90, which it introduces into
computations.
Another system, called the centesimal system, was proposed
in France a little over a century ago. In this system the
right angle is divided into 100 equal parts called grades, the
grade is divided into 100 equal parts called minutes, and
the minute is divided into 100 equal parts called seconds.
The centesimal system has been used to some extent in France,
but its use has never been looked upon with favor in other
countries. If its use were to become general, an enormous
amount of labor would have to be expended in the re-computa-
tion of existing tables. For this reason the centesimal system,
in spite of its intrinsic advantage over the sexagesimal system,
will probably never come into general use.
EXERCISE I
Express the following angles in terms of a right angle :
1. 30. 3. 68 14'. 5. 228 46'.
2. 120. 4. 114 38' 12". 6. 321 14' 22".
7. The angles of a right triangle are in arithmetical progres-
sion, and the greatest angle is three times the least ; what
is the number of degrees in each angle ?
12
PLANE TRIGONOMETRY
Show by a figure the position of the revolving line when it
has generated each of the following angles :
8. | rt. angle. 11. 2^ rt. angles. 14. -150.
9. - 1 rt. angle. 12. 4| rt. angles. 15. 275.
10. -1| rt. angles. 13. 17| rt. angles. 16. 1225.
17. The angles of a triangle are such that the first contains
a certain number of degrees, the second 10 times as many min-
utes, and the third 120 times as many seconds ; find each
angle.
18. How many degrees are passed over by each of the hands
of a watch in one hour ?
Represent by a figure each of the following angular magni-
tudes :
19. l|-+2^ rt. angles. 23. 4 rt. angles.
20. 2| 1^ rt. angles. 24. 4n rt. angles (n integral).
21. - 4 rt. angles. 25. (4 n -f 1) rt. angles.
22. 6^ rt. angles. 26. (4 n 2) rt. angles.
8. Circular measure. Another system for the measurement
of angles has, in modern times, come into vogue. It is exten-
sively used in work connected with
higher branches of mathematics, and
is the almost universal unit employed
in theoretical investigations.
The unit of circular measure is the
radian, which is obtained as follows :
On the circumference of a circle lay
off an arc, AB, equal in length to the
radius of the circle, OA. The angle
AOB is called a radian. Accordingly:
A radian is an angle at the center of a Circle, subtended by an
arc equal in length to the radius of the circle.
In order to use the radian as a unit of measure, it is necessary
to prove that it is a constant angle ; or, in other words, it is
necessary to prove that the magnitude of the radian is the same
for all circles.
THE MEASUREMENT OF ANGULAR MAGNITUDE 13
9. THEOREM. The radian is a constant angle.
By definition the radian is measured by an arc equal in length
to the radius. Also,
An angle of two right angles is measured by an arc equal to
one half the circumference.
Therefore, since angles at the center of a circle are to each
other as the arcs by which they are subtended (Geom.),
a radian radius R 1
= =
2 rt. angles semi-circumference irR TT
.-. a radian = ~ of 2 right angles = 1 x 180 = 57.2958
= 57 17' 44. 8" nearly.
Therefore the radian is a constant angle.
10. The reason for the use of this unit may now be readily
understood.
Since 1 radian = ?_L^,
7T
.. TT radians = 2 rt, A = 180.
Similarly, - radians = 1 rt. Z = 90.
- radians = | rt. Z = 30.
6
radians = 60.
o
1 TT radians = 120.
f TT radians = 270.
2 TT radians = 4 rt. A = 360.
5 TT radians = 10 rt. A = 900.
18 TT radians = 36 rt. A = 3240.
This gives a method for the expression of the value of an
angle that is often far more convenient than that furnished
by the sexagesimal system. It is especially useful in dealing
with angles of great magnitude, and it greatly simplifies many
of the investigations and formulas of trigonometry.
14 FLAKE TRIGONOMETRY
11. The symbol r is often used as the symbol to denote
radians. Thus, 6 r would stand for 6 radians, O r for 6 radians,
7r r for TT radians, etc.
When the value of the angle is expressed in terms of TT, and
when the unit is the radian, it is customary to omit the r and to
give the value of the angle in terms of TT alone, the r being
understood. Thus, when referring to angular magnitude, TT
means TT radians, ~ means radians, 6 TT means 6 TT radians,
2
etc. When the word "radians" is omitted, the student should
mentally supply it, or he may readily fall into the error of sup-
posing that TT alone means 180. The value of TT is the same
here as in geometry, i.e. 3.14159. Neither TT nor any multiple
of TT can by itself ever denote an angle. It simply tells how
many radians the angle contains. Too great care cannot be
exercised in keeping this distinction clear.
12. To find the number of degrees in an angle containing a
given number of radians, and vice versa.
Since 180 = IT radians,
1 = of a radian,
180
180 , ,
and l r = ol a decree.
Hence,
To convert radians into degrees, multiply the number of radians
i 180
% '
To convert degrees into radians, multiply the number of degrees
by -E-. '
y 180
EXERCISE II
1. How many degrees are there in 3 radians ?
= 3 x = = m.89 nearly
7T 7T
= 171 53' 24" nearly.
THE MEASUREMENT OF ANGULAR MAGNITUDE 15
2. How many radians are there in 113 15' ?
113 15' = 113.25.
Since 1 = ^,
180
113.25 = 11 3.25 x -^
180
_ 113.25 x 3.14159
180
= 1.976 + radians.
Express in degrees, minutes, and seconds the following
angles:
5. ^- 7. ^- 9. 3?r r .
Express in radians the following angles :
11. 45. 14. 225. 17. 286 38'. 20. A.
Q0
12. 120. 15. 60 30'. 18. 684 26'. 21. .
7T
13. 135. 16. 115 45'. 19. n. 22. 78.126.
23. The difference between two acute angles of a right tri-
angle is ^ radians; find the value of each of the angles in degrees.
5
24. If one of the angles of a triangle is 56 and a second
angle is ^-^ radians, find the value of the third angle.
5 ,
25. The angles of a triangle are in A. P., and the smallest
is an angle of 36 ; find the value of each in radians.
26. The value of the angles of a triangle are in A. P., and
the number of degrees in the least is to the number of radians
in the greatest as 60 : TT ; find each angle in degrees.
27. The value of one of the interior angles of one regular
polygon is to the value of one of the interior angles of another
regular polygon as 3 : 4, and the number of sides in the first is
to the number of sides in the second as 2 . 3 ; find the number of
sides in each.
16
PLANE TRIGONOMETRY
28. Find the number of radians in one of the interior angles
of a regular pentagon ; a regular heptagon ; a regular nonagon.
29. The angles of a triangle are in A. P., and the number
of radians in the least angle is to the number of degrees in
the mean angle as 1:120; find the value of each angle in
radians.
30. The angles of a quadrilateral are in A. P., and the
greatest is double the least; find the value of each angle in
radians.
31. Express in degrees and in radians the angle between the
hour hand and the minute hand of a clock at (1) five o'clock ;
(2) quarter-past nine ; (3) half -past ten.
32. At what time between four and five o'clock are the hour
and the minute hands of a clock 90 apart ? At what time are
they 180 apart ?
13. THEOREM. The circular measure of an angle whose vertex
is at the center of a circle is the ratio of its intercepted arc to the
radius of the circle.
By geometry,
arc AC
Z.AOB arc A B a radius'
arc AC ' X/ _ AOB
radius
arc A
: x a radian.
radius
Hence, the number of radians in any angle is found by dividing
the arc which subtends that angle by the radius of the circle.
The formula just obtained is often expressed in the following
convenient, though somewhat incorrect, form :
arc = angle x radius. (1)
The meaning of this formula is, that the length of any arc of a
circle is equal to the length of the radius of the circle multiplied
by the number of radians in the angle subtended by the arc.
THE MEASUREMENT OF ANGULAR MAGNITUDE 17
EXERCISE III
1. Find in degrees the angle subtended at the center of a
circle whose radius is 30 ft. by an arc whose length is 46 ft. 6 in.
In this circle the arc which subtends an angle equal to a radian is 30 ft.
in length, and the required angle is subtended by an arc whose length is
46.5 ft.
,. ^radians = *?? x i* = 88.8'. Ans.
30 300 TT
2. In a circle whose radius is 8 ft., what is the length of
the arc subtended by an angle at the center, of 26 38' ?
Let x = the length of the required arc.
ber of radians in 26
. (See Art. 13.)
Then, - = the number of radians in 26 38'
8
x = 3.72 ft. nearly.
3. In running at a uniform speed on a circular track, a man
traverses in one minute an arc which subtends at the center of
the track an angle of 3| radians. If each lap is 880 yd., how
long does it take him to run a mile ?
Let x the number of yards traversed during each minute.
Then, x = 3^ x R. (See Art. 13.)
99
= x 140 = 440 yards.
Since Y& = 4,
therefore he can run a mile in 4 min.
4. The radius of a circle is 15 ft. ; find the number of radians
in an angle at the center subtended by an arc of 26^ ft.
5. The radius of a circle is 32 ft.; find the number of
degrees in a central angle subtended by an arc of 5 TT ft.
6. The fly wheel of an engine makes 3 revolutions per
second ; how long will it take it to turn through 5 radians ?
7. The minute hand of a tower clock is 2 ft. 4 in. long ;
through how many inches does its extremity move in half an
hour ?
CONANT'S TRIG. 2
18 PLANE TKIGONOMETRY
8. A horse is picketed to a stake ; how long must the rope
be to enable the horse to graze over an arc of 104.72 yd., the
angular measurement of this distance being 150 ?
9. What is the difference between the latitude of two places,
one of which is 150 mi. north of the other, the radius of the
earth being reckoned as 4000 mi. ?
10. The angle subtended by the sun's diameter at the eye of
an observer is 32' ; find approximately the diameter of the sun,
if its distance from the observer is 92,500,000 mi.
NOTE. In this example the diameter of the sun, which is really the chord
of an arc of which the observer's eye is the center, may be regarded as
coinciding with the arc which it subtends.
11. Calling the earth a sphere, and the arc of a great circle
on its surface subtended by an angle of 1 at the center 69 J mi.,
what is the radius of the earth ?
12. A railway train is traveling at the rate of 60 mi. an
hour on a circular arc of two thirds of a mile radius ; through
what angle does it turn in 10 sec. ?
13. The radius of a circle is 3 m.; find approximately, in
radians, the arc subtended by a chord whose length 'is also 3 m.
14. How many seconds are there in an angle at the center
of a circle subtended by an arc one mile in length, the radius
of the circle being 4000 mi. ?
15. In the circle of Ex. 14, what is the length of an arc that
subtends an angle of 3' at the center?
16. What is the ratio of the radii of two circles, if the semi-
circumference of the greater is equal in length to an arc of the
smaller which subtends an angle of 225 at the center?
17. If an arc 1.309 m. long subtends at the center of a
circle whose radius is 10 m. an angle of 7 30', what is the ratio
of the circumference of a circle to its diameter ?
18. The circumference of a circle is divided into four parts
which are in A. P., and the greatest part is twice the least ; find
the number of radians in the central angle subtended by each
of the respective arcs into which the circumference is divided.
THE MEASUREMENT OF ANGULAR MAGNITUDE 19
19. The diameter of a circle is 80 m., and an arc whose
length is 15.75 m. subtends a central angle of 22 30'; find the
value of TT to four decimal places.
20. How many radians are there in a central angle subtended
by an arc of 20" ?
21. The semicircumference of a certain circle is equal to its
diameter plus a given arc ; find the central angle subtended by
that arc.
22. Find the radius of a globe such that the distance of 3 in.
on its surface, measured on an arc of a great circle, may subtend
at the center an angle of 1 45'.
23. At what distance does a telegraph pole, 24 ft. high, sub-
tend an angle of 10', the eye of the observer being on the same
level as the foot of the pole ?
NOTE. The suggestion made in connection with Ex. 10 applies to this
problem also. When a chord and its arc differ but little from each other
it is often convenient to use the arc in place of the chord.
24. At what distance will a church spire 100 ft. high subtend
an angle of 9', the angle being measured from the level on
which the church stands?
25. The difference between two angles is - - radians, and
y
their sum is 76 ; what is the value of each of the angles ?
26. If an incline rises 5 ft. in 300 ft., find the angle it makes
with its projection on the horizontal plane.
27. How many radians are there in an angle of a?
28. How many radians are there in an angle of 10" ?
CHAPTER II
TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE
14. In the present chapter only acute angles will be con-
sidered. In Chapter V the definitions here given will be
extended to angles of any magnitude.
Let any line having a given initial position OA begin to
revolve on as a pivot, in a direction opposite to the direction
A ' in which the hands of a
clock move. Let the angle
which it generates be the
acute angle A OA' .
A From any point in
either side of the angle,
asP in the side OA', let fall a perpendicular PM to the other
side of the angle.
The trigonometric functions, or ratios, of the angle AOA
are then denned as follows :
The sine of the angle A OA is the ratio ^ = side opposite
OP hypotenuse
The cosine of the angle AOA' is the ratio M = side adjacent
OP hypotenuse
The tangent of the angle A OA is the ratio = e opposite
OM side adjacent
The cotangent of the angled OA is the ratio ^ = side adjacent
MP side opposite
The secant of the angle AOA f is the ratio = hypotenuse >
OM. side adjacent
The cosecant of the angle A OA is the ratio QL = h .Ypotenuse
MP side opposite
20
TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 21
In addition to these there are two other functions, less
frequently used,
versed sine of A OA' = 1 cosine of AOA',
coversed sine of A OA 1 = 1 sine of A OA'.
In writing, it is customary to abbreviate the words " sine,"
"cosine," "tangent," etc., and to express the functions of any
given angle, a?, as follows :
sin x, cos x, tan a?, cot a?, sec #, esc #, vers a?, covers x.
It should be noted at the very beginning that these functions
are mere numbers, and their values can be expressed numerically
whenever the angle to which they belong is known. Thus,
sin x may equal J, J, or any other proper fraction ; tan x
may equal 2, 5, 18, or any other real number whatever. The
expression sin a;, for example, is a single symbol, and is to
be regarded as the name of the number which expresses the
value of the particular ratio in question. The expressions
sin, cos, etc., have no meaning unless some angle is asso-
ciated with them.
15. The trigonometric functions are always constant for the
same angle.
From any points in either side of the angle x, as A,
A f , A", drop perpendiculars AB, A'B', A"B" to the other
side. Then, by geometry, the triangles A OB, A' OB 1 , A" OB"
are similar, and their homologous sides are proportional.
Therefore, A
BA_B'A' _B"A" _
OA~ OA' = '' OA" =
OB OB' OB"
OA OA' OA"
= cos a:,
U B" A B
and similarly for the other functions.
Hence, the value of any function of x remains unchanged as
long as the value of the angle itself remains unchanged.
Any increase or decrease in the size of the angle pro-
duces a change in the value of the function, or ratio. From
this it is readily seen why these ratios are called functions of
the angle.
22 PLANE TRIGONOMETRY
From the -last paragraph the following important results may
now be stated :
(1) To every acute angle there corresponds one and only one
value of each trigonometric function.
(2) Two unequal acute angles have different trigonometric
functions.
(3) To each value of any trigonometric function there is but
one corresponding acute angle.
16. Fundamental relations between the trigonometric func-
tions of an acute angle. From the definitions given in Art. 14
it follows immediately that the sine of the angle x is the recip-
rocal of the cosecant x ; also that cosine x is the reciprocal of
secant #, and that tangent x is the reciprocal of cotangent x.
That is, .
cscrr'
cos# = , or cos # sec # = 1,
sec #
tan x = , or tan ^ cot a: = 1.
cot x
Also, it follows from the definitions that
sin x T cos a:
tan#=- , and cot x
cosx sin a;
In the right triangle ABC, a 2 + b 2 = c 2 . Therefore,
c ,
and _ = ! +
a 2 a 2
From these equations it follows that ^
sin 2 a? + cos 2 x - 1 , (3)
sec 2 x 1 + tan 2 x, (4)
esc-' JT, = 1 + cot 2 x. (5)
TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 23
17. From the definitions of the trigonometric functions,
p. 20, it follows that in any right triangle any function of
either of the acute angles is equal to the corresponding co-
function of the other -acute angle. For example,
sinJ. = -, and cos=-. .-. sin A cos B= cos(90 A}.
G c
Similarly, cos A = sin B = sin (90 J.),
tan A = cot B = cot (90 - J.),
cot A = tan B = tan (90 - A),
sec A = esc B = esc (90 - A),
esc A = sec B = sec (90 - A),
vers A = covers B = covers (90 A),
covers A = vers B = vers (90 A).
Hence,
Any function of an acute angle is equal to the corresponding co-
function of its complement.
The meaning of the prefix co, in cosine, cotangent, cosecant,
and coversed sine appears from the above. The cosine of an
angle is the complement-sine, i.e. the sine of the complement of
that angle: the tangent of an angle is the cotangent of its com-
plementary angle; and a similar statement may be made for
the secant and for the versed sine of an angle.
ORAL EXERCISES
Prove the following relations :
1. sin A cot A = cos A.
SOLUTION. Using only the left number of the equation, we proceed as
follows : ___ A
sin A cot A = sin A -. (Art. 16, (2).)
sin A.
= cos A .
.*. sin A cot A = cos A .
2. cos A tan A = sin A.
3. (sec A tan A) (sec A + tan A) = 1.
4. (esc A cot ^4)(csc A -f- cot A) = 1.
5. (tan A + cot A) sin A cos ^4 = 1.
24 PLANE TRIGONOMETRY
6. (tan A cot A) sin A cos A = sin 2 A cos 2 A.
7. sin 2 -*- esc 2 6 = sin 4 0.
8. sin 4 0- cos 4 0=sin 2 0-cos 2 0.
9. (sin - cos 0) 2 = 1 - 2 sin cos 0.
10. (sin - cos 0) 2 + (sin + cos 0) 2 = 2.
11. sec cot = esc 0.
12. (tan + cot 0) 2 = sec 2 + esc 2 (9.
13. cot 2 9 cos 2 = cot 2 - co., 2 0.
14. sin 2 + esc 2 6 + 2 = (sin (9 + esc 0) 2 .
15. vers 6 (1 + cos 0) = sin 2 0.
16. sin 2 (9 + vers 2 (9 = 2(1- cos (9).
17. sec sin tan = cos 0.
18. esc 6 cos cot 6 = sin 0.
19. sec 2 (9 - tan 2 = sin 2 + cos 2 0.
20. esc 2 (9 -cot 2 = sin 2 (9 + cos 2 0.
EXERCISE IV
Prove the following identities :
1. cos 4 d - sin 4 = 2 cos 2 6-1.
SOLUTION. Using only the left member of the equation, we proceed as
= (1) (cos 2 B - sin 2 0)
= cos 2 - (1 - cos 2 0) (Art. 16, (3).)
= 2cos 2 0- 1.
2. sin 3 + cos 3 = (sin + cos 0)(1 - sin cos 0).
3 smA +1+008^2080^.
1 + cos A sin A
4. (1 + sin a + cos a) 2 = 2(1 + sin a)(l + cos a).
5. ( COS 3 - sin 3 0) = (cos - sin 0)(1 + sin cos 0).
6. cos 2 /3 (sec 2 /3- 2 sin 2 /3) = cos 4 /3 + sin 4 /3.
.<
t
TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 25
sin (S 1 4- sin /3 9 ,.,
7. :: :~ 5 + H = sec 2 ff (csc/3 + 1).
1 sin /8 sin
8. tan a + tan /3 = tan a tan /:? (cot a + cot /3).
9. cot a 4- tan /3 = cot a tan /3 (tan a + cot /3).
10. cos 6 a 4- sin 6 a = 1 3 sin 2 a cos 2 a.
/I sin A
11. \/ 7 = sec A tan A.
*1 4- sin A
12. sin 2 <9 tan 2 04-cob 2 cot 2 = tan 2 4- cot 2 <9- 1.
13 . CSC / + CSC / =2 sec 2 A.
esc JL 1 esc J. -h 1
esc A
14. = cos ^L.
cot A + tan A
15. (1 sin a cos ) 2 (1 + sin a + cos a) 2 = 4 sin 2 a cos 2 a.
16. (sec A H- cos A) (sec A cos J.) = tan 2 A + sin 2 ^.
17. = sin A cos A.
cot ^4. H- tan A
1 tan A _ cot J. 1
1 + tan A ccff-A+1'
19. sjn 3 J. cos J. + cos 3 A sin ^4 = sin A cos A. *
20. sin 2 J. cos 2 ^L + cos 4 A = 1 sin 2 A.
esc a sec a cot a tan a
cot a -f- tail a esc a + sec a
1 + tan 2 A sin 2 A
21.
22.
23. sec A-tauA l _ 2sQGA tan A + 2 tan2 ^
sec J. 4- tan ^4.
24. tan 2 a sec 2 a 4- cot 2 a esc 2 a
= sec 4 a esc 4 a 3 sec 2 a esc 2 a.
tan .A cot A
25. T + ^ = sec ^4. esc y 4- 1.
1 cot A 1 tan ^1
cos A sin ^4. ,, , ,,
26. - = sin A + cos A.
1 tan A 1 cot A
26 PLANE TRIGONOMETRY
27. COt 4 A + COt 2 A = CSC 4 A CSC 2 A.
28. Vcsc 2 A I = cos A esc ^4.
29. tan 2 ^4 - sin 2 A = sin 4 .A sec 2 A.
30. (1 + cot^4. csc A) (I + tan J. + sec A) =2.
1 11 1
31.
32.
csc A cot A sin .A sin A csc .A + cot A
cot .A cos A. cot A cos vl
cot A + cos A cot A cos ^4.
33. 2 - vers 2 = sin 2 0+2 cos 0.
34. sin 8 A cos 8 .A = (sin 2 A cos 2 A)(l 2 sin 2 A cos 2 A).
cos ^1 esc A - sin A sec ^
3g
cos ^ + sin ^.
tan A + sec ^4. 1 _ 1 + sin A
tan A sec A + 1 c< >s A
37. (tan a + csc ) 2 (cot fi sec a) 2
= 2 tan a cot /3(csc a.+ sec /3)
38. 2 sec 2 a sec 4 a 2 csc 2 a + csc 4 a = cot 4 a tan 4 Tx.
39. (sin a + csc a) 2 + (cos a + sec a) 2 = tan 2 a + cot 2 a -f 7.
v x . sec A csc, A
40. (1 + cot A + tan A) (sin J. cos A)= -
csc 2 J. sec 2 A
2
41. 2 vers^l -f cos 2 J. = 1 + vers
42. S6C x ~ tan * = 1 + 2 tan x (tan g - sec x).
sec a; + tan x
2 sin 6 cos cos 6
43. - - - - r = COL (7.
44. (sin a cos /3 + cosa sin /3) 2 4- (cos a cos /S sin a sin/^) 2 =l
45. (ta,^ + sec^ = l^.
1 sin 6
TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 27
18. Limits of the values of the trigonometric functions of an
acute angle.
Since sin 2 A + cos 2 A = 1,
and since each term, being a square, is positive, neither sin 2 A
nor cos 2 A can be greater than unity. Hence, neither sin A nor
cos^l can be numerically greater than unity.
Since esc A is the reciprocal of sin A, and sec A is the recip-
rocal of cos A, both sec A and esc A can have any values numeri-
cally greater than unity, but neither can ever be numerically
less than unity.
Since sec 2 A=\ + tan 2 J.,
tan A = Vsec 2 ^. 1.
Hence, tan A can have any value between and oo. And
since cot A is the reciprocal of tan A, therefore cot A can have
any value between oo and 0.
These results are summarized as follows :
When A is an acute angle,
sin^L can take any value between and + 1,
cos A can take any value between -f- 1 and 0,
tan A can take any value between and +00,
cot A can take any value between + GO and 0,
sec A can take any value between -f 1 and -foo,
esc A can take any value between +00 and + 1.
These results also follow directly from the definitions of the
functions of an acute angle, p. 20.
19. To express all the trigonometric functions in
terms of any one of them.
From any point in either side of the angle
A let fall a perpendicular upon the otlu r
side. Let the hypotenuse of the right
triangle thus formed be taken as unity,
28 PLANE TRIGONOMETRY
and call the perpendicular a. Then the remaining side of the
right triangle is Vl a 2 . Then,
sin A = - a = sin A,
cos A = Vl a 2 = Vl sin 2 A,
sin A
tan A =
Vl - a 2 Vl - sin 2 A
cot A =
Vl - sin 2 A
sin A
sec . =
Vl - a 2 Vl - sin 2
csc . = -=
,
sin .A
This gives the value of each of the functions, except the vers A
and the covers A, in terms of sin A.
To express the values of the functions in terms of cos J.,
tan A, or of any other given function of A, proceed in a similar
manner. It is not best, however, to assume the hypotenuse
equal to unity for all cases. Sometimes the side opposite the
given angle should be taken as unity, and sometimes the side
c adjacent. For example, to find the
other functions of A in terms of
tan A, assume the side adjacent A
equal to unity, and let the side oppo-
A B site the same angle equal a ; then the
hypotenuse of the right triangle equals VI + a 2 , and the* required
values are found as follows :
tan A = - = a = tan A,
Vl + a 2 Vl + taii 2 ^.
-V
TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 29
cos .4 = - = ,
tan
a tan A
In this work it will be noticed that the side adjacent to A is
assumed equal to unity, while in the preceding the hypotenuse
was assumed to be unity. Any other supp6sition might be
made with equal correctness, but no other would be equally
convenient.
EXERCISE V
1. Express all the other functions of 6 in terms of cos 6.
This problem can be solved, and the required values found,
in a manner similar to that employed in finding the values of
each of the other functions in terms of sin 0, or tan 0, which
has just been illustrated. Or the values can be found by means
of the relations deduced in Art. 16. Thus :
= vi- cos 2 6,
si ii# Vl cos
tan =
cos cos
A ~~
: ,etc.
Sill
2. Express all the other functions of in terms of cot 6.
3. Express all the other functions of in terms of sec 0.
4. Express all the other functions of 6 in terms of esc 6.
5. Given sin 6 = f , find cos 6 and tan 0.
cos
B = Vl - sin 2 = Vl - ^ = i V21.
cos0 5 5 5 V21 V21 21
30
PLANE TRIGONOMETRY
6. Construct the angle 6 if tan 6 = f .
The angle 6 may be considered one of the acute angles of a right triangle.
Hence, to construct 6 we have only to construct a right triangle whose legs
are respectively 2 and 7. Since tan = f , is the acute angle opposite the
Bide 2 ' 11.
7. If sin = -j3_, find sec 0.
8. If sin A = J-J, find vers A.
9. If cos = f, find esc 6 and tan 0.
10. If cos = |, find cot and sec 0.
11. If tan A = J-J, find sec J. and cos A.
12. If tan. A = |, find esc A and cos .4.
13. If cot A = | , find sin J. and cos A.
14. If sec B = 5, find siri J5 and tan .#.
15. If sec B = |^, find tan .5 and vers B.
16. If esc A = 8, find cos A and tan A.
17. If esc JL = f , find sin A and sec A.
18. Find all the functions of each of the acute angles, ^4, B,
of the right triangle whose sides are 8, 15, 17.
19. Find all the functions of each of the acute angles, A, B,
of the right triangle whose sides are x + y,
2 xu
x y x-y
20. . If sin 2 + cos = 2|, find tan 0.
21. If tan 2 - sec 6 = f>, find cos 0.
22. If 10 sin 2 6 - 5 cos = - |, find esc 6.
23. If sin + cot = 4y, find 0080.
24. If sin 0, = a and tan 6 = ft, prove (1 - a 2 )(l - ft 2 ) = 1 .
'
\ \
CHAPTER III
VALUES OF THE FUNCTIONS OF CERTAIN USEFUL
ANGLES
20. Functions of an angle of 0. If the angle A is very
small, then in considering the value of sin A, that is, the ratio
CB
- , it is at once seen chat the numerator, CB, is very small in
comparison with the denominator, AB. Hence, the numerical
value of sin A is very small when the angle A is very small.
Also, if A decreases, the numerator of the fraction will also
decrease, while the denominator will
remain constant ; and as the angle
approaches as a limit, the sine of the
angle will also approach as a limit. When the angle becomes
0, that is, when A B coincides with A C, we shall have CB = 0,
and AB = AC. Hence,
sin 0= -=0,
When we say that sin A when A = 0, we simply mean
that, if A is made small enough, we can make the value of CB,
and hence the value of sin A as small as we please ; or, to ex-
press the same statement in different words, we can make sin A
smaller than any assignable quantity.
Hence, as stated above, sin A approaches as a limit when
A approaches as a limit.
In a similar manner, we interpret the statements cosO = l,
tan 90 = oo , etc,, as meaning that cos A approaches 1 as a limit,
tan A approaches oo as a limit, etc., when A approaches as a
limit, when A approaches 90 as a limit, etc.
31
32
PLANE TRIGONOMETRY
21. Functions of an angle of 30. Let OAC be an equilateral
triangle ; then is it also equiangular. From the vertex draw
OB perpendicular to AC. Then in the right triangle GAB the
angle A = 60, and the angle A OB = 30. Also,
theleg AB=IAC=
Let AB = a. Then OA = 2 a, and
= V 6U 2 - AB* = V4 a 2 - a 2 = V3 a 2 = a V3.
The trigonometric functions of 30 can now be found as
follows :
OA
tan 30= B A = -A_ = JL = 1 V3,
av /3 V3 3
BA
V3
esc 30 = = = 2.
BA a
22. Functions of an angle of 45. Let OAB be an isosceles
right triangle. Each of the acute angles is 45, and the leg OB
equals the leg AB. ^4
Let AB = a. Then OB = a and OA = a V2,
and we have :
sin 45'
cos45=^ =
-
2
FUNCTIONS OF CERTAIN USEFUL ANGLES
- BA - a --\
~ ~'
33
CSC 45=^= = V2.
BA a
23. Functions of an angle of 60. Let OAC be an equilateral
triangle. Then is it also equiangular. From the vertex A
draw AB perpendicular to 00. Then in the right triangle
OAB, angle 0= 60, and angle OAB= 30. Also, OB = OC
= \ OA.
Let OB = a. Then OA = 2 a, and AB = V OA* - OB*
= V42_ a 2 = V3a2 = 6? V3.
The trigonometric functions of 60 can now be found as
follows :
*.!. V 5 ,i -
OB a
cot60 = ^^ = -^=- J L = i
BA aV3 V3 3
fO 4 9 /, 9 9
csc 60 - ^ = ^^ =-^- =
O a
24. Functions of an angle of 90. Let the angle AOB (p. 34)
be very nearly a right angle. Then the angle A is very small,
and B the foot of the perpendicular from A to OB is very near
the vertex 0. When the angle approaches a right angle, AB
CON A NT'S TKIO. - 3
34
PLANE TRIGONOMETRY
will approach coincidence with AO, and B will approach coinci
dence with 0. Hence,
. o BA OA -
YA
one
cos 90 = =
OA OA
BA OA
0,
_
BA OA
* =-^=ir
OA OA
The real meaning of these equations is that, as the angle ap-
proaches 90 as a limit, the sine of the angle approaches 1 as a
limit, the cosine approaches as a limit, the tangent approaches
oo as a limit, etc. It is, however, customary to say sin 90 = 1,
cos 90 = 0, tan 90 = oo , etc.
A more complete discussion of the values of the trigonometric
functions for limiting cases such as the above is given later.
See Art. 41, p. 59.
25. In the following table are collected the results obtained
in the last five sections. These results are exceedingly im-
portant, and the student should become thoroughly familiar
with them before proceeding further.
30
45
60
90
sine
1
JvS
|V3
1
cosine
1
JV3
*V2
i
tangent
iV3
1
V:]
CO
cotangent
CO
V3
1
*V3
secant
1 I jVS
V2
o
-a
cosecant
cc j 2
V2
|v3
1
FUNCTIONS OF CERTAIN USEFUL ANGLES 35
It is necessary to commit to memory only one half of this
table. The remainder can be obtained at any time by means
of the relations which were found in Art. 17, of which the
-following is a condensed statement : Any trigonometric function
of an acute angle is equal to the corresponding co-function of its
complement.
EXERCISE VI
Verify the following :
, 1. cos + sin 30 + sin 90 = 2.
2. cos cos 60 -f sin sin 60 -f sin 30 = 1.
3. tan 2 30+ sec 2 30 = If.
4. cos 2 60 + cos 2 45 4- cos 2 30 = f .
5. sin 60 cos 30 -f- cos 60 sin 30 = 1.
6. sin 2 30 tan 2 45 + sec 2 60 sin 2 90 = 4 J.
7. (sin 30 + cos 60) (sec 45 + c.sc 45; - 2 V2.
8. sin 30 sin 45 sin 60 tan 60 = f V2.
9. cot 30 tan 60 sin 45 cos 45 = f .
10. tan 2 45 + sin 2 30 - cos 2 30 - f tan 2 30 = J.
Prove the following identities :
11. sin A cos (90 - A) sec (90 -A) = sin A.
12. cos A cos (90 - A) sin (90 - A) esc A == cos 2 A.
13. tan (90 - A) cot (90 - A) tan A
= cos (90- A) esc (90- A).
cos (90 -A) cot(90-J.)
14. = sin A.
esc (90 - A) sin A
15. cos 2 A sec 2 (90 - A) tan 2 A cot 2 (90 -A) = tan 2 A.
16 tan 2 (90-^) cos 2 .4 csc 2 (90-^) ^ 4 A
sin 2 A ' cot 2 (yO - A) ' sec 2 (yo - A) "
17. cos (90 - A) 1 - cos (90 - A) _ t . m A
covers A sin ( ( JO A)
18. secMEEl +cos 2 (90-^)csc 2 (90-7l).
19. csc 2 ^ = 1 + sin 2 (90 - A) sec 2 (90 - A).
20 cot 2 (90 - A) tan 2 (90 -A')^ 1
esc 2 (90 - A) ' sin 2 (90 -A)
CHAPTER IV
THE RIGHT TRIANGLE
26. In order to solve a right triangle, two parts besides the
right angle must be given, of which at least one must be a
side. The known parts may be :
1. An acute angle and the hypotenuse.
2. An acute angle and the opposite leg.
3. An acute angle and the adjacent leg.
4. The hypotenuse and either leg.
5. The two legs.
27. In the preceding sections we have seen that the trigono-
metric functions are pure numbers ; and in the case of the angles
0, 30, 45, 60, and 90, the values of these functions have been
ascertained. From a trigonometric table the values of the
functions of any angle can be found ; and by the aid of these
values the solution of any triangle can be effected.
The method for each case arising under right triangles is
illustrated by the following examples :
CASE 1
Given A = 61 22', c = 46.2; find B, a, b.
fit 22'
B = 90 - 61 22' = 28 38'.
b
(1)
(2) sin A--- .-.a=csiuA
c
= 46.2 x 0.8777.
.-. a = 40.54.
(3) cos^l =-. .-. b = coos ,4
c
= 46.2 x 0.4792.
.-. b = 22.14.
30
THE RIGHT TRIANGLE
CASE 2
Given A = 31 17', a = 321 ; find B, c, b.
(1) B = 90 - 31 17' = 58 43'.
(2) sin.4 = ^ .' = -;?
321
= 618.14.
0.5193
.-. c = 618.14.
(3) tan J. = - /. b =
b tan A
321
0.6076
.-. b = 528.31.
CASE 3
Given A = 43 42', b = 38.6 ; find B, a,
, 43'42'
(1)
= 90 -43 42' =46 18'.
(2) tan /I = - .. a b tan A
(3)
= 38.6 x 0.9556.
.-. a = 36.89.
*
cos A
_ 38.6
0.7230*
.-. c = 53.39.
CASE 4
Given a = 36. 4, <? = 48.4; find A, B, b.
(1) sin A = -
= 36.4
~48.4
= 0.7521, nearly.
... A = 48 46'.
(2)
B = 90 - 48 46' = 41 14'.
(3) tanJ =- .-.6 =
b
36.4
tan
1.141
.-. b = 31.9.
38
PLANE TRIGONOMETRY
The value of b could also be found directly by means of the familiar
geometric relation
from which we have b = Vc 2 a 2 .
CASE 5
Given a = 34.9, b = 38.6 ; find A, B, c.
(1) tan 4 = 2
= 34.9
38.6
= 0.9041.
.-. A =42 7'.
(2) = 90 -42 7' = 47 53'.
b-38.6
(3) sin,4 =?. .-. c = -^
c sin ^1
34.9
0.6706
.-. c = 52.04.
The value of c can also be found directly by means of the relation
c 2 = Va* + b' 2 .
From the methods of solution illustrated in the examples
given in Art. 27, we deduce the following general rule :
Rule for the solution of right triangles. From the equation
A + B = 90, and from the equations that define the functions
of an acute angle of a right triangle, select an equation in which
the required part is the only unknown quantity. From this equa-
tion find an expression for the required part, and compute the value
of this part from the expression thus obtained.
If a and c, or b and c, have values that differ but little
from each other, the methods here given will yield inaccurate
results. In such cases the method of Art. 101, p. 144, should
be employed.
The student will find it advantageous to check his results in
all cases, to avoid numerical errors as far as possible. Any
method of checking can be employed that involves a process
of solution different from the one used in first obtaining the
required part.
THE RIGHT TRIANGLE
39
EXERCISE VII
In the following examples, use the first two parts as the given
parts, and solve for the three remaining parts :
~i. A = 21 19', c = 18. =68 41', a =6.5,
= 49 16',
2.
^ = 40 44',
=31.
3.
^ = 71 38',
=5.4.
4.
=13 14',
= 92.
5.
.A = 63 11',
a = 12.
6.
B = 43 52',
6 = 70.
7.
A = 19 36',
6 = 42.
8.
5 = 56 17',
a = 9.
9.
a = 12.6,
= 26.
10.
6 = 42.6,
= 46.
6 = 16.8.
= 20.2, = 23.5.
= 18 22', a =5.12, 6 = 1.7.
^4 =72 46', 6 = 21, a = 89.6.
,6 = 26 49', 6 = 6.1, c = 13.4.
^ = 468', a =72.8, c = 101.
B =70 24', a =15, =44.6.
^4 = 33 43', 6 = 13.5, = 16.2.
A = 28 59', B = 61 1', 6 = 22.7.
A = 22 10', B = 67 50', a = 17.4.
SOLUTION BY LOGARITHMS
28- Problems in the solution of triangles can usually be per-
formed quite as expeditiously by the use of logarithms as by the
use of the actual values of the trigonometric functions, and in
many cases the amount of labor is very greatly reduced by the
use of logarithms.
The method of solution by logarithms in the different cases
that arise in connection with right triangles is illustrated by
the following problems :
CASE 1
Given ^1 = 59 17', =42.68; find , a, ft,
(1) B = 90 - 59 17' = 30 43'.
(2) sin A=-- .-.a = csiuA. n
log a = log c + log sin A.
logc = 1.63022
log sin A = 9.93435 - 10
log a = 1.56457
.-. a = 36.69.
(3) cos A =--, b = ccosA.
log c = 1.63022
log cos A = 9.70824 - 10
log b = 1.33846
.-. b = 21.8.
40
PLANE TRIGONOMETRY
Given A = 55
55 //'
CASE 2
a = 68. 34; find B, b, c.
(1) B = 90 - 55, 11' = 34 49'.
(2) tanX=2. ...&=__
b tan yl
log ft = log a -f colog tan A.
log = 1.83467
colog tan A = 9.84227 - 10
log b = 1.67694
.-. b = 47.527.
(3) sin A = 1
sin A
log c = log + colog sin A.
log a = 1.83467
colog sin A = 0.08567
logc= 1.92034
.-. c = 83.242.
CASE 3
Given A = 49 13', b = 72.3 ; find B, a, e.
(1) 73 = 90 -49 13' = 40" 17'.
(2) tan A =-. .-. a = ft tanA.
log a = log ft + log tan A.
log ft = 1.85914
log tan .4 = 10.06116 - 10
logo= 1.92330-10
a = 83.81.
(3) cos 4 = -
c
cos 4
log c = log b 4- colog cos A .
log ft =1.85914.
colog cos A = 0.18495
logc = 2.04409
.-. c = 110.68
CASE 4
Given c= 61.14, a= 48.56; find ,4,^,
(1) sin A = a --
c
log sin A = log a + colog c.
loga = 1.68628
colog c = 8.21367 - 10
log sin /I = 9.89995 - 10
.-. A = 52 35'.
*72 3
THE RIGHT TRIANGLE
41
(2) cos A -.
.'. b = c cos A .
log b = log c + log cos A,
logc = 1.78633.
log cos A = 9.78362 - 10
log b = 1.56995
b = 37.149.
(3) tan B = -
lo
tan B = log b -f colog a.
log b = 1.56995
colog = 8.31372 - 10
log tan B = 9.88367 - 10
.-. B = 37 C 25'.
CASE 5
Given a = 126, b = 198 ; find A, B, c.
(1)
log tan ^4 = log a + colog 5.
log a = 2.10037
colog b = 7.70333 - 10
log tan A = 9.80370 - 10
.-.4 = 32 28'.
(2) tanB=-
log tan B = log 6 + colog a.
log 6= 2.29667
colog a= 7.89963-10
log tan B = 10.19630 - 10
.-. B = 57 32'.
sin A = -
C = ~ - 7'
sm -4
log c = log a + colog sin A.
log a = 2.10037
colog sin A - 0.27018
log c = 2. 37055
.-. c = 234.72.
NOTE. In the last two. cases the angle B might have been found directly
by subtracting A from 90. It is, however, better to determine the value of
the second angle independently, as a means of checking the work.
42 PLANE TRIGONOMETRY
AREA OF THE RIGHT TRIANGLE
29. The area of any triangle is equal to one half the product
of the base and the altitude. In the case of the right triangle
either of the legs can be regarded as the base and the other as
the altitude. Hence the area of a right triangle can be found
when any two parts are known, provided one or both the known
parts are sides, by computing, if necessary, the legs of the tri-
angle, and then taking one half their product. That is,
If a, 6, denote the legs of a right triangle, and A the area,
then A = i- ab. (1)
Ex. l. In the right triangle ABO, given .4 = 36 14',
a = 26. 8; to find the area.
First find log b by the method of Case 2, p. 40. Then we have
log A = log a -f log b + colog 2.
log a = 1.42813
log b = 1.56315
colog 2 = 9.69897 - 10
log A = 2.69025
.-. A = 490.06.
Ex. 2. In the right triangle ABC, given ^. = 40 23,
c = 39.6; to find the area.
First find log a and log& as in Case 1, p. 89. Then we have
log A = log a + log b -f colog 2.
log a = 1.40921
log b= 1.47950
colog 2 = 9.69897- 10
log A = 2.58768
.-. A = 386.97.
EXERCISE VIII
Solve the following right triangles, finding the angles to the
nearest minute :
1. Given A = 34 10', a = 21 ;
find ^ = 55 50', 5 = 30.939, c= 37.39.
2. Given 5=50 12', a =65;
find .4=39 48', b = 78?15, c= 101.55.
THE RIGHT TRIANGLE 43
3. Given 5 = 47 15', c= 54.39;
find .4 = 42 45', a = 36. 92, 5 = 39.94.
4. Given A = 31 25', c = 45.62 ;
find B = 58 35', b = 38.93, a = 23.78.
5. Given A = 29 17', c=31.68;
find =60 43', a = 15.495, 6=^=27.63.
6. Given 4 = 49 17', c= 36.48;
find JB=4043', a = 27.65, 6=23.796.
7. Given J. = 41 9', b = 156;
find B =48 51', a =136.33, c= 207.17.
8. Given B = 59 11', 6 = 221 ;
find ^1=30 49', a =131. 83, ^ = 257.33.
9. Given B = 62 55', c=92.4;
find ^1 = 27 5', a = 42.068, 6 = 82.268.
10. Given A = 29 31', a = 290.6;
find B= 60 29', b = 513.29, c= 589.85.
11. Given ^ = 45 20', a = 41. 46;
find ^. = 44 40', 5 = 41.946, c = 58.979.
12. Given a =20. 08, c?=28.26;
find A = 45 17', ^ = 44 43', b = 19.885.
13. Given B = 55 13', a = 72.96 ;
find ^1 = 34 47', 6 = 105.04, c = 127.89.
14. Given B = 51 19', 6 = 106.8;
find A =38 41', a =85.512, c= 136.81.
15. Given B = 59 49', a = 254.36 ;
find 4 =30 11', 6 = 437.33, c = 505.92.
16. Given A = 51 50', 6 = 6.813;
find B = 38 10', a = 8.668, c= 11. 025.
17. Given B = 57 46', 6 = 0.0688;
find A = 32 14', a = 0.04338, c = 0.08134.
44 PLANE TRIGONOMETRY
18. Given 6 = 963.3, c=1465;
find ^1 = 48 53', =41 7', a = 1103.7.
19. Given a = 691, e= 877.62;
find .4 = 51 56', j5=384', 6 = 541.05.
20. Given a = 62. 36, 6 = 33.823;
find A = 61 32', ^ = 28 28', c = 70.96.
In the following examples find the required angles to the
nearest second :
21. Given A = 41 38' 20", b = 262.38 ;
find .g = 4S 21' 40", a=233.27, <?=351. 08.
22. Given ^=71 14' 12", <?= 129.3;
find ^=18 45' 48", a = 122.43, 6 = 41.6.
23. Given A = 41 17' 30", a = 29.41;
find B = 48 42' 30", 6 = 33.486, c = 44.568.
24. Given B = 61 12' 15", c = 382.6 ;
find A = 28 47' 45", a = 1 84. 29, 6 = 335. 29.
25. Given 6 = 1426, c = 2291.2;
find A = 51 30' 38", B = 38 29' 22", a = 1793.38.
26. Given B = 54 2' 28", a = 49.628 ;
find A = 35 57' 32", 6 = 68.41, c = 84.514.
27. Given = 35.421, 6 = 18.168;
find A = 62 50' 40", B = 27 9' 14", c = 39.81.
28. Given a = 39.313, 6 = 19.852;
find A = 63 12' 26", J5 = 26 47' 34", c = 44.036.
29. Given a = 126. 43, 6=131.52;
find A = 43 52' 9", B = 46 7' 51", c = 182.44.
, *
30. Given a = 476.32, c = 812.36;
find .4 = 35 53' 53", B = 54 6' 7", 6 = 658.05.
31. Given ,4 = 68 17' 22", c = 269.4;
find B = 21 42' 38", a = 250.29, 6 = 99.658.
THE RIGHT TRIANGLE 45
111 the following ten examples find the area of the triangle in
each case, having given :
32. a = 10, 6 = 12. 37. .4 = 42 27', 6 = 50.
33. a = 268, b = 316. 38. A = 54 24', c = 90.
34. a = 3, <?=5. 39. =39 55', a =294.
35. b = 20. 7844, ^=24. 40. B = 66 36', b = 48.
36. J.= 35, a = 16. 41. ^=70 52', <? = 582.
42. Find the value of A in terms of a and c.
43. Find the value of A in terms of a and A.
44. Find the value of A in terms of a and B.
45. Find the value of A in terms of c and A.
46. Given A = 72, a = 9 ; find A.
47. Given A = 72, 5 = 9; find A.
48. Given A = 250, A = 40 ; find a.
49. Given A = 250, B = 29 30' ; find a.
50. Given A = 254.2, <?= 32; find B.
51. The hypotenuse of a right triangle is 28 and one of the
legs is 13. Find the angle opposite the given leg.
52. The legs of a right triangle are 36 and 39, respectively.
Find the angle opposite the shorter leg.
53. The tangent of one of the acute angles of a right tri-
angle is 2\. Find the angle.
54. The cotangent of one of the acute angles of a right
triangle is \\. What is the angle?
55. One of the acute angles -of a right triangle is 49 38'
and the adjacent leg is 68.42. Find the hypotenuse and the
other leg.
56. The legs of a right triangle are 41625.3 and 11362.7,
respectively. Find the larger angle.
57. The hypotenuse of a right triangle is 262.46 and one of
the acute angles is 28 15' 42". Find the opposite leg.
46 PLANE TRIGONOMETRY
58. The legs of a right triangle are 515.38 and 221.34, re-
spectively. Find the hypotenuse.
59. One of .the acute angles of a right triangle is 46 21' and
the adjacent leg is 26.38. Find by natural functions the
other leg and the hypotenuse.
Angle of elevation and angle of depression. The angle of
elevation of an object above the point of observation is the
D angle between a line from the eye
angle of depression \^ of the observer to the object and
a horizontal line in the same ver-
tical plane. The angle of depres-
sion of an object below the point
angle of elevation of observation is the angle be _
tween a line from the eye of the
observer to the object and a horizontal line in the same vertical
plane.
In the figure, BA C is the angle of elevation of the point B
above the point A ; and DBA is the angle of depression of the
point A below the point B.
60. The angle of elevation of the top of a tower 80 ft.
high is 41 49'. What is the distance of the point of observa-
tion from the foot of the tower ?
61. At a distance of 31.15 ft. from the foot of a vertical
cliff the angle of elevation of the top of the cliff is 56 18'.
What is the height of the cliff?
62. From the top of a monument the angle of depression of
a point on the ground, on the same level as the foot of the
monument, is 43 41'. The point is found by measurement to
be 128.29 ft. distant from the foot of the monument. What
is the height of the monument?
63. From the top of a hill 304 ft. 9 in. in height the angle
of depression of an object on the ground is 40 37'. What is
the distance of the object from a point directly below the point
of observation and on the same level with the object?
64. What is the height of a tree that casts a shadow 42.6 ft.
long when the angle of elevation of the sun is 60 11' ?
THE RIGHT TRIANGLE 47
65. What must be the length of a ladder set at an angle of
71 14' with the ground to reach a window 21.88 ft. high ?
66. To find the width of a river a point P is selected on
one bank, and a distance of 138.2 ft. is measured parallel to
the course of the river from the given point P to a point Q.
Directly opposite (), on the other side of the river, is the point
#, and the angle SPQ is found to be 66 11'. What is the
width of the river ?
67. A guy rope 49.11 ft. long is attached to the top of a
mast, and makes an angle of 50 56' with the level of the
ground. What is the height of the mast ?
68. The top of a flag pole, broken by the wind, falls so that
it touches the ground at a distance of 19. 73 ft. from the foot of
the pole, and is inclined to the ground at an angle of 65 40'.
What is the height of the portion that remains standing, and
what was the total height of the pole?
69. What is the angle of elevation of an inclined plane that
rises 26 ft. in a horizontal distance of 31.9 ft. ?
70. A man walking on a level plain toward a tower observes
that at a certain point the angle of elevation of the top of the
tower is 30 ; on walking 300 ft. directly toward the tower the
angle of elevation of the top is found to be 60. What is
the height of the tower ?
SOLUTION. Let x = the height of the tower and y = the distance from
the second point of observation to the foot of the tower.
From the triangle A CD = tan 30 = ,
300 + y V3
.-. y = V3x - 300;
from the triangle BCD - = tan 60 = V3,
y
equating these values of y, we have
V3z-300= -*-,
Vo
300 B y
2 x = 300 x 1.732,
x = 259.8.
rd
1 " "' ; '
48
PLANE TRIGONOMETRY
! >
NOTE. In solving problem 70 natural functions have been employed.
On p. 156 a method will be given by means of which problems of this kind
can be solved by the use of logarithms. In the following problems it is
recommended that natural functions be employed.
v> 71. At a point on a level plain the angle of elevation of the
top of a church spire is 45, and at a point 50 ft. nearer, and in
the same straight line with the first point and the church, the
corresponding angle of elevation is 60. What is the height of
the spire?
^ 72. From the top of a cliff 150 ft. high the angles of depres-
sion of the top and bottom of a tower are 30 and GO , respec-
tively. What is the height of the tower?
73. The angles of elevation of the top of a tower, taken at
two points 268 ft. apart and in the same straight line with the
tower, are 21 14' and 53 4G', respectively. What is the height
of the tower?
74. At the foot of a mountain the angle of elevation of the
summit is 45 ; one mile up the slope of the mountain, which
rises at an inclination of 30, the angle of elevation of the sum-
mit is 60. What is the height of the mountain ?
75. At a certain point south of a tower the angle of eleva-
tion of the top of the tower is 60, and at a point 300 ft. east
of the point the corresponding angle of elevation is 30. What
is the height of the tower ?
30. The isosceles triangle. The
perpendicular from the vertex, (7, of
an isosceles triangle to the base
divides the triangle into two equal
right triangles.
Any two parts of either of these
right triangles being given, one or
both of which are sides, the right
triangle can be completely determined. Therefore the isosceles
triangle also can be completely determined.
Denoting the base of the isosceles triangle by <?, and the
altitude by h, the area, A, is given by the formula
A = I ch. (1)
-
THE RIGHT TRIANGLE
49
31. The regular polygon. A regular polygon is divided into
equal isosceles triangles by lines drawn from the center to the
vertices of the polygon. Each of
-the isosceles triangles is divided
into two equal right triangles by
the apothem of the polygon.
Any side of either of these right
triangles being given, the polygon
can be completely determined if
the number of sides is known.
For the angles at the center of
the polygon can be found when
the number of sides, n, is known, by dividing 360 by n.
Taking one half of this angle as one of the acute angles of the
right triangle, and combining it with the given side, we have
at our disposal two parts of a right triangle, one of which is a
side. The remaining parts can then be found by the methods
already given for the solution of right triangles.
Denoting the perimeter of the polygon by p and the apothem
by h< the area of the polygon can be found by the following
formula :
A =
(1)
It should be remembered that the legs of the isosceles tri-
angles are radii of the circumscribed circle, and the apothem
is the radius' of the inscribed circle of the polygon.
EXERCISE IX
Solve the following isosceles triangles, finding the part
indicated in each case :
-jC 1. Given c=83.2, h = 56.9; find C.
2. Given c= 92.56, 7i = 59.72; find C.
3. Given c=252.64, 0= 62 28' 36"; find a.
4. Given <7= 142 27' 44", a = 92.452 ; find c.
5. Given C= 102 44' 42", h = 92.96 ; find a.
6. Given c = 85.32, A = 49.84; find A.
y- 7. Given c = 136.48, A = 60.51; find a.
L 8. Given A = 1426.3, = 2291.2; find A.
CON A NT'S TRin 4.
50 PLANE TRIGONOMETRY
9. Find the value of A in terms of a and 0.
10. Find the value of A in terms of a and A.
11. Find the value of A in terms of h and A.
Solve the following regular polygons, having given :
~^12. n=10, c='3. 14. rc = 6, tf = 12.
13. n=S, h = 2. 15. n = 20, a = 10.
16. What is the area of a regular octagon formed by cut-
ting away the corners of a square whose side is 6?
* 17. What is the area of a circle inscribed in an equilat-
eral triangle whose side is 20?
*^- 18. What is the area of a regular polygon of 18 sides if
the radius of the circumscribed circle is 2?
^ 19. One of the diagonals of a regular pentagon is 12.15.
What is the area of the pentagon? a*|. 0|ri>*"
20. Compute the area of a regular heptagon if the length
of one of its sides is 13.88.
v 21. The radius of the circumscribed circle of a regular
dodecagon is 27. What is the area?
CHAPTER V
THE APPLICATION OF ALGEBRAIC SIGNS TO TRIGO-
NOMETRY
32. In the preceding work no attempt has been made to
apply the definitions of any of the trigonometric functions to
any except positive acute angles.
These definitions will now be extended so as to apply to
negative as well as to positive angles, and to angles of any
magnitude whatever.
33. The coordinate axes. The location of a point or a line
lying in a given plane is often described by referring it to two
intersecting straight lines in that plane, called coordinate axes.
These lines are usually drawn perpendicular to each other.
Let the two lines XX' and YY' intersect at right angles.
Then the plane of these lines is divided into four quadrants,
designated as the first, second, third, and fourth quadrants,
respectively. These quadrants are numbered as indicated in
the figure.
34. Coordinates of a point in a plane. The location of any
point in the plane determined by the axes XX' and YY' is
described by means of its perpendicular distances from these
axes. Y
The distance of a point from
YY' measured along a line parallel
to XX' is called the abscissa of
the point ; and the distance of a
point from XX', measured on a X-
line parallel to YY' is called the
ordinate of the point.
The abscissa of a point is usu-
ally designated by the letter x, y'
51
52
PLANE TRIGONOMETRY
and the ordinate by the letter y. These two distances, taken
together, are called the coordinates of the point.
The line XX' is called the axis of abscissas, and the line
YY' is called the axis of ordinates. These axes are, for the
sake of brevity, often called the #-axis and the #-axis, respec-
tively. Their point of intersection, 0, is called the origin.
Any abscissa measured to the right of YY' is considered
positive, and any abscissa measured to the left of YY' is con-
sidered negative.
Any ordinate measured above XX' is considered positive, and
any ordinate measured below XX' is considered negative.
The coordinates of a point determine its position completely.
For example, if the point A is 4 units from YY 1 and 6 units
from XX', its position can be
located as follows : measure off
on XX' a distance equal to 4
units, and through the point
thus found draw a line parallel
to YY'., Also, measure off on
YY 1 a distance equal to 6 units,
and through the point thus deter-
mined, draw a line parallel to
XX'. The intersection, A, of
these two lines is the required
Y
12
point. The abscissa of A is 4, and its ordinate is 6, and this
point, whose location is given by means of its coordinates, is
called the point (4, 6).
The point B, located in a similar manner, has for its coordi-
nates x = 3 and y = 4 ; and this point B is called the point
(3, 4). The point is called the point (4, 5); and the
point D is called the point (6, 3). In a similar manner we
can locate any other point (#, 5), where a and b are any real
quantities whatever, either positive or negative.
35. Trigonometric functions of any angle. Let the line OA
(p. 53) start from OX and revolve in a positive direction until
it occupies a position in any one of the four quadrants. From
any point P in the revolving line draw a perpendicular PM to
the axis of abscissas, XX' . In each of the four figures we have
THE APPLICATION OF ALGEBRAIC SIGNS
53
CM= x and MP = y. Let the distance OP = r. The trigono-
metric functions of the angle XOA, which may be represented
by 0, are then, for all positions of OA, defined as follows :
M
x
X
X
x x
sill =
ordinate
revolving line r
/j abscissa x
cos u = - : = ->
revolving line r
= =
abscissa x
, a abscissa x
cot u = - = ->
ordmate y
a revolving line r
sec 6 = - = ->
x
abscissa
esc
a __ revolving line _ r
ordinate y
54 PLANE TRIGONOMETRY
The functions vers and covers are denned in a manner
similar to that employed in the case of the right triangle, as
follows: T ers 0=1 -cos*,
covers 6 = 1 sin 6.
NOTE. In the case of cot 0, esc 0, tan 90, sec 90, cot 180, esc 180, tan
270, sec 270, cot 360 and esc 360, these definitions fail. For, taking as an
illustration the tangent of 90, we have in that case a fraction whose numera-
tor is r and whose denominator is 0. The value of tan 90 is, then, if we
attempt to use the above definition, given by this fraction whose numerator
is r, and whose denominator is 0. But there is no such thing as division by
0, hence, according to the definition given, the symbol tan 90 has no mean-
ing. This and other similar cases will be discussed later. (See pp. 57-63.)
36. In a manner precisely similar to that employed in Art.
16 it can be proved that, for any value whatever of the fol-
lowing relations are true :
sin 2 6 + cos 2 6 = 1$ (1)
sec 2 6 = 1 + tan 2 6 ; (2)
esc 2 = 1 + cot 2 6. (3)
Also, from the definitions of the functions, the following rela-
tions are immediately derived :
sin 6 = -^-r, .-. sin esc 6 = 1, (4)
CSC U
-^
cos0 = - -, .-. cos 6 sec = 1, (5)
sec#
- -
cot u
, .-. tan6cot0 = l. (6)
Also, since, cos# = -, . . a? = reos6, (7)
(8)
(9)
THE APPLICATION OF ALGEBRAIC SIGNS
55
37. Signs of the trigonometric functions. In dealing with
the functions of an acute angle of a right triangle (Art. 14,
p. 20), no attention was paid to the question of positive
or negative signs. All lines employed in that connection
were considered positive; hence the value of each of the
functions was considered positive. But in dealing with the
general angle we have to consider both positive and nega-
tive lines, and as a result the signs of the functions undergo
certain changes as the revolving line passes from quadrant to
quadrant.
First Quadrant. Assume that the revolving line is always
positive, and let it occupy any position in the first quadrant.
In this position both x and y are positive; hence, since r is
also positive, both numerator and denominator are positive in
the case of each of the functions. Therefore all the trigono-
metric functions are positive for the angle in the first quadrant.
Y
Second Quadrant. Let the revolving line occupy any posi-
tion in the second quadrant. In this case x is negative and y
is positive; and we have the following results:
The sine is a fraction whose numerator and denominator are
both positive ; therefore the sine of an angle in the second
quadrant is positive.
The cosine is a fraction whose numerator is negative and
whose denominator is positive; therefore the cosine of an angle
in the second quadrant is negative.
The tangent is a fraction whose numerator is positive and
whose denominator is negative; therefore the tangent of an
angle in the second quadrant is negative.
56
PLANE TRIGONOMETRY
The cotangent is a fraction whose numerator is negative and
whose denominator is positive; therefore the cotangent of an
angle in the second quadrant is negative.
The secant is a fraction whose numerator is positive and
whose denominator is negative; therefore the secant of an
angle in the second quadrant is negative.
The cosecant is a fraction whose numerator and denominator
are both positive; therefore the cosecant of an angle in the
second quadrant is positive.
Third Quadrant. Let the revolving line occupy any position
in the third quadrant. In this case both x and y are negative;
therefore the following results can at once be obtained:
The sine is negative.
The cosine is negative.
The tangent is positive.
The cotangent is positive.
The secant is negative.
The cosecant is negative.
Fourth Quadrant. Let the revolving line occupy any position
in the fourth quadrant. In this case x is positive and y is
negative; therefore the following results can at once be
obtained: The gine . g negative<
The cosine is positive.
The tangent is negative.
The cotangent is negative.
The secant is positive.
The cosecant is negative.
THE APPLICATION OF ALGEBRAIC SIGNS
57
The above results are conveniently grouped together by
means of the following table:
sine +
sine -f-
cosine
cosine +
tangent
cotangent
secant
tangent +
cotangent -f
secant +
cosecant +
cosecant +
Y
sine
sine
cosine
cosine +
tangent +
cotangent +
secant
tangent
cotangent
secant +
cosecant
cosecant
Y'
From the definitions of the versed sine and of the coversed
sine it follows that these two functions are always positive.
38. Changes in sign and magnitude of the trigonometric func-
tions as the angle increases from to 360.
As before, we assume for the revolving line a constant length,
r. As the revolving line starts from its initial position we
have x = r, and y = 0. As the angle 0, which is generated by
the revolution of this line, increases from to 90, y increases
and x decreases; and when OA coincides with OY, we have
x = 0, and y = r. Hence, as the angle increases from to 90,
x decreases from r to 0, and y increases from to r.
As the angle increases from 90 to 180, x decreases in-
creases numerically from to r and y decreases from r
to 0.
As the angle increases from 180 to 270, x increases de-
creases numerically from r to and y decreases increases
numerically from to r.
As the angle increases from 270 to 360, x increases from
to r and y increases decreases numerically from r to 0.
Inasmuch as all changes in sign and magnitude among the
trigonometric functions are directly dependent on the changes
just noted, the following results are now obtained without
difficulty.
58
PLANE TRIGONOMETRY
X-
X X-
Y
Y
Y
Y
39. Sine. As the angle increases from to 90 the numera-
tor of the fraction that expresses the value of the sine increases
from to r, and the denominator r remains constant. Hence
the sine increases from to 1. As the angle increases still
further, the numerator begins to decrease, the denominator still
remaining constant, and at 180 the numerator becomes 0.
Hence as the angle increases from 90 to 180 the sine decreases
from 1 to- 0. As the revolving line enters the third quadrant,
y becomes negative and continues to decrease algebraically,
becoming r when the angle equals 270. Hence in the third
quadrant the sine is negative, and as the angle increases from
180 to 270 the sine decreases from to - 1. In the fourth
quadrant y continues negative ; but as the angle increases y
increases algebraically, and when the revolving line reaches its
original position, y again becomes 0. Hence as the angle
increases from 270 to 360 the sine is negative, and increases
from 1 to 0.
Collecting the above results for the sake of convenience we
have the following statement :
THE APPLICATION OF ALGEBRAIC SIGNS 59
In the first quadrant the sine increases from to 1; in the
second it decreases from 1 to ; in the third it decreases from
to 1 ; in the fourth it increases from 1 to 0.
40. Cosine. In a manner similar to that employed in the
case of the sine, the following results are obtained:
As the angle increases from to 90 the cosine decreases
from - to -, i.e. from 1 to 0. As the angle increases from
r r
90 to 180 the cosine decreases increases numerically from -
T
to ^^, i.e. from to 1. As the angle increases from 180
to 270 the cosine increases decreases numerically from'
to -, i.e. from 1 to 0. As the angle increases from 270
r r
to 360 the cosine increases from - to -, i.e. from to 1.
r r
41. Tangent. The value of the tangent is the value of the
fraction ^ When the angle is very small, the numerator of
x
this fraction is very small, and the denominator is very nearly
equal to r. Hence the tangent of the angle is very small ; or,
as it is commonly expressed, when the angle equals 0, the
tangent of the angle is also equal to 0.
As the angle increases the numerator y increases and the
denominator x decreases. Hence the tangent of the angle
increases. When the angle is nearly 90, the numerator is
very nearly equal to r\ and as the angle approaches 90 the
value of the numerator continually increases, approaching r as
its limit. At the same time the value of the denominator con-
tinually decreases, approaching as its limit. Hence, as
approaches 90 the value of tan can be made to exceed any
finite number previously assigned, no matter how great that
number may be. This is usually expressed by saying that
when the angle is equal to 90, the tangent of the angle is equal
to infinity. Hence, ~
In the first quadrant the tangent increases from - to -, i.e.
from to QO .
In the second quadrant the denominator x becomes negative
while the numerator y remains positive. Hence the tangent
60 PLANE TRIGONOMETRY
of an angle in the second quadrant is negative. When the
angle is but little greater than 90, the numerator is very
nearly equal to r and the denominator is very small, and nega-
tive. Therefore, as the revolving line enters the second
quadrant, the numerical value of the tangent can be taken to be
greater than any negative finite limit previously assigned.
That is, when the angle is in the second quadrant and differs from
90 by an amount that is less than any finite number assigned
in advance, no matter how small that number may be, the
tangent of the angle is negative and is numerically greater than
any finite limit assigned in advance. To express this we shall
'say that tan 90 = oo. It is thus seen that tan 90 will be
called equal to either +00 or oo according as the angle is
approaching the limit 90 from the positive direction, or as the
revolving line is leaving the position at which the angle equals
90 and is just entering the second quadrant. As the angle
increases, the numerator decreases and the denominator, which
is negative, increases numerically. Hence, the tangent decreases
numerically increases algebraically and when the angle
becomes equal to 180, the tangent becomes equal to 0. Hence,
In the second quadrant the tangent increases from - to ,
i.e. from oo to 0.
In the third quadrant both numerator and denominator are
negative. Hence the tangent is positive. The numerator in-
creases numerically from to r, and the denominator de-
creases numerically from r to 0. Hence,
In the third quadrant the tangent increases from - to -^,
i.e. from to oo.
In the fourth quadrant the numerator is negative and the
denominator is positive. Hence the tangent is negative. The
numerator decreases numerically from r to 0, and the de-
nominator increases from to r. Hence,
In the fourth quadrant the tangent increases from -^ to ,
i.e. from oo to 0.
The same restriction is to be observed with respect to the
value of tan 270 as was noted in connection with tan 90.
That is, if the angle is in the third quadrant and is approaching
270 as its limit, the tangent of the angle can be made to exceed
THE APPLICATION OF ALGEBRAIC SIGNS 61
in magnitude any finite positive limit previously assigned. If
it is in the fourth quadrant, the tangent is negative and can
be made to exceed in numerical magnitude any finite limit
_previously assigned. For this reason it is customary to say
that tan 270 = oo .
42. Cotangent. The value of the cotangent is the value of
the fraction -. When the angle is very small, the numerator
y
is nearly equal to r and the denominator is nearly equal to 0.
Hence the value of the cotangent of is infinity. Then,
letting the angle increase, and reasoning in the same manner
as in the case of the tangent, we obtain the following
results :
In the first quadrant the cotangent is positive and decreases
T
from to , i.e. from GO to 0.
r
In the second quadrant the cotangent is negative and de-
r
creases from to , i.e. from to -co.
r
In the third quadrant the cotangent is positive and decreases
r
from to , i.e. from oo to 0.
-r
In the fourth quadrant the cotangent is negative and de-
T
creases from - to -, i.e. from to oo.
-r
Remarks similar to those made in connection with tan 90 and
tan 270 apply to cot 0, cot 180, and cot 360.
43. Secant. The value of the secant is the value of the
fraction -. The numerator remains constant for all positions
x
of the revolving line, while the denominator varies. When the
angle is very small, the numerator and the denominator are
approximately equal. Hence the secant of is equal to unity.
As the angle increases the denominator x decreases, thus caus-
ing the value of the secant to increase. When the angle is
nearly equal to 90, the denominator is nearly equal to 0,
and approaches as its limit. Therefore the secant can be
62 PLANE TRIGONOMETRY
made to exceed any finite limit previously assigned. We shall
express this by saying that sec 90 = <x> . Hence,
As the angle increases from to 90 the secant increases
from - to , i.e. from +1 to +00 .
r
When the revolving line enters the second quadrant the
denominator x becomes negative and begins to increase numeri-
cally decrease algebraically becoming equal to r when
the angle becomes 180. Hence, beginning with a negative
value numerically greater than any finite limit assigned in
advance, the secant increases decreases numerically until it
reaches the value 1. Hence,
As the angle increases from 90 to 180 the secant increases
T T
from -^ to , i.e. from oo to 1.
r
In the third quadrant the denominator continues negative,
but begins to decrease increase numerically as soon as the
revolving line enters the quadrant. At 270 the denominator
becomes 0. Hence,
As the angle increases from 180 to 270 the secant decreases
from to -, i.e. from 1 to oo.
r
In the fourth quadrant the denominator again becomes posi-
tive, and increases from to r as the angle increases from 270
to 360, returning to its original value when the revolving line
completes one entire revolution. Hence,
As the angle increases from 270 to 360 the secant decreases
from -^ to -, i.e. from oo to 1.
r
The same restriction is to be observed with respect to the
value of sec 270 as was noted in connection with sec 90.
That is, if the angle is in the third quadrant and is approaching
270 as its limit, the secant of the angle can be made to exceed
in numerical magnitude any finite negative limit assigned in
advance. If the angle is in the fourth quadrant, the secant is
positive, and can be made to exceed in magnitude any finite
positive limit assigned in advance. We shall express this by
saying that sec 270 = cc .
THE APPLICATION OF ALGEBRAIC SIGNS
63
44. Cosecant. The value of the cosecant is the value of the
fraction -. Remembering that the numerator remains con-
y
stant, and tracing out the changes in sign and magnitude of
the denominator, as in the case of the secant, we obtain the fol-
lowing results :
As the angle increases from to 90 the cosecant decreases
from - to -, i.e. from oo to 1.
r
As the angle increases from 90 to 180 the cosecant increases
7* 7*
from - to -, i.e. from 1 to oo.
r
As the angle increases from 180 to 270 the cosecant increases
y v*
from - to , i.e. from oo to 1.
r
As the angle increases from 270 to 360 the cosecant decreases
'/* '/*
from to -, i.e. from 1 to oo.
r
Remarks similar to those made in connection with sec 90
and sec 270 apply to esc 0, esc 180, and esc 360.
The changes that take place in the sign and magnitude of
the different trigonometric functions are conveniently grouped
together in the following table :
r
FIRST QUADRANT
increases from to 1
SECOND QUADRANT
sine decreases from 1 to
cosine decreases from to 1
tangent increases from GO to
cotangent decreases from to oo
secant increases from x to 1
cosecant increases from 1 to oo
X'
sine
cosine decreases from 1 to
tangent increases from to oo
cotangent decreases from oo to
secant increases from 1 to oo
cosecant decreases from oo to 1
THIRD QUADRANT
sine decreases from to- 1
cosine increases from 1 to
tangent increases from to oo
cotangent decreases from oo to
secant decreases from 1 to oo
cosecant increases from -co to 1
FOURTH QUADRANT
sine increases from 1 to
cosine increases from to 1
tangent increases from oo to
cotangent decreases from to oo
secant decreases from oo to 1
cosecant decreases from 1 to -co
T'
45. After the changes in sign and magnitude have been
obtained for the first three functions, the corresponding changes
for the last three can be found by remembering that the
64
PLANE TRIGONOMETRY
cotangent, secant, and cosecant are the reciprocals of the
tangent, the cosine, and the sine respectively. The student
should verify the above results by obtaining them in this
manner also.
In connection with the general definitions of the trigonometric
functions given on p. 53, it was noted that these definitions
failed in the case of certain functions for certain values of the
angle. These cases have been explained in some detail in Arts.
41-44, and we now have definitions of the tangent, the cotan-
gent, the secant, and the cosecant of any angle from to 360
inclusive ; and hence, by the usual considerations, Arts. 50-57,
definitions of these functions for any angle whatever.
In order that the relations between tan 90 and cot 90, tan
270 and cot 270, sec 90 and cos 90, etc., may be the same as
that between the same functions in the case of other angles we
shall say that = 0, and - = oo . But the student is cautioned
oo
that "oo " is not a number in the usual sense of the word, and
that these two equations are not to be taken literally. They
are used merely for the sake of expressing concisely the result
of a definite limiting process, a process much more complicated
than that of ordinary division.
46. Geometrical representation of the trigonometric functions.
The trigonometric functions are pure numbers, the value in each
case being a ratio between two given magnitudes. These
magnitudes are represented by lines, and if the length of
the revolving line is properly chosen, it is possible to represent
the values of the functions themselves by lines.
Let the revolving line be the
radius of a circle, and let its value
be assumed to be unity.
The sine of the angle AOB is
CD
-. But since OD=\, we may
OD
say
CD CD
Similarly,
THE APPLICATION OF ALGEBRAIC SIGNS
65
vers <9 = 1 - cos = (L4 - 00= AC,
covers = 1 - sin (9 = OG - OE = GE.
For an angle of the second quadrant the so-called "line
values" of the trigonometric functions are obtained as follows:
vers = 1 - cos = 0.4 - 00= OA+00= CA,
covers = 1 - sin = OG- OE=E&.
The change in sign when 00 is replaced by 00 in obtain
ing the value of the versed sine should be noted carefully.
CONANT'S TRIG. 5
66
PLANK TRIGONOMETRY
For angles of the third and fourth quadrants the line values
are obtained in a manner similar to that employed in connec-
tion with angles of the first two quadrants. The figures are
lettered so that the following values hold for both :
oc an
= =
0(J OA
an
esc
OI> OH OH
= - = - OH,
CD oa i
vers 0=1- cos 0=OA- OC= OA,
covers 0=1 - sin = GO- - OE=Ea.
The signs of the trigonometric functions when used as lines
are, of course, the same as when they are used as ratios. It
will be noticed that when the line that represents the sine
extends upward from the axis of abscissas, or horizontal di-
ameter, the sine is positive ; when it extends downward, the
THE APPLICATION OF ALGEBRAIC SIGNS 67
sine is negative. The cosine is positive when the line that
represents its value extends toward the right from the origin,
negative when it extends toward the left. The tangent is
positive when its line extends upward from the axis of abscissas,
or horizontal diameter, negative when it extends downward.
The cotangent is positive when its line extends toward the
right from the axis of ordinates, or vertical diameter, negative
when it extends toward the left. The secant and the cosecant
are positive when their respective lines extend in the same
direction from the origin as the revolving line, negative when
they extend in an opposite direction. The versed sine is con-
sidered as extending toward the right from the foot of the sine,
and the coversed sine upward from the foot of the perpen-
dicular dropped from the extremity of the revolving line to the
vertical diameter. Both are always positive.
The trigonometric functions were originally used as lines ;
and the numerical value was, in each case, the length of the
line in terms of the revolving line, or the radius of the circle,
taken as a unit. There are certain advantages connected with
the use of these line values, but for general purposes the ratios
are so much more convenient than the line values that they
have now come into almost universal use.
47. Limiting values of the trigonometric functions. In dis-
cussing the variation in the values of the different functions
the following limits were found. In the case of the sine the
positive limit was 1, and the negative limit was 1. For the
cosine also these limits were -f 1 and -- 1 respectively. For
the tangent and the cotangent the limits were +00 and -co.
For the secant and the cosecant it was found that the positive
values that these functions could take were comprehended
between + 1 and + oo, and the negative values between 1
and co. Hence, we can make the following definite state-
ment respecting the limits between which the different func-
tions can vary :
The sine can take any value between -f- 1 and 1.
The cosine can take any value between + 1 and 1.
The tangent can take any value between -f oo and oo.
The cotangent can take any value between + oo and GO.
68
PLANE TRIGONOMETRY
The secant can take any value between + 1 and + oo, and
between 1 and oo.
The cosecant can take any value between -f- 1 and + oo, and
between 1 and oo.
From the definitions of the versed sine and the coversed sine
it follows that each of these functions can take any value be-
tween and -f 2.
48. Graphs of the trigonometric functions. The graphs of
the trigonometric functions can be plotted in the ordinary
manner if the values of the angles are taken as ordinates and
the corresponding values of the functions as abscissas.
Sine. For the sine we form the following table of values
from the equation y ~ s j n Xt
In this table the values of the sine are, for convenience in
plotting, given decimally, instead of in the ordinary common
fractious.
/scf
30 00 QO 120 f 50
-A'
X
y
30
.5
45
.71
60
.87
90
1
120
.87
135
.71
150
.5
180
225
- .71
270
- 1
315
- .71
360
415
.71
450
1
495
.71
etc.
etc.
Continuing this table, and plotting the points thus deter-
mined, we find that the graph is a curve consisting of an
infinite number of waves like those in the figure. By using
negative values of the angle we obtain similar waves at the left
of the origin. The curve is called the sine curve, or sinusoid.
THE APPLICATION OF ALGEBRAIC SIGNS
69
Cosine. The graph of the equation
y = cos x
is found in a similar manner. Forming a table of values, and
plotting the points determined by these values, we find that
the cosine curve has the following form.
90
30 60
Y
Tangent. The table of values for x and y formed from the
e( l uation # = tan*
is as follows.
X
y
30
.58
45
1
60
1.73
90
00
120
- 1.73
135
- 1
150
- .58
180
210
.58
225
1
240
1.73
270
CO
300
- 1.73
315
-1
330
-.58
360
390
.58
etc.
etc.
-270
-96
30 60 90
I 2O 150'
I /ISO 270
70
PLANE TRIGONOMETRY
Continuing the table, and plotting the points determined by
the values thus found, we obtain the tangent curve, which con-
sists of an infinite number of branches, each like one of those
in the figure. Negative values of the angle give an infinite
number of like branches at the left of the origin.
Cotangent. The graph of the equation
y = cot x
is similar to that of y = tan #, except that the points where the
different branches cross the #-axis are 90 to the right of those
where the tangent curve branches cross, and the curvature is
toward the right instead of toward the left. The form of the
graph is shown in the following figure.
X
y
CO
30
1.73
45
1
60
.58
90
120
-.58
135
- 1
150
- 1.73
180
CO
210
1.73
225
1
240
.58
270
300
-.58
315
-1
330
-1.73
360
CO
390
1.73
etc.
etc.
Secant. The table of values for the equation
y sec x
can readily be found if it is remembered that sec# is the recip-
rocal of cosx. The graph has the form shown in the first figure
on p. 71.
Cosecant. The graph of the cosecant is similar in form to
that of the secant, but the relative position of the various
THE APPLICATION OF ALGEBRAIC SIGNS
Y
71
-36O -i
branches with respect to the #-axis is different. The graph is
shown in the following figure.
49. Periods of the trigonometric functions. In considering the
changes in value through which the functions pass as the angle
increases, it is seen that the sine, for example, takes all its pos-
sible values, in both increasing and decreasing order of change,
while the angle is increasing from to 360. As the angle
increases from 360 to 720 the values of the sine which were
obtained in the first 360 are repeated, this repetition of values
occurring in the original order. The same cycle of values will
again occur in the next 360, and so on, for each complete
revolution of the generating, or revolving line. The angle
formed by the generating line while this regular recurrence
of values takes place is called the period of the sine; and in
accordance with this result we may say that
The period of the sine is 360, or 2 IT.
A similar course of reasoning shows us that 360 is also the
period of the cosine, of the secant, and of the cosecant.
72 PLANE TRIGONOMETRY
The values of the tangent repeat themselves completely with
each increase of 180 in the angle. Hence,
The period of the tangent is 180, or IT.
The period of the cotangent is the same as the period of the
tangent.
EXERCISE X
1. Trace the changes in sign and magnitude of sin 6 as
varies from - ^ to - ^ ; from - 270 to - 450.
2. Trace the changes in sign and magnitude of cos A as A
varies from TT to 2 TT.
3. Trace the changes in sign and magnitude of tan A as A
varies from 180 to 540.
4. Trace the changes in sign and magnitude of sec A as A
varies from - 90 to -270.
Find the value of each of the following:
5. sin 6 + cos 6 when 6 = 60.
6. sin 2 6 + 2 cos when = 45.
7. sin A 4- tan A when A = 135.
8. sin 60 + tan 240.
9. cos cos 30 + tan 135 cot 315.
10. cos tan 60 - sec 2 30 cot 225.
11. 2 sin 90 sec 2 30 + cos 180 tan 315.
12. 2sec 2 7rcos0 + 3sin 3 ^-csc^.
2 2t
13. COSTT tan !L - sec 2 11^ tan 2 ii
464
14. For which of the following values of 6 is sin 6 cos 6
positive and for which is it negative?
(9 = 0; (9=60; = 120 ; (9=210; (9=240; (9=300;
6 = 330.
THE APPLICATION OF ALGEBRAIC SIGNS 73
15. For which of the following values of is sin -f cos
positive and for which is it negative?
(9 = 135; (9 = 210; 0=300; 0=315; 0=330.
16. Prove that sec 6 - tan 6 0=3 sec 2 tan 2 + 1.
9 79
17. If cos = a ~ _ , find sin and tan 0.
a 2 4- 6 2
18. If tan = a + a , find cos and sin 0.
2a + 1
19. Prove the equation sin0 = # + - impossible for all real
values of x.
20. Prove the equation sec 2 = ^ . impossible unless
CHAPTER VI
TRIGONOMETRIC FUNCTIONS OF ANY ANGLE
50. Functions of an angle 9 in terms of functions of 6.
Let the revolving line OA generate an angle 0, of any mag-
nitude. The final position of OA is, then, in any one of the
four quadrants, as shown in the figures. Also, let the line
OA' generate an angle 0, equal in magnitude to the positive
angle 0, generated by OA.
Take OB= OB 1 , and from B and B' draw perpendiculars
BC, B'C', to XX'. Then are the triangles OBC, OB'O', equal
geometrically, since they are right triangles having the hypote-
nuse and an acute angle of one equal respectively to the hypote-
nuse and an acute angle of the other. Hence the points 6 Y , 6 Y/ ,
coincide, BO=B'Q'< and 00= 0' C' .
TRIGONOMETRIC FUNCTIONS OF ANY ANGLK 75
For convenience, let OB = r, OB' = r f , BC = y, B'C'=y',
00 = x, OC' = z' -, then for each of the four figures we have
sin ( - 0) = ^ = 1 == - sin (9,
r r
cos ( - ) = - = - = cos 0,
r r
tan (-0)=^ = = - tan0,
COt ( 0) = ^ = =r COt
y -y
A\ ^ r
sec ( u) = = = sec
# a;
CSC ( 0) = = ^- = CSC 0.
y ~ y
EXAMPLES.
1. sin ( - 30) = - sin 30 = J,
A/2
2. cos (-45)= cos 45 = -^>
40
3. tan (- 60) = - tan 60 = - V.
51. Functions of an angle 90 6 in terms of functions of 0.
Let the revolving line OA (p. 76) generate an angle 0, of any
magnitude, and at the same time let OA' generate an angle
whose magnitude is 90 0.
As before, take OB OB' , and from B, B', draw perpendic-
ulars BO, B'C', to XX'. The triangles OB C, OB'C', are, in
each of the four figures, equal geometrically. The proof should
be supplied by the student.
With the same notation as in the previous figures we have,
considering only the actual lengths of the lines, and paying
no attention to positive and negative signs, r = >', y = or',
x y' .
76
PLANE TRIGONOMETRY
The following equations then hold true for all possible cases:
sin (90 - 0) = -' = - = cos 0,
r r
cos (90 - 0) = - = =
r r
tan (90 - 0) = V- = - = cot 0,
x 1 y
y
sec (90 - (9) = - = - = esc 0,
x' y
esc (90 - 0) = - t = - = sec 0.
'
NOTE. For the special case that occurs when 6 is an acute angle, these
relations were established independently in connection with the definitions
of the functions of an acute angle of a right triangle (Art. 17, p. 23).
TRIGONOMETRIC FUNCTIONS OF ANY ANGLE
77
52. Functions of an angle 90 + 6 in terms of functions of 6.
Let the revolving line OA generate an angle 0, of any magni-
tude, and at the same time let OA' generate an angle whose
^nagnitude is 90 + 6.
As in each of the previous cases, take OB = OB', and from
B, B', draw perpendiculars BC, B 1 C', to XX'. The triangles
OBC, OB' C' are, in each of the four figures, equal geometri-
cally. The proof should be supplied by the student.
With the notation used in the previous cases we have, con-
sidering only the actual lengths of the lines, and paying no
attention to positive and negative signs, r = r',x = y f , y = x f .
If positive and negative signs are taken into account, these equa-
tions become r = r', x = y', y = x'.
The following equations then hold true for all possible cases:
sin (90 + 6) = '4 = - = cos 0,
r' r
cos (90 + 0) = ^ = ^ = - sin 0,
r r
78
PLANE TRIGONOMETRY
tan (90 + 0) = = = - cot 6,
x' -y
cot (90 + 0) = - = ^ = - tan 6,
'
sec (90 + 9) = r - t = = - esc 0,
x -y
esc (90 + 0) = -= - = sec0.
y x
EXAMPLES.
1. sin (9(T 4- 30) = cos 3CP =
2. cos (90 + 45) = -sin45 = -|
3. tan (90 + 60) = - cot 60 =-
4. cot (90 + 120) = - tan 120 = - (- V3) = V3,
5. sec (90 + 135) = ~ esc 135 = - V2,
6. esc (90 + 150) = sec 150 = - f V3.
53. Functions of an angle 180 9 in terms of functions of 9.
Let the revolving line OA generate an angle 0, of any magni-
tude, and at the same time let OA' generate an angle whose
magnitude is 180 - 0.
TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 79
As in the previous cases, take OB = OB', and from B, B',
draw perpendiculars BO, B'C', to XX'. .The triangles OBC,
OB'C', are, in each of the four figures, equal geometrically.
The student should supply the proof.
With the notation used in the previous cases we have, con-
sidering only the actual lengths of the lines, and paying no
attention to positive arid negative signs, r = r', x == x', y = y' .
If positive and negative signs are taken into account, the second
equation becomes x = x' .
The following equations then hold true for all possible cases :
sin (180 - 0) = ^ = 2 = sin 0,
r' r
cos (180 - 0) = t = - = - cos 6,
r ' r
tan (180 - 0) = tf- = -- = - tan 0,
x' x
cot (180 -6) = - = - = - cot 0,
y y
sec (180 - 0) = r , = - - - - sec 0,
x' x
csc (180 - (9) = - t = - = esc 0.
1 1 y y
EXAMPLES.
1. sin (180 - 80) = sin 30 = \.
2. cos (180 - 60) = - cos 60 - -|>
3. tan (180 - 45) = - tan 45 - 1,
4. cot (180 - 120) = - cot 120 = - f--^2)=4
\ o / o
5. sec (180- 135) = - sec 135 = - (- V2)= V2,
6. esc (180 - 150) = esc 150 = 2.
80
PLANE TRIGONOMETRY
54. Functions of an angle 180 -f in terms of functions of 6.
Let the revolving line OA generate an angle #, of any magni-
tude, and at the same time let OA' generate an angle whose
magnitude is 180 + 6.
As in the cases already considered, take OB = OB', and from
B, B', draw perpendiculars BC, B'C', to XX'. The triangles
OB 0, OB' C' , are, in each of the four figures, equal geomet-
rically. The student should supply the proof.
X-
With the notation used in the previous cases we have, con-
sidering only the actual lengths of the lines, and paying no
attention to positive and negative signs, r = r r , x = x\ y = y*.
If positive and negative signs are taken into account, the last
two equations become x = a;', y = y* respectively.
The following equations then hold true for all possible cases :
sin (180 + 0) = tf- = ^ = - sin <9,
r r
cos (180 + 0) = ~ = = - cos 0,
TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 81
tan (180 + 0) = - = = tan 0,
x' x
cot (180 + 0) = ^ = = cot 0,
y' -y
sec (180 + 0) = ^ = ^~ = - sec 0,
x' -y
esc (180 + 0) = r - t = -- = - esc 0.
y -*
EXAMPLES.
1. sin (180 +30) = - sin 30 = -l
2. cos (180 + 45) = -cos 45=-iV2,
3. tan (180 +60) = tan 60= V3,
4. cot (180 + 120)= cotl20 = -iV3,
5. sec (180 + 1 35) = - sec 135 = - ( - V2) = V2,
6. esc (180 + 150)= -esc 150 = -2.
55. In a manner precisely similar to that employed in the
preceding sections, we can determine the functions of an angle
270 6 in terms of functions of 6. The figures for each
quadrant should be constructed by the student, and the values
obtained, as in the cases which have just been considered.
These relations, true for all values of 0, are as follows :
sin (270 -0) = - cos 0,
cos (270 -0)=- sin 0,
tan (270 -(9)= cot0,
cot (270 -6)= tan0,
sec (270 - 0) = - esc 0,
esc (270 -0)=- sec 0.
56. The corresponding values of functions of an angle
270 -f in terms of functions of can also be obtained in a
manner similar to that employed in the cases already discussed
(Art. 50-54). These values are as follows:
sin (270 + 0) = - cos 0,
cos (270 +0)= sin0,
CONANT'S TRIG. 6
. vV
82 PLANE TRIGONOMETRY
tan (270 + #) = -cot (9,
cot (270 4- 0)=- tan 6,
sec (270 + #)= csc0,
esc (270 + 0)= -sec (9.
EXAMPLES.
sin (270 - 210) = - cos 210 = - ( - \ V3) = J V3,
cos (270 - 150):= - sin 150= - |,
tan(270 + 185) = -cot 135 =-(-l)=l,
cot (270 -2 10 = tan240 = V3
sec (270 + 30) = esc 30 = 2,
esc (270 + 60) = - sec 60 = - 2.
57. Functions of an angle 360 + 9 in terms of functions of 6.
When the revolving line lias generated an angle 360 + #, its
position is the same as that occupied after it has generated the
angle 6. Hence,
The functions of an angle 360 + 6 are the same as the cor-
responding functions of 6.
Also, since the revolving line returns to its initial position
after any number of complete revolutions, in either a positive
or negative direction, it follows that, when n is any positive or
negative integer or zero,
Functions of an angle n x 360 4- 6 are equal to the correspond-
ing functions of 6.
In a similar manner it may be shown that the functions of
n x 360 9 are equal to the corresponding functions of 6.
58. By means of the equations contained in Arts. 50-57,
pp. 74-82, the functions of any angle can be found in terms
of functions of an angle less than 90.
For example, gin 2m > = ^ ( - x m , + 3510)
= sin :JJ31
= sin (270 + 81)
= - cos 81.
Similarly, cos ( - 2058) = cos 2008
= cos (5 x 860 + 258)
= cos -_>:>s
= cos (270 - 12)
= - sin 12.
TRIGONOMETRIC FUNCTIONS OF ANY ANGLE
80
o
By reductions of this kind it is easy to find the values of func-
tions of any large angle, either positive or negative. Multiples
of 360 should first be subtracted, and the remainder of the
reduction performed by the theorems of this chapter.
59. The following table contains the values of the functions
of the angles between and 360 which are of most frequent
occurrence in elementary mathematics.
sine
cosine
tangent
cotangent
secant
cosecant
30
45
GO
90
120
135
150
180
270
1
2
M
ivs
1
H
H
1
2
- 1
1
H
H
1
2
i
2
-I*
-H
-1
()
|V3
i
V3
00
- >/3
-i
>
CO
00
V3
i
H
-H
- 1
-Va
*>
1
N
V2
2
co
-2
-V2
o
-5 VI
- 1
co
00
2
V2
h
1
fvs
V2
2
CO
_ 1
NOTE. In the above table the double sign, which is used wherever the
value co occurs, signifies that either the positive or the negative value is
obtained according as the revolving line approaches the given position from
the one or from the other side. For example, tan 90 = + co if the revolving
line approaches the positive portion of the y-axis from the right, i.e. through
positive rotation; tan 90 = co if the revolving line approaches the same
position from the left, i.e. through negative rotation.
EXERCISE XI
Prove that
1. sin 210 tan 225 + cos 300 cot 315 = - 1.
2. cos 240 cos 120 - sin 120 cos 150 = 1.
3. tan 120 cot 150 + sec 120 esc 150 = - 1.
4. tan 675 sec 540 + cot 495 esc 450 = 0.
84 PLANE TRIGONOMETRY
5. For what values of A between and 360 are sin A and
cos .4 equal? For what values are tan A and cot A equal?
6. What sign has sin A -{-cos A for the following angles?
A = 120; A=135 ; A = 150; A = 300; A=315; A = 690 ;
x 7. What sign has sin A cos A for each of the following
angles? A = 210; A =225; A = 240; A = 300; A =625;
A = .
o
8. What sign has tan A cot A for each of the following
angles? A =60; A = 120; A =135; A = 150; A = 210;
A = 225; A = il^.
6
9. Find all the angles less than 360 that satisfy the fol-
lowing relations :
=_-^?; (6) cos 6 = - ^?; <V) tan 6 = - 1 ;
V3. 2
10. Prove sec (A - 180) = - sec A.
- 11. Prove cot (A - 270) = - tan A.
12. Prove cos A + cos (90 + A) + cos (180 - A)
-cos (270- A)=0.
13. Prove
---- '
tan (180 + A} cot (270 - A) sec (360 -A) A
tan (180 -A) ' cot (270 + A) ' esc (360 + A)
14. Find the value of
sin( A) cos ( A) tan( A)
cos(90 + A) sin (90 + A) cot (90 + A) '
15. Express in simplest form
cos (180 - A) tan (270 - A) t gec ^
sin (180 + A) ' cot (270 + A) '
\
CHAPTER VII
GENERAL EXPRESSION FOR ALL ANGLES HAVING A
GIVEN TRIGONOMETRIC FUNCTION
60. From the definitions of the trigonometric functions it is
evident that a given angle can have but one sine, one cosine,
one tangent, etc.
But the converse statement is not true. A given sine may
belong to any one of an infinite number of angles. The same
is true of the cosine, the tangent, or of any of the other trigo-
nometric functions. This has already been alluded to inci-
dentally (Arts. 50-57, pp. 74-82). Expressions will now be
found for all angles that have a given sine, a given cosine, a
given tangent, etc.
i
61. When the revolving line has made one complete revolu-
tion in either direction, it has generated an angle of 2 TT
radians ; when it has made two complete revolutions, it has
generated an angle of 4 TT radians ; and, in general, when it
has made three, four, five, etc., revolutions, it has generated
an angle of 6 TT, 8 TT, 10 TT, etc., radians.
These statements may be combined into a single expression
by means of the following statement :
When the revolving line has made any number of complete
revolutions in either direction, it has generated an angle of 2 nir
radians, where n is some positive or negative integer or zero.
62. General expression for all angles that have the same sine.
Let XOA (p. 86) be any convenient angle, a, and let XOA!
be equal to TT a. By Art. 53, the sine of XOA = the sine of
XOA' ; or sin a = sin (TT a). Also, TT a is the only other
angle between and 360, or between and 2 TT whose sine is
equal to the sine of a. But (Art. 61) any angle whose initial
line coincides with OX and whose terminal line also coincides
85
86
PLANE TRIGONOMETRY
with OX is represented by the expression 2 mr. Hence, all
angles whose initial lines coincide with OX and whose terminal
lines coincide with OA are represented by the expression
2 ntr + .
Any angle whose initial line coincides with OX and whose
terminal line coincides with OX' is represented by the expres-
sion 2n7r-h TT, or (2w 4- 1) TT. Hence, all angles whose initial
lines coincide with OX and whose terminal lines coincide with
OA' are represented by the expression (2w + I)TT a. These
two expressions, 2 mr + a and (2 n+ I)TT , are both included
in the more general expression ynr -f( l) n ; that is, a is to be
added to any even multiple of TT, and subtracted from any odd
multiple of TT. This will be understood if successive values
are substituted for 71, and the resulting positions of the ter-
minal line noted. This is conveniently done by means of the
following table :
If w = 0, mr +(!)"= ,
n = 1, UTT 4- ( 1 )" = TT a,
71 = 2, W-7T + ( 1 ) 7i = 2 7T + 0,
n = 3, .WTT + ( - !)" = 3 TT - a,
71 = 4, WTT-f- ( !)" = 4-7T -|- ,
n = 5, 7i?r 4- ( l) n a = 5 TT a,
71 = 6, TiTT -f ( 1 )" = 6 7T + ,
n = l, UTT -f ( 1 )" = 7 TT a,
W=8, /Z7T +(-!)= 8 7T+a,
7i = 9, mr + ( 1 ) n a = 9 TT a,
71= -1,
71= -2,
H7T + ( l) n = 7T ,
rc-TT + ( - l) n a = - 2 TT + a.
In this table we observe that
whenever n is an even number, the
expression ( l) n =-f-l, and the
angle that the revolving line has
then generated is (Art. 61, p. 85) a
certain number of complete revolu-
tions plus the angle . If n is an odd
number, the expression ( 1)"= 1,
and the angle that the revolving
GENERAL EXPRESSION FOR ALL ANGLES 87
line has generated is a certain number of complete revolutions
plus a half revolution, minus the angle a. That is,
The expression n-rr +(!)"<* is a general expression for all
-angles that have the same sine as the angle a.
63. In this connection it should be noted that, when n is
any positive or negative integer or zero, 2 n is, by definition, an
even number, and 2 n -f- 1 is an odd number.
64. General expression for all angles that have the same cosine.
The cosine of the angle 360 a, or 2 TT , is equal to the
cosine of the angle a ; and 2 TT a
is the only angle between and 2 TT
that has the same cosine as the
angle a.
But, reasoning in the same manner X-
as in Art. 62, all angles whose initial
lines coincide with OX and whose
terminal lines coincide with OA are
included in the expression 2 mr + a ;
and all angles whose initial lines coincide with OX and whose
terminal lines coincide with OA' are included in the expression
2 mr a. Hence,
The expression 2 mr a is a general expression for all angles
that have the same cosine as the angle a.
65. General expression for all angles that have the same
tangent. The tangent of 180 + , or TT -f- , is equal to the
tangent of a ; and TT + a is the only angle between and 2 TT
that has the same tangent as the angle a (see fig. p. 88).
All angles whose initial lines coincide with OX and whose
terminal lines coincide with OA are included in the general ex-
pression 2 mr + a, and all angles whose initial lines coincide
with OX and whose terminal lines coincide with OA' are in-
cluded in the general expression (2 n + I)T + <* But, since 2 n
signifies only even integers, and 2 w -f 1 only odd integers, while
n includes all integers, both even and odd, the two expressions,
2 HTT + a and ('2 n + I)TT -|- , can be combined as follows :
The expression HTT -{- a is a general expression for all angles
that have the same tangent as the angle a.
88
PLANE TRIGONOMETRY
66. Since cot is the reciprocal of tan a, the general expres-
sion for all angles that have the same cotangent as .the angle a
is HTT + a.
Since sec a is the reciprocal of
cos a, the general expression for all
angles that have the same secant as
-X the angle a is 2n7ra.
Since esc a is the reciprocal of
sin , the general expression for all
angles that have the same cosecant
as the angle a is mr + ( l) n a.
67. In the following examples, and in practical work generally,
the smallest positive value of a is taken. This is done simply
for convenience, the results just obtained being perfectly
general.
EXERCISE XII
1. What is the general expression for all angles whose sine
is J?
The smallest positive angle whose sine equals is 30, or -.
6
.-. = is the smallest positive angle,
and (Art. 62) 0= WTT+( 1)" is the general expression for all angles
whose sine is }.
2. What is the general expression for all angles whose
tangent is V3 ?
The smallest positive angle whose tangent is x/3 is 60, or ^ .
3
.-. ^ is the smallest positive angle,
o
and (Art. 65) 6 = mr + ^ is the general expression for all angles whose
tangent is V3.
3. What is the general expression for all angles whose cosine
is J, and whose tangent is V3?
The only angles between and 360 whose cosine is -\ are 120 and 240.
The only angles between and 360 whose tangent is - V3 are 120 and
300 .
The only angle that satisfies both these conditions is 120, or | TT.
.-. 0= 2n>rr+ ITT.
Another general expression for the same angles is (2 n -f 1) TT | TT.
GENERAL EXPRESSION FOR ALL ANGLES 89
Find the general value of which satisfies each of the follow-
ing equations :
4. sin0=lV2. 13. tan0 = JV3.
5. sin 0=1. 14. sin20 = .
6 . sin0=-|V8. 15 COS 20 = |.
7 ' s in * = -J- 16 . 8 tan 2 = 1.
8. COS0=W3.
17. 3 sec 2 = 4.
9. cos0 = - 1 |V2.
18. cot 2 = 1.
10. cos = 0.
11. cos0=-l. 19 - tan 2 0=2 sin 2 0.
12. tan = 1. 20. 2 tan 2 = sec 2 0.
21. What is the general value of that satisfies both of the
following equations ?
sin = J V3, and cos = J.
22. What is the general value of that satisfies both of the
following equations?
sin = l, and cos = J V3.
In the following five examples, show that the same angles
are indicated by both the given expressions.
23. mr + -, and
24. W7r +(_
25. M7T ^, and WTT + TT
6 6
26. WTT + ^, and mr +
o o
27.
\\
90 PLANE TRIGONOMETRY
68. An equation involving trigonometric functions of an
unknown angle is called a trigonometric equation.
The solution of a trigonometric equation involves the de-
termination of all angles that satisfy the equation.
In solving a trigonometric equation, the smallest positive
angle that satisfies it should first be determined, and then the
general value should be found for all angles that satisfy it.
This has been illustrated in the examples of Exercise XII, and
will be still further shown in those of the following set.
EXERCISE XIII
1. Solve the equation cos 2 6 + 2 sin 2 6 = J.
This may be written
2-2cos 2 = .
.-. cos' 2 = f .
The smallest positive angle whose cosine is - V3 is 30, or -.
Therefore, using the positive result, = 2 rnr -
Alp, the smallest positive angle whose cosine is - \/3 is 150, or f TT.
Therefore, using the negative result, = 2 rnr | TT, or (2 n + 1) TT -.
6
These two sets of values may be combined in the single expression
2. Solve the equation 2 cos 2 V3 sin + 1 = 0.
This may be written
2-2 sin 2 - V3 sin 6 + 1 = 0.
2 sin 2 + V3 sin 0-3 = 0.
Factoring, (sin + V3) (2 sin - V3) = 0.
.-. sin = - V3, or sin = | V3.
The sine of an angle cannot be numerically greater than 1; therefore, the
first equation gives no solution.
The smallest positive angle that satisfies the equation
sin = \ V3,
is = 60, or 5,
o
and (Art. 62) the general expression for the value of all angles that have the
same sine as 60 is TT
8 = n7r+ (~l) n |-
Therefore, the most general expression for all angles that satisfy the
original equation is mr + ( l) n f
GENERAL EXPRESSION EOR ALL ANGLES 91
3. Solve the equation tan 4 cot 3 6.
This may be written tan 4 = tan I * - 3 0\ by Art. 51,
= tan (nir + f - 3 6\ by Art. 65.
.-. 4 = riTT + - - 3 0,
Solve the following equations, finding the general value of
in each case :
4. 2 sin 2 0- cos = 1. 16. sin 30 = sin 90.
5. tan 2 + sec = 1. 17. cos 6 = cos 2 0.
6. cot 2 0- esc = 1. 18 C os40 = cos50.
7. cos 2 - sin 0=1.
19. cos mv = cos r&0.
8. 2 sin 2 + 3 cos = 0.
20. cos 4 = sin 2 0.
9. 2 cos 2 + cos 0= 1.
21. sin 4 .0 = cos 2 0..
10. sin 2 - 2 cos + I = 0.
11. 3sin 2 0-2sin0=l. 22. "tan 2 = tan 30.
12. 23> cot 5 6 = COt 2
13. csc20-cot0=3. 24 - tan 4 = cot 5 0.
14. tan 2 + cot 2 = 2. 25. tan ^0 4- cot nO = 0.
15. sin 50= sin 2 0. 26. tan 2 tan = 1.
M
CHAPTER VIII
RELATIONS BETWEEN THE TRIGONOMETRIC FUNCTIONS
OF TWO OR MORE ANGLES
69. Sine and cosine of the sum of two angles. Let x and y
be acute angles, and let x -f y be either acute or obtuse. In
both figures the lettering is so arranged that the following
demonstrations apply to either case,
o
D F
o F
= Z
From C, any point in OB, draw CD _L XX', and CE JL OA ; and from E
draw EH \\ XX' and EFXX'.
Since Z x - Z OEH = 90 -
..-. Z.x = Z.HCE.
T\f~V
Then we have sin (x + y) = =- : -
^ J) OC
Also,
OC
= oc + oc
~ ~OE ~OC + ~CE OC
= sin x cosy + cosZ HCE sin y.
. . sin (a? + ?/) = sin x cos y + cos a? sin y.
OP
OC
OF - DF
(1)
cos (x + y} =
OC
= OF HE
OC OC
= OF OE HE CE
OE OC CE OC
cos x cos y sin Z HCE sin y.
cos (x + y) = cos as cos y sin a? sin ?/.
92
(2)
RELATIONS BETWEEN TWO OR MORE ANGLES 93
70. The above proofs are given only for the case when both x
and y are acute.
To prove the formulas true for all values of x and y we
_proceed as follows :
Let x and y be acute angles, and let x l = 90 + x; .then we have (Art. 52),
sin x } = cos x, and cos a^ = sin x* (1)
Then, sin (aj + #) = sin (90 + x + y)
= cos (z + ?/), (Art. 52) (2)
where # and y are both acute angles.
But (Art. 69, p. 92) when # and y are both acute angles,
cos (x + y) cos x cos y sin # sin #.
Substituting in this equation the values given in (1) and (2), we have
sin (a?i + y} sin a?i cos y + cos a?i sin ?/. Q.E.D.
In like manner, cos (x^ + ?/) = cos (90 + a; + ?/)
= -sin(a; + y), (Art. 52) (3)
where x and y are both acute angles.
But (Art. 69, p. 92) when x and y are both acute angles,
sin (a: + ?/) = sin # cos ?/ cos x sin y.
Substituting in this equation the" values given in (1) and (3), we have
cos (a?i + t/) = cos xi cos ?/ sin oc\ sin ?/. Q.E.D.
Formulas (1) and (2) (Art. 69, p. 92) have now been
proved for the case when x is obtuse and y is acute.
Letting y^ = (90 + /), and proceeding in the same manner,
we can establish these formulas for the case when both angles
are obtuse.
Then, letting a; 2 =90 + a: 1 , ^ = 90 + ^, x s =90-\-x^ etc.,
and proceeding in a precisely similar manner, we can establish
the formulas for all possible values of x and y.
71. Sine and cosine of the difference of two angles. Let x and
y be two acute angles, placed as represented in the figure. It
is here assumed that x > y.
From C, any point in the final position of the
generating line OA, draw CD 1 OX and CE A. OB.
Prolong DC, and from E draw EH \\ OX, inter-
secting DC produced in H.
Since /
X
94 PLANE TRIGONOMETRY
Then, sin (x - y) = -
= FE- HC
OC
= FE OE HC EC
OE OC EC OC
sin x cos y cos Z ECU sin y.
.-. sin (a? -y}= sin a? cosy cosa?siny. (1)
In like manner, nn
OF +
06'
= OF EH
OC OC
= OFOEKH EC_
OE OC EC OC
= cos x cos ij + sili Z EC/7 sin y.
. : cos (a? y ) = cos . cos ?/ + sin x sin ?/. (2)
These proofs have been given on the assumption that x > y.
To prove that they are true when x < y, we proceed as follows :
sin (x -, y} = sin [ - (y - x)]
= sin (?/ #), (Art. 50, p. 75)
= sin ^ cos -f cos y sin x,
or, rearranging the terms and the factors in each term,
sin (a? y) = sin a? cos 2/ cos a? sin?/. Q. E. D. (3)
In like manner,
cos (x - y} = cos \_-(y - x~}~\
- cos (y - x) (Art. 50)
= cos y cos x -f- sin ?/ sin #,
or, rearranging the factors in each term,
cos (x y} = cos x cos y + sin a? sin y. Q. E.D. (4)
72. The formulas of Art. 71 have now been proved for all
cases when x and y are both acute angles. To prove that they
are true for all possible values of x and ?/, we proceed as
follows :
Let x and y be acute angles, and let x\ 90 + x. Then,
sin X L = cos x, and cosa,^ = sin a:. (1)
Then we have sin (x l y) = sin (90 + x y)
= cos(x - y). (Art. 52) (2)
RELATIONS BETWEEN TWO OR MORE ANGLES 95
But since .r and // are acute angles,
cos (x y} = cos x cos y + sin x sin y. (3)
Substituting in (3) the values given in (1) and (2), we have
sin(a?i y} = sin a?i cosy - cos^isiny. Q. E. D. (4)
In like manner,
cos (.r t - y) = cos (90 + x - y) = - sin (x - y). (5)
But since x and y are acute angles,
- sin (x - y} = - (sin x cos y - cos x sin y).
(Art. 71, p. 94) (6)
Substituting in (6) the values given in (1) and (5), we have
cos (a?i y) = cos a?i cos y + sin a?! sin t/. Q. E. D. (7)
Formulas (1) and (2) (Art. 71, p. 94) have now been proved
for the case when x is an obtuse angle and y is an acute
angle.
Letting y 1 = 90 + ?/, and proceeding as before, we can es-
tablish these formulas for the case when both angles are
obtuse.
Then, letting x 2 = 90 + ^, # 2 = 90 + y v x 8 = 90 + x v etc.,
and proceeding in a precisely similar manner, we can establish
the formulas for all possible values of x arid y.
EXERCISE XIV
1. Find the value of sin 75.
sin 75 = sin (45 + 30)
= sin 45 cos 30 + cos 45 sin 30
= _L ^1 + JL1
V2 2 \/:2 2
= V3 + 1
2\/2
2. Find the value of sin 15.
si nl 5 = sin (45 -30)
= sin 45 cos 30 - cos 45 sin 30
1 \/3 1 1
~ \/2 2 V2 2
= V3-1
2V2
96 PLANE TRIGONOMETRY
3. Find the value of cos 105.
cos 105 = cos (60 + 45)
= cos 60 cos 45 - sin 60 sin 45
= i i Va i
2 V2 2 V2
2 "s/2
4. If sin a = | and sin /3 = ^|, find sin ( /3).
5. If sin a = | and cos ft = if, find cos (a -f /:?).
^ 6. If cos a = &, and cos ft = -|, find cos (a /3).
Prove that
7. si
8. sin 105 + cos 105 = cos 45.
9. sin 75- sin 15 = cos 105 + cos 15.
10. sin (45 - 6) cos (45 -</>)- cos (45 - 0) sin (45 - 0)
= sin ((/> 6).
HINT. Let # = 45 - and y = 45 - <. Then compare with (1),
Art. 66. The converse application of the x-y formulas, as illustrated by
this example, is of frequent occurrence.
11. sin (45 + 6) cos (45 - () + cos (45 + 0) sin (45 - 0)
= cos (6 $).
13. cos (30 + ) cos (30- a) + sin (30 + ) sin (30 - )
= cos 2 a.
14. cos a cos (/3 ) sin sin (/3 a) = cos /3.
15. sin (n + 1) sin (w 1) + cos (n + 1)<* cos (w- 1)
= cos 2 .
16. sin (n + l)a sin (n + 2)a -f cos (n + 1) cos (n + 2)
= cos a.
17. sin (a - jB + 15) cos (/3 - + 15)
- cos (a 0+ 15) sin (/ a + 15) = sin (2 2 yS).
RELATIONS BETWEEN TWO OR MORE ANGLES 97
The following examples are of especial importance, and are
often used as standard formulas.
18. sin 75 = cos 15 =
_
2V2
19. sin 15 - cos 75 -
2V2
20. cos (a? + y) cos (a? y) = cos 2 a? sin2 y.
21. sin (a? + -*/) sin (ac y} = cos 2 y cos 2 sc.
73. Tangent of the sum and of the difference of two angles.
For all values of x and y we have (Art. 69)
sin (x + y) = sin a; cos y + cos a: sin y,
and cos (a; + y) = cos # cos y sin a; sin y.
tan Q + y) ^ sin *' cos ^ + cos x sin ^ .
cos x cos y sin x sin ?/
Dividing both numerator and denominator by cos a: cosy, we have
sin x cos y cos ar sin y
,. cos a: cos \i cos a: cos y
tan (a? -f y) =
1 tan a? tan y
In like manner, .
sn arsn
cos a: cosy
(1)
tan * -
cos (a: - y)
_ sin x cos y cos x sin y
cos x cos y + sin x siti y
sin a; cos y cos a: sin y
_ cos a; cos y cos a: cos y
~~ cos a: cos y sin a: sin y
cos a; cosy cos a: cosy
sin a: _ sin y
cos a: cos y
. (2)
1 -f tan 35 tan i/
CONANT S TRIG.
98 PLANE TRIGONOMETRY
74. Cotangent of the sum and of the difference of two angles.
For all values of x and y we have
Expanding cos (x -f ?/) and sin (x -f- ^), dividing both
numerator and denominator by sin x sin ?/, and reducing, we
- (1)
coty
In a similar manner it can be proved that
uin*^ \j\rii y ~t /"O\
coty cota?
75. Formulas (1) and (2), Art. 69, (1) and (2), Art. 71,
(1) and (2), Art. 73, and (1) and (2), Art. 74, are often re-
ferred to as the addition and subtraction formulas. The addi-
tion formulas are sometimes known as the x + y formulas, and
the subtraction formulas as the x y formulas. When refer-
ence is made to both groups together, the general expression,
"the x-y formulas," is often employed.
76. From the formulas for the functions of the sum of two
angles the formulas for the functions of the sum of three angles
are at once obtained, as follows :
sin ( x + y + z) = sin [(x + y) + z]
sin\(.r + y) cos z -f cos (x + /y) sin z\
= (sin x cos y + cos x sin y) cos z
+ (cos x cos// sin a: sin y} sin z.
.*. sin (x + y + z) = sin a; cos// cos z -f cosx siny cosz
-f cos x cosy sin z sin x siuy sinz. (1)
In like manner it can be proved that
cos (x + y + z) cos x cosy cosz - cosx siny sinz
sin x cosy sin z sin x sin y cosz, (2)
and that
cos (x + y + z)
_ tan x + tan y + tan z tan x tan /y tan z ,.,,
1 tan x tan y - tan x tan z tan y tanz
^
RELATIONS BETWEEN TWO OK MORE ANGLES 99
EXERCISE XV
1. If tan = | and tan ft 1, iind tan ( + ft).
2. If tan a = | and tan /3 = f f, find tan (_).
3. If tan = f and cot ft = -f%, find cot (a -f /8).
4. If tan = | and ft = 45, find tan ( + /3).
5. If tan = -|- and tan /8 = , find tan (2 a -f ft).
6. If tan a = ^ and tan /9 = - - - , find tan (a + ft).
n + l 2 W -f 1
7. If tan a = | and tan ft = Jj, prove that a -f ft 45.
The next four examples are of especial importance, and are
tan 75" = cot 16 = 2
often used as standard formulas.
8.
, , cot (9-1
12. cot - + =
10. tanl5-cot75=2-V3.
14. tan [ -f
15. Prove the identity cos ( 4- ft) cos ft -+- sin (a + ft) sin ft =
cos a.
HINT. Let a + /? = x and /? = y. Then compare with (2), Art. 69.
Many of the remaining examples can be worked without difficulty by
applying the addition or subtraction formulas directly.
16. sin 2 a cos a -f- cos 2 a sin a = sin 3 a.
17. sin 3 a cos cos 3 a sin = sin 2 a.
18. cos 3 a. cos 2 a sin 3 a sin 2 a = cos 5 a.
sec esc (x.
20. sin (60 + ) cos (30+ a) - cos (60 + a) sin (30 + a) = J.
21. tin2+tan = tftn 3 g>
1 tan 2 tan a
22. - =tan2c .
1 Un( + /8)tan( - ft)
100 PLANE TRIGONOMETRY
tana- tu .-
23 .
1 + tan a tan (a
cot 3 a cot 2 a 4- 1
24. = cot a.
cot 2 a cot 3 a
25. tan 2 - tan 6 = tan sec 2 0.
26. sec 2 6 = 1 + tan 2 tan 0.
27. csc20 = cot0-cot20.
tan 3 6 tan 2 tan 4 6 tan 3
tan 3 tan 20 1 + tan 4 tan 3
4tan0
29. tan (45 4- 0) - tan (45 - 0) =
JL tan u
sin (# 4- y) cot # 4- cot y
3O. - ^ - = -
cos (x T- y) 1 4- cot # cot y
77. The algebraic sum of two sines or of two cosines in the
form of a product. For all values of x and y we have (Arts. 69
sin (x 4 ?/) = sin x cosy 4 cos x sin y,
and sin (a; y) = sin a: cos y cos # sin y.
Adding and subtracting, we have
sin (x 4 y) 4 sin (a; y} = 2 sin # cos y, (1)
and sin (x 4 #) sin (x #) = 2 cos x sin y. (2)
Also (Arts. 69 and 71),
cos (a: 4 y) = cos x cosy sin a; sin y,
and cos (a? y) = cos a: cos y 4 sin re siny.
Adding and subtracting, as before, we have
cos (x 4 y) 4 cos (ar ?/) = 2 cos a: cos y, (3)
and cos (a; 4 y) cos (a; y~) = 2 sin a; sin y. (4)
Let a: 4- y u, and x y = v.
Solving these two equations for x and y,
.
Substituting these values of a: and ?/ in (1), (2), (3), and (4), we have
smu 4 sinv = 2sm^^cos^^$ (5)
**
A^--\r
sin w - sin v = 2 cos ^ + ^ sin^-^T (6)
cost* + cost? - 2cos^pcos^p; (7)
cost* - cos v = - 2sin^^ sin-^. (8)
RELATIONS BETWEEN TWO OR MORE ANGLES ; I01
These formulas are among the most important of all' the
formulas of trigonometry. The student should commit them
carefully to memory, and become perfectly familiar with their
-application. They will sometimes be referred to as the u-v
formulas.
As illustrations of the manner in which certain expressions
can be simplified by the application of one or more of these
processes, the following examples are given :
2.
sin 75 - sin 1 5 _
2 2
= 2 cos 40 sin 30
= cos 40.
75 + 15 . 75 -15
2 cos sin
cos 75 + cos 15 n _ _75 + 15 75 - 15
cos _
2
_ 2 cos 45 sin 30
~ 2 cos 45 cos 30
= tan 30
= iV3= 0.57735.
(sin 6 + sin 2 0)(cos 2 - cos 4 6)
(sin 5 6 + sin 0)(cos 3 - cos 5 0)
_ (2 sin 4 cos 2 0)(2 sin 3 sin 0) =
~ (2 sin 3 cos 2 0) (2 sin 4 9 sin 0)
EXERCISE XVI
Prove the following relations :
l. sin 70 + sin 50 = V3 cos 10.
2 sin80-sin60 = tan ^ 3 . sin 2 + sin 6
cos 8 6 + cos 6 6 cos 2 6 + cos 6 6
sinSg-rintf 8g 4g-
sin 2 ^ + sin 2 fl = fan ^ ^ ^ _ B
sml A- sin 2 5
K.)L' . PLANE TRIGONOMETRY
cos cos 20 2 * cosJ.+ cos
cos B cos ^4. 2
9. sin ( J. + JB) + cos (A - J5) = 2 sin(45 + 5) cos (45 -
cos b A cos 3 ^4 , cos 2 A cos 4 ^4 si n A
10.
sin <) A sin 3 .A sin 4^4 sin ^ A cos 4 ^4 cos 3 A
11. sin (60 + A)- sin (60 - A) = .sin A
12. cos (30 - 0) + cos(30 + 0; - V^ cos 0.
13.
14 sin 6 + sin 3 + sin 5 + sin 7 = t 4 ^
cos H- cos 30 + cos 50 + cos 7
15 sin - sin 50 + sin 9 - sin 18 = cot Q
cos cos 5 cos 90 + cos 13
16 _
sin x sin ^/
cos x -\- cos v ^ ^
17. = cot ' cot
cos x cos y 2 2
18. cos 3 + cos 5 + cos 7 + cos 150=4 cos 4 cos 5 cos 6 0.
19.
20. sin 50 + sin 10 -sin 70 = 0.
2
sin (3 A + i?) + sin(J. 3 ^
22. sin 80 + sin 70 - sin 10 - sin 20 = -t sin 40 4- sin 50.
23. cos x + cos 2 # + cos 4 # + cos 5 # = 4 cos ~ cos cos 3 r.
RELATIONS BETWEEN TWO OR MORE ANGLES 103
24. sin O + P + 7) + sin (a - yS - 7) + sin (a + /3 - 7)
-f- sin ( @ -f 7) = 4 sin a cos /? cos 7.
25. sin 2 a + sin 2 /3 + sin 2 7 sin 2 (a -f ft -f- 7)
= 4 sin (/3 + 7) sin (7 + a) sin (a
= cos 3 6
cos 3 6 + 2! cos 5 6 + cos 7 ~ cos 5
sin 30 + 2 sin 50 + sin 10 . c
27. __^ =sm 5
S1H0 + 2 sin 30 + sin 00
sin J + # - 2 sin A + sin Qi -
28.
cos (^4 + ^) 2 cos J. + cos ( A B}
29.
cos(
sin (x + y -|- 2) + sin( ^ + y + ^; sin (2: y-\-z)+ s\
= cot ^.
so. cos 20 + cos 100 + cos 140 =0.
78. The product of two sines, of two cosines, or of a sine and a
cosine expressed in the form of an algebraic sum.
In (1), (2), (3), and (4), Art. 77, the u-v formulas are ex-
pressed in a form which is quite as important as that already
considered, and which is so convenient, and of such frequent
application that the formulas are here reproduced in that form.
Using the left for the right and the right for the left members,
they are
2 sin oc cos y = sin (& + y} + sin (oc - y) ; (1 )
2 cos ac sin y = sin (x + y} - s :n (& - y} ; (2)
2 cos w cos y = cos (oc + y} + cos (x y} ; (3)
2 sin x sin y = cos (a? + y) - cos (x y). (4)
These formulas are the converse of the u-v formulas, and
may be conveniently referred to by that name. The two
groups taken together are useful in solving problems and in
performing investigations which, without them, could be
handled only with the greatest difficulty.
104 PLANE TRIGONOMETRY
EXERCISE XVII
1. Express in the form of a sum or difference 2 sin 6 sin 4 6.
2 sin 60 sin 4 = - (cos(60 + 40)&cos(60 - 40))
= - (cos 100 -cos 20)
= 00820- COS 100,
2. Express in the form of a sum or difference cos (A 2B)
sin (04 + 2 B).
cos(^ -2B)sin(A +25)= | (sin 2/1 - sin (-45))
= (sin 2^4 + sin 45).
3. Find the value of 2 sin 75 sin 15 .
2 sin 75 sin 15 = cos (75 - 15) - cos (75 + 15)
= cos 60 - cos 90
= i-o
= *
Express as a sum or difference the following:
4. 2 sin 60 cos 26. 30
8. COS - COS .
5. 2 cos 40 sin '20.
6. cos ? sin ?*. 9- 2 sin (2 ,1 + 10 cos (A-*).
2 2
-* ** 10. 2 cos 3 J. cos 01 2*).
o u 7 "
1 T C "T* 11. sin (60 + 0) cos (60 - 0).
Prove the following identities :
12. cos (120 + 0) cos (120 - 0) = (2 cos 2 - 1).
13. cos (30 - 0) cos (60 - 0) = | (2 sin 20 + V 3).
14. sin (120 - 0) cos (60 + 0) = J- (sin 60 - 2 0).
15. sin (0 + 45) sin (0 - 45) = - | cos 2 0.
16. cos 3 sin 2 - cos 4 sin = cos 2 sin 0.
17. sin 3 sin 6 + sin sin 2 = sin 4 sin 5 0.
18. sin 20 cos + sin 6 cos = sin 3 cos 2 + sin 50 cos 20.
19. cos (40 - 0) cos (40 + 0) + cos (50 + 0) cos (50 - 0) =
cos 2 0.
RELATIONS BETWEEN TWO OR MORE ANGLES 105
20. sin A cos {A + B) cos A sin (A B) = cos 2 -4 sin B.
3 7T 4 7T . 47T, 10 7T A
21. 2 cos - cos -f cos - -f- cos = 0.
22. 4 sin A sin ^ sin Q = sin (5 + (7 J.) -f sin ((7+ .A B)
+ sin ( j. + B - Q) - sin (A + ^ -f 6 Y ).
cos 3 ^4. sin 2 A cos 4 J. sin A _ _ t o >4
cos 5 A cos "2 A cos 4 ^4. cos 3 .A
24. 4 sin sin (60 + 0) sin (60 - (9) = sin 3 0.
25. 4 cos cos + cos Z - = cos 3 6.
v 8 / \ o
26. sin 20 sin 40 sin 80 = \ V3.
27. cos 20 cos 40 cos 80 =i.
CHAPTER IX
FUNCTIONS OF MULTIPLE AND SUBMULTIPLE ANGLES
79. Functions of an angle in terms of functions of half the angle.
If in the addition formulas, Arts. 69, 71, 73, and 74, we put
x = y, we have
sin (x 4- x) = sin x cos x -\- cos x sin a?,
cos (x + x) = cos x cos x sin x sin #,
tan (* + *)= *" * + **"*,
1 tan x tan #
and c
cot x -f- cot x
sin 2 x = 2 sin a? cos a? 5 (1)
cos 2 = cos 2 a? - sin 2 a?; (2)
; (3)
(4)
In these formulas 2# may have any value whatever; or, in
other words, 2 # is any angle whatever.
Hence, these formulas are to be regarded as formulas for
expressing the values of functions of an angle in terms of
functions of half the angle. They may also, of course, be re-
garded as formulas for expressing the functions of twice an
angle in terms of functions of the angle itself.
80. If we let 2x= 0, we have the formulas in the following
useful form : ~ ^
sin6 = 2 sin - cos -; (1)
10<5
MULTIPLE AND SUBMULT 1PLE ANGLES 107
cos = cos 2 - - sin 2 - (2)
2 2
2cos 2 --l.
2tan-
(3)
1 - tan 2 -
2
cot 2 - - 1
cot 6 = -
2cot-
2
81. Functions of an angle 3 & in terms of functions of oc.
If in the addition formulas we put y = 2 x, we obtain expres-
sions for the value of functions of 3 # in terms of functions of
#, as follows :
sin (a; + 2 a;) = sin x cos 2 x 4- cos x sin 2 x
= sin x (cos 2 a; sin 2 a:) + cos a: 2 sin a: cos a;
= sin x (1 2 sin 2 a;) + 2 sin x (1 sin 2 a;)
= sin x 2 sin 3 a, 1 + 2 sin a: 2 sin 3 a:.
.-. sin 3 3C = 3 sin a? 4 sin 3 cc. (1)
In like manner,
cos (a; + 2 a:) cos a: cos 2 a; sin x sin 2 a:
= cos a: (cos 2 a; sin 2 a:) sin x 2 sin a? cos a:
= cos a: (2 cos 2 a; 1) 2 (1 cos 2 a;) cos a;
= 2 cos 3 x cos x 2 cos x + 2 cos 3 a:.
.. cos 3 a; = 4 cos 3 x 3 cos a% (2)
Also, ten 3*= -
1 - tan z tan 2 x ^ _ fcftn ^ 2 tan x
1 tan 2 a;
3
3 tan a? ^tan 3 x XON
/. tan 3 a? = - - . (3)
In a similar manner it is possible to obtain formulas for the
functions of higher multiples of x in terms of functions of x.
108 PLANE TRIGONOMETRY
82. Functions of an angle expressed in terms of functions of
twice the angle.
Since cos 2 x = 1 - 2 sin 2 x,
we have 2 sin 2 ;r = 1 cos 2 x.
.. sin a? =
Also, cos 2 x = 2 cos 2 a: 1,
2 cos 2 s = 1 4- cos 2*.
1 + cos2ag *2\
Dividing (1) by (2) we have
(3)
These formulas are often given in the following form, where
*=-
(5)
In this form they are to be regarded as formulas for express-
ing the values of functions of a half-angle in terms of functions
of the angle itself.
The magnitude of the angle determines which of the two
signs preceding the radical is to be employed.
EXERCISE XVIII
1. If sin 6 = 1, find sin 2 9 and sin 3 0.
2. If sin = \, find cos 2 d and cos 3 0.
3. If cos 6 = |, find sin 20 and cos 30.
4. If tan 6 == l, find tan 2 and tan 3 0.
5. If tan 6 = 1, find sin 2 and tan 3 6.
MULTIPLE AND SUBMULTIPLE ANGLES
Prove the following identities :
6. cos 4 sin 4 = cos 2 0. cot tan
1
a an COt + tail
7. tan 4- cot = 2 esc 2 0. )t
j
^T
tan -2 cot 2 0-1
H5j
T
^
The next six equations are especially important, and may^e
regarded as standard formulas.
1
11 f S in 6 4 D S 6 Y 11 -in 9 18 r*e 2 ~ sec2 * s\\
^
' ^"g h *ij - sec 20
12. tan 9^ 8in26 ;,* sec0-.l
J
l + cos29 1 2 - 2sec
l
TO * A S * n ^ " A f A\
, XI
i oA* 2O tanf 77 "-!- 1 2DI 1
i T\r
1 COS 2 U ^ u - ^ /i ttl11 I i 1 "c\ I' i
p
1 - sm \4 2/ ^| V <*
9 i _ eos Q
}
1 4- t*in i i Z> ; ***
1
1
6
2 sin 9 \ & ta "V9
?-
-|
i
, 9 _ 1 4- cos 9 eos (Jby ^/\ ^\^jf
2 sin 9 2
/A e\ 2 cos 20 _, ,, ro ^
16. [ sin cos ] =1 sin 9. ^ i i ^i o /}
4
\ " */ ^ s^^ \
\*
/i &i
/-> o0 1 -h sec sin 3 cos 30 /-
*
2 sec sin cos
11
^
1 cos J. + cos B cos (A + ./?) ^4 ^^-B
&
1 + cos 4 cos B cos ( A + .#) " 2 ~ 2
r
~j
25. tan (45 + 0) + tan (45 - 0) = 2 ^ '
N^
26. tan 2 - sec sin = tan sec 2 0.
^x
v i
^*.
1
\
T*,
siii 2 -sm 2 /3 Hn / rt + N T
* . /i /-> """ Lail V 1 r*x
sin a cos a sin p cos p ** \
f
i
OQ cos + sin cos sin _ 9 ^ o ^
h
'-O
cos sin cos + sin i'
V ^
- L
110 PLANE TRIGONOMETRY
2g cos(0 + 15) _ sin (9 - 15) = 4 cos 2
sin (0 + 15) cos (6 - 15) 1 + 2 sin 20
30.
32
sin 20 + sin
;- pos :; +si "^=ta,
1 + cos 20 + sin 20
- tan + 1 = 1 - sin 2
tan + 1 cos 2
33 sin 2 1 - cos =
1 - cos 2 " cos '" 2 '
34 sin (n + 1)0 + sin (ft- 1)0 + 2 sin w0 = cot
cos (n 1)0 cos (w +1)0 2
35.
.
COS Sill
36. sin 6 + sin 4 - sin 2 = 4 sin 2 cos cos 3 0.
37. (sec 20 + 1) Vsec 2 - 1 = tan 2 0.
38. 4 cos cos (60 - 0) cos (60 + Q = cos 3 #
39. 16 cos 20 cos 40 cos 60 cos 80 = 1.
40. tan (45 + 0) = X /f
M sin 2
41 sin (n + 1)0 - sin (n - 1)0 _ t 0.
cos (ft + 1) + cos (n 1) + 2 cos w0 ~~ 2
42. cos 2 O + 1)0 - cos 2 nO = - sin (2 n + 1)0 sin 0.
83. Identities that are true for angles whose sum is 180 or
90. When three angles are involved whose sum is either 90
or 180, many relations are found to exist that do not hold true
for angles in general.
For, if A + B + C= 180, we have (Art. 53, p. 78),
sin (A + B) = sin C, cos (^4 + B) = - cos (7, tan (A+B)=- tan (7,
MULTIPLE AND SUBMULT1FLE ANGLES 111
and similar relations hold between functions of the sum of any
two of the given angles, and the corresponding functions of the
third angle, since the sum of any two is the supplement of the
third.
Also, if (- +- = 90, the sum of any two of these angles
is the complement of the third. Therefore,
in (I + f ) = cosf , cosg H- f ) = sinf , tan (f + f)= cotf ,
sn
and similar relations hold between functions of the sum of any
two of the angles and the corresponding co-functions of the
third.
Ex. 1. If A + B + O= 180, prove that
sin 2 A -f sin 2 B sin 2 = 4 cos A cos B sin O.
Left member = 2 sin (A + 5) cos (4 - B) - 2 sin C cos C
= 2 sin C cos(.4 - ) + 2 sin C cos (A + )
= 2 sin C [cos (.1 + B) + cos (.4 - 5)]
= 2 sin C (2 cos A cos 7?)
= 4 cos -4 cos R sin C.
Ex. 2. If A -h 5 + # = 180, prove that
ABO
cos A + cos # + cos (7= 1 + 4 sin sin sin .
Left member = 2 cos -- cos - +1-2 sin a
/^ A T> SI
= 1 + 2 sin cos ^- - 2 sm 2 .
= 1 + 2 sin
112 PLANE TRIGONOMETRY
Ex.3. If A + B+ (7=180, prove thai
tan A + tan B + tan (7 = tan A tan B tan 0.
Since A + B = 180 - C, tan (^4 + 5) = - tan C ;
tan A + tang = _ tftn c<
1 tan /I tan B
Clearing of fractions, tan A + tan B = tan C + tan A tan
.-. tan .4 + tan B + tan C = tan /I tan B tan C.
EXERCISE XIX
If A + B + C= 180, prove that
1. sin 2 A 4 sin 2 B + sin 2 C = 4 sin A sin B sin C.
2. cos 2 A -f- cos 2 .B 4 cos 2 (7= 1 4 cos A cos .B cos
3. cos 2 A - cos 2 4 cos 2 (7= 1 -4 sin A cos B sin (7.
4. sin 2 A sin 2 B sin 2 "(7= 4 sin A cos B cos (7.
A .B (7
5. cos, A + cos B cos 0= 1 + 4 cos cos sin .
A B C
6. sin A 4 sin B 4 sin (7=4 cos cos cos .
7. sin A -f sin I? sin C 4 sin mt cos .
i- ^ '
8. sin 2 A + sin 2 B sin 2 (7=2 sin A sin .Z? cos (7.
9. cos 2 A + cos 2 5 - cos 2 (7= 1-2 sin A sin .B cos (7.
14 - CW - '- : 0j f a,/;.:
sin A 4 sin .B sin (7 A, .Z? * A*<I Pr***
10. - = tan tan .
sin A 4 sin B 4 sin C 2
sin 2 A + sin 2 J9 4 sin 2 (7
sin 2 A 4 sin 2B sin 2 f
1 4 cos A cos J5 4 cos C B , C
12. = tan cot .
I 4 cos A 4 cos ^ cos (7 22
sin J. + sin 5 -f- sin (7
* -
14. cot A cot B 4- cot jB cot (7 4- cot (7 cot A
MULTIPLE AND SUBMULT1PLE ANGLED 113
M^iif,
1.
j 7? /7 (7 ^4.
15. tan tan + tan tan - + tan - tan = 1.
A 7? C* ABC
16. cot + cot 4- cot = cot cot cot .
17. sin 2
ABO A+ B B+ 0^
18. cos 4- cos 4- cos = 4 cos cos ^ cos -
2 2 "2 4 44
19. sin (A + B- (7)
4- C- A) + sin ((7+ A -
= 4 sin A sin B sin 0.
. sin (^1 + 2 ) + sin (B + 2 (7) 4- sin ((74- 2'
4 sm
sin
-C C-A
sin -
CONANT'S TRIO. 8
CHAPTER X
INVERSE TRIGONOMETRIC FUNCTIONS
84. If sin 6 = a, where a is any known quantity, 9 may
have any one of an infinite number of values. The symbol
" sin" 1 a " is used to denote the angle whose sine is #, and is,
accordingly, read "the angle whose sine is a." It is some-
times called the inverse sine, or the anti sine of a.
To illustrate the use of this notation, let us take the equation
cos0 = J. (1)
We know from this that may equal 60, 300, 420, 660, ....
To state the fact that may equal any one of these angles, we
employ the equation /, _-, ^
6 = cos 1 1, (2)
which is read "0 = the angle whose cosine is |."
We are then to understand that (1) and (2) are inverse state-
ments, the former asserting that the cosine of some angle, #, is
equal to ^, and the latter asserting that 9 is the angle whose
cosine is |.
From (2) we also understand that 60, 300, 420, , are
angles that satisfy the equation, since the cosine of each of
these angles is J. In other words, (2) is satisfied by any of
the angles (Art. 64, p. 87) included in the general expression
7T
2^ f .
Similarly, if tan 0=1,
then the equation 9 = tan' 1 1
asserts that 9 may equal 45, 225, 405, . That is, 9 may
have any one of the values represented by the expression
+ , 7T
H7T +
114
INVERSE TRIGONOMETRIC FUNCTIONS 115
It is strongly urged that the student become familiar at the out-
set with the idea that the expressions sin" 1 1, cos" 1 J, tan~ J V3,
etc., are single symbols, and denote angles. They represent
angles just as definitely as do the symbols 0, <, A, B, x, y, etc.,
which are used so frequently for that purpose. The only point
to be noted is, that the angle which is represented in this man-
ner is described by means of one of its trigonometric functions.
85. Angles expressed by the symbols sin" 1 ^, cos'^VS,
tan" 1 1, etc., are called inverse trigonometric functions, or in-
verse circular functions.
Since a central angle has the same magnitude in degrees as
the intercepted arc, these functions are used to represent arcs
as well as angles. The notation arc sin #, arc cos J, arc tan
JV3, etc., is often used instead of sin' 1 a, cos" 1 !, tan~ 1 ^V8, etc.
In using the notation here adopted, the student should note
that the symbol 1 is not an algebraic exponent. That is,
sin" 1 a is not the same as (sin a)" 1 .
The former expression denotes the angle whose sine is a, and
the latter denotes - , or esc a.
sin a
86. The smallest numerical value of an angle whose sine,
cosine, tangent, etc., have given values, is called the principal
value of the angle.
Thus, the principal values of
are 30, 120, -45, 60.
In a case like the second, where two values are given, which
are numerically equal but have opposite signs, the positive
value is usually understood. Thus, the principal value of
cos -i(_l) i s usually considered to be 120.
To avoid ambiguity, it will be understood that, when any of
these symbols are employed, the principal values of the angles
are referred to.
If a is positive, the principal values of all the inverse func-
tions except vers" 1 ^ and covers" 1 a lie between and 90.
The principal value of vers" 1 ^ lies between and 180, and the
116 PLANE TRIGONOMETRY
principal value of covers" 1 a lies between and 90, or between
180 and 270.
If a is negative, the principal values of sin" 1 a and csc" 1 ^ lie
between and 90, or between 180 and 270. The principal
values of cos" 1 a and sec" 1 a lie between 90 and 180, or be-
tween 90 and 180. The principal values of tan" 1 a and
cot" 1 a lie between 90 and 180. As stated above, the positive
values of these angles are usually employed. Since vers and
covers 6 are always positive, vers~ 1 a and covers' 1 a are impossi-
ble when a is negative.
87. Ex. 1. Prove that sin" 1 ! + cos" 1 if = cos' 1 ! f. (1)
Let sin- 1 f = a, cos- 1 1| = ft, cos- 1 f f = y.
Then, sina=f, cos/3={f, cosy=ff.
We are to prove that a + ft = y. (2)
This can be done by proving that any function of a + ft is equal to the
same function of y, since, if two sines, two cosines, two tangents, -, are
equal, the principal values of the angles are also equal.
In this case we select the cosines ; and we are now to prove that
cos ( + /?) = cosy. (3)
Expanding, cos cos ft sin a sin ft = cos y. (4)
The values of cos ft, sin ct, and cos y are already known ;
3 and, obtaining the values of cos a and sin ft from the figures
in the margin, and substituting in (4), we have
t* - * = li
.. cos( + ft) = cosy.
Ex. 2. Prove that cos' 1 ! + sin" 1 1\ + sin' 1 I
Let cos- 1 1 = a, sm- 1 f s = p, si
Then, cos a = f, sin/8 = T \,
We are to prove that a -f ft + y = ,
Selecting in this case the sines, we proceed as follows:
= cos y.
sin a cos ft -f cos a sin ft = cos y.
INVERSE TRIGONOMETRIC FUNCTIONS
Substituting numerical values, we have
117
Ex. 3. Prove that
2 sin" 1 + tan" 1 -- cos" 1 - = 0.
VlO ? V2
Let
Then,
We are to prove that 2 a + fi y = 0,
v/10 7
sin a = - , tan B = - ,
VlO 7
V2
cos y =
V2
or, 2 a + j8 = y.
Selecting the tangents as convenient functions to deal with in this case,
we proceed as follows :
tan (2 a + ft) = tan y
1 -
The left member = tan 2 + ten fl <
1 - tan 2 tan 3
1 -
1- 2ta " tan/?
l-tan 2
But
tan y = 1.
.. tan (2 a + /#) = tan y.
2 + ^8 - y,
63
Ex. 4. Prove that
2 sin- 1 -A- + cos" 1 ~ + i tan- 1 ^ = TT.
Via ^52 7
,
65 7
2 16 24
Then, sin= ' = , cos (3=, tan y = -
via 6o 7
V13
sin
We are to prove that
2 ce + ^8 - TT - | y.
118 PLANE TRIGONOMETRY
Selecting the sines as the most convenient functions with which to work
in this case, we proceed as follows:
sin (2 a + /3)= sin (TT - \ y).
sin 2 a cos ft + cos 2 a sin ft = sin \ y. (1)
All the functions of a, ft, and y can be determined at once from the
proper figures ; and the values of sin 2 a, cos 2 ft, and sin | y must be
computed. 2 3 12
sin 2 a = 2 sin a cos a = 2 __ __ = --
V13 V13 13
945
cos 2 a = cos 2 sin 2 a = --- =
13 13 13
Substituting in (1), we have
tt-tf + A-ti=!
192 + 315 = 3
845 5'
* = *
2 a + /3 = TT - \ y.
$y = TT.
EXERCISE XX
Prove that
1. sin" 1 ^ = 008-! if. 3. cos" 1 3f = esc" 1 ff.
2. sin-i T \ = tan-i T 5 2 . 4. sin-i-=2sin- 1 --L
5 VlO
5. tan-i J-tan-i} = tan-il.
^N 6. sin' 1 if + cos" 1 if = sin- 1 f .
7. sin-if-f tariff =tan-iff
8. tan- 1 T 2 T + cot- 1 - 2 Y 4 - = tan' 1 J.
9. tan-i^ + tan-i|= sin-ii.
o V2
10. sin' 1 = + tan' 1 - = cos" 1 - -
V5 3 V2
11. COt- 1 f I + Cot" 1 J/- = COt- 1 1.
12. 2 tairi 1=00^1^2.
13. 2 tan- 1 + tiiir 1 J = tan- 1 .
\\
INVERSE TRIGONOMETRIC FUNCTIONS 119
14. tan' 1 1 + cot' 1 4 = | cos- 1 f .
is. sin-if + cot-i|-tarriJU|.
16. tan" 1 | + tan -1 ^ = tan" 1 1 + tan" 1 ^.
18. 2 cos- 1 If = tan-iifl.
19. cos" 1 x 2 cos" 1 \/ -
\ f)
20. tan" 1 x -f tan" 1 y tan" :
_ 1 xy
\J\ 21. tan' 1 x + cot- 1 O 4- 1) = tan- 1 (y? + x+ 1).
22. tan' 1 - - tan- 1 ^-^ = -
y x+ y 4
23. sin l a + cos" 1 b = cos - 1 (b Vl a 2
24. tair 1 - \ + tan- 1 ^ ^ -f tan' 1 - - = 0.
1 + ab 1 -f bo 1 -}- ca
25. sin (2 sin- 1 a) = 2 a Vl - a 2 .
26. sin f cos" 1 - J = tan f sin" 1
\ O/ V
27. sin (sin- 1 a + sin" 1 b) = a Vl b 2 4- Vl a 2 .
28. tan (tan" 1 a -f tan" 1 5) =
1-
29. tan (2 tan" 1 a) = -
30. cos (2 tan- 1 j) = sin (4 tan' 1 1).
88. Solution of equations expressed in the inverse notation.
The method of solution of equations that are expressed in terms
of inverse functions is best illustrated by means of examples.
Ex. l. Solve the equation
tan- 1 (x + 1) + tan- 1 (x - 1) = tan' 1 ^-.
Let tan- 1 (* + !) = , tan- 1 (*-!) = # tan" 1 ^ = y.
Then, tan = a;+l, t&u fi = x 1, tany = ^ v .
To find what values of x will satisfy the equation
120 PLANE TRIGONOMETRY
when tan a, tan (3, and tan y have the above values, we proceed as follows :
tan (a + /?) = tan y.
Then the left member = *an a + tan ft
1 tan a tan p
2-z 2
Equating this to tan y, we have
2x = 8
2-z 2 31'
Solving, we have a: = J, or 8.
The second value is inadmissible as long as we use the principal value of
the angles. Therefore, x _ l
Ex. 2. Solve the equation
tan" 1 x + tan" 1 (1 x} = 2 tan" 1 V# x 2 .
Let tan-^^a, tan" 1 (1 - x)= (3, tan-Vz - x* = y.
Then, tan a = x, tan /? = 1 - ar, tan y = Vz - x 2 .
To find what values of x will satisfy the equation we proceed as follows :
tan (a + ft) = tan 2 y,
tan ft + tan /3 _ 2 tan y
1 tan a tan /2 1 tan 2 y'
l_a;(l-a;) 1 - x + x 2 '
1 = 2V^^2.
Solving, x = %.
EXERCISE XXI
Solve the following equations :
2. sin" 1 ^ = cos" 1 ( x).
3. tan" 1 x = cot" 1 x. _, 1 2?r
7. tan * x 4- ^ tan x - =
4. tan" 1 a: = cot" 1 ( a?) . ^
- + sm = 2' : ~T'
Ml
INVERSE TRIGONOMETRIC B^UNCTIONS
9. tan" 1 (x 4- 1) cot" 1 - = tan~ 1 -.
x 1 *j
10. tan" 1 2 a; 4- tan- 1 3 a =
11. cos" 1 x cos" 1 VI x 2 = cos" 1 a; VS.
12. sin" 1 (3 x 2) 4- cos" 1 x = cos" 1 VI x 2 . \ * -"S
121
13.
x-2
2 4
14.
"2 5 x
15. sin (cot" 1 J)= tan (cos" 1 Vie).
16. tan (cos" 1 x) = sin (cot" 1 J ) .
17. sin" 1 - = sin" 1 - 4- sin" 1 -
x a o
18. tan" 1 -7- = tan- 1 (cos
Vsm a?
19. esc" 1 x = sec" 1 -
COS
20.
21. tan" 1 ^4 4- tan - i = tan~ 1 (-7).
; - /-/ <V,
/( 1
>>;*>; 4
>l 1 I T-
' 1 1 < !
CHAPTER XI
THE GENERAL SOLUTION OF TRIGONOMETRIC EQUA-
TIONS
89. A trigonometric equation is an equation in which the un-
known quantity or quantities appear in the form of trigono-
metric functions.
These equations have been used with the utmost freedom in
previous chapters, though no formal definition has been given
until the present time. They have been used in many differ-
ent ways, involving one or more functions, one or more angles,
and one or more values of the given angles in any single
equation.
At first the only angles used were acute angles, and an equa-
tion was understood to involve functions of an acute angle
only. Then the idea was introduced of an angle unrestricted
in magnitude ; and after this had been done, all results were
freed from the restraints which had previously been imposed
by the fact that we were dealing with acute angles only.
A large class of the equations with which we have previously
been concerned consist of trigonometric identities, that is, equa-
tions in which both sides had the same value for all possible
values of the angles employed, though the form might be
different.
Examples of these are the formulas that have been proved
from time to time, as, sin 2 6 + cos 2 1 ; sin (x -f /) = sin
x cosy + cos x sin y ; etc. Equations of this kind are true for
all possible values of the angle or angles involved.
But trigonometric equations are, of course, not ordinarily
true for all values of the angles involved. For example, if we
consider the equation cos Q _. \
we see at once that we can assign but two values of 6 between
and 360 that satisfy this equation. In other words,
122
GENERAL SOLUTION OF TK1GONOMETIUC EQUATIONS 123
cos 6 = is true only for = 60 and 6 = 300, as long as is
restricted to values between and 360. If angles of unre-
stricted magnitude are allowed, cos 6 = J is satisfied by all
values of 6 that are included in the general expression
0-Siirdbg,
O
and by no other values.
In like manner, the equation
is satisfied by all values of that are given by the general
expression
0=W7T+^,
o
and by no other values ;
sin 6 = J V2
by those values of 6 that are given by the expression
and by no other values ; and so on for other examples that
might be given. In all these illustrations it is to be under-
stood that n is any positive or negative integer or zero.
The solution of an equation is the determination of the value
of the angle or angles that satisfy the equation. In Art. 67,
p. 88, a method of solution was given by means of which some
of the simpler forms of trigonometric equations could be treated.
But at that time only a limited number of the formulas of
transformation were at our disposal. Hence, the number of
classes of equations that could be handled was necessarily quite
limited.
The methods of reduction and transformation that are now
available make it possible to solve many classes of equations
that were formerly quite out of our reach, and also to sim-
plify some of the methods previously employed. The present
chapter will illustrate some of the simpler cases of this kind.
This work should be looked upon as an extension of that
given in Art. 68, p. 90.
124 PLANE TRIGONOMETRY
90. Solution of equations of the form
a cos 6 + b sin 6 = c. (1)
A simple method of solving equations of this form is fur-
nished by the introduction of what are termed auxiliary angles,
as follows :
Assume a right triangle whose legs are a, 6, and designate
by (/> the angle lying opposite the leg b. The hypotenuse of
this right triangle is Va 2 -}- 2 , and we now have
cos < = , and sin <f> =
Dividing each member of the original equation by Va 2 -}- 5 2 ,
we have
cos + sin 6 = (2)
Substituting cos </> and sin <p for their respective values in this
equation we have
cos <j> cos H- sin <f> sin = : G
or, cos (0 <) =
Since a, b, and c are known, cos (6 <) is known, and 9 <
can at once be found from the tables. Calling this angle ,
for convenience we have
cos (# <) = cos a.
. . 6 <t> = 2 MTT a, Art. 64, p. 87.
# = 2 ftTT + ^ a.
The cosine -of an angle can never be numerically greater than
unity. Hence, in dealing with the equation cos (0 </>) -
it is to be remembered that we must have c ^ Va 2 + 2 . If
c > Va 2 -f- b' 2 , there is no real value of <$> which will satisfy
the equation.
GENERAL SOLUTION OF TRIGONOMETRIC EQUATIONS 125
Ex. i. Solve the equation V3 cos + sin 6 = V2.
Dividing both sides of the equation by V3 + 1, i.e. by 2, we have
In this case we have a = V3, 6 = 1, and Va' 2 + 6 2 = 2. Hence, the auxiliary
angle <f> is equal to 30. The original equation then becomes
cos 30 cos 6 + sin 30 sin = |V2.
cos(0-30)=|V2.
But \/2 is the cosine of 45. Hence, we write
cos (0-30) = cos 45.
4'
6 4*
Ex. 2. Solve the equation 5 cos 6 4- 2 sin = 4.
In this problem we have a = 5, and 6 = 2. Dividing both sides of the
equation by Va 2 + ft 2 , we have
JL cos + -?= sin = -t= (1)
A/29 V29 A/29
In the preceding example we were able to find the value of < from the
familiar coefficients and -, which we already knew were the cosine and
sine respectively of 30. But in this example we have unfamiliar values to
consider.
From the figure on the margin of the page we see that <f> is an
o i^i
angle whose cotangent is ~. Turning to the tables, we find that the
o
value of <f) is 68 12' ; and (1) can now be written
cos 68 12' cos + sin 68 12' sin = -4=
V29
Letting a equal the angle whose cosine is this becomes
V29
cos (0-68 12') = cos a.
Reducing the value of ^ to a decimal, we find it to be 0.7428 ; and,
V29
consulting the tables, we find that the angle whose cosine is 0.7428 is 42 2'.
Therefore, cos (0 _ 68 12') = cos 42 2',
- 68 12' = 2 mr 42 2',
NOTE. Each of the foregoing examples could have been solved by
replacing either sin or cos by its value in terms of the other, then
obtaining the value of the single function involved, and finally obtaining
the value of from the value of this function. But the process just ex-
plained is much simpler and better.
126 PLANE TRIGONOMETRY
91. Solution of equations involving multiple angles. The
simplest forms of equations involving multiple angles have
already been considered (Art. 68, p. 90). But these, and
also many other forms of equations in which multiple angles
appear, are more conveniently treated by means of the various
reduction formulas that are now available.
The following problems will illustrate some of the methods
of most frequent application.
Ex. l. Solve the equation sin 3 6 -f sin 7 6 = sin 5 0.
By (5), Art. 77, p. 100, we have
2 sin 50 cos 20 = sin 50.
.-. sin 5 = 0, or cos 2 6 = |.
If sin 50 = 0, then 5 = HIT.
If cos 2 = -, then 20 = 2 mr
Therefore, the general values of that satisfy the equation
sin 30 + sin 7 = sin 5 0,
are = ^?, and 0= mr .
o o
Ex. 2. Solve the equation cos 4 cos 6 sin 20 = 0.
Applying the proper reduction formulas, we have
2 sin 5 sin - 2 sin cos = 0.
.-. sin (sin 5 cos 0) = 0.
From the first factor we have
sin = 0.
.-. = n7r. (1)
From the second factor we have
cos = sin 5
= cos r -
..0=2mr(|-50Y
Using the positive sign, we have
12
GENERAL SOLUTION OF TRIGONOMETRIC EQUATIONS 127
Using the negative sign, we have
[the sign of n being left unchanged because n denotes all negative as well
as all positive integers] mr t IT
*--j+g- (3)
Collecting the values given in (1), (2), and (3), we have as the general
values of that satisfy the given equation
Ex. 3. Solve the equation cos 2 x = cos x + sin x.
Expressing cos 2 # in terms of functions of x we have
cos 2 x sin 2 x = cos x + sin x ;
(cos x + sin x) (cos x sin x) = cos x + sin x.
.-. cos x + sin x 0, (1)
or, cos a; sin ar = 1. (2)
From (1) we have tan x 1.
From (2) we have
7T
x = nit
1 1.1
cos x sin x = -,
V2 V2 V2
or, cos ? cos x sin - sin x = cos ?,
444
f *\
cos ar + ) = cos
V 4/
x = 2 /ITT, or # = 2 n?r
EXERCISE XXII
Solve the following equations :
1. cos x V3 sin#= 1. 7. cos ex. -f sin a = V2.
2. sin x V3 cos # = 1. 8. sin m0+ sin nO = 0.
3. sin ^ + V3 cos ^ = V2. 9. cos w^ + cos n^ = 0.
4. V3 sin cos ^ = V2. 10. 3sin# + 2cos^ = 2.
5~ sin 6 + cos ^ = V2. 11. 6 cos 3 sin = 3.
6. cos a sin a = -|- V2- 12. 4 cos # 3 sin = 5.
128 PLANE TRIGONOMETRY
~~ 13. sin 7 x sin 4 x + sin x 0.
- 14. sin 5 x sin 3 x + sin a; = 0.
15. sin 1 x sin x = sin 3 a?.
16. sin 4x sin 2 x = cos 3 #.
17. cos 6 + cos 2 + cos 3 = 0.
18. sin + sin 2 + sin 3 6 = 0.
19. cos 7 cos = sin 4 0.
20. cos 2 - cos - sin 2 + sin = 0.
21. sin 4 sin 3 + sin 2 sin = 0.
22. cos70 + cos50+eos30+cos0 = 0.
23. 2 cos 2 = cos 3 + sin 0.
24. cos &<^ cos (k 2) = sin 0.
25. sin 5 cos sin 6 cos 2 = 0.
26.
27. COS 3 + 2 COS = 0.
28. cos20-f-sin30 = 0.
29. cos 5 + cos = V2 cos 3 0.
30. sin + V3 cos 4 = sin 7 0.
31. cos20-cos 2 = 0. 36. cot 0- tan 0=2.
32. cos 3 + 8 cos 3 = 0. 37. sec esc = 2 V2.
33. sin 3 - 8 sin 3 = 0. 38. cot 2 0- cot = - 2.
34. sin20 + 3sin0 = 0. 39. sec 4 0- sec 20= 2.
35. esc cot = V3. 40. sec + esc = 2 V2.
41. tan 3 + tan 2 + tan 0=0.
42. tan 30- tan 20 -tan 0=0.
43. tan 3 + tan = 2 tan 2 0.
44. sin 5 cos - sin 4 cos 2 = 0.
vr
GENERAL SOLUTION OF TRIGONOMETRIC EQUATIONS 129
92. Changes in sign and magnitude of the expression a cos a?
4- & sin x. In connection with the solution of equations of the
form . , .
a cos x 4- o sin x = U,
it is often useful to trace the changes in sign and magnitude
of the left member of the equation as x increases from
to 360.
The simplest case occurs when a I and 6 = 1; in which
case we have simply sin x 4- cos x to examine. Proceeding as
in Art. 90 we have
cos x 4- sin x = V2 sin x 4 cos x
LV2 V2 J
= V2(sin x cos 45 4- cos x sin 45)
= V2sin<>4-45 ).
For convenience we replace cos x 4- sin x by y, and then, form-
ing the equation y = V2 sin (x + 45), we form the following
table of values.
Plotting the graph by the method explained in Art. 48, we
have the following result.
X
y
1
45
V2
90
1
135
180
-1
225
-V2
270
-1
315
360
1
/T\
273
~Kd
Since the greatest value that the sine of any angle can
have is 1, the maximum value of this expression occurs when
sin (x 4- 45) = 1, i.e. when z = 45. This gives V2 as the
maximum value of the expression sin x 4- cos x.
In like manner, the minimum value of the expression is
- V2, which corresponds to the angle x = 225.
COKANT'S TRIG. 9
130 PLANE TRIGONOMETRY
If the table of values is extended, and the graph is plotted for
values of x greater than 360, the values of ?/, i. e. of cos x + sin #,
will be repeated in their original order ; that is, cos x + sinx
is a periodic function with a period of 360. (See Art. 49, p. 71.)
93. When a or 5, or both a and , are different from unity,
the process is slightly modified, as follows :
a cos x+ b sin x = Va 2 + b 2 l ^ - cos x -\ - sin x ]
= Va 2 -f- b 2 (cos x cos a + sin x sin a)
= Va 2 -f- b 2 cos (x a) .
Here, as is readily seen from the figure on the
b margin of the page, it has been assumed that a is
the angle whose cosine is a and whose sine is
When a and b are known, a can be found, as in
Va 2 + b 2
Art. 90, p. 124.
The table of values can then be obtained and the graph con-
structed, as in the preceding case.
Since cos (x a) has 1 for its maximum value and 1 for its
minimum value, the expression a cos x + b sin x has Va 2 + b 2
for its maximum value and Va 2 b 2 for its minimum value.
NOTE. In computing the table of values for the purpose of constructing
the graph, the values of y can always be obtained directly from the expres-
sion as it is originally given, without any reduction whatever. This is
sometimes preferable; and in certain cases, as for example the functions
given in Examples 7, 9, 10, and 11 in the following set, it is easier to com-
pute the values directly than to compute them after transforming the
expression.
EXERCISE XXIII
Trace the changes in sign and magnitude of the following
expressions as x increases from to 360. Find the period
and construct the graph in each case.
1. sin x cos x. 5. sin x + V3 cos x. 9> cos 3 9.
10 ' Sm 8 '
2. V3sinz + cosz. 6 . 2 sin x + 3 cos x.
11. tan 20.
3. sin* + V3cos*. 7. cos 20. ^ sin 20- sin*
4. V3 sin x cos x. 8. sin 6 cos 6. cos 2 + cos 6
CHAPTER XII
THE OBLIQUE TRIANGLE
94. The law of sines. Let A, B, denote the angles of a
triangle, and a, b, c respectively the sides opposite.
From any vertex, as (7, draw CD perpendicular to AB, meet-
ing AB, or AB produced, in D.
A D B A B
From the first figure we have
Also,
= b sin A.
- = sm B.
a
.. h = a sin B.
Equating these values of h we have
b sin A = a sin B.
From the second figure we have
- = sm A.
b
= sn
A.
Also,
whence as before,
- = si
b sin A = a sin
131
132 PLANE TRIGONOMETRY
Therefore in either case we have the same result,
b sin ^4.= a sin J9;
a - sin A
In like manner drawing perpendiculars from the vertices A
and B to the opposite sides respectively we can prove that
b _ sin
G sin
and
c sm
The results obtained in (1), (2), and (3) enable us to state
the law of sines as follows :
The sides of a triangle are proportional to the sines of the
opposite angles.
Equations (1), (2), and (3) are often combined and written
in the following manner :
sin A sin B sin
95. The geometric meaning of each of the three ratios just
stated will be understood from the following :
Let ABO be any triangle, and let a circle be circumscribed
about the triangle. From the center to the vertices of the
triangle draw the radii OA, OB, 00, respectively, and also
draw OD perpendicular to AB.
By geometry
From this we have
= r sin C.
.*. c = 2 r sin C.
In like manner it can be proved that
a= 2rsin A*
and b = 2 r sin B.
.
THE OBLIQUE TRIANGLE
133
Equating the values of 2 r obtained from these three equa-
tions we have a i c
2r = - - = -r = -^-~ - That is,
Bin .4 sin If sin O
The ratio of any side of a triangle to the sine of the opposite
angle is equal to the diameter of the circumscribed circle.
96. The law of cosines. Let ABO be any triangle, and let
(7Z), the perpendicular from the vertex to the opposite side,
meet AB, produced if necessary, in D.
D
B A
From the first figure we have
= 52 + ^2 _ 2 c - b cos A.
2 be
From the second figure we have
2 = h* + BD*
= h? + (AD - c)
= 2 + AD*-2c
= b 2 + c 2 2 c - b cos A.
2 be
Therefore, the same result is obtained for both triangles.
In like manner, drawing perpendiculars from A and B to the
opposite sides respectively, we can prove that
and
cos B =
cos C' =
(2)
(3)
134 PLANE TRIGONOMETRY
Equations (1), (2), and (3) are often useful when expressed
in the following form :
(4)
C.
The law of cosines can now be stated as follows :
The square of any side of a triangle is equal to the sum of the
squares of the other two sides minus twice their product into the
cosine of the included angle.
yv
97. The law of tangents. We have already proved that, in
. . i a sin A
any triangle, - =
Therefore, considering this equation as a proportion, and
taking the four quantities by division and composition,
a b__ sin A sin B
a + b sin A + sin B
2 sin *** cos ^p?
~ L
cot^i^tanAzi^.
a-b 2
2
In like manner it can be proved that
A-C
tan
and
THE OBLIQUE TRIANGLE 135
The law of tangents can now be stated as follows :
The difference of two sides of a triangle is to their sum as the
tangent of half the difference of the opposite angles is to the tan-
gent of half their sum.
NOTE. In using the formulas of this section it is better to let the greater
side and the greater angle precede the smaller in all cases. The formulas
are true, whichever order is used ; but if the smaller side and the smaller
angle precede the greater side and the greater angle respectively, negative
numbers are introduced, and if logarithms are to be employed, these num-
bers should be avoided whenever it is possible to do so.
98. The given parts. In the solution of oblique plane tri-
angles four cases occur. In each case three parts are given, as
follows :
1. One side and two angles.
2. Two sides and the angle opposite one of them.
3. Two sides and the included angle.
4. Three sides.
The formulas developed in Arts. 94-97 are sufficient for the
solution of every possible case that can arise. These cases will
now be considered separately.
99. CASE 1. Given one side and two angles. Let the given
angles be A and J5, and the given side a. The formulas for
solution are as follows :
b _ sin B 7 _ a sin B
2. - -, . . - ;
a sin A.
c sin
>
a sin JL' sin A
Ex. i. Given a = 467, A = 56 28', B = 69 14'; find the re
maining- parts.
The work may be conveniently arranged as follows :
C = 180 - (.4 + B) = 54 18'.
(1) By natural functions.
b = a x sin B - sin A = 467 x 0.9350 t 0.8336 = 523.8.
c = a x sin C *- sin .4 = 467 x 0.8121 -* 0.8336 = 454.95.
\\
136 PLANE TRIGONOMETRY
(2) By logarithms.
log b = log a + log sin B log sin A
= log + log sin B + colog sin A.
log c = log a + log sin C log sin A
= log a + log sin C + colog sin A.
log a = 2.66932 log a = 2.66932
log sin B = 9.97083 - 10 log sin C = 9.90960 - 10
colog sin A = 0.07906 colog sin A = 0.07906
2.71921 2.65798
b = 523.85 c = 454.97
NOTE. To insure accuracy the student should check all results by solving
each problem by a second method, entirely independent of the first ; or by
the same method, using a different combination of parts. In the case under
consideration it is usually sufficient to employ the same method, i.e. the law
of sines, combining the parts in a manner different from that employed in
the first place. For example, after c has been found we can solve again for
b by the formula b = csm / f , as follows :
sin C
log c = 2.65798
log sin B = 9.97083 - 10
colog sin C = 0.09040
log b = 2.71921 b = 523.85 check.
EXERCISE XXIV
Solve the following triangles :
1. Given a = 438.3, A = 43 50' 24", B= 69 30'
An*. C= 66 39' 24", b = 592.74, c = 580.*.
1*1
2. Given b = 421, A = 31 12', B = 36 20'.
Ans. (7=112 28', a = 368.08, c = 656.63.
{ 3. Given a = 412, 4 = 58U', B = 65 37'.
Ans. 0= 56 9', 5 = 441.37, c = 402.45.
4. Given 6 = 81.5, B = 43 44' 18", 0= 75 2' 42".
4 =61 13', a =103.32, c= 113.89.
5. Given c = 77.93, B = 22 15' 20", O= 41 50' 30".
Aw. A = 115 5& 10", a = 105.07, 5 = 44.23.
6. Given c = 6.98, A = 25 7' 10", (7= 36 12' 24".
Ans. B = 118 40' $", 4 = 5.016, b = 10.37.
THE OBLIQUE TRIANGLE 137
7. Given a = 928.4, A = 61 17' 15", 6 V = 58 18' 40".
Am. B = 60 24' 5", c = 900.78, ft = 920.45.
8. Given a = 328.4, A = 29 41' 12", B = 37 50' 24".
Ana. C =11 2 28' 24", 5 = 406.77, c = 612.73.
9. Given A = 64 35', 0= 73 49', a = 213.47.
Ans. B = 41 36', 5=156.92, c= 226.98.
10. Given ^1 = 41 23' 47", B = 124 49', 5 = 65.536.
Am. 0= 13 47' 13", a = 52.788, c = 19.023.
11. Two points, A and ^, are separated by a body of water.
In order to find the distance between them a line AQ is meas-
ured 612.3 ft. in length, and the angles BAG, ACB are meas-
ured and are found to be 49 17' and 68 11' respectively.
What is the distance from A to B ?
12. It is desired to find the distance of a lighthouse A to
each of two stations B, C, situated on shore, and in the
same horizontal plane with the base of the lighthouse. BC
is 21 miles, Z.ABO is 39 38', and ZACB is 74 56'. Find AB
and AC.
13. The angles of elevation of a balloon that has ascended
vertically between two stations one mile apart on a level plain,
and in the same vertical plane with the balloon, are 29 41' and
37 17' respectively. What is the distance of the balloon from
each station, and what is its vertical height above the plain ?
14. Solve the preceding problem on the supposition that
both the stations are on the same side of the balloon.
15. To find the width of a stream a line AB, 351 ft. long, is
measured on one side, parallel to the bank. On the opposite
side of the stream a stake C is set, and the angles CAB, CBA,
are observed and are found to be 38 17' and 31 29' respec-
tively. What is the width of the stream ?
16. From the top and bottom of a column the angles of
elevation of the top of a tower 236 ft. high standing on the
same horizontal plane are 44 27' and 61 31' respectively.
What is the height of the column ?
138 PLANE TRIGONOMETRY
17. When the altitude of the sun is 49 52', a pole standing
on the slope of a hill inclined 16 55' to the level of the plain
casts a shadow directly down the hill a distance of 45 ft. 8 in.
What is the height of the pole ?
18. An observer in a balloon measures the angle of depres-
sion of an object on the ground and finds it to be 63 58'. After
ascending vertically 582 ft. he finds the angle of depression of
the same object 74 49'. What was the height of the balloon
at the time of the first observation ?
19. From a ship the bearings of two objects were found to
be N.N.W. and N.E. by N., respectively. After sailing due
east 10 miles the two objects were in a line bearing W.N.W.
How far apart were the objects ?
NOTE. For an explanation of the term "bearing," and for instruction in
reading angles by means of the compass, see p. 176.
20. From a ship a lighthouse bears N. 21 12' E. After the
ship has sailed S. 25 12' E. 2| miles the lighthouse bears due
north. Find the distance of the lighthouse from the last point
of observation.
100. CASE 2. Given two sides and the angle opposite one of
them. Let the given parts be the sides a and &, and the angle
A. The required parts can be found in the following manner :
By the law of sines
(1)
sin A a a
From this equation the angle B can be found.
Then, C= 180 - (4 + B).
Also, ^ = ^4, .'.c = ^?. (2)
a sin A sin A
In solving for the angle opposite the second side, in this
case the angle B, it is to be noted that two solutions are pos-
sible, since the sines of supplementary angles are equal (Art.
53, p. 79).
The following considerations will determine the number of
solutions for any given set of conditions.
THE OBLIQUE TRIANGLE
139
If a > b, then A > B, and B is necessarily an acute angle,
since a triangle can have but one obtuse angle. Therefore
there is one and only one solution.
If a = b, then A = B, and both
A and B are acute angles. There-
fore there is one and only one solu-
tion, an isosceles triangle.
If a < b, then A < B, and A is an FlG - 1-
acute angle. In this case B may One solution, a> 6
be either acute or obtuse, and there will be two solutions if
a > CD, the perpendicular drawn from the vertex C to AB,
produced if necessary. That is, either of the two triangles
AB l C, AB 2 C, will satisfy the given conditions. But the perpen-
dicular CD = b sin A.- Therefore, if A is acute and #<&, and
c
c
b sin A
FIG. 2.
Two solutions, a > b sin A
FIG. 3.
One solution, a = b sin A
if a > b sin A, there are two solutions. The angles
AB^C, are supplementary, since /.AB 1 C=/.B 1 B^C. Both
angles are given by the formula
If a = b sin A, that is, if a is equal to the perpendicular CD,
there is but one solution, a right triangle. This is also seen from
the fact that when a= b sin A, the value of sin B reduces to
unity. This gives B = 90.
If a < b sin A, that is, if a is less
than the perpendicular CD, there is
no solution, and the triangle is impos-
sible. This is also seen from the fact
that when a<bsinA, the fraction
FIG. 4. b sin A is ter than unit But
No solution, a < 6 sin A a
140 PLANE TRIGONOMETRY
this fraction is in all cases equal to sin B ; and as the sine of an
angle can never exceed unity the triangle is therefore impossible.
These results may be summarized as follows :
Two solutions.
A acute, a < 6, and a > b sin A.
One solution.
(0) A obtuse and a > b.
(5) A acute and a = b sin A.
(c?) A acute and a > b.
No solution.
(a) A acute and a < b sin A.
(b) A obtuse and a = b or a < b.
To determine the number of solutions, first note whether A
is acute or obtuse. Then, on examining the different cases just
studied, it is seen that there can never be more than one solu-
tion unless A is acute and the Me opposite A is less than the side
adjacent. In this case there may be two solutions, one solution,
or no solution.
The comparison between a and b sin A is often most con-
veniently made by means of the natural value of sin A. In
many cases the computation can be performed mentally ; for
all that is now desired is to determine whether a is less than,
equal to, or greater than b sin A.
If logarithms are used, we compute log sin J5. The results
are as follows.
(a) log sin 1?>0, no solution.
(b) log sin B = 0, one solution, a right triangle.
(V) log sin B < 0, one solution if a > 5, and two solutions if
a< b and A is acute.
The student should bear in mind that the given parts are
not necessarily a, b, and A ; they, may be any two sides and
the angle opposite one of them. If other parts are given than
those mentioned above, the proper modifications should be
made in the formulas for determining the number of solutions.
Ex. 1. Given a = 26, b = 72, A = 30 ; find the remaining
parts.
Since sin A \, we have b sin A = 36. Hence, the triangle is impossible
as a < 36.
THE OBLIQUE TRIANGLE 141
Ex. 2. Given a = 88, b = 103, A = 120; find the remaining
parts.
Here A is obtuse and a < b ; therefore the triangle is impossible.
Ex. 3. Given a =738.4, 6 = 1185.7,. ^ = 79 38' 40"; find
the remaining parts.
Solving by logarithms we proceed as follows :
a
logb = 3.07397
log sin A = 9.99287 - 10
colog a ="7.13171-10 Since log sin5 >> there is no
log sin 5 = 10.19855 -10
Ex. 4. Given a = 28.2, e = 45.65, A = 38 1' 7.5" ; find the
remaining parts.
Proceeding as in Ex. 3 we have ,
a
logc = 1.65944 ... C = 90, and the triangle is a
log sin A = 9.79081 - 10 right triangle.
colog a = 8.54975 - 10
log sin C - 10.00000 - 10
Solving for B and b by the usual methods employed in the case of right
triangles (Arts. 26 and 27, pp. 36-38), we find B = 51 50' 52.5", b= 35.998.
Ex. 5. Given a = 67.53, b = 56.82, A = 77 14' 19" ; find the
remaining parts.
Here a > b and A is acute; therefore there is but one solution.
The unknown parts are found in the following manner :
log b = 1.75450
log sin A = 9.98914 - 10
colog a = 8.17050 -10 C = 180-(A+B)
__ 4gO 00 1 KAff
log sin B = 9.91414- 10
*= 55 8 ' 47 "' Check:
log b = 1.75450 log = 1-82950
log sin C = 9.86843 - 10 log sin C = 9.86843 - 10
colog sin A = 0.08586 colog sin A = 0.01086
log c = 1.70879 log c = 1.70879
.-. c= 51.143. c = 51.143
142 PLANE TRIGONOMETRY
Ex.6. Given = 168.32, 5=221.46, 4 = 33 39' 16"; tind
the remaining parts.
In this case the simplest method of finding the number of solutions is to
obtain the value of b sin A by multiplying the value of b, 221.46, by the
natural value of sin A, and comparing the result with 168.32, the value of a.
The sine of 33 39' 16" is approximately 0.55. Hence, it is seen at a glance
that b sin A is a trifle over one half of 221.46; that is, much less than a
Hence, since A is acute and a < &, there are two solutions.
The work of computation, exhibited in compact form, is as follows :
log b = 2.34529 log a = 2.22613
log sin A = ,9.74365 - 10 log sin C = 9.99396 - 10
colog a = 7.77387 - 10 colog sin A = 0.25635
2.22613
9.35729 - 10
0.25635
log sin B = 9.86281 - 10 log c = 2.47644 1.83977
.-. B l = 46 48' 50", .-. c x = 299.53, c 2 = 69.147.
B 2 = 133 11' 10".
.-. C = 99 31' 54", or, 13 9' 34".
NOTE. The method of checking results is the same as that used in con-
nection with Case 1. In Ex. 5 above the check' work for c is given. After
a little practice this work can be performed with great rapidity. Every
result obtained by the student should, be subjected to a check of some kind.
EXERCISE XXV
1. Determine the number of solutions in each of the follow-
ing cases :
(1)
a = 30,
5 = 60,
4 = 30.
(2)
a = 20,
5 = 60,
4 = 30.
(3)
= 40,
5=<;o,
4=30.
(4)
a = 750,
5 = 638,
A = 69 30'.
(5)
a = 38. 8,
5 = 45.5,
4 = 60.
(6)
a = 226,
5 = 196,
4 = 123 40'.
2. Given
a=l 09.68,
e = 467,
A= 13 35';
find
(7=90',
^ = 76 25',
5 = 453.94.
3. Given
a =392,
5 = 124,
A = 36 41' 42";
find
.5=10 53' 45"
<7=13224'33"
= 484.37.
4. Given
a = 168.2,
5 = 218.6,
4 = 3422 ; 50";
\f r~"
find .
g 1 = 4712'49",
6\ = 9824'21",
e 1 = 294.67.
5o=13247'll",
(7 9 =12 49' 59",
<? 9 = 6(:>.16.
THE OBLIQUE TRIANGLE 143
5. Given 6 = 8472.2, c = 3211.7, (7=16 33' 45";
find ^ = 114 40' 42", ^ = 48 45' 33", 1 = 10238.
^ 2 = 32 11' 48", B 2 =UIU' 21", 2 =6003.4
6. Given a = 506, 6 = 432, ^ = 367'12";
find ,6 = 30 13', 6 7 =113 39' 48", c= 7-86.22.
7. Given a = 36.27, 6 = 23.96, 5=30 26' 14";
find ^4 1 = 50 C 4'24", ^ = 99 29' 22", ^ = 46.65,
A 2 =129 55' 36", (7 2 =1938'10", c a =l
'
8. Given = 283.4, 5 = 268.5, JL = 60 40' 26";
find J5=5541 / 23", (7= 63 38' 11, c= 291.25.
9. Given a = 158, 6 = 179, J. = 2117' 22";
find ^ = 24 17' 18", 6\ = 13425' 20", ^=310.8,
5 2 = 15lf p .42'42", <? 2 = 2 59' 56", 2 = 22.767.
10. Given a = 628. 2, 6 = 234.4, 4 = 119 40' 40";
find ^=18 54' 58", (7=41 24' 22", ^=478.22.
11. Given a = 86. 14, 6 = 97.82, ^ = 38 15' 14";
find ^ = 44 40' 42", C\ = 974'4", c? 1 =138.07,
6 7 2 = 625'28", ^ 2 = 15.57.
12. Given a = 158, 6 = 179, ^ = 21 17' 22";
find j5=2417'18", 0= 134 25' 20", c= 310.8,
5' = 155 42' 42", 0' = 2 59' 56", c' = 22.77.
13. Given a = 36. 38, 6 = 23.92, A = 39 2' 14";
find J5=2427'49", 0= 116 29' 57", <? = 51.69.
14. Given a = 0.09593, 6 = 0.16864, 5=125 33';
find ^1=27 34' 12", 0= 26 52' 48", e= 0.09375.
15. Given a = 354.16, 6 = 433.86, .A = 361'4";
find ^ = 46 5' 5", ^ = 97 53' 51", ^ = 596.57,
R 2 =133 54' 55", O 2 = 10 4' 1", ^ 2 =105.26.
144 PLANE TRIGONOMETRY
16. Given a = 25.675, 6 = 50.139, = 68 4' 14";
find 4=28 21' 42", C=8334'4", e=53.709.
17. Given a=542.99, 6 = 310.71, ^=122 49' 17";
find 5=28 44' 34", (7= 28 26' 9", e=307.66.
18. Given a= 346.66, <?=412.33, J.= 2419' 51" ;
find ^ = 126 19' 31", ^ = 29 20' 38", ^ = 677.87,
2 = 50'47", <7 2 =15039'22", 6 2 = 73.524.
19. Given a = 56.82, 6 = 67.53, ^=77 14' 19";
find ^L = 558'47", (7=47 36' 54", c = 51.14.
101. Given two sides and their included angle.
First method. When one angle O is given, the remaining
angles can be found by the law of tangents (Art. 97, p. 134),
which can be expressed in the following manner :
2 a + b 2
The angle - - = 90 Hence, its value is known, and
' "2
the value of - can be obtained from the above equation.
2
The values of A and B can then be found as follows :
A+B , A-B_
~~~ ~~
, nd
The remaining side c can now be found by the law of sines
in either of the two following ways :
a sin C b sin
c = , or c = :
sin A. sin f
/Second method. The third side c can be found directly by
the law of cosines (Art. 96, p. 133), as follows :
THE OBLIQUE TRIANGLE 145
and the angles A and B can then be found by the law of sines,
as follows : . n -L - n
A a sm ' 7? Sln
c c
Third method. In the triangle ABC let the given parts be
a, 0, C. From the vertex B draw BD perpendicular to AC.
Then,' BD = a sin (7,
and 1)0= a cos (7.
Now
Substituting in this equation the values of BD and D C, we
have
a cos (7
In like manner, drawing a perpendicular from A to the side
BO it can be proved that
5 sin O
tan
cos C
The third side can now be found by the law of sines, as
under the first method.
NOTE. The first method is the best for use when all the unknown parts
are desired. If only the third side is desired, the second method can be
used to advantage. The second and third methods are not suitable for
computation by means of logarithms.
Ex. 1. Given a= 138.65, = 226.19, (7=59 12' 54"; find
the remaining parts.
b-a= 7a,54 log(b-a)= 1.94221
B:t:!f^rv> ^.^^4=10^46-10
B + A = 60 o 23/33" colog(6 + a)= 7.43790 - 10
Q 75 A
n " A log tan 9.62557 10
**~ A = 22 53' 31" 2
2 A= 37 30' 2" ^r''~- 22 58 ' 31 "
B= 83 17' 4"
CONANT'S TRIG. 10
146 PLANE TRIGONOMETRY
Check:
loga= 2.14192 log b = 2.35447
log sin C = 9.93494 - 10 . log sin C = 9.93494 - 10
colog sin A = 10.21554 - 10 colog sin B - 0.00299
logc= 2.29240 log c = 2.29240
c= 196.06 c= 196.06
NOTE. In the solution of this problem b precedes a since b > a. (Compare
Art. 97, p. 134.) In finding c we use A rather than B, because B is so near
90 that any solution obtained by means of its sine is likely to be inaccurate.
NOTE. In Ex. 1 the check solution gives a result exactly equal to that
obtained by the original solution. In the work near the top of p. 136 the
check solution also gave a result exactly equal to that obtained in the origi-
nal solution. In general, however, the check solution may be expected to
differ slightly from the original.
Ex. 2. Given a = 7, c = 9, B = 60 ; find the third side 6.
In this problem the second method furnishes the solution with the
smallest amount of labor.
fe 2 = a 2 + c 2 2 etc cos B,
b = V49 + 81 - 2 .7 9 = VtJ7.
.-. b = 8.1854.
EXERCISE XXVI
1. Given a = 426, 6 = 582, 0= 52 18';
find A = 46 21' 16", ^=81 20' 44", c = 465.8.
2. Given 6 = 123, c = 211, 4 = 115 22';
find ^ = 41 46' 45", 0= 22 51' 15", a = 286.16.
3. Given a = 121. 6, c = 192.2, B =114 .42';
find ^=24 26' 49", 0= 40 51' 11", 6 = 266.94.
4. Given a = 619, 6 = 515, 6^=39 17';
find A = 84 46' 10", B= 55 56' 50", c= 393.56.
<
5. Given 6 = 35.218, c = 54.176, A = 32 48' 14";
find ^=37 49' 35", 0= 109 22' 11", a =31.112.
6. Given a = 46.792, c = 61.234, ^=45 29' 16";
find ^ = 49 34' 5", 0= 84 56' 39", 6 = 43.836.
THE OBLIQUE TRIANGLE 147
7. Given b = 718.01, c = 228.88, A = 68 55' 2";
find B = 92 30' 47", (7= 18 3-1' 11", a = 670.61.
8. Given 5 = 2478.1, c = 5134.8, A = 137 8' 49";
find 5 =13 37' 43. 5", 0= 29 13' 27. 5", a =7152. 5.
9. Given a = 4.1203, 5 = 4.9538, O= 65 38' 52";
find A = -&4' 18", B = 65 16' 50", c = 4. 9683.
10. Given a = 0.59217, 5 = 0.21833, (7= 41 37' 4";
find ^1=119 42' 18", ^=18 40' 38", c = 0.4528.
11. Two objects A and B are separated by a body of water.
In order to find the distance between them a third point C is
chosen from which each of these points is visible, and the
following measurements are made: CA = 2560 ft., (7.5=3120
ft., and Z ACB = 105 35'. Find the distance from A to B.
12. If two sides of a triangle are 68.6 ft. and 92.2 ft.
respectively and the included angle is 112 42', what is the
third side ?
13. Find the distance between two points A, B, which are
separated by a marsh, when the distances of these points from
a third point C are 4214 ft. and 6932 ft. respectively, and the
angle A CB is 51 11.
14. In an isosceles triangle each of the equal sides is 9 and
the included angle is 60. Find the third side.
15. In an isosceles triangle each of the equal sides is 9 and
the included angle is 120. Find the third side.
16. There are two points, A, B, on the bank of a river, but
owing to a curve in its course it is impossible to measure the
distance between them directly. A third point C is chosen
such that the distances AC=l6Q ft. and 5(7=1680 ft. can
be measured, and the angle ACB is found to be 68 42' 30".
What is the distance from A to B?
17. In a given triangle two of the sides are 6 and 9 respec-
tively, and the included angle is 38. What is the third side?
18. The diagonals of a parallelogram are 8 and 10 respec-
tively, and they intersect at an angle of 60. What are the sides
of the parallelogram?
148 PLANE TRIGONOMETRY
19. If two sides of a triangle are 1468 and 2136 respectively
and the included angle is 72 21' 14", what are the values of
the other angles?
20. There are two points, A, B, so situated that they are not
visible from each other, and there is no other point from which
both can be seen. To find the distance from A to B two other
points (7, .Z), are selected so that A and D are visible from (?,
and B and are visible from D\ and the following measure-
ments are made: CD = 826.5 ft., ZACD = 121 12',ZOZ) =
58 55', ^ADC= 49 12', ^ADB = 62 38'. What is the dis-
tance from A to B?
102. Given the three sides a, b, c. When the three sides of
a triangle are given, the angles can be found directly from the
formulas proved in Art. 96, p. 133.
* <
In order to obtain a form suitable for computation by means
of logarithms we proceed as follows :
Let the sum of the sides of the triangle # + >-h<?=2s.
Then we have a + _ c =2 (s <?)
b + e a = 2 (s a),
Then, 1 cos A = I
2 be
2 be
2 be
b c)(a b + c)
2 be
-b}(s-e)
be
THE OBLIQUE TRIANGLE 149
Also (Art. 82, p. 108), 1 - cos A = 2 sin 2 ^.
NOTE. Since A < 180, being one of the angles of a triangle, < 90 ;
\ A A
therefore sin , cos, and tan are positive. Hence the radical in (4),
and the corresponding expressions in (5) and (6) below, are always positive.
-, , 5 2 4- c 2 a?
In like manner, 1 + cos A = 1 H
2 be
2 be
2 be
2 be
_ 2 s(s - a)
be
Also (Art. 82, p. 108), 1 + cos A = 2 cos 2 ^.
COS
ca
(7
o = \
*
.
6)
Dividing (4) by (5), we have
tap: f = \ 7(g -a) ^
In like manner it can be proved that
150 PLANE TRIGONOMETRY
Any one of the three formulas just given can be used in
finding the angle required. If the half angle is very small, the
cosine formula will not give a result as accurate as either the
sine formula or the tangent formula, since the cosines of angles
that are very small differ but little from each other ; and for
a similar reason the sine formula should not be used when the
half angle is near 90. In general the tangent formula is better
than either of the others.
To insure as great a degree of accuracy as possible, it is
better to solve for all the angles rather than solve for two
angles and then subtract their sum from 180. If each angle
is computed separately and their sum is found to be within
two or three seconds of 180, the work of solution is probably
correct.
If all the angles are to be computed, the following variation
of the tangent formula may be found useful.
tan =
2 * s(s-a)*
1 /( s a ) ( s b)(s c)
~~
Putting V- ~ = *
we have Un i = 7^'
In like manner, tan = ; (9)
2 s b
tan !=--. (10)
Ex.1. Given a = 79. 3, 5 = 94.2, c>=66.9; find all the
angles.
The work of solving for A and B is as follows :
a =79.3 s- a = 40.9
b = 94.2 s - b = 26
c = 66.9 s-c = 53.3
2 s = 240.4 s = 120.2
s = 120.2
THE OBLIQUE TRIANGLE
151
log (*-&)= 1.41497
log - c ) = 1 .72673
colog (s- ) = 8.38828 -10
colog 6- = 7.92010-10
2)19.45008 -20
log tan ^ = 9.72504-10
.-. |- = 2757'56".
A = 55 55' 52".
log (s-c)= 1.72673
log (s- a) =1.61172
colog (s - b) = 8.58503 - 10
colog .s=^.92010 -10
2)19.84358-20
log tan = 9.92179-10
... ^ = 39 52' 6.9".
B = 79 44' 13.8".
^+B = 13540'5.8' / .
.-. (7 = 44 19' 54.2".
If the value of C is found by logarithms in the same manner as were the
values of A and B, it will be found to be 44 19' 56.8", which is 2.6" larger
than the value found by subtracting the sum of A and B from 180. The
sum of the three angles, when all are found independently, is 180 0' 2.6".
The sum of the three angles determined in this manner is rarely equal to
exactly 180. This is due to the fact that logarithmic computation is
almost always slightly inexact. It is customary in practical work to divide
the error among the three angles according to the probable amount for each
angle.
Ex. 2. Solve the preceding example by the use of formulas
(8), (9), arid (10).
In solving by this method it is best to find all the logarithms
at the outset, and then to subtract the logarithms of s a,
s b, s c, respectively, from the logarithm of r. A com-
pact arrangement of the work can be secured by following the
model below.
log (s- a) = 1.61172
log (s- b) =1.41497
log (s-c) = 1.72673
colog s = 7.92010 - 10
log r 2 = 2.67352
log r= 1.33676
s = 120.2 Check.
Check.
log tan ^ = 9.72504 -10
log tan | = 9.92179 - 10
log tan = 9.61003 -10
A
2
B_
2
C _
2
A =
B =
C =
27 57'
39 52'
22 9'
55 55'
79 44'
44 19'
56"
6.9"
58.4"
52"
13.8"
56.8"
152 PLANE TRIGONOMETRY
EXERCISE XXVII
1. Given a = 56, ft = 58, c = 64 ;
find ^=54 22' 43", = 57 20' 32", (7= 68 16' 44".
If fe^ ' 1
2. Given a = 54, 5 = 52, e = 68 ;
find 4 = 51 24' 3.8", B = 48 48' 52.8", O= 79 47' 7.6".
3. Given a = 35, ft = 41, c = 47 ;
find .4 = 46 15' 5", =57 48' 16", C = 75 56' 41.5".
4. Given a = 73, b = 82, c = 91 ;
find A = 49 34' 58", ^=58 46' 58", C=7138'4".
5. Given a = 47, ft = 51, c = 58;
find .4 = 50 35' 18", .B = 56 58' 4", 6 Y = 72 26' 40".
6. Given a = 286, ft = 321, c = 463 ;
find J. = 37 33' 57", = 43 10' 46", G 7 = 99 15' 23".
7. Given a = 138, ft = 246, c == 321 ;
find ^=23 47' 23", ^=45 58' 41", 0=110 14'
8. Given a = 196, ft = 211, <?=173;
find vl = 60 25'31", =69 26', 6^=50 8' 36".
9. Given a = 48.3, ft = 53.2, <? = 62.7;
find ^ = 48 24' 24", ^=55 27' 44", C= 76 7' 55".
10. Given a = 226.4, ft = 431. 6, c= 316.8;
find ^=30 35' 53", 5=103 58' 55" C =45 25' 8".
/ll. Given a = 262.43, ft = 514.36, c = 556.25 ;
find A = 50 59' 18"
12. Given a = 2243. 8, ft = 2469.2, c = 3125.6;
find ^1 = 45 26' 3", ^ = 51 37' 42", (7= 82 56' 19".
\ f '
'
- 7
THE OBLIQUE TRIANGLE 153
13. Given a = 25617, 6 = 34178, c = 23194;
find .4 = 48 31' 56", 5= 88 44' 34", (7= 42 43' 30".
14. Given a = 0.34177, b = 0.45623, c = 0.58216 ;
find A = 35 54' 30", B = 51 31' 34", = 92 33' 56".
15. Given a = 11.682, = 14.468, c= 20.386;
find ^ = 34 6' 13", = 43 58' 47", (7= 101 54' 58".
16. Given a = 1.9141, 6 = 1.8365, c= 1.2854;
find A = 73 14' 32," B =66 44' 22", <7=401'5".
17. The sides of a triangle are respectively 36.92, 31.84,
26.14. Find the smallest angle of the triangle.
18. The sides of a triangle are in the ratio of 29 : 21 : 38.
Find the medium angle. |> t \\^^ ({
19. The sides of a triangle are to each other as 3:4:5. Find \(L I ! - (&
all the angles.
i S ^ ~~ ~~
20. In a given triangle a =8, 6 = 8, c = S. Find all the
angles.
21. Three cities are respectively 22.6, 21.4, 19.6 miles apart.
If the curvature of the earth is disregarded, what angles are
made by the lines joining the cities?
22. In discussing the solution of a triangle when two sides
and the angle opposite one of them are given, it was noted that
two solutions were possible when an angle was found by means
of its sine. Why does not a similar ambiguity exist when an
angle is found by means of formula (4), p. 149?
23. The sides of a triangle are a = 7, b = 8, c 5. Find the
angle A.
24. The sides of a triangle are a = 7, 6 = 5, c 3. Find the
angle A.
25. An object 16.2 ft. in length is so situated that one end
is 17J ft. and the other is 11.9 ft. from the eye of an observer.
What angle does the object subtend at the eye?
ff# -'
>i I a *
154
PLANE TRIGONOMETRY
103. Area of a triangle. In geometry it was proved that the
area of a triangle (A) can be found by either of the following
formulas: A = | base x altitude,
or, A = Vs (s a)(s b)(s <?).
The work of finding the area of a triangle can be greatly
simplified by trigonometry, as will be seen from the following
section.
104. CASE 1. Given two sides and the included angle. The
area of any triangle is equal to one half the product of the base
and the altitude. Therefore, using either of the following
figures,
But
A = 1 c CD.
CD = a sin B.
c
sin A.
Substituting this value of c in (1), Case 1, we have.
a 2 sin B sin C
A =
2 sin A
But since A + B + 6'= 180, sin A = sin (B + <7) ;
. ^ __ a 2 sin B sin C
= 2 sin (B + (7)
CD
(2)
In like manner it can be proved that
A = * be sin A,
and A = -*- ab sin C.
CASE 2. Given a side and the two adjacent angles. By the
law of sines (Art. 94, p. 131),
a : c = sin A : sin C.
a sin C
(4)
THE OBLIQUE TRIANGLE 155
CASE 3. Given the three sides. In Art. 80, p. 106, it was
proved that , ,
sin A = 2 sin cos .
L '
But (Art. 102, p. 149),
be
and cos -^ '
Substituting these values in the above equation, we have
2
sin A = T-S(S a)(sb)(s c).
Substituting this value of sin A in (2), we have
A = V*( -)(_ 6)( s - c). (5)
CASE 4. Given three sides and the radius of the circum-
scribed circle. By Art. 95, p. 132, we have
where r is the radius of the circumscribed circle. Substituting
this value in (2), we have
A=f (6)
CASE 5. Given three sides and the radius of the inscribed
circle.
Let r be the radius of the inscribed circle. The triangle
can be divided into three triangles whose bases are a, 6, <?, re-
spectively, and whose common altitude is r. Then
A = r( + ft + c). (7)
CHAPTER XIII
MISCELLANEOUS PROBLEMS IN HEIGHTS AND
DISTANCES
105. In this chapter certain problems will be considered
that are frequently met in land surveying, railroad work,
etc. The degree of accuracy required in practical problems of
this kind can only be known after the nature of the special
problem under consideration is known. Hence, in the examples
that are here considered no attempt is made to conform to
the ordinary practice of field surveyors. In many classes of
problems that they are called upon to solve a .sufficient degree
of accuracy is, secured if the angles are measured to single
minutes and the computations are performed by means of four-
place tables of logarithms; while in others the measurements
are made with the greatest possible accuracy and the computa-
tions are performed with the aid of eight-, ten-, or twelve-place
tables. For this reason it is quite impracticable for an elemen-
tary text-book in trigonometry to attempt to conform to field
usage.
The tables used in the solution of the problems in this
chapter are five-place tables.
106. The height of an object by means of observations made
at distant points.
Let AB represent the height of an object, and let (7, JJ,
be two points of observation on the same level with A, so
situated that A, 0, Z), are in the
same straight line. Let the angle
of elevation of B at C be , and at
D be /3, and let DC=a. Then from
the triangle ABC
./ / ~t *\
/,v< = s1 ""'
150
PROBLEMS IN HEIGHT- L FIANCES 157
and from the triangle DCB
BC
a sin (a
Substituting this value of 1M? in (1). and reducing we have
a formula which gives the value of x in a form suitable for
computation either by logarithms or by natural functions.
Ex. 1. What is the height of a tower when the angles
of elevation of the top of the tower from two points 250 ft.
apart on the ground and in the same straight line with the foot
of the tower are 30 and 60 respectively?
= 60% and = 3G. Therefore
>in 60 sin 30
x =
sin 3IT
50-1 x ^ = 218.5 ft.
107. If the height of an object is to be determined, and no
two points can be found that are in the same st might line,
and at ilie same time conveniently situated for observation, the
following method is often employed :
From A measure AB in any convenient direction. Let the
angle of elevation of the top of the object D, measured at A*
be , and let the distance AB be a. At A and B measure the
angles DAB=& and DIM = 7, respectively. Then in the
triangle AD*
Therefore,
AD _ _ 81117 _ _ siu 7
~o~ " sin (1W - ( + 7)) ~ sin ( + 7)
1 -ing the value of AD obtained
from this equation, we have
.
sin ( + 7)
158 PLANE TRIGONOMETRY
MISCELLANEOUS EXAMPLES
THE RIGHT TK I ANGLE
1. The angle of elevation of the top of a vertical cliff
426.28 ft. high, taken from a point on the same level as the
foot of the cliff, is 59 51' 14". What is the distance from the
foot of the cliff to the point where the observation was taken?
2. A pole 36 ft. high casts a shadow 39 ft. long. What is
the angle of elevation of the top of the pole, measured at the
extremity of the shadow?
3. The height of a room is 12.62 ft. and its length is
14.44 ft. What is the angle of e! of one of the upper
corners of the room taken at the lower corner on the same
side?
4. What is the elevation of the sun when a tree 31.6 ft. high
casts a shadow 42.9 ft. in length ?
5. What angle does a ladder 25.2 ft. long make with the
ground when it just reaches the sill of a window 18.95 ft.
above the level on which the foot of the ladder rests ?
6. The angle of depression of a point on the ground, meas-
ured from the top of a building 49.27 ft, high, is 34 6' 36".
What is the distance from the foot of the building to the given
point ?
7. The length of the diagonal of a rectangular field is
247.39 ft., and the angle between the diagonal and the
shorter side of the field is 29 40' 36". What is the width
of the field?
8. A path measuring 256.4 ft. in length leads diagonally
across a rectangular plot of ground, making with one of the
sides an angle of 61 12' 52". What is the length of the
side ?
9. The angle of elevation of a balloon measured at a certain
point is 71 14' 12", and from this point to a point directly
below the balloon the horizontal distance is 415.9 ft. What
is the height of the balloon and its distance from the point of
observation ?
PROBLEMS IX HEIGHTS AND DISTANCES 159
10. A kite is fastened to a string 483.2 ft. long, and the string
makes an angle of 63 19' 28" with the level of the ground.
What is the vertical height of the kite above the ground, no
allowance being made for the sagging of the string ?
11. To ascertain the width of a river a distance AB is meas-
ured along one of the banks 262.38 ft. Directly across the
river from B is a point (7, and the angle BAG is found upon
measurement to be 41 38' 20". Required the width of the
river.
12. Two forces, of 198.52 Ib. and 393.13 Ib. respectively,
are acting at right angles to each other. What is the resultant
of the two forces, and what is the angle which the direction
of each force makes with the resultant ?
13. What is the radius of the parallel passing through a
point on the earth's surface whose latitude is 43 15', the radius
of the earth being reckoned as 3956 mi. ?
14. The angle of elevation of the top of aitill viewed from
a certain point is 29 17', and from a point 362.4 ft. nearer,
measured directly toward the hill, the angle of elevation is
48 12'. Required the height of the-MH-. -
15. From the top of a mountain the angles of depression of
two milestones 2 mi. apart and in the same vertical plane with
the top of the mountain are 10 14' 42" and 5 38' 46" respec-
tively. What is the height of the mountain?
16. A flagstaff which is broken at a certain distance above
the ground falls so that its tip touches the ground at a distance
of 13.5 ft. from the foot of the portion which remains standing.
The length of the part broken over is 35.1 ft. What was the
total height of the staff before it was broken over ?
17. If the angle of depression of the visible horizon, observed
from the top of a mountain 3 mi. in height, is 2 13' 59", what is
the diameter of the earth ?
18 A ladder 30 ft. long when set in a certain position
between two buildings will reach a point 20 ft. from the
ground on one of the buildings, and on being turned over
without having the position of its foot changed it reaches a
1HO PLANE TRIGONOMETRY
point on the other building 15 ft. from the ground. What is
the angle between the two positions of the ladder ? (Solve by
natural functions.)
19. A lighthouse 50 ft. high stands on the top of a rock.
The angle of elevation of the top of the rock and of the top
of the lighthouse, measured from the deck of a vessel, are 6 5'
and 6 58" respectively. What is the height of the rock, and
the distance from the vessel to the foot of the rock ? (Solve
by natural functions.)
20. At any point on the earth's surface a line is drawn tan-
gent to the surface at that point. If the earth is considered a
sphere whose diameter is 7912.4 mi., how far from the surface
will the line be at the end of 1 mi.?
21. A building 50 ft. high stands at the foot of a hill, and
from the top of the hill the angles of depression of the top
and of the bottom of the building are 45 15' and 47 12'
respectively. What is the height of the hill ?
22. The angles of a triangle are 1:2:3, and the perpendicu-
lar from the greatest angle to the side opposite is 15 ft.
Required the sides of the triangle.
23. A bridge of five equal spans crosses a river, each span
measuring 100 ft. from center to center. From a boat moored
in line with one of the middle piers the length of the bridge
subtends a right angle. What is the distance from the boat to
the bridge? (Solve by natural functions.)
24. An observer on a vessel at anchor sees another vessel
due north of him; at the end of fifteen minutes it bears E.,
and half an hour later it bears S.E. How long after it is first
seen is it nearest the observer, and in what direction is it sail-
ing, its course being supposed to be in a straight line from the
time of the first to the time of the last observation? (Solve by
natural functions.)
25. A statue on a column subtends the same angle at dis-
tances of 27 and of 33 ft. from the column. If the tangent of
the angle equals T ^, what is the height of the statue ? (Solve
by natural functions.)
PROBLEMS IN HEIGHTS AND DISTANCES 161
26. A tower 145 ft. high stands on an elevation 75 ft.
high. At what point in the plain on which the elevation
stands must an observation be made in order that the tower
and the height of the elevation may subtend equal angles?
(Solve by natural functions.)
27. A flagstaff 50 ft. high stands in the center of a plot
of ground in the form of an equilateral triangle. Each side
of the triangle subtends at the top of the staff an angle of 60.
What is the length of one of the sides of the triangle ? (Solve
by natural functions.) *
28. A tower stands on the slope of a hill that has an
inclination of 15 to the level of the plain. At a point 80 ft.
farther up the hill it is found that the tower subtends an angle
of 30. Prove that the tower is 40(VJ- V) ft. in height.
29. At a distance of 300 ft. from the foot of a tower the
angle of elevation is one third as great as it is at a distance of
60 ft. What is the height of the tower?
THE OBLIQUE TRIANGLE
30. The angles of elevation of a balloon measured at the
same instant at two points on level ground and in the same
vertical plane as the balloon are 41 56' and 28 14' respectivel} T .
The two points from which the angles are measured are 3462
ft. apart and on the same side of the balloon. Required its
height at the time of observation.
31. The angle of depression of an object viewed from the
top of a tower is 50 12' 56", and the angle of depression of
a second object 250 ft. farther away, and in a straight line with
the first object and the foot of the tower is 31 19' 54". What
is the height of the tower ?
32. The angles of depression of two objects on a level plain,
viewed from an elevation in the same vertical plane with the
objects, are 48 12' and 29 17' respectively, and the distance
between the two points is 362.4 ft. Required the height of
the point of observation.
CON ANT'S TKIG. 11
162 PLANE TKIGONOMETKY
33. The sides of a triangular plot of ground are 138 ft.,
246 ft., and 321 ft. respectively. What is the greatest angle
formed by the sides?
34. Two objects are separated by a building, and it is re-
quired to find the distance between them. At a third point,
distant 268 ft. and 315 ft. respectively from the given ob-
jects, the angle subtended by the line connecting the objects
is measured and is found to be 108 17'. What is the distance
between the objects ?
i^.
35. What is the distance between two points 4, B, when
the distances from these points to a third point C are 6282 ft.
and 2344 ft. respectively, and the angle :S*UL is 119 40' 40"?
Is more than one solution possible ? Why ? (See Art. 100,
p. 138.)
36. The distance between two points A, B, cannot be ob-
tained directly by the use of the chain or tape because of an
intervening body of water. A third point C is chosen from
which both A and B are visible, and the following measure-
ments are then made: 4(7=3101.8 ft., Z CAB = 51 28',
Z. ABQ = 70 37' 33". What is the required distance ?
37. In a system of triangulation the sides of a triangle con-
necting the stations on the tops of three hills have been com-
c puted and have been found to be 54,692.73 ft., 61,284.39 ft.,
and 42,798.64 ft. respectively. What are the values of the
angles of this triangle as computed from the sides ?
38. An observation station A is set up in a field along one
side of which runs a straight, level road. Two points of ob-
servation on the road, J5, (7, one fourth of a mile apart, are
*o chosen, on opposite sides of the first station and the angles
ABO, ACB, are measured and found to be 46 20' 28" and
38 24' 48" respectively. What is the distance from the station
A to the road ?
39. The distances from a point on shore to two buoys are
known to be 1286 ft. and 2466 ft. respectively, and the angle
subtended at that point by the line connecting the buoys is 42
14' 16". What is the distance between the buoys?
PROBLEMS IN HEIGHTS AND DISTANCES 163
40. A tripod is set up on a rock, and to find the distance
from the tripod to the shore a line 8500 ft. in length is meas-
ured along the shore, and at each extremity of the line the
angle is measured which subtends the line connecting the
tripod with the other end of the line. The angles are found
to be 46 28' and 43 32' respectively. Find the distance from
the tripod to the line of measurement along the shore.
41. Two vessels lying at anchor 1 mi. apart are observed
from a third vessel sailing east to be in a straight line due
north. After sailing an hour and a half one of the vessels
bears N.W. and the other W.N.W. Find the rate at which
the vessel is sailing. \ \ ^ ^
42. The distance between two points A, B, is to be deter-
mined, where B is accessible and -4Ms not. A kite is sent up
and made fast, and its position is determined to be 517.3
yd. vertically above D, which is on the same level with A and
B. The following angles are then measured: A 6^5 = 13 15'
15", CAD= 21 9' 18", DBC=<2& 15' 34". \What is the dis-
tance from A to B? J $ v % $~ I
43. Two forces, of 410 Ib. and 320 Ib. respectively, are act-
ing at an angle of 51 37'. Required the direction and in-
tensity of the resultant.
44. A kite A has been sent up and is fastened to the ground
at a point Q. The kite has drifted a certain distance and now
stands directly above a point B, which is on the same level as
(7, but is separated from it by obstacles which render direct
measurement impracticable; and the height of the kite is de-
sired. To ascertain this a line is measured from to a point
Z), 4262.4 ft. in length, and the following angles are meas-
ured: ACB= 31 17' 14", ACD= 66 14' 52", CDA = 52 51'
38". Required the vertical height of the kite above the point B.
(See Art. 107.)
45. Two rocks are to be charted. To ascertain the distance
between them the angles of elevation of a point at the top of a
cliff 527.4 ft. high are taken and are found to be 21 8' 16"
and 23 14' 20" respectively, and the angle subtended by the
164 I'LAM-: TUGOXOMHTKY
line connecting the rocks, measured at a point at the top of
the cliff, is 16 3' 30". Required the distance between the
rocks.
46. A balloon, J., is sighted at the same instant from two
points, B, C, which are on the same level, and are 262.4 ft.
apart. The angle of elevation of the balloon at B is 41 15'
24", ZABC = 62 48' 14", ZACB=59 14' 21". What is the
height of the balloon at the instant of observation? ^ ^\ ^ t *^ \
47. A tower stands on the slope of a hill which makes an
angle of 16 with the horizon. At a distance of 95 ft. from
the foot of the tower, measured directly up the side of the hill,
the height of the tower subtends an angle of 38. What is the
height of the tower?
48. A tree stands E.S.E. of an observer, and at noon the
extremit}^ of the shadow of the tree is directly N.E. of the
position in which he is standing. The length of the shadow is
60 ft., and the angle of elevation of the top of the tree viewed
from the position of the observer is 45. What is the height of
the tree? (Solve by natural functions.)
49. It is required to find the distance between two points,
A, B, neither of which is accessible. For that purpose a base
line, (7Z>, 4968 ft. long, is measured, and the following angles
are observed: ACD=W8 14', 6^Z) = 41 15', 7X7=11,5
21', ADO= 39 42'. What is the distance from A to B>
50. Two points are so situated that it is not possible to
measure directly from one to the other, but observations can
be taken at either point. Two other points, (7, D, are chosen,
5226 ft. apart, and the following angles are measured: ACB
= 15 18' 24", DAO= 21 12' 46", DBC= 23 18' 42", AT)C =
BDC= 90. What is the distance from A to #?
51. To find the distance between two inaccessible points, A,
B, two other points, (7, D, are chosen,' so situated that from
either of them the three other points can be seen; and the fol-
lowing measurements are then made: 6 Y .Z) = 826.5 ft., ZACD
= 121 12', Z BOD = 58 55', Z.ADC= 49 12', Z.ADB = 2
38'. What is the distance from A to B'>
PROBLEMS IN HEIGHTS AND DISTANCES
165
52. Two points, A, B, are so situated that only one point, (7,
can be found which is conveniently situated for observation,
from which both can be seen. A fourth point, D, is found from
which A and Q can be seen, and a fifth point, J, from which B
and C can be seen. The following measurements are taken,
from which it is required that the distance from A to B shall
be computed: CD = 6428.72 ft., OE = 5872.54 ft.,
= 64 21'.
53. Two points, A, B, are so situated that no point can be
found from which both can be seen. Two other points, (7, 1), are
found, so placed that A and D can be seen from C and B from D,
and also two additional points, E, F, so placed that A and
can be seen from F, and B and D from E. The following data
can now be obtained for the determination of the distance from
A to B: (7Z)=1254 ft,, OF =1216 ft., 7)^=1216 ft., Z.AFC
= 78 14' 15", ZFCA = 53 51' 40", Z.4CZ) = 52 17' 18", Z. CDB
= 155 24' 20", ZBDE=53 49' 8", ZD^ = 82 57'. What
is the length of the line AB?
CHAPTER XIY
FUNCTIONS OF VERY SMALL ANGLES HYPERBOLIC
FUNCTIONS TRIGONOMETRIC ELIMINATION
108. Trigonometric functions of very small angles. Let
A OB be any angle less than 90* With as a center and any
radius OA describe a circle.
Draw BQ perpendicular to OA, and
produce it to intersect the circle in B 1 ' .
Draw tangents to the circle at B, B'.
These tangents will, by geometry, inter-
sect OA produced in the same point D.
Then
chord BB' < arc BB' < BD + B' D.
Dividing by 2, CB < arc AB < DB.
CB arc AB BD
'OB OB " OB '
CB . /j BD ;\rcAB ,
But - = sm0, = tan 0, and = the circular
OB OB OB
measure of the angle 0, or of the arc AB (Art. 13, p. 16).
Therefore, sin < < tan 0.
This important result may be expressed as follows :
When 6 < 90, sin 6, 6, and tan are in the ascending order
of magnitude.
109. Dividing the inequality just obtained by sin 0, we have
1<
sin<9
< sec 0,
or,
166
FUNCTIONS OF VERY SMALL ANGLES 167
Therefore, lies between 1 and cos 6 for all values of 6
u
between and -
But as approaches as its limit, cos approaches 1 as its
limit; and at the same time approaches 1 as its limit.
COS0
Therefore, when is very small, and is approaching as its
limit, lies between 1 and a quantity that may be made
to differ from 1 by a quantity e which may be made as small as
we please ; and as approaches as its limit, e also approaches
as its limit. t, . * ; , t - ^<( l^ e^ . &-^
In other words, when fl x approaches as its limit, S1 " ap-
u
proaches 1 as its limit. This fact is often expressed by the
statement that when is very small, sin = 0, approximately.
In like manner it can be shown that as approaches as its
limit, tan will also approach the limit ; that is, when is
very small, tan = approximately.
From the above it follows also that when is very small, sin
= tan 0, approximately.
In this discussion it should be remembered that is ex-
pressed in^circular measure ; i.e. is the number of radians
in the angle or arc under consideration.
EXERCISE XXVHI
1. Find the sine and the cosine of V.
Let x be the circular measure of 1'.
Therefore, since x > sin x > 0, (Art. 108)
sin 1' lies between and 0.000290889.
Also, cosl' = Vl-sin 2 l
> Vl - (0.00029088^) 2
^O QQQQQQQ
>0.9999999.
.-.cosT = 0.9999999+. (1)
But (Art. 108, p. 166), sin x > x cos x.
.-. sin l'> 0.000290888 x 0.9999999
.kO.000290887. (2)
168 PLANE TRIGONOMETRY
Therefore, sinl' lies between (1) and (2); i.e.
sin 1' = 0.00029088+,
and the next decimal place is either 7 or 8.
Find approximately the values of the following :
2. sin 10'. 4. sin 1'. 6. cos 15'.
S.^cosTO'. .5. sin 15'. 7. sin 8".
HYPERBOLIC FUNCTIONS
110. In the differential calculus it is proved that the follow-
ing equations are true for all values of x :
ein ~ _ ~ X X X> i . . . -fl\
bill X JU -f -T- ' , {1J
cos ^_^_^__|_?: ^__^_...j (2)
where e = 2.7182818 is the base of the natural system of log-
arithms. In (1) and (2) x is the value of the angle or arc
expressed in radians.
If in (3) x is replaced by ix, where i = V 1, we have
X 2
The series in the first parenthesis is the same as the right
member of (2), and that in the second parenthesis is the same
as the right member of (1). Hence, replacing these series by
their values, we have equation (4) in the following form :
e** = cos x + i sin x. (5)
In a precisely similar manner it may be shown that
e~ lr = cos x i sin x. (6)
HYPERBOLIC FUNCTIONS
Adding (5) and (6), and dividing by 2, we have
. (7)
Subtracting (6) from (5) and dividing by 2 i, and the cor-
responding value for sin x is obtained :
(8)
These equations give the values of the sine and the cosine of
any angle whatever in exponential form.
111. If in (5) and (6) of the preceding section we replace x
by ix, the following equations are obtained :
e~ x = cos ix-\- i sin ix\ (1)
e x = cos ix i sin ix. (2 )
By addition and subtraction we obtain from these the results
below : e x , e -x
cos ix = - -~ - ; (3)
It will be noticed that the exponential functions which occur
in the right-hand members of (3) and (4) possess a striking
similarity to those which appear in (7) and (8) of the preced-
ing section. It has been found convenient to make use of this
similarity, and, corresponding to the exponential values of
sin x and cos x given in those equations, to give the following
definitions :
- is called the hyperbolic cosine of x,
oX p X
and is called the hyperbolic sine of x.
ft
These functions are written in abbreviated form cosh x and
and sinh x respectively. Accordingly we have
g.r I g X
cosh x y " = cos ix ; (5)
sinh x = '- = i si nia?. (6)
170 PLANE TRIGONOMETRY
The name hyperbolic is applied to these functions because
they bear to the equilateral hyperbola a relation analogous to
that which sin a; and cos x bear to the circle. (Art. 46, p. 64.)
The other hyperbolic functions are denned as follows :
-
cosh#
sinh x
sech x = 1 ; (9)
cosh x
- ..
sinli x
112. Ex. l. Prove the relation sinh = 0.
By (6), Art. Ill, we have
(10)
.
2 2
Ex. 2. Prove the relation
sinh (x -f y) = sinh x cosh y + cosh x sinh y.
By definition
sinh (x + y) = - i (sin (ix + iy) )
= i (sin t.r cos /# + cos ix sin z/y)
= i (i sinh x cosh ?/ + i cosh x sinh #)
= sinh x cosh y + cosh x sinh ?/.
Ex. 3. Prove the relation
sinh x + sinh y = 2 sinh ^^ cosh
By definition
sinh x + sinh y i (sin ix + sin z
HYPERBOLIC FUNCTIONS 171
EXERCISE XXIX
Prove the following identities :
1. cosh = 1. 9. sin ( ix) = sin ix.
. i TTI . 10. cos( iz} = uosix.
2. smn = i.
11. tan ix = i tanh x.
3. cosh = 0. 12 - sinh (-2:)= -sinh a;.
13. cosh ( x) = cosh x.
4. sinh7n'=0. , , .
14. coth ( x) = coth #.
5. cosh9r* = -l. 15 sec h(-:r)=sech*.
6. sinh2mr = 0. 16> C sch ( - ^) = - csch a;.
7. cosh2mr = l. 17.
8. tanh = 0. 18. sech 2 x + tanh 2 x = 1.
19. csch 2 x coth 2 a; = 1 .
20. cosh (x + /) = cosh x cosh y -}- sinh x sinh ^.
21. sinh 2x 2 sinh a: cosh x.
22. cosh 2 # = cosh 2 x + sinh 2 #.
23. sinh # sinh y = 2 cosh ^^ sinh H
24. cosh a; + cosh y= 2 cosh ^-^ cosh ~
2i 2
25. cosh x cosh y = 2 sinh x ^ sinh ' y ~^
2 2
113. The notation for inverse hyperbolic functions is the
same as for inverse circular functions (Art. 84, p. 114).
If y = sinh x,
then, x = sinli" 1 ?/.
But by (6), p. 169, y = e ll.
Solving this equation for #, we have
1).
172 PLANE TRIGONOMETRY
In like manner, cosh' 1 ^ = log (?/ -f V/ 2 1) ; (2)
tanh- 1 *, = 1 log i^; (3)
J- y
coth- 1 y = tanh- 1 - = \ log 1 ; (4)
y * y -*
sechr 1 / = cosh a - = log *- ; (5)
csch- 1 y = sinh- 1 = log A ^. (6)
EXERCISE XXX
Prove the following relations :
1. tanh- 1 -^_ = 2 tanh- 1 2:.
2. sinh" 1 2 # = 2 sinh" 1 a; cosh" 1 a;.
3. sinh" 1 z = cosh" 1 Vl + x z .
4. sinh" 1 a; = tanh" 1
VI
T -4-
5. tanh" 1 x+ tann" 1 y = tanh" 1 -
ELIMINATION
114. It often happens that two or more equations are given
that contain trigonometric functions of some angle, or perhaps
of more than one angle. From these equations a single equa-
tion is to be obtained from which all trigonometric functions
have been eliminated.
In theory the required elimination can always be performed,
but in practice this often involves processes that are some-
what complicated ; and the desired results are obtained with a
greater or less degree of difficulty.
No general rule for work of this kind can be given ; and the
process is best illustrated by a few examples.
TRIGONOMETRIC ELIMINATION 173
115. Ex. i. Find the values of r and 6 from the equations
r sin 6 = a ; (1)
r cos d = b. (2)
Squaring and adding,
r 2 (sin 2 + cos 2 0) = a 2 + 6 a ,
r 2 = 2 + b 2 ,
r = \ / a 2 + b*.
Also, dividing (1) by (2),
tan0 = ,
6
= tan- 1 ? .
o
Ex. 2. Find the equation of relation between a and b if
sin 3 = a, and cos 3 = 6.
From the values here given we have
sin = , and cos = 6*.
But for all values of 0, sin 2 + cos 2 0=1.
Therefore, substituting, a ^ + 6 7 = 1,
which is the equation desired.
Ex. 3. Eliminate 6 from the equations,
a cos + b sin # = c,
d cos + e sin =/.
Solving by any of the ordinary methods of elimination,
d c<7 a/*
sm0 = - J-,
od ae
bd ae
Substituting these values of sin and cos in
sin 2 + cos 2 0= 1,
and reducing, we have
- *
(j/_ ce y + (c,i ._ a f) 2 = (bd - ae)
Ex. 4. Eliminate from the equations
cot + tan = a ; (1)
sec cos = b. (2)
From (1) a = - 1 - + tan = 1 + ta f .
tan tan
S6C \/ xo\
a= te^- (3)
174 PLANE TRIGONOMETRY
From (2) 6 = sec - =
sec sec
.
sec0
From (3) and (4) 2 6 = sec 3 0, and a& 2 = tan 3 0.
But sec 2 0- tan 2 0=1.
(4)
= 1,
or, aV - aV = 1.
Ex. 5. Eliminate from the equations
- cos 6 - & sin 6 = cos 20; (1)
#
-sin<9 + ^cos<9:=2sin20. (2)
a 6
Multiplying (1) by cos and (2) by sin and adding the resulting equa-
tions, we obtain x
- = cos cos 20 + 2 sin 2 sin
a
= cos cos 2 + sin sin 20+ sin sin 2
= cos + 2 sin 2 cos 0. (-1)
In like manner, multiplying (1) by sin and (2) by cos0 and subtracting,
= 2 sin 2 cos - cos 2 sin
= sin0+2sin0cos 2 0. (4)
we obtain 1t
= 2 sin 2 cos - cos 2 sin
b
Adding (3) and (4),
- + 1 - cos + sin + 2 sin cos (cos + sin 0)
= (cos + sin 0) ( 1 + 2 sin cos 0)
= (cos + sin 0)(cos 2 + sin 2 + 2 sin cos 0)
= (cos0 + sin0) 3 .
i
(5)
By subtracting (4) from (3) and reducing the result, we find that
()
Squaring (5) and (6) and adding the results, we obtain the following,
which is the desired equation :
x y
__ ^
a b
TRIGONOMETRIC ELIMINATION 175
Ex. 6. From the following simultaneous equations, find the
values of r, </>, 6 :
r sin 0cos< = a; (1)
r cos cos < = & ; (2)
r sin < = c. (3)
Dividing (1) by (2), ten0 = |. .-. = tan-*S. (4)
Squaring (1) and (2) and adding,
r 2 cos 2 < = a 2 + 6 2 . (5)
Taking the square root of (5), and then dividing (3) by this result,
c (6)
Va 2 + b 2 Va 2 + 6 2
Squaring (3) 'and adding the result to (5),
r 2 = a 2 + b' 2 + c 2 ,
r = Va 2 + 2 + c 2 .
EXERCISE XXXI
1. Find r and if r sin = 1.25 and r cos = 2.165.
Eliminate from the equations following :
2. cos + 6 sin = c, and 6 cos 6 a sin = c?.
3. - cos + f sin = 1, and - sin ^ cos = 1.
a b a b
4. a sec 06 tan = <?, and c? sec + tan 0=6.
5. a cos 20 = 6 sin 0, and e sin 2 = 6? cos 0.
6. cos + sin = a, and cos 2 = 6.
7. sin + cos = a, and tan + cot 0=6.
8. cot + cos = a, and cot cos = 6.
9. sin cos = , and esc sin = 6.
10. sin + cos sin 2 = a, and cos + sin sin 20=6.
11. sec cos = a, and esc sin = 6.
Eliminate and $ from the following equations :
12. tan + tan $ = a, cot + cot ^> = 6, and + c = a.
13. sin + sin <$> = a, cos + cos $=b, and $ = <*.
14. a cos 2 + 6 sin 2 = ccos 2 (, #sin 2 + 6cos 2 = c?sin 2 <,
and c tan 2 - d tan 2 <j> = 0.
176
PLANK TRIGONOMETRY
SPHERICAL TRIGONOMETRY
CHAPTER XV
GENERAL THEOREMS AND FORMULAS
116. Spherical trigonometry is that branch of trigonometry
which treats of the solution of spherical triangles.
117. The following definitions and theorems are to be found
in works on solid geometry. For a discussion of the defini-
tions and for proofs of the theorems the student is referred to
any text-book on that subject.
DEFINITIONS AND THEOREMS
1. The curve of intersection of a plane and a sphere is a
circle.
2. A great circle is a circle formed by a plane that passes
through the center of the sphere.
3. A small circle is a circle formed by a plane that inter-
sects the sphere without passing through its center.
4. Through any two points on the surface of a sphere one
and only one great circle can be passed, unless these points
are at opposite extremities of a diameter of the sphere.
5. A spherical angle is the angle between two arcs of great
circles. It is equal to the angle between the tangents to the
two circles drawn at their point of intersection ; it is also equal
in angular magnitude to the dihedral angle formed by the
planes of the two great circles.
6. A spherical polygon is a portion of the surface of the
sphere bounded by three or more arcs of great circles.
7. A spherical trian'gle is a spherical polygon of three sides.
CONANT'S TRIG, 12 177
178 SPHERICAL TRIGONOMETRY
118. Let ABC be any spherical triangle, and the center
of the sphere on whose surface the triangle is drawn. The
vertices are represented geometri-
cally by the letters A, B, C, and the
same letters are used to designate
the angles lying at these vertices
respectively. The sides opposite
these angles are designated by the
corresponding letters a, b, c. Since
is the center of the sphere, OA = OBOC, each being a
radius of the same sphere. Also, the arcs a, b, c, are the meas-
ures of the central angles BOC, AGO, A OB, respectively.
THEOREMS. The following theorems on spherical triangles
were proved in solid geometry.
I. The sum of any two sides of a spherical triangle is greater
than the third side.*
II. In any spherical triangle the greatest side is opposite the
greatest angle, and conversely. Also, equal sides are opposite
equal angles.
III. Any angle of a spherical triangle is less than 180.
IV. The sum of the angles of a spherical triangle is greater
than 180 and less than 540 ; i.e. 180 < A -f B + 0< 540.
V. Any side of a spherical triangle is less than 180.
VI. The sum of the sides of a spherical traingle is less than
360 ; i.e. a + b + c< 360.
VII. The difference of any two angles of a spherical triangle
has the same sign as the difference of the corresponding opposite
sides ; e.g. A B and a b are of the same si</n.
VIII. If from the vertices of a spherical triangle as poles, arcs
of great circles are drawn, a second spherical triangle will be
formed which is called the polar of the first triangle.
Let ABC be any spherical triangle, and a', &', c' be arcs of great circles
drawn with A,B, C, respectively as poles. If these arcs are extended and
the great circles are fully drawn, the surface of the sphere is divided into
* Three great circles intersect on the surface of a sphere in such a way as to
form eight triangles ; and one of these triangles always satisfies the theorems of
this section. Only such triangles are considered in this work.
GENERAL THEOREMS AND FORMULAS
179
eight spherical triangles. That triangle A'B'C' is called the polar of ABC
which is so situated that A and A' lie on the same side of BC ; B and B' on
the same side of A C ; C and C" on the same side of AB.
Any angle of a spherical triangle is the supplement of the side opposite
in its polar.
HABC is the polar of A'B'C', then conversely A'B'C' is the polar of
ABC.
Let ABO arid A' B' 0' be two polar triangles, and let a, b, <?,
and a', 6', <?', be the sides opposite the like-named angles in the
two triangles respectively.
A' -180 -a,
B 1 = 180 - b,
Then, .l = 180 -a',
B = 180 - b r ,
(7=180-</.
Spherical triangles are called isosceles, equilateral, equiangular,
right, and oblique under the same conditions as are the corre-
sponding plane triangles.
It is to be remembered, however, that a spherical triangle
may have one, two, or three right angles. If it contains two
right angles, it is called a bi-rectangular spherical triangle ; and
if it contains three right angles, it is a tri-rectangular spherical
triangle.
NOTE. The length of a side of a spherical triangle, expressed in linear
measure, can not be determined until the radius of the sphere is known.
119. FUNDAMENTAL THEOREM. To express a side of a
spherical triangle in terms of the other two sides and of the angle
opposite :
Let ABQ be a spherical triangle and the center of the
sphere.
180
SPHERICAL TRIGONOMETRY
E
From D, any point in the radius
OA, draw DE, DF, perpendicular
to OA, in the planes OAB, OAC,
respectively. Connect EF.
Then is the plane angle EDF
equal to the angle A (Art. 117,
p. 177).
In the plane triangles DEF, OEF, we have (Art. 96,
P- 133 ) EF 2 = DE 2 + DF 2 -2DE- DF cos A,
and EF 2 = OE 2 + OF 2 - 2 OE - OF cos a.
Equating these values of EF 2 and transposing, we have
L-2 OE.
Substituting OD 2 for OE 2 - DE 2 , and also for OF 2 - DF 2 , this
becomes
20D 2 + 2DE-DFcosA-2 OE OFcosa=0.
Dividing by 2 OE OF,
OD OD DE DF
i.e.
cos a cos b cos c -f sin b sin c cos A.
*J.
120. An examination of the figure which accompanies the
demonstration in the preceding article shows that the implied
supposition is there made
that both b and c are less
than 90, but that no restric-
tion is placed on a. ti/ \ V \*>
In order to establish the
truth of the theorem for all
values of a and b we pro-
ceed as follows :
Let b be greater than 90. Produce the arcs CA and OB
until they intersect again in C'.
Since AC > 90, we have AC' < 90. Therefore in the tri-
angle ABC', A( j, 90 o
and, by hypothesis, AB < 90,
while BC' is unrestricted.
GENERAL THEOREMS AND FORMULAS 181
Applying (1), Art. 119, to the triangle ABC', we have
cos a' = cos b' cos c -f- sin b r sin c cos Z C'AB. (1)
But (Art. 53, p. 78), cos a' = cos a,
cos b r = cos b,
and cos Z C'AB = cos A.
Substituting these values in (1), we have
cos a = cos b cos c -f- sin b sin c cos ^4.
In like manner it can be shown that the theorem remains
true if a and b are both greater than 90. Hence, it is true
for all spherical triangles which come within the scope of
our work.
Also, by drawing the perpendiculars DE< DF, from some
point in the radius OB in the planes BOC, BOA, respectively,
in the figure of Art. 119, we can obtain a corresponding formula
for expressing the value of cos b ; and by drawing these per-
pendiculars from some point in the radius 00, in the planes
OOA, COB, respectively, a similar formula for the value of
cos c. Therefore,
cos a = cos b cos c 4- sin b sin c cos A,
cos b = cos c cos a-\- sine sin a cos B, (2)
cos c = cos a cos b + sin a sin b cos C.
The above are relations involving the sides and one of the angles
of a spherical triangle.
From these equations the following are at once derived :
cos a cos b cos c
cos A =
T> cos b cos c cos a
cos -o =
COS
sin c sin a
cos c cos a cos b
sin a sin b
sin b sin c
(3)
These relations express the values of the cosines of the angles of
a spherical triangle in terms of the sides of the triangles.
182 SPHERICAL TRIGONOMETRY
121. After the first of the three formulas in (2) or in (3)
in the preceding article has been obtained the others can be
derived from it by a cyclic interchange of the letters a, >, c,
replacing at the same time A by B, and B by 0.
122. The law of sines. From plane trigonometry we have
the relation . .
sin 2 A = 1 cos 2 A.
Replacing cos 2 A by its value from (3) in the preceding
section,
sin 2 A = 1 ( cos a ~ cos fr cos g ) 2
sin 2 b sin 2 c
_ sin 2 b sin 2 c (cos a cos b cos <?) 2
sin 2 b sin 2 c
_ (1 cos 2 #)(! cos 2 (?) (cos a cos b cos e) 2
sin 2 b sin 2 c
Expanding, reducing, and rearranging terms, we have
. 2 A _ 1 c s 2 c s 2 & cos2 + 2 cos cos cos <?
sin ^i . , : r
snr 6 sm j c
Dividing both sides of the equation by sin 2 a and extracting
the square root, we obtain
sin A Vl cos 2 a cos 2 b cos 2 c+2 cos a cos 6 cos x-, >.
sin a sin a sin o sin c
In a precisely similar manner it can be proved that and
. * sin b
also that : have the same value. Therefore, since each of
sin c
these ratios has the same value, they are equal to each other.
sin A _ sin B __ sin ^^\
sin a sin b sin c '
which is the law of sines. It may be stated in words as
follows :
The sines of the sides of a spherical triangle are to each other as
the sines of the opposite angles.
GENERAL THEOREMS AND FORMULAS
183
An inspection of (1) shows that a cyclic interchange of the
letters a, 5, c, and A, J5, (7, leaves the right member of the equa-
tion unchanged, while the left member is changed into and
. Q sin b
successively. Hence, after (1) has been proved, (2) can
sin c
be established by cyclic interchange of letters.
123. To derive a relation involving the angles and one of the
sides of a spherical triangle.
Let A'JS 1 '0' and ABC be two spherical triangles polar to each
other. Then (Art, 118, p. 1T9),
a' = 180 - A,
b f = 180 - B,
c' = 180 - 0.
By (1), Art. 119, p. 180,
cos a' = cos b' cos c' + sin b f sin c'cos A! .
But, by (1),
cos a' = cos A,
cos b' = cos B,
cos c 1 = cos C.
sin b 1 = sin _Z?,
sin c' = sin (7,
cos A' = cos a.
Substituting these values in (2), we have
cos A = cos B cos C sin .Z? sin (7 cos a.
In like manner we can obtain corresponding values for cos B
and for cos 0.
Therefore,
cos A = cos B cos (7+ sin B sin C cos
cos B = cos (7 cos ^4 + sin (7 sin A cos 6,
cos = cos J. cos B + sin .A sin B cos <?.
From these equations the following are at once derived
cos A + cos B cos C
(3)
cos a =
cos b
cose
sin J5 sin
cos B+ cos (7 cos J.
sin C sin ^1
cos (7 + cos A cos .g
sin A sin .B
(4)
184 SPHERICAL TRIGONOMETRY
124. To derive a relation involving two angles and the sides
of a spherical triangle.
Resuming (1), Art. 119, p. 180, we have
cos a = cos b cos e + sin b sin c cos A.
Substituting in this equation the value of cos c obtained
from (2), Art. 120, p. 181,
cos a = cos b (cos a cos b -f sin a ski b cos (7) -f- sin b sin c cos A
= cos a cos 2 b + sin a sin > cos b cos (7 + sin b sin <? cos A.
cos a (1 cos 2 6) = sin a sin 6 cos b cos (7 4- sin b sin <? cos A.
Substituting for 1 cos 2 b its value, sin 2 6, and dividing both
sides of the equation by sin 6, we obtain the desired relation,
cos a sin b = sin a cos b cos C + sin c cos A. (1)
In like manner we can obtain corresponding expressions for
the value of cos a sin c, of cos b sin c?, etc. Therefore,
cos a sin b = sin a cos b cos (7 + sin c cos .A, "
cos a sin c = sin a cos c cos 5 -f sin b cos A,
cos 5 sin a = sin 6 cos a cos (7 + sin c cos .#,
cos b sin <? = sin b cos <? cos A 4- sin a cos .B,
cos c sin 6 = sin c cos 5 cos A + sin a cos (7,
cos c sin # = sin c cos a cos B -f- sin 5 cos (7, .
(2)
125. To derive a relation involving two sides and the angles
of a spherical triangle.
Resuming the first equation under (3), Art. 123, p. 183, we
cos A cos B cos C + sin B sin cos a.
Substituting in this equation the value of cos C obtained
from the third equation of the same set,
cos A = cos B ( cos A cos B -f sin A sin B cos <?)
4- sin B sin C cos a
= cos A cos 2 J5 sin A sin i? cos B cos c 4- sin B sin (7 cos a.
Transposing and factoring,
cos A (1 cos 2 B) = sin A sin B cos B cos 6> + vsin B sin (7 cos a.
GENERAL THEOREMS AND FORMULAS
185
Replacing 1 cos 2 B by its value, sin 2 B, and dividing both
sides of the equation by sin B, we obtain the desired relation,
cos A sin B cos a sin cos c sin A cos B. (1)
In like manner we can obtain corresponding expressions for
the value of cos A sin, (7, cos B sin J., etc. Therefore,
cos A sin 1? = cos a sin (7 cos c cos ^ sin A,
cos A sin (7 = cos a sin .5 cos b cos (7 sin A,
cos (7 sin 5 = cos c sin A. cos a cos jB sin (7,
cos (7 sin A = cos <? sin B cos 5 cos A sin (7,
cos J5 sin A = cos 6 sin cos <? cos A sin 5,
cos B sin (7 = cos 6 sin A. cos a cos (7 sin ^.
(2)
126. From the formulas in Art. 124 a group of important
relations is derived, as follows :
From the first of the six formulas there given we have
cos a sin b = sin a cos b cos (7 + sin c cos A.
Dividing both sides of the equation by sin a,
sin
sin a
T> T . sin c , , sin C .-, i
Replacing - by its equal -, tins becomes
sin a sin -4
cot a sin b = cos b cos (7+ sin C
sin A
. *. cot a sill b = cos 5 cos C -f- sin (7 cot A.
In like manner we can obtain corresponding expressions for
the value of cot a sin <?, cot b sin #, etc. Therefore,
cot a sin b = cos 5 cos (7+ sin (7 cot A.,
cot a sin c cos c cos B + sin ^ cot A,
cot 5 sin c = cos <? cos A -\- sin -A cot B,
r (^)
cot 5 sin a = cos a cos (7 + sin (7 cot B,
cot c sin a = cos a cos 5 + sin B cot (7,
cot c sin 5 = cos b cos .A + sin A cot (7.
186 SPHERICAL TRIGONOMETRY
127. The values of sin ^, cos ^, tan -, etc., in terms of the
sides of the triangle.
From (3), Art. 120, p. 181,
cos A =
cos a cos b cos
sin b sin c
From this we have
1 , A S ^ n ^ s ^ n C + COS ^ cos c cos
X COo */l
sin b sin c
by Art. 68, = cos (b ~ c) .~ cos a .
sin b sin c
Dividing by 2, and applying (8), Art. 77, p. 100,
. a-4- b c . a b 4- c
sin - - sin -
1 cos A 2
sin b
sn 6-
Putting #-f5-h& = 2s, and replacing - ^ ()S by its equal
- %A_ sin (g 5) sin (> c)
2 sin 6 sin c
. .. sn =
2 sin 6 sin c
In like manner,
1 . j _ sin b sin c cos b cos <? + cos a
\. -f- COS J\. --
sin b sin c
1 + cos A _ cos a cos (6 + c)
2 2 sin b sin
S = _ 2 _ :
2 sin 6 sin
cos = - . (2)
2 * sin i sin c
GENERAL THEOREMS AND FORMULAS 187
Dividing (1) by (2), we have
tan =
2 * sin 8 sin (s a)
Since any angle of a spherical triangle is less than 180, all
the functions of the half angles are positive; i.e. sin ,
A A ^
cos, tan , are all positive. Therefore the signs of the
'2. A
radical expressions in (1), (2), and (3) are positive.
Since s, a, 5, c, s a, s b, s c are severally less than 180
and positive, the values obtained in (1), (2), and (3) are real.
128. The values of sin |, cos ^, tan |, etc., in terms of the
angles of the triangle.
From (4), Art. 123, p. 183,
cos A 4- cos B cos O
cos a =
Therefore, 1 cos a =
sin sin C
sin B sin C ' cos B cos cos A
sin B sin C
cos (B + (7) cos A
sin B sin C
-, i . sin B sin (7 -f cos 5 cos (7+ cos A
and 1 + cos a =
sin B sin (7
_ cos (B O) 4- cos A
sin .2? sin (7
Putting A + B -f- 0=2 S, and proceeding as in the last sec-
tion, we obtain
. a lGOStScoafS A)
- sin J sing '
sin 5 sin C
cos iS cos T/S 7
188 SPHERICAL TRIGONOMETRY
Since any side of a spherical triangle is less than 180, all
the functions of the half sides are positive; i.e. sin ^, cos-,
tan -, are all positive. Therefore the signs of the radical
2
expressions in (1), (2), and (3) are positive.
To prove that these expressions are real we proceed as follows :
Let A'B'C' be the polar triangle of ABC, and let a', b 1 , c',
be the sides of A'B' 0' which lie opposite the angles A, B, C,
respectively of the original triangle.
Then, since a', 6', <?', are supplements of A, B, (7, respectively,
and since a 1 < b' + e', we have
180 -A< (180 - B) + (180 - (7).
Transposing, B + C - A < 180 ;
i.e. S-A<90.
Therefore, cos (S A) is positive.
Also, since A + B + lies between 360 and 540, 8 lies
between 180 and 270. Hence cos 8 is negative ; i.e. cos 8
is positive.
Therefore the radical expressions in (1), (2), and (3) are
real.
129. Gauss's equations. From Art. 69, p. 92,
(A , B\ A B A B
C S U ~2 J = C S 2" C S "2 ~ S1U "2 Sm "2 '
A A
Substituting in this equation the values of cos and sin ,
T) T>
and corresponding values for cos-- and sin-' (Art. 127,
p. 186), we have
cos fA+.lP\ m /sin g sin Q - a} ^ /sin * sin (
\2 2y ^ sin b sin c ^ sin a si
8-5)
Sill
/sin (8 b) sin (s <?) /sin (s #) sin (s
^ sin b sin <? ^ sin a sin <?
sin s sin ( s
sin
sin (s c) A /sin (s a) sin (s 5)
in c * sin 6 sin a
GENERAL THEOREMS AND FORMULAS 189
But by Art. 77, p. 100, and Art. 80, p. 106,
o 2 s c . c
2 cos - sm -
sin s sin (s <?) __ 2 2
!smj
a + b
sin c n . c c
2 sm - cos -
cos
COS 9
and by Art. 127, p. 186,
/sin (s a) sin (s b) _ . O
* sin a sin b 2
Substituting these values in (1), and reducing, we have
A+B o
008 -3- = r~ sm 2' (2)
cos-
Tn like manner corresponding values can be obtained for
sin - , sin ^ , and cos . These four relations,
which are commonly known as Gauss's Equations, are as follows :
a + b
A + B ~2~ .
cos ^~- = - sin -g- ; (3)
cos-
a-b
A + B COS ^~ O
sm g = - cos ^-5 (4)
cos-
. a + b
sn
. a-b
sin -
. A-B 2 C
sm - = - T~ cos "o'
sin-
190 SPHERICAL TRIGONOMETRY
130. Napier's analogies. From Gauss's Equations the follow-
ing are derived. The method of derivation is obvious, and the
work is left as an exercise for the student.
a b
cos -
COS
(3)
sin
131. Special formulas for the solution of spherical right tri-
angles. If one of the angles of the triangle, as (7, is a right
angle, the following special formulas are derived from those
established in the preceding sections :
From (2), Art. 120, p. 181,
cos c = cos a cos b + sin a sin b cos C. (1)
But, since (7= 90, cos 0= 0. Therefore the second term of
the right member becomes zero. Therefore,
cos c = cos a cos b.
In a manner similar to that just employed, the following
formulas are derived for the special case when C is a right angle.
From (2), Art. 122, p. 182, formulas for finding either of the
oblique angles when the hypotenuse and the opposite leg are
S iven - sin a-
sin A = - ,
sin c
j - T> sin b
arid sin B -
sin c
GENERAL THEOREMS AND FORMULAS 191
From (3), Art. 123, p. 183, formulas for finding either of
the oblique angles when the opposite leg and the other oblique
angle are given.
cos A = cos a sin B, 1
cos B = cos b sin A. J
From (2) and (3) are derived the following formulas for
finding an oblique angle when the hypotenuse and the adjacent
leg are given.
cos A = tan b cot <?,
cos B = tan a cot c.
From (2), Art. 126, p. 185, formulas for finding the oblique
angles when the legs are given.
,, tan a
tan A = ,
sin b
tan B =
tan b
sin a
(5)
From (3), Art. 123, p. 183, formulas for finding the legs
when the two oblique angles are given.
cos
,
sin B
sin A
Multiplying together the two formulas just obtained, and
replacing the left member of the product, cos a cos 5, by its
value given in (1), we have the following formula for finding
the hypotenuse when the two oblique angles are given :
cos c= cot A cot B. (7)
132. Napier's rules. The formulas of the last section are
sufficient for the solution of every possible case that can arise
under spherical right triangles. But it is often better to solve
the various cases that arise under right triangles by two con-
venient and simple rules devised by Napier, the inventor of
logarithms.
192 SPHERICAL TRIGONOMETRY
These rules are constructed by supposing that a right tri-
angle has five parts. These parts, which are usually called
Napier's parts, are
(1) The two legs.
(2) The complement of the hypotenuse.
(3) The complements of the two oblique angles.
The right angle is not considered, and plays no part what-
ever in the solution of a triangle by this method.
Any one of the five parts may be regarded as the middle
part. The two parts immediately adjacent to this are called
the adjacent parts, and the other two are called the opposite
parts.
Napier's rules for the solution of spherical right triangles
are as follows :
1. The sine of the middle part is equal to the product of the
tangents of the adjacent parts.
2. The sine of the middle part is equal to the product of
the cosines of the opposite parts.
The similarity of the vowel sounds in the syllables tan-, ad-
and co-, op- renders it easy to remember these rules, and also
to distinguish them from each other.
To test the correctness of these rules, assume any three parts
as the given parts. For example, let the given parts be a, b,
and co-A. In this case b is the middle
part, and a, co-A, are to be considered
adjacent parts. Hence we have
sin b = tan a tan (co-A)
= tan a cot A.
This is the same as the first of the two
formulas under (5), Art. 131, p. 191,
which has already been proved to be
true.
As another illustration, let the given parts be a, co A, co-B.
Here co-A is the middle part, and a, co-B are to be considered
opposite parts. Hence
sin (co-A) = cos a cos (co-B),
cos A cos a sin B.
GENERAL THEOREMS AND FORMULAS 193
Tliis is the same as the first of the two formulas under (3),
Art; 130, p. 190, which has already been proved to be true.
In like manner Napier's rules as applied to any other group
of three parts will be found to reduce to one of the formulas
already proved. ^
133. DEFINITION. Two angles, or an angle and a side, are
said to be of the same species when both are greater or both are
less than 90; they are said to be of opposite species when
one is greater and the other is less than 90.
In any right triangle if a and b are of the same species, the
hypotenuse c is less than 90 ; if a and b are of opposite species,
c is greater than 90.
This follows from (1), Art. 130, p. 190. For if a and b are
both greater or both less than 90, the product cos a cos b is posi-
tive. Therefore cos c is positive ; therefore c is less than 90.
But if a and b are of opposite species, the product cos a
cos b is negative. Therefore cos c is negative ; therefore c is
greater than 90.
EXERCISE XXXII
1. Prove that in any right triangle a leg and the angle oppo-
site are of the same species.
2. By the aid of Napier's rules derive the formulas in (6),
Art. 131, p. 190.
3. If the hypotenuse of a right triangle is equal to 90, what
must be the values of a and b? Why?
4. Prove tan2 =
2 sin (e -f- a)
5. Prove
2 cos(B-A)
6. If a = 90 and b = 90, what must be the values of the
remaining parts of the right triangle?
7. In a right triangle a side and the hypotenuse are of the
same or of opposite species according as the included angle is
less or greater than 90.
CONANT'S TRIO. 13
CHAPTER XVI
SOLUTION OF SPHERICAL TRIANGLES
134. A spherical triangle is determined when any three of
its parts are known. That is, when any three parts are given,
the remaining parts can be computed.
In the solution of spherical triangles we have six cases to
consider, as follows : having given,
(1) The three sides.
(2) Two sides and the included angle.
(3) Two sides and the angle opposite one of them.
(4) Two angles and the side opposite one of them.
(5) Two angles and the included side.
(6) The three angles.
135. The right triangle. We proceed first to the considera-
t ion of the right triangle. We shall suppose that is the right
angle ; and here, as in Plane Trigonometry, only two parts are
known in addition to the right angle.
136. Ambiguous cases. Whenever a solution is obtained by
means of the sine or the cosecant, the solution is ambiguous,
because, both sine and cosecant being positive in the second
quadrant as well as in the first, a given value of either of these
functions is, in general, satisfied by two angles, one in the first
and the other in the second quadrant.
Hence, whenever a required part of a spherical triangle is
found by means of the sine or the cosecant, it is necessary to
test the result, and to determine whether or not both solutions
are admissible.
When a solution is found by means of the cosine, tangent,
cotangent, or secant, there is no ambiguity, since each of these
functions is positive in the first quadrant and negative in the
second quadrant.
194
SOLUTION OF SPHERICAL TRIANGLES 195
For this reason it is of great importance that the student
should note carefully the sign of each of the functions that
appear in an equation.
137. CASE 1. Given two legs, a and 6; to find c, A, B.
The formulas for solution are contained in (1) and (5), Art.
131, p. 190, or are obtained directly from Napier's Rules, and
are as follows :
cos c = cos a cos b ; (1)
(2)
tan 5-^. (3)
sm a
For a check formula use cos c = cot A cot B.
Ex. 1. Given a = 46 50', b = 31 15'; find c, A, B.
log cos a = 9.83513 - 10
log cos b = 9.93192 - 10
log cos c = 976705 10. ]og ^ , = ^ _ w
log sin a = 9.86295 - 10
log tan a = 10.02781 - 10 lo S tan B = 9 ' 9 o 20U ~ 10 -
colog sin b = 10.28502 - 10 B ~ 39 45 ' 32 "'
log tan A = 10.31283- 10.
A = 64 3' 9".
Since c is obtained by means of its cosine and A and B by
means of their tangents, there is no ambiguity respecting the
result. Both a and b are in the first quadrant ; therefore cos a
and cos b are positive. It follows from this that the right mem-
ber of (1) is positive when applied to this particular problem ;
therefore cos c is positive, and consequently c is in the first
quadrant.
In like manner it can be shown that A and B are in the first
quadrant.
When only one solution exists that will satisfy the conditions
of a problem, the solution is said to be unique.
Ex. 2. Given a = 38 44' 40", b = 42 26' 28" ;
find c = 54 51' 37", A = 49 56' 12", B = 55 36' 44".
196 SPHERICAL TRIGONOMETRY
138. CASE 2. Given the hypotenuse c, and one of the legs
a; to find b, A, B. The formulas for solution are (Art. 131,
p. 190)
cos 5 =
sin A =
cos a
sin a
sin c
n tan a
cosj5 = .
tanc
For a check formula use
cos B = cos b sin A (Art. 131, p. 191).
The solutions for b and B, being obtained in each case by
means of a cosine, are unique.
The solution for A, being obtained by means of its sine, is
apparently ambiguous. But by Art. 133, p. 193, a and A are
of the same species. Hence, as a is given, the species of A
becomes known at once, and the ambiguity disappears.
Ex.1. Given c = 54 36' 30", a = 23 17' 40";
find b = 50 54' 30", A = 29 1' 5", =72 11' 20".
Ex. 2. Given c = 98 15' 12", a = 133 40' 24" ;
find b = 78 0' 7", A = 133 2' 30", B = 81 15' 40".
139. CASE 3. Given one of the legs a and the opposite angle
A; to find ft, c, B. The formulas for solution are as follows,
(Art. 131, p. 190):
sm a
sin c =
sin b =
sm f =
sin A*
tan a
tan A
cos ^4.
cos a
For a check formula use sin b = -^ . (Art. 131, p. 191)
tan A
SOLUTION OF SPHERICAL TRIANGLES 197
The solution is ambiguous, being obtained in each case by
means of a sine. The different cases that may arise are as
follows :
(1) If a = A, then sin a = sin A, tan a = tan A, and cos a = cos
A i therefore sin <? = !, sin 6 = 1, and sin .5=1. Hence the
solution is unique.
(2) If c and a are of the same species, then B < 90 ; there-
fore b < 90 (Ex. 7, p. 193).
(3) If c and a are of opposite species, then B > 90; there-
fore b > 90 (Ex. 7, p. 193).
After c has been computed b and B may be found, if other
formulas than those given above are desired, by the following
(Art. 131, p. 191):
cos b =
COS Jt5 =
cos a
tan a
tan c
These formulas give unique solutions for b and B, but for
obtaining <? it is necessary to make use -of the sine. As any
given value of the sine is satisfied by two supplementary values
of the angle, this case often gives two solutions.
Ex.1. Given a= 70 55' 50", ^=82 58' 6";
Find Cj = 72 13' 45", ^ = 20 54' 18", B l = 22 0' 19".
or, c 2 = 107 46' 15", b 2 = 159 5' 42", B 2 = 157 59' 41".
Ex. 2. Given a= 76 59' 59", JL = 39 50' 56".
The triangle is impossible. Why V
140. CASE 4. Given one of the legs a and the adjacent
angle B, to find c 9 6, A. The formulas for solution are
(Art. 131, p. 191) _ tan a
tun c j ,
cos B
cos A = cos a sin B,
tan b = sin a tan B.
For a check formula use 7
cos A = !=; (Art. 131, p. 191)
tan c
The solution is unique. Why?
198 SPHERICAL TRIGONOMETRY
Ex. l. Given a = 21 5' 15", B = 39 8' 10" ;
find c = 26 26' 6", A = 53 55' 13", b = 16 19' 5".
Ex. 2. Given a = 59 27' 32", B = 36 24' 25" ;
find c = 64 35' 56", b = 32 25' 17", A = 72 26' 47".
141. CASE 5. Given the hypotenuse c and one of the oblique
angles A; to find a, b, B. The formulas for solution are
(Art. 131, p. 190) .
sin a = sin c sin A,
tan b = tan c cos A,
cot .Z? = cos c tan .A.
For a check formula use
sin a = tan b cot .B (Art. 131, p. 191).
The solution for a, being obtained by means of its sine, is
apparently ambiguous. But since A is given, and since a and
A are of the same species, the proper value of a can always be
determined. Hence the solution is unique.
Ex. i. Given c= 117 39' 48", A -127 20' 25";
find a = 135 14' 18", b = 49 9' 58", B = 58 40' 37".
Ex.2. Given e = 68 50' 31", 4 = 55 11' 17";
find a=4958', 5 = 55 51' 53", 5 = 62 33' 58".
142. CASE 6. Given the two oblique angles A, B; to find
, b, c. The formulas for solution are (Art. 131, p. 191)
sin
7 cos B
cos b = - -
sin A" 1
cos c = cot A cot B.
For a check formula use
cos c = cos a cos b (Art. 131, p. 190).
The solution is unique.
SOLUTION OF SPHERICAL TRIANGLES 199
Ex.l. Given 4 = 63 25' 32", 5 = 136 1' 27";
find a = 4953'16", b = 143 34' 30", c= 121 13' 34".
Ex. 2. Given A = 119 20' !!",.# = 114 V 35";
find a=12228'6", = 117 57' 42", <? = 75 25' 16".
143. The isosceles spherical triangle. An isosceles spherical
triangle can always be solved by means of the formulas em-
ployed in the solution of spherical right triangles ; for, by pass-
ing an arc of a great circle through the vertex and the middle
point of the side opposite, the isosceles triangle can always be
divided into two symmetrical right triangles.
EXERCISE XXXIII
1. In a right spherical triangle given c = 20 50' 52",
a =15 12' 44"; find 6, A, B.
2. In a right spherical triangle given a = 75 28' 24",
b = 33 37' 8" ; find c, A, B.
3. In a right spherical triangle given a = 66 9' 9",
^ = 155 49' 46"; find b. c, B.
4. In a right spherical triangle given a = 122 5',
B = 125 40' ; find 5, c, A.
5. Iii a right spherical triangle given <? = 115 35' 4",
J. = 5729'; find a, b, B.
6. In a right spherical triangle given A = 45 23' 8",
B = 58 17' ; find a, b, e.
7. In a right spherical triangle given c = 80 28' 44",
A = 33 20' 24" ; find a, 6, B.
8. In a right spherical triangle given e = 139 42',
a = 21 47' 46" ; find 6, A, B.
9. In a right spherical triangle given a ="110 38',
B = 153 55' 40" ; find 5, c, A.
10. In a right spherical triangle given a = 112 49',
^ = 100 27'; find 5, c, B.
200 SPHERICAL TRIGONOMETRY
11. In a right spherical triangle given a = 55 52'
b = 34 46' 42" ; find <?, A, B.
12. In a right spherical triangle given A 54 20',
^=64 49' 51"; find a, b, c.
13. In a right spherical triangle if a =6, prove that
cos 2 a = cos c.
14. In a right spherical triangle prove that
sin b = cos c tan a tan B.
15. In a right spherical triangle prove that
sin 2 A + sin 2 B = 1 + sin 2 a sin 2 ^.
16. In a right spherical triangle prove that
. sin (5 + c) = 2 cos 2 cos 5 sin <?.
THE OBLIQUE SPHERICAL TRIANGLE
144. In solving oblique spherical triangles we have six cases
to consider, as follows :
CASE 1. Given the three sides , &, c; to find A, B 9 c.
The formulas for solution are (Art. 127, p. 186)
sin g sin ( a)
tan =
2 sin s sin ( - 5)
tan ^= Jain (- a) sin 0-i)^
2 sin s sin (s (?)
The corresponding formulas for the sines or for the cosines
of the half angles may be employed (Art. 127, p. 186), but in
general the tangent formulas are to be preferred.
If all the angles are to be found, a saving of labor can be
effected in the following manner.
SOLUTION OF SPHERICAL TRIANGLES 201
Multiply both numerator and denominator of the fraction
under the radical sign in (1) by sin (s a). Then let
tan r = -J sin ( * - a) sin (s - 6 ) sin Q - c)
sin s
and we may write
2 sin (s a)
Making the corresponding changes in (2) and (3), we have
the three equations :
tan =
2 sin (s a) '
tanr
2 sin -fi'
(7 tan r
tan =
2 sin (s c')
If these formulas are employed, it will be found that the
work of solution can be more compactly arranged and more
conveniently carried out than by the use of any other method.
Ex. 1. Given a = 51 43' 18", b = 38 2' 20", c = 75 11' 30" ;
find A.
a = 51 43' 18" log sin (s -b) = 9.84518 - 10
6 = 38 2' 20" log sin (-c) = 9.10311 -10
c = 75 11' 30" colog sin s = 0.00375
2 s = 164 57' 8" colog sin (s-a)= 0.29127
= *rwu ^331-20
s-a = 30 45' 16" Jog ^ n A = ^IQQ - 10
s - b = 44 26' 14"
s - c = 7 17' 4" 4.= 22 42' 27.4
s = 82 28' 34" Check.
A = 45 24' 55
202 SPHERICAL TRIGONOMETRY
Ex. 2. Given a = 125 40' 14", b = 53 56' 12", c = 98 51' 16";
find A, B, C.
"~ZZ *-*-
c= 98 51' 16" log tan | = 9.65185 - 10
2s = 278 27' 42" p
s = 139 13' 51" log tan 2 = 9.83894 - 10
s - a - 13 33' 37" 4. _ 62 o ^ ^,
s-b = So 17' 39"
j-c = 4022'85" =24 9' 38
4Q/ ,
log sin (* - a) = 9.37008 - 10 C = M
log sin (s- 6)= 9.99854-10 2
J _ 1010 QQ' fi'f
logsin(s-c)= 9.81145-10
= 48 19' 16"
cologsin= 0.12071 C = 69 13' 20"
log tan 2 r = 19.30078 - 20
log tan r = 9.65039 - 10
EXERCISE XXXIV
1. In a spherical triangle given a=11922' 27", 6 = 60 44'40",
c=1083T' 3"; find A, B, 0.
2. In a spherical triangle given a = 53 42', b = 118 39' 28",
c = 130 38' 20" ; find A, B, 0.
3. In a spherical triangle given a = 129 11' 36", b = 109 29'
18", <?= 83 14'; find the largest angle.
4. In a spherical triangle given a = 22 56' 46", b = 60 47',
c = 69 49' 32" ; find B and C.
145. CASE 2. Given two sides , 6, and the included angle,
C; to find A, B, c. The angles A, B, may be found by the
first two of Napier's Analogies (Art. 130, p. 190) :
a b
A + B cos a
tan = ot
cos
A-B
SOLUTION OF SPHERICAL TRIANGLES
From the values of " - and - - obtained from these
equations the values of A and B can at once be found.
The value of c can then be obtained from any one of Gauss's
Equations (Art. 129, p. 189) ; for example,
The solution is unique.
EXERCISE XXXV
1. In a spherical triangle given a = 8554' 16", 6=125 1' 27",
<?=526' 26"; findJ,j&,o.
2. In a spherical triangle given a = 119 32' 30", 6=8631'35",
0=49 40' 22"; find A, B, c.
3. In a spherical triangle given b = 61 23' 18", c = 48 30' 6",
A = 60 53' 24" ; find B, <7, a.
4. In a spherical triangle given a = 7240'40", c = 11033' 38",
=53 50' 20"; find -4, 0, b.
5. In a spherical triangle given b = 68 20' 25", o= 52 18' 15",
4 =117 12' 20"; find., (7, a.
146. CASE 3. Given two sides , 6, and the angle opposite
one of them A; to find B, C , c. The value of B can be found
by means of the law of sines (Art. 122, p. 182), from which
we have A
sin = ^_^ s i n 6. (1)
sin a
After B has been determined C and c can be found by the
use of the first and the third of Napier's Analogies.
a + b
"~
___tan. (3)
cos -
204 SPHERICAL TRIGONOMETRY
Since is determined by means of its sine, the solution is
ambiguous.
The following tests may be conveniently employed to deter-
mine the number of solutions.
If sin A sin b > sin a, there is no solution ; for in that case
sin B > 1, which is impossible.
If sin A sin b < sin #, (1) is satisfied by two supplementary
values of B. But - - and - - are necessarily of the same
2 2
species. Therefore, if both these values of B satisfy this con-
dition, there are two solutions ; if not, there is but one.
NOTE. To make use of the test just given it is necessary that we first
solve for B. There are several methods of testing for the number of solu-
tions without first finding B, but it is not thought best to include any of
them in this work. For a full explanation of them the student is referred
to more extended treatises on the subject of Spherical Trigonometry.
Ex. i. Given a = 56 30', b = 31 20', A = 105 11' 10";
find B, C, c.
Since in this case sin A sin b < sin a, there may be either
one or two solutions. To test for the number of solutions
we find the possible values of B.
log sin A = 9.98456 - 10
log sin b = 9.71602 -10
colog sin a = 0.07889
log sin B = 9.77947 - 10
B = 37 0' 3",
or, B = 142 59' 57".
We have from data given, ! < 90.
This shows that only the smaller of the two values of B is
admissible.
Therefore there is but one solution.
SOLUTION OF SPHERICAL TRIANGLES 205
The work of solution may be compactly and conveniently
arranged as follows :
2
a - b = 25 10'
^=12 35
A+B = U2U f 13"
A + B ,
A-B = 68 II' 7" 2
log ain = 9.1*501 -10
lo S sin = 9-34112- 10
colog sin ^ ^ = 0.25140
log tan ^Lzi = 9.34874 - 10 col S sin ^T = ' 66182
log tan ^9.57605 -10 log tan ^ = 9.83053 - 10
log cot ^ = 0.33347
- = 20 38' 38" r
= 24 53' 31"
c = 41 17' 16" C = 49 47' 2"
EXERCISE XXXVI
1. In a spherical triangle given a = 71 14', b = 122 27' 18",
'4 = 77 23' 24"; find B, (7, c.
2. In a spherical triangle given a = 80 5' 16", b = 82 4',
.4 = 83 34' 12"; find B, <7, c.
3. In a spherical triangle given a = 151 22' 30", b = 133 31'
25", A= 143 32' 28"; find B, 0, c.
4. In a spherical triangle given a =30 38', b = 31 29' 45",
A = 87 53' 20" ; find the remaining parts.
147 CASE 4. Given two angles A, B, and the side oppo
site one of them a; to find C, 6, c. As in the preceding case
one of the parts, in this case 6, can be found by means of the
law of sines, from which we have (Art. 122, p. 182)
~~ sin A &1D a '
206 SPHERICAL TRIGONOMETRY
The values of c and O can then be found by means of the
fourth and the second of Napier's Analogies :
. A + B
sin -
c
tan 2 =
sn
C A-B
-*" ' (3)
The solution is ambiguous, the value of b being determined
by means of its sine.
If sin B sin a > sin A, there is no solution; for in that case
sin b > 1, which is impossible.
If sin B sin a < sin A, (1) is satisfied by two supplementary
values of b. To ascertain whether or not both these values are
admissible we proceed in a manner similar to that employed
in the last case. If both values of b satisfy the condition im-
posed by the fact that - -^ and a are of the same species,
there are two solutions ; otherwise there is but one.
NOTE. The number of solutions can always be determined by forming
the polar of the given triangle and then determining by the tests under
Case 3 the number of solutions of that triangle. The number of solutions of
the given triangle is always the same as the number of solutions of its polar.
Ex. i. Given A = 29 43' 12", B = 45 4' 18", a = 36 19' 32";
find 6, c, C.
In this case sin B sin a < sin A ; therefore there may be either
one or two solutions. Solving for 5, we proceed as follows :
log sin B = 9.85003 - 10
log sin a = 9.77260 - 10
colog sin A = 0.30173
log sin b = 9.92736 - 10
b = 57 48' 38",
or, b = 122 13' 22".
We have from data given, A +
SOLUTION OF SPHERICAL TRIANGLES 207
%
Both of the values of b just found satisfy this condition.
Hence, there are two solutions. The values of and c can
now be found in the ordinary manner, both values of b being
employed.
EXERCISE XXXVII
1. In a spherical triangle given A = 109 20' 10", .#=134
16' 24", a= 148 48' 40"; find 6, c, 0.
2. In a spherical triangle given J. = 113 30', .8=125 31'
34", a = 66 44' 40"; find 5, c, 0.
3. In a spherical triangle given A = 28 35' 5", J5 = 47 51'
15", a = 38 41 '32"; find b, c, 0.
4. Iii a spherical triangle given A = 24 30', 5=38 15',
a = 65 22'; find 5, c, 0.
148. CASE 5. Given a side c and the two adjacent angles
A, B ; to find a, 6, C. The third and fourth of Napier's
Analogies may be used for determining the values of a and b
(Art. 130, p. 190) :
A-B
cos -
2
A-B
From these formulas the values of a and b can be obtained.
The value of C can then be found by means of the first of
Napier's Analogies :
a-b
cos
a 2 .A + B
tan - = -- cot
2 a + b 2
COS ~2~
The solution is unique.
208
SPHERICAL TRIGONOMETRY
Ex. l. Given A =108 28' 55", B = 38 11' 27", c = 52 29';
find a, ft, 0.
= 35 8' 44
= 73 20' 11'
r = 26 14' 30"
log cos - - = 9.91259 - 10
log tan = 9. 69282- 10
colog cos
= 0.54250
log tan - = 10.14791 - 10
^-^ = 54 34' 24.4"
= 16 30' 1.3"
2
a-b
a = 71 4' 26"
& = 38 4' 23"
log sin
= 9.76016 - 10
log tan | = 9.69282 - 10
colog sin A + B . = 0.01863
log tan ^~ = 9.47161 - 10
^-=^=16 30' 1.3"
log cos = 9.98174 - 10
lo cot
colog cos
= 9.47599 - 10
= 0.23682
log tan ^ = 9.69455 - 10
| = 26 19' 56"
C = 52 39' 52"
EXERCISE XXXVIII
1. In a spherical triangle given ^. = 126 40' 50", j5=81
45' 42", c = 51 56' 12"; find a, b, 0.
2. In a spherical triangle given B= 27 27' 36", C = 40 44'
20", a =155 16'; find 6, c, A
3. In a spherical triangle given J. = 127 19' 38", (7=108
41' 30", b = 125 22' 32"; find a, c, .5.
4. In a spherical triangle given A = 154 20' 42", B == 79
16' 22", c = 85 24' 28"; find a, b, C.
149. CASE 6. Given the three angles A, B, C; to find the
three sides a, b, c. Any of the three groups of formulas in
Art. 128, p. 187, can be used. The formulas for the tangents
are recommended in preference to those for the sines or for the
cosines.
SOLUTION OF SPHERICAL TRIANGLES 209
- cos -
o
If all three of the sides are to be found, it is convenient to
proceed in a manner similar to that employed in Art. 144, p.
200, where three sides were given and three angles were to be
found.
Multiplying both numerator and denominator of the fraction
under the radical sign in (1) by cos ($ A) we have
Putting tan R = "
^, A)t!OS(CO8( ^, g)
we may write
tan - = tan R cos (S A) .
Making the corresponding changes in (2) and (3), we have
the three equations
tan - = tan R cos ($ J5),
2
tan | = tan R cos (# - (7).
The solution is unique.
COXANT'S TRIG. 14
210 SPHERICAL TRIGONOMETRY
Ex. l. Given A = 221, B = 128, 0= 153 ; to find a.
The formula for tan -, with the algebraic sign of each factor written
above it for convenience, is as follows :
tan fl = /
cos 5 cos (S A )
2 A
_
\c
os(S-B)cos(S-
C)
A
= 221
log-
cos S = 9.51264 -
10
B
= 128
log cos(S
-.-0 = 9.93753-
10
C
= 153
colog cos (S
- B) = 0.26389
2 S
= 502'
colog cos (5
- C) = 0.85644
2)20.57050-
20
S
= 251
s
-A
= 30
log
tan ^=10.28525 -
-10
s
-B
= 123
- = 62 35' 35
o
s
-C
= 98
a = 125 11' 10"
The result is real (Art. 128, p. 187), the four negative signs under the
radical producing a positive quantity.
Ex. 2. Given A = 21 26' 20", B = 56 46' 28", (7=115 23'
4"; find a, 5, c.
Proceeding by the second method, we first find the value of log tan R.
The following is suggested as a convenient arrangement of the work:
tan R=
\ cos (S- A) cos (S - B) cos (S - C)
A _ 910 9f>' ()f\r>
log tan 2 = 9.30865 - 10
B = 56 46' 28" 2
C = 115 23' 4" log tan* =9.70007 -10
25 = 193 35' 52" 2
5 = 96 47' 50" log tan c - = 9.87055 - 10
S - A = 75 21' 36"
iS'- = 40 T28" 5 = 11 30' 17.5"
S-C = -19 24' 52"
log cos S = 9.07330 - 10 o = :}l 39/ 43 "
colog cos (5 - /I ) = 0.59732
^_ Qf>o o-/ 4 f>//
colog cos (S -B) = 0.11590 2 "
colog cos (S - C) = 0.02542 a = 23 0' 35"
log tan 2 R = 9.81 194 - 10 b = 63 19' 26"
log tan R = 9.90597 - 10 c = 73 10' 9"
SOLUTION OF SPHERICAL TRIANGLES 211
EXERCISE XXXIX
1. In a spherical triangle given A = 121 40' 24", B= 60 12'
22", O= 105 40'; find a, b, c.
2. Iii a spherical triangle given ,4 = 58 20' 27", =8430'30",
(7=61 35' 10"; find a, 6, c.
3. In a spherical triangle given A = 105 14' 4", B=55 31'
24", =88 51 '6"; find a, 5, <?.
4. In a spherical triangle given A = 171 49' 33", B= 5 15' 23",
=9 18' 28"; find a, 6, c.
THE AREA OF A SPHERICAL TRIANGLE
150. In considering the problem of finding the area of a
spherical triangle we have two principal cases to consider.
I. Given the three angles A, B, C.
Let r = radius of sphere.
E = spherical excess = A + B + - 180.
A = area of triangle.
It is proved in geometry that the area of a spherical triangle
is to the area of the surface of the sphere as its spherical excess,
in degrees, is to 720. Hence, we have
A : 4 Trr 2 - E : 720.
180
II. Given the three sides a, &, c.
The problem is to express the value of E in terms of the sides.
(1) CAGNOLI'S THEOREM.
sin -= sin
+ B . A + B C
sin - - cos cos -
212 SPHERICAL TRIGONOMETRY
C
sin cos
(cos ^ - cos ^} (Art. 130, p. 190)
C \ ' A J
cos-
: C Of n - a . V
"- cos- " "-
2 2
v \_, / /-> flf ' \
sm cos ( 2 sin- sin - )
c
cos-
(Art. 77, p. 100)
.
sin - sin - m .
_____ ^ V sin s sin ( s a) sin (g 6) sin ( s c}
cos c_ sin a sin 5
2 (Art. 127, p. 186)
Replacing sin a and sin b by their values (Art. 80, p. 106)
and canceling, we have
E _ Vsin 8 sin (g a) sin (g 6) sin (s c)
Sill . '
2 n a b c
L COS - COS - COS -
(2> L'HUILIER'S THEOREM. This theorem, which expresses
the value of E by means of its tangent, is derived as follows :
~Ei A
tan =
A + , TT- (7
cos - - + cos -
(Art, 77, p. 100)
SOLUTION OF SPHERICAL TRIANGLES 213
a-b c
cos cos - cos
^_ 2 2 (Art> 129 , p . 189)
a -\-b c C
cos-^- + cos- sm-
sin sin
__JL_ -JLcotS. ( Art - 77 > P- 10 )
(Art. 127, p. 186)
(8) All other cases may be solved by first finding the three
sides or the three angles, and then applying the proper formula.
ANSWERS
1. 1.
2. If
3. 0.7581+.
PLANE TRIGONOMETRY
Exercise I. Pages 11, 12
4. 1.2737+.
5. 2. 54 19-.
6. 3.5693+.
7. 40, 60, 80.
17. 5, 25, 150.
18. 30, 360, 21600.
4.
5.
6.
7.
8.
9.
10.
30.
120.
36.
54.
270.
150.
540.
2700.
11. =
3' T
Exercise II. Pages 14-16
12
2?r
17 8599 TT
23. 27,
63.
3
5400
24. 52,
66,
72.
3?r .
lg 20533 TT
7T
7T
7 7T
13.
5400
25. f,
3'
Is"
STT
19 W7r .
26. 30,
60,
90.
14.
180
4
20 ^-
27. 4, 6.
121 7T
180
oa 3?r
57T
77T
360
21. J.
28. --,
7
9
16.
463 TT
oo 13021 TT
OQ 1 T
2?r
1
720
30000
' 2' 3
' 3
/ 2
5?r 2ir
31. 150, ; 82
30', 11^; 135.
37T
9 ' 3 '
6
' 24 '
4
32.
minutes past four ; 54 T 6 r minutes past four.
5. 1.77.
6. 28 7' 30".
7. 0.265 sec.
8. 40yd.
9. 2 8' 52.8".
10. 861,031 mi.
(approximately) .
11. 3962.95.
12. 14 19' 26.2".
13. 1.047 radians,
Exercise III. Pages 17-19
14. 51.56. 21. 65 24' 30.4".
15. 102 ft.
(approximately) .
16. 5:4.
17. 3.1416.
18. -, -T, -
399
19. 3.1416.
20. 0.000097+.
215
22. 98 ft.
23. 1 mi. 908 ft. nearly.
24. 7 mi. 1237.2 ft.
25. 18 and 58.
26. 19.099'.
27 60 ^
10800
28. 0.00004848.
216
ANSWERS
Exercise V. Pages 29, 30
7. T 4 r V7. 11. Ii, $?. 15. U> e
8. |f. 12. |, |. 16. f A/7,
9. f, f. 13. ft A/61, ft A/61. 17. f, ft A/14
10. & A/16, f 14. |Vfl,2VO.
18. sin A = T 8 7 , cos A = tf , etc. ; sin 5 = jf , cos J5 = T 8 7 , etc.
1. Sill ^1
x* + y'
- , ws ^i , CIA;. , iii jj ,
V>V_/0 J-f ~
20. f.
21. i. 22. f.
^23T}f
Exercise VIII. Pages 42-48
32. 60.
43. $ a 2 cot A. 53. 23 11' 55".
63. 355.34.
33. 45188.
44. \a?\,*\\B. 54. 38 9' 25".
64. 74.335.
34. 6.
45. c 2 sin A cos A. 55. 80.49,105.64.
65. 42.838.
35. 124.71.
46. 29 22'. 56. 74 43' 54".
66. 313.1.
36. 182.8.
47. 60 38'. 57. 124.27.
67. 38.13.
37. 1143.4.
48. 20.48. 58. 560.88.
68. 43.03.
38. 1916.64.
49. 33.64. 59. 25.165, 36.458.
69. 39 11'.
39. 36157.5.
50. 41 36'. 60. 89.44.
71. 118.3.
40. 498.51.
51. 24 54' 16". 61. 46.71.
72. 100.
41. 52444.44.
52. 42 42' 34". 62. 122.53.
73. 145.58.
42. iaVc 2 -a 2 .
Exercise IX. Pages 49, 50
1. 64 20' 26".
2. 75 32' 50".
9. M 2 sin-cos .
16.
A = 309.01.
A = 29.82.
3. 243.57.
10. wrt 2 sin A cos A. 17.
A = 104.71.
4. 175.068.
11. nh' 2 cotA. 18.
A = 12.312.
5. 148.91'.
6. 80 17'.
12. A = 69.24. 19.
13. A = 1325.46. 20.
A = 115.92.
A = 700.616.
7. 91.204.
14. A = 3741.18. 21.
A = 2186.95.
8. 3 34' 8".
Exercise X. Pages 72, 73
K V3 + 1
V2-2 Q V3 + 2
11. L^.
12. -2.
2
2 2
c 1 + 2 V2
3 A/3 1Q 3 A/3 -4.
13. -|.
6 2 "
8> 2 3
, etc.
ANSWERS 217
14. Positive for 60, 120, 210, 330 ; negative for 0, 240, 300.
15. Positive for 330; negative for 210, 300; zero for 135.
2ab 2ab 2a + l 2 a 2 + 2 a
' a 2 + 6 2 ' a 2 - 6 2 2 a 2 + 2 a + 1 ' 2 a- + 2 a + 1
Exercise XI. Pages 83, 84
5. 45 and 226 ; 45, 135, 22o, 315.
6. Positive for 120 and 690 ; negative for 150, 300, and ; zero for 135
and 315.
7. Positive for 210 and 780; negative for 240, 300, 625 and ; zero
for 225.
8. Positive for 60, 150, and ^^ ; negative for 120 and 210 ; zero for 135
and 225.
9. (a) 240 and 300; (6) 210 and 330; (c) 135 and 315; (d) 30 and
210.
14. 3.
15. cot 2 A esc A.
Exercise XII. Pages 88, 89
I. 14. * = i.
4 6
15. = n7r--
2 ^4
s\ /j ( I \ n ^T
6'
7. e =mr-(-\y\.
6 17. e = nr -.
8. 6 = 2mr^- 18 ^ = W7r 7r >
7T
9. ^ =(2 w + !)TT - v
19. ^ WTT or mr -
2 20. 6= mr -
4
11. 0=(2W + 1)7T.
21. <9^2w7r+-.
12. f = iw + 7- 3
4
22. *<1.-H), + S.
Exercise XIII. Pages 90, 91
4. 2 nir , or (2 n + I)TT. 5. 2 mr, or 2 nTr |.
3
g W7r _)_ (_ 1) ^, or MTT + ( 1) M ^-^-
6 2
218 ANSWERS
7. W T+(-l)?.
2 W7T
8. 2 mr ~ 18. 2 HIT, or
9. 2 WTT *, or (2 w + I)TT. 19. -1^-, O r 2 r?r
8\ y " -
m n m + n
10. 2 nir -. 20. nir -, or + .
3 4 3 12
11. 2 nir + -, or sin = - |. 21. WTT + ^, or 7 -^ + -JL.
^j
12. W7T -. 22. 7i7T.
13. WTT + -, or cot 6 = 2. 23. .
4 3
14. HT^. 24. 5? + -.
15. (2n + l)or. 25. r
1 3 2(m
16. + ,orH. 26 . (2)1 +
Exercise XIV. Pages 95-97
4 - -if- 5- ;;:; 6. ^
Exercise XV. Pages 99, 100
1. 1. 2. &\. 3. - II 4. - 4. 5. 3.
Exercise XVIII. Pages 108-110
- >, u.
, 4V2
23
2 7
3\/15
3. it, - Hi
3VI6
9 '
27*
8'
16
4. |, 6.
5
Exercise
XXI.
Pages 120, 121
1.
^V2,
6 V ^
9. I.
13.
iv
2.
\ V2.
2V?'
10. 1 or - i.
14.
If-
3.
1.
7. V3.
11. or .].
15.
1-
4.
x imaginary.
-3Vl7
12. 1 or \.
16.
*V5.
5.
13.
4
17.
aft
19
ab
20.
V3.
vV
1 -f Vft
Va 2 -
1 + V6 2 -1
21.
2.
18.
WTT or n
r+J,
4
ANSWERS 219
Exercise XXII. Page 127, 128
1. 2r, or2*w- 11. mr +(- 1) W 36 52', or 2 mr -.
3 2
. 12. 2n7r-3652'.
2. 2W7T + -, or (2n
13. , or .
439
14. f,or,|.
-'
7. 2W7T + , or2w7r- .
16.
O
17. ^ + |,
18.
19. S;qr-2!:+(--I)-i.
20. 2n7r, or
10. 2/i7r, or 2 WIT + 112 38'.
21. 2 mr, (2 n + 1) - , or (2 n + 1) -
22. (2w+l)^,(2rc + l)^,or(2N + l).
'2 4 o
23. 2 7Z7T, 01' ?I7T y o _
35. 2 mr, or 2 7i?r +
25. ttr,or(2 + l). ^ B)r _ ^ or .yr +( _ 1)n jr .
26. -
27. , ,r 2 , ,
,
4 2
29. n7r^, or(2n+l)-
41. 5E f or-|
30. (2* + l)f,or^.f(-l)*|.
42. WTT, or nir-
31. 7i7T. 3
32. (2w+ 1)|, or WIT |-
43. nr, or
33. ?ITT, or WTT 44. WTT, or ~ +
220
ANSWERS
Exercise XXIV. Pages 136-138
11. 640.65ft.
12. AC = 8332.2 ft., AB = 12163.53 ft.
13. Distances 2841.2 ft., 3475.46 ft. Height 1721.08 ft.
14. Distances 11975.68 ft., 24182.77 ft. Height 19769.54 ft.
15. 121.04ft. 16. 171.15ft. 17. 110.39ft. 19. 4.588 mi. 20. 4.506 mi.
Exercise XXVI. Pages 146-148
11. 4536.4 ft. 13. 5402.6 ft. 15. 15.6.
12. 134.49 ft. 14. 9. 16. 1781.2 ft.
19. A= 39 46' 0.4", B = 68 2' 45.6". 20. 4494.3 ft.
17. 5.65.
18. 4.58: 9.81
Exercise XXVII. Pages 152, 153
17. 43 55' 13". 20. 60, 60, 60.
18. 49 8' 46". 21. 66 44' 2", 60 26' 53", 52 49' 9".
19. 30, 60, 90. 23. 60. 24. 120. 25. 73 44'.
Miscellaneous Examples. Pages 158-165
1. 247.56 ft. 3. 41 9' 7". 5. 48 45' 44". 7. 122.48 ft.
2. 42 42' 34". 4. 36 22' 21". 6. 72.75ft. 8. 123.47ft.
9. Height = 1224.3 ft.; distance = 1292.9 ft. 10. 431.78 ft.
11. 233.27 ft. 12. 440.36 Ib. ; 63 12' 26", 26 47' 34".
13. 2881.46 mi. 15. 2304.52ft. 17. 7912.8 mi.
14. 407.61ft. 16. 67.5ft. 18. 108 11'.
19. Height = 350.67 ft., distance =3205.15 ft. 20. 8.0076 in.
21. 746 ft. 22. 17.32, 30, 34.64. 23. 244.95.
24. tan- 1 f ; 3 \ of an hour. 25. 6ft.
26. 136.13 ft. from the foot of the tower. , 27. 61.24ft.
29. 109.9ft. 31. 308.66ft. 33. 110^'.^" 35. 4782.2ft.
30. 4621.1ft. 32. 407.61ft, 34. 473.3ft. 36. 2785.6ft.
37. 60 20' 8", 76 49' 18", 42 50' 29". 38. 595.84 ft.
39. 1743.36 ft. 40. 4244.4 ft. 41. 9.1 mi. arc-hour. 42. 383.37 yd.
43. Resultant = 658.36 Ib. ; angle bet. resultant and greater force 22 23' 43".
44. 2019.62 ft. 47. 63.08. 50. 3883 ft. 52. 13451.52 ft.
45. 410.35ft. 48. 45.92ft. 51. 4494.3ft. 53. 1949.77ft.
46. 178.88 ft. 49. 10520.49 ft.
Exercise XXVIII. Pages 167, 168
2. 0.0029089. 4. 0.002036. 6. 0.99999.
3. 0.9999958. 5. 0.004363. 7. 0.00003878.
ANSWERS 221
Exercise XXXI. Page 175
1. r = 2.5, = 2.165. 5. 2 c 2 - ad 2 = bed.
2.
3.
4.
9.
10.
11.
a 2 + b' 2 = c 2 + cf 2 . 6. a 4 2 a 2 + & 2 = 0.
z 2 , y 2 _ 2 7. 2 & - b = 2.
2 6 2 8. (a 2 - & 2 ) 2 - 16 ab = 0.
a 2 + ^2 _ 6 - 2 + C 2.
(a 2 + I) 2 + 2 &(a 2 + l)(a + 6) - 4(a + ft) 2 = 0.
(a + 6)* + f a _ 5)1 - 2 12 - ( ~ & ) tan + & = -
| i _| 13. a 2 - & 2 - 2 cos a - 2 = 0.
SPHERICAL TRIGONOMETRY
Exercise XXXIII.
Pages 199, 200
1.
6 =
14 25' 20",
A
47 30' 46",
B
= 44 25' 26"
,
2.
c =
77 56' 37",
A =
81 50' 9",
B
= 34 28' 58"
.
3.
Impossible.
4.
c =
69 55' 18",
b =
130 15' 58",
A = 115 33' 51".
5.
a =
49 30' 54",
b =
131 41' 29",
B
= 124 6' 53"
,
6.
a =
34 20' 53",
b =
42 23' 40",
c
= 52 25' 39"
7.
a =
80 28' 44",
b =
78 38' 54",
B = 83 47 '23".
8.
b =
145 13' 27"
, A =
35 2' 7",
B
= 118 8' 2".
9.
b =
155 23' 47"
, c =
71 18' 48",
A
= 98 54' 34".
10.
&i =
153 59' 53"
, Ci =
69 36',
BI
= 152 6' 47"
.62 =
26 0' 7",
Co =
110 24', B-2
= 27 53' 13"
11.
c =
62 33' 19",
A =
68 51' 35",
B
= 39 59' 48".
12.
a =
49 53' 28",
b =
58 26',
c
= 70 17' 27".
Exercise XXXIV.
Page 202
1.
A =
113 51' 22"
,B =
66 17' 20",
C
= 960'18".
2.
A =
65 10',
B =
98 50' 37",
C
= 125 17' 48".
3.
A =
129 22' 58",
, B =
109 41 '38",
c
= 97 21' 36".
4.
A =
23 16' 48",
B =
62 13' 34",
G
= 107 54' 18'
Exercise XXXV.
Page 203
1.
A =
117 33' 50", B =
46 37' 46", c
62 36' 45".
2.
A =
116 0' 7",
B =
51 34' 15", c
57 51' 26".
3.
B =
101 4' 47",
C =
40 8' 22", a
=
57 31' 43".
4.
A =
35 18' 32",
c =
126 39' 6", b
77 10' 36".
5.
B =
69 28' 26",
c =
42 13' 34", a
=
76 17' 36".
Exercise XXXVI.
Page 205
1.
B = 119 34' 43", C = 96 55' 26",
c = 105 36' 14".
2.
7? = 631'40",
C = 84 50' 28",
c = 80 51' 28".
3.
BI = 63 55' 10".
Ci = 3351'5", ci =
= 26 41' 4".
B = 1 16 4' 50",
Impossible.
222
ANSWERS
Exercise XXXVII. Page 207
1. 6 = 156 51' 40", c = 30 57' 43",
2. b = 125 22' 40", c = 155 48' 12",
3. bi = 75 38' 40", ci = 102 0' 42",
4. b. 2 = 104 21' 20", c 2 = 134 30' 27",
Exercise XXXVIII. Page 208
1. a = 129 29' 29", b = 107 45' 45",
2. 6 = 36 23' 38", c = 122 53' 23",
3. = 123 21' 30", c = 84 15' 24",
4. a = 153 51' 21", 6 = 89 26',
Exercise XXXIX. Page 211
1. a = 142 5' 25", b = 38 47' 39",
2. a = 4920'39", 6 = 62 31' 13",
3. a = 107 45' 46", b = 54 27' 19",
4. a = 118 52' 50",
b = 34 20' 45",
(7=69 37 '20".
C= 155 50' 58".
Ci = 48 27' 53".
C 2 = 146 55' 13".
C = 54 54' 16".
A = 161 1' 28".
5 = 129 4' 47".
(7= 78 21' 23".
c = 135 57' 44".
c = 5137'5".
c = 99 18' 46".
c = 84 53' 32".
FIVE-PLACE
LOGARITHMIC AND TRIGONOMETRIC
TABLES
BASED OX THE TABLES OF F. G. GAUSS
ARRANGED BY
LEVI L. CONANT, PH.D.
PROFESSOR OF MATHEMATICS IN THE WORCESTER
POLYTECHNIC INSTITUTE
NEW YORK :. CINCINNATI : CHICAGO
AMERICAN BOOK COMPANY
COPYRIGHT, 1909, BY
AMERICAN BOOK COMPANY.
ENTERED AT STATIONERS' HALL, LONDON.
CONANT TRIG. TAliLES.
W. P. I
INTRODUCTION
1. A logarithm is the exponent by which a number a must be
affected in order that the result shall be a given number m. That
is, if a x = m, then x is called the logarithm of m to the base a. The
above equation written in logarithmic form is log a m = x.
Any positive number except 1 may be used as the base of a
system of logarithms. In practical work involving numerical
computation 10 is the base that is universally employed.
All computations by means of logarithms are based on the
following theorems :
2. The logarithm of a product is equal to the sum of the logarithms
of the factors.
PROOF. Let m and n be any two positive numbers, and let x
and y be their logarithms respectively. Then
m-n = 10*-1 O^IO^.
/. log(w^i) = x -f- y = log m + log n.
3. The logarithm of a quotient is equal to the logarithm of the
dividend minus the logarithm of the divisor.
PROOF. = ! =10*-".
n I0 y
.'. log = x y = log m log n.
n
4. The logarithm of any power of a number is equal to the loga-
rithm of the number multiplied by the index of the power.
PROOF. m y = (10*)* = 10**.
.'. log m y = xy y log m.
5. The logarithm of any root of a number is equal to the logarithm
of the number divided by the index of the root.
PROOF. Vm = ^10^ = 10*.
]ow m
6. The logarithm of any integral power or root of 10 is an
integral number. The logarithms of all other positive numbers
are fractions.
Negative numbers have no logarithms. If any logarithmic
computation is to be performed which involves negative numbers,
the problem should be solved as though the numbers were all posi-
tive ; and the algebraic sign of the result should then be deter-
mined by the usual methods of algebra.
7. The logarithm of a number consists of two parts, an integral
part and a decimal. The integral part is called the characteristic,
and the decimal part the mantissa. As logarithms are usually
printed the mantissa is always positive. The characteristic may
be positive, negative, or zero. The characteristic of the logarithm
of any number may be found by one of the following rules :
I. The characteristic of the logarithm of a number greater than
one is positive, and is one less than the number of digits in the integral
part of the number.
II. The characteristic of the logarithm of a decimal fraction is
negative, and is numerically one greater than the number of ciphers
immediately after the decimal point.
For example, the characteristic of the logarithm of 3286 is 3 :
of 294645 is 5 ; of 0.0241 is -2 ; of 0.000649 is -4.
For the sake of convenience a negative characteristic is often
changed in form by adding to it and subtracting from it the number
10. For example, if the characteristic of a logarithm is 2, and
the mantissa is .38416, the logarithm may be written 8.38416 10.
If the characteristic is 1 and the mantissa is .74925, the logarithm
may be written 9.74925 10. If the negative forms of the charac-
teristics are retained, the above logarithms are written 2.38416 and
1.74925 respectively. When it is remembered that the mantissas
are positive, the reason for writing the negative sign of a charac-
teristic above instead of before it will be readily understood.
In all work connected with the logarithms in the following
tables the characteristics, when negative, are to be understood as
being increased and diminished by 10.
TABLE I
Directions for finding the logarithm of a number.
8. When the number is between i and 100.
The entire logarithm, including both characteristic and man-
tissa, is given on p. 9.
9. Numbers containing one or two significant figures.
The mantissa is found on p. 9. It is the same for all numbers
containing the same significant figures arranged in the same order,
no matter where the decimal point is placed.
The characteristic is found by means of the rules given above.
For example,
log 53 = 1.72428, log .53 = 9.72428 - 10,
log 5.3 = 0.72428, log .053 = 8.72428 - 10.
10. Numbers containing three significant figures.
The number, no attention being paid to the decimal point, is
found at the left of the page in the column headed No. The
mantissa is found on a line with the number, and in the column
headed 0. The characteristic is found as before, by one or the
other of the rules on p. 4.
For example,
log 763 = 2.88252, log .0763 = 8.88252 - 10,
log 76.3 = 1.88252, log .00763 = 7.88252 - 10.
11. Numbers containing four significant figures.
The first three significant figures are found in the column
headed No., and the fourth is at the top of the page. On a
line with the first three figures, and in the column headed by the
fourth figure, the mantissa is found. The characteristic is deter-
mined as in the previous cases.
For example,
log 296300 = 5.47173, log .2963 = 9.47173 - 10,
log 29,63=1.47173, log .0002963 = 6.47173 - 10.
12. Numbers containing more than four significant figures.
Let the number whose logarithm is required be 61487. Since
the number lies between 61480 and 61490, the logarithm of the re-
quired number lies between the logarithms of those numbers, i.e.
between 4.78873 and 4.78880.
Now log 61490 = 4.78880
and log 61480 = 4.78873
giving a difference of .00007
Hence, we see that an increase of 10 in the number produces
an increase of .00007 in the logarithm. But the actual increase
we have to consider in the number is 7. Now if an increase of
10 in the number produces an increase of .00007 in the logarithm,
an increase of 7 in the number will produce an increase of -^ of
.00007, or .000049. Calling this correction .00005, we have
log 61480 = 4. 78873
correction = .00005
.-. log 61487 = 4.78878
It is here assumed that an increase in the number is accom-
panied by a proportional increase in the logarithm of the number.
This is not true ; but in obtaining logarithms from a table, that
assumption is always made. If greater accuracy is desired, it
will be necessary to use tables containing a greater number of
figures.
Directions for finding the number corresponding to a given
logarithm.
13. Logarithms whose mantissas are found in the table.
When the exact mantissa of a logarithm is found in the table,
the first three significant figures of the number corresponding to
the logarithm are found in the column headed No., and on a
line with the given mantissa. The fourth significant figure is at
the top of the column in which the given mantissa is found.
For example,
2.68529 is the logarithm of 484.5. See p. 17.
9.68529-10 is the logarithm of 0.4845.
7.68529 - 10 is the logarithm of 0.004845.
5.68529 is the logarithm of 484500.
14. Logarithms whose mantissas are not found in the table.
When the exact mantissa of the given logarithm is not found
in the table, the first four significant figures of the number corre-
sponding to the logarithm are the same as the first four significant
figures of the number corresponding to the next smaller logarithm.
The remaining figures are found by interpolation, as illustrated in
the following.
To find the number corresponding to the logarithm 3.44127.
Number corresponding to 3.44138 is 2763 See p. 13.
Number corresponding to 3.44122 is 2762
.00016 ~T
Thus we see that an increase of .00016 in the logarithm corresponds
to an increase of 1 in the number. But the given logarithm,
3.44127, is .00005 greater than the logarithm of the number 2762.
Therefore, the increase in the required number is ;$$$}f, or, more
simply, -^g- of 1. This gives .31 as the required increase. Hence
2762.31 is the number whose logarithm is 3.44127.
Similarly,
78.565 is the number whose logarithm is 1.89523.
58317.5 is the number whose logarithm is 4.76580.
.17532 is the number whose logarithm is 9.24383 - 10.
15. Cologarithms.
The cologarithm of a number is the logarithm of the recipro-
cal of that number.
Since the reciprocal of a number is unity divided by that num-
ber, and since the logarithm of unity is 0, it follows that the
cologarithm of a number is found by subtracting the logarithm of
the number from 0, or from 10 10.
For example,
colog 256 = log 2 iff = log 1 - log 256 = - 2.40824 = - 2.40824.
To avoid the use of negative logarithms the above work is
performed, and the value of the above result is expressed as
follows: log 1 = 10. 00000 -10
log 256= 2.40824
.-. colog 256= 7.59176-10.
From the definition of a cologarithm it follows that the effect
of subtracting the logarithm of a number is the same as that of
adding its cologarithm. For example, finding the logarithm
of HI by each of the two possible methods, we have :
BY LOGARITHMS BY COLOGAKITHMS
log 293 = 12.46687 - 10 log 293 = 2.46687
log 478= 2.67943 colog 478= 7.32057 - 10
Subtracting, 9.78744 - 10 Adding, 9.78744 - 10
The result is the same in both cases.
TABLE III
This table contains the logarithmic sine and tangent for every
second from 0' to 3', and the logarithmic cosine and cotangent for
every second from 89 57' to 90 ; the logarithmic sine, cosine, and
tangent for every ten seconds from to 2, and the logarithmic
sine, cosine, and cotangent for every ten seconds from 88 to 90 ;
and the logarithmic sine, cosine, tangent, and cotangent for every
minute from 1 to 89.
I
16. The logarithmic sine, cosine, tangent, or cotangent of an
angle less than 90.
If the angle is less than 45, use the column having the name
of the proper function at the top, and the column of minutes at
the left of the page; if the angle is between 45 and 90, use the
column having the name of the proper function at the bottom,
and the column of minutes at the right of the page.
To illustrate the use of this table, let us find the logarithm of
sin 26 28' 12".
% P- 48 > log sin 26 28' = 9.64902 - 10.
The difference between log sin 26 28' and log sin 26 29' is .00025.
Increasing the former by ^| of this difference, or .00005, we have
log sin 26 28' 12" = 9.64907 - 10.
As a further illustration, find log tan 71 3/ 10".
% P- 44 > log tan 71 38' = 10.47885 - 10.
Increasing this by J J of .00042, i.e. by .00013, we have
log tan 71 38' 19" = 10.47898 - 10.
If the logarithm of a cosine or of a cotangent is to be found,
the correction for seconds must be subtracted, since these functions
decrease as the angle increases from to 90.
17. The angle corresponding to a logarithmic sine, cosine, tan-
gent, or cotangent.
Find the angle whose log tan = 9.65647 10.
The next smaller logarithmic tangent is (p. 47) 9.65636 10,
which corresponds to an angle of 24 23'. The difference between
this logarithm and the log tan 24 2i' is .00033, and the difference
between log tan 24 23' and the given logarithm is .00011. There-
fore, the angle corresponding to the next smaller logarithm, i.e.
24 23', must be increased by 1J of 60", i.e. by 20". Hence,
9.65647 - 10 = log tan 24 23' 20".
In the case of the logarithm of the cosine or of the cotangent
we work from the next larger logarithm to the next smaller, in-
stead of from the smaller to the larger as in the case of the sine
and the tangent.
TABLE IV
This table contains the numerical or natural values of the sine,
cosine, tangent, and cotangent for every minute from to 90.
-
TABLE I
THE COMMON OR BRIGGS
LOGARITHMS
OF THE NATURAL NUMBERS
FEOM 1 TO 10000
MOO
No. Log.
No. Log.
No. Log.
No. Log.
No. Log.
2O 1.30103
21 1.32222
22 1.34242
23 1.36173
24 1.38021
4O 1.60206
41 1.61278
42 1.62325
43 1. 63 347
44 1.64345
6O 1.77815
61 1.78533
62 1 . 79 239
63 1. 79 934
64 1. 80 618
8O 1.90309
81 1.90849
82 1.91381
S3 1. 91 908
84 1.92428
1 0. 00 000
2 0. 30 103
3 0.47712
4 0.60206
5 0. 69 897
6 0.77815
7 0.84510
8 0.90309
9 0. 95 424
25 1. 39 794
26 1.41497
27 1.43136
28 1.44716
29 1.46240
45 1. 65 321
46 1.66276
47 1.67210
48 1.68124
49 1.69020
65 1. 81 291
66 1. 81 954
67 1.82607
68 1.83251
69 1. 83 885
85 1. 92 942
86 1.93450
87" 1.93952
88 1.94448
89 1.94939
MBh i.ooooo
11 1. 04 139
12 1.07918
13 1. 11 394
14 1. 14 613
3O 1.47712
31 1. 49 136
32 1.50515
33 1,51851
34 1.53148
5O 1.69897
51 1.70757
52 1.71600
53 1. 72 428
54 1. 73 239 *
7O 1.84510
71 1.85126
72 1.85733
73 1.86332
74 1.86923
9O 1.95424
91 1.95904
92 1. 96 379
93 1.968-JS
94 1.97313
15 1. 17 609
16 1.20412
17 1. 23 045
18 1.2-5527
19 1.27875
35 1.54407
36 1. 55 630
37 1. 56 820
38 1.57978
39 1. 59 106
55 1. 74 036
56 1.74819
57 1.75587
58 1.76343
59 1.77085
75 1. 87 506
76 1. 88 081
77 1.88649
78 1.89209
79 1. 89 763
95 1. 97 772
96 1. 98 227
97 1.98677
98 1 . 99 123
99 1. 99 564
2O 1. 30 103
4O 1.60206
6O 1.77815
8O 1.90309
1OO 2.00000
MOO
It)
100-149
No.
01234
56789
1OO
00000 00043 00087 00130 00173
00217 00260 00303 00346 00389
101
00432 00475 00518 00561 00604
00647 00689 00732 00775 00817
102
00860 00903 00945 00988 01030
01072 01115 01157 01199 01242
103
01 284 01 326 01 368 01 410 01 452
01494 01536 01578 01620 01662
104
01 703 01 745 01 787 01 828 01 870
01912 01953 01995 02036 02078
105
02 119 02 160 02 202 02 243 02 284
02325 02366 02407 02449 02490
106
02531 02572 02612 02 653 02 694
02735 02776 02816 02857 02898
107
02938 02979 03019 03060 03100
03141 03181 03222 03262 03302
108
03342 03383 03423 03463 03503
03 543 03 583 03 623 03 663 03 703
109
03743 03782 03*822 03862 03902
03941 03981-04021 04060 04100
110
04139 04179 0-1218 04258 04297
04336 04376 04415 04454 04493
111
04532 04571 04610 04650 04689
04727 04766 04805 04844 04883
112
04922 04961 04999 05038 05077
05 115 05 154 05 192 05 231 05 269
113
05 308 05 346 05 385 05 423 05 461
05500 05538 05576 05614 05652
114
05 690 05 729 05 767 05 805 05 843
05881 05918 05956 05994 06032'
\
115
06070 06108 06145 06183 06221
06258 06296 06333 06371 06408
116
06446 06483 06521 06558 06595
06633 06670 06707 06744 06781
117
06819 06856 06 893 06930 06967
07004 07-041 07078 07115 07151
118
07 188 07 225 07 262 07 298 07 335
07372 07408 07445 07482 07518
119
07555 07591 07628 07664 07700
07737 07773 07809 07846 07882
12O
07918 07954 07990 08027 08063
08099 08135 08171 08207 08243
121
08279 08314 OS 3"50 08386 08422
08458 08493 08529 08565 08600
122
08636 08672 08707 08743 08778
08814 08849 08884 08920 08955
123
08991 09026 09061 09096 09132
09167.09202 09237 09272 09307
124
09342 09377 09412 09447 09482
09517 09552 09587 09621 09656
125
09691 09726 09760 09795 09830
09864 09899 09934 09968 10003
126
10037 10072 10106 10140 10175
10209 10243 10278 10312 10346
127
10380 ]0415 10449 10483 10517
10551 10585 10619 10653 10687
128
10721 10755 10789 10823 10857
10890 10924 10958 10992 11025
129
11 059 11 093 11 126 11 160 11 193
11227 11261 11294 11327 11361
13O
11394 11428 11461 11494 11528
11561 11594 11628 11661 11694
131
11727 11760 11793 11826 11860
11893 11926 11959 11992 12024
132
12057 12090 12123 12156 12189
12222 12254 12287 12320 12352
133
12385 12418 12450 12483 12516
12548 12581 12613 12646 12678
134
12710 12743 12775 12808 12840
12872 12905 12937 12969 13001
135
13033 13066 13098 13130 13]62
13194 13226 13258 13290 13322
136
13354 13386 13418 13450 13481
13513 13545 13577 13609 13640
137
13672 13704 13735 13767 13799
13830 13862 13893 13925 13956
138
13988 14019 14051 14082 14114
14145 14176 14208 14239 14270
139
14301 14333 14364 14395 14426
14457 14489 14520 14551 I45^H
140
14613 14644 14675 14706 14737
14768 14799 14829 14860 1489^
141
14922 14953 14983 15014 15045
15 076 15 106 15 137 15 168 15 198
142
15229 15259 15290 15320 15351
15381 15412 15442 15473 15503
143
15534 15564 15594 15625 15655
15685 15715 15746 15 776 15806
144
15836 15866 15897 159^7 15957
15987 16017 16047 16077 16107
145
16137 16167 16197 16227 16256
16286 16316 16346 16376 16406
- 146
16435 16465 16495 16524 16554
16584 16613 16643 16673 16702
147
16732 16761 16*791 16820 16850
16879 16909 16938 16967 16997
148
17026 17056 17085 17114 17143
17173 17202 17231 17260 17289
149
17319 17348 17377 17406 17435
17464 17493 17522 17551 17580
No.
O 1*2 3 4
56789
100-149
150-199
ii
No.
1
2
3
4
5
6
7
8
9
150
17609
17638
17667
17696
17725
17754
17 782
17811
17840
17869
151
17898
17926
17 955
17984
18 013
18041
18070
18099
18127
18156
152
18184
18213
18241.
18270
18298
18327
18355
18384
18412
18441
153
18469
18498
18526
18554
18 583
18611
18639
18667
18 696
18724
^154
18752
18780
18808
18837
18865
18893
18921
18949
18977
19005
155
19033
19061
19089
19117
19145
19173
19201
19229
19257
19285
156
19312
19340
19368
19 396
19424
19451
19479
19507
19535
19562
157
19590
19618
19645
19673
19700
19728
19756
19783
19811
19838
158
19866
19893
19921
19 948
19976
20003
20030
20 058
20085
20112
159
20140
20167
20194
20222
20249
20276
20303
20330
20358
20385
16O
20412
20439
20466
20493
20520
20548
20575
20602
20629
20 656
161
20683
20710
20737
20763
20790
20817
20844
20871
20898
20925
162
20 952
20978
21005
21 032
21059
21085
21 112
21139
21 165
21 192
163
21219
21 245
21272
21299
21325
21 352
21378
21405
21 431
21458
164
21484
21511
21537
21 564
21590
21617
21643
21669
21696
21722
165
21 748
21775
21801
21 827
21 854
21880
21906
21932
21 958
21985
166
22011
22 037
22063
22 089
22115
22 HI
22167
22194
22220
22246
167
22272
22298
22324
22 350
22376
22 401
22427
22453
22479
2T2505
168
22531
22557
22 583
22608
22634
22660
22686
22712
22737
22763
169
22789
22814
22840
22866
22891
22917
22943
22968
22994
23019
17O 23045
23070
23096
23121
23 If 7
23172
23198
23223
23249
23274
171
23300
23325
23 350
23376
23401
23426
23 452
23477
23,502
23528
172
23553
23378
23603
23629
23 654
23679
23 704
23729
23754
23779
173
23805
23830
23855
23 880
23905
23930
23955
23980
24005
24030
174
24055
24080
24105
24130
24155
24180
24204
24229
24254
24279
175
24304
24 329
24353
24378
24403
24428
24452
24477
24 502
24527
176
24551
24576
24601
24625
24650
24674
24699
24724
24748
24773
177
24797
24822
24846
24871
24895
24920
24944
24969
24993
25018
178
25 042
25066
25091
25115
25 139
25 164
25188
25 212
25 237
25261
179
25285
25310
25 334
25358
25 382
25406
25431
25455
25479
25503
180
25527
25 551
25575
25600
25624
25 648
25672
25696
25720
25744
181
25768
25 792
25816
25 840
25864
25888
25 912
25 935
25 959
25983
182
26 007
26031
26055
26079
26102
26126
26150
26174
26198
26221
183
26245
26269
.26293
26316
26340
26364
26387
26411
26435
26458
184
26482
26 505
26529
26 553
26576
26600
26623
26647
26670
2 694
185
26717
26741
26764
26788
26811
26834
26858
26881
26905
26928
1B6
26951
26975
26998
27021
27045
27068
27091
27114
27138
27161
^gjl
^27184
27207
27231
27254
27277
27300
27323
27346
27370
27393
M: 416
27439
27462
27485
27508
27531
27554
27577
27600
27623
i V; 646
27669
27 692
27715
27738
27761
27784
27807
27830
27852
W^ 27 875
27898
27921
27944
27967
27989
28012
28035
28058
28081
191 1 28 103
28126
28149
28171
28194
28217
28240
28262
28285
28307
192
28330
28 353
28375
28398
28421
28443
28466
28488
28511
28533
193
28556
28 578
28601
28623
28646
28668
28691
28713
28735
28758
194
28780
28803
28825
28847
28870
28892
28914
28937
28959
28981
195
29003
29026
29048
29070
29092
29115
29137
29 159
29181
29203
196
29226
29248
29270
29292
29314
29 336
29 358
29380
29403
29425
197
29 447
29469
29491
29 S13
29535
29 557
29579
29601
29623
29645
198
29667
29688
29710
29732
29754
29776
29798
29820
29842
29 863
199
29885
29907
29929
29951
29973
29994
_3a?i?^
30016
30038
30060
30081
No.
O
1
2
3
4
5
6
7
8
9
150-199
12
200-249
No.
O
1
2
3
4
5
6
7
8
9
2OO
30103
30125
30146
30168
30190
30211
30233
30255
30276
30298
201
30320
30341
30363
30384
30406
30428
30449
30471
30492
30514
202
30 535
30557
30578
30 600
30621
30643
30664
30685
30707
30728
203
30750
30771
30792
30814
30835
30 856
30878
30899
30920
30942
204
30963
30984
31006
31027
31048
31069
31091
31112
31 133
31154
205
31 175
31 197
31218
31239
31260
31281
31302
31323
31345
31366
206
31387
31408
31 429
31450
31471
31492
31 513
31534
31 555
31576
207
31 597
31618
31 639
31660
31 681
31 702
31 723
31 744
31765
31 785
208
31806
31 827
318-18
31869
31 890
31911
31931
31 952
31973
31994
209
32015
32035
32056
32077
32098
32118
32139
32160
32181
32201
21O
32222
32 243
32263
32284
32305
32 325
32346
32 366
32387
32408
211
32428
32449
32469
32490
32510
32 531
32 552
32572
32593
32613
212
32634
32654
32 675
32 695
32715
32736
32 756
32777
32797
32818
213
32838
32 858
32879
32 899
32 919
32940
32 960
32980
33001
33021
214
33041
33062
33082
33102
33122
33143
33 163
33183
33203
33224
215
33244
33264
33284
33304
33325
33 345
33 365
33385
33405
33425
216
33445
33465
33 486
33 506
33 526
33 546
33 566
33 5S6 V
33 606
33 626
217
33646
33666
33 686
33 706
33 726
33 746
33766
33786
33 806
33826
218
33 846
33866
33885
33 905
33 925
33 945
33 965
33985
34005
34025
219
34044
34064
34 084
34104
34124
34143
34163
34183
34 203
34223
22O
34242
34 262
34282
34301
34 321
34 341
34361
34380
34400
34 420
221
344^9
34 459
34479
34498
345ft
34 537
34 557
34 577
34 596
34 616
222
34635
34 655
34674
34 694
34713
34 733
34753
347^2
34 792
34811
223
34830
34850
34869
34889
34 908
34 928
34 947
34 967
34 986
35 005
224
35025
35044
35 064
35083
35102
35 122
35 141
35160
35 ISO
35 199
225
35 218
35 238
35 257
35 276
35295
35315
35334
35353
35 372
35 392
226
35411
35430
35 449
35 468
35 488
35 507
35 526
35 545
35564
35583
227
35603
35 622
35 641
35 660
35 679
35 698
35717
35736
35 755
35 774
228
35 793
35813
35 832
35851
35 870
35 889
35908
35 927
35 946
35965
229
35 984
36003
36021
36040
36059
36078
36 097
36116
36135
36 154
23O
36 173
36192
36211
36229
36248
36 267
36 86
36305
36324
36342
231
36361
36380
36399
36418
36 436
36 455
36474
36 493
36511
36 530
232
36 549
36 568
36 586
36605
36 624
36 642
36 661
36 680
36698
36 717
233
36736
36 754
36 773
36 791
36810
36 829
36847
36 866
36 884
36 903
234
36922
36940
36959
36^977
36 996
37014
37033
37051
37070
37088
235
37107
37 125
37 144
37162
37 181
37 199
37218
37236
37254
37 273
236
37 291
37310
37328
37346
37365 37383
37401
37420
37 438
37457
237
37475
37493
37511
37 530
37 548 37 566
37 585
37 603
37621
238
37 658
37 676
37 694
37712
37 731
37 749
37 767
37 785
378031
239
37840
37858
37876
37 894
37912
37931
37949
37967
37 98|
24O
38021
38 039
38057
38 075
38093
38112
38130
38148
38166
241
38202
38220
38 238
38 256
38274
38 292
38310
38328
38 346
38 364
242
38382
38399
38417
38435
38 453
38471
38 489
38 507
38 525
38 543
243
38 561
38578
38 596
38614
38 632
38 650
38668
38686
38703
38721
244
38739
38 757
38775
38792
38810
38828
38846
38863
38881
38899
245
38917
33 934
38952
38970
38987
39005
39023
39041
39 058
39076
246
39 094
39111
39129
39 146
39164
39182
39199
39217
39 235
39 252
247
39^70
39287
39305
39 3-22
39 340
39 358
39375
39 393
39410
39428
248
39 445
39 463
39480
39498
39515
39 533
39 550
39 568
39 585
39602
249
39620
39637
39655
39672
39690
39707
39724
39742
39 759
39 777
No.
O
1
2
3
4
5
6
7
8
9
200-249
250-299
13
No.
O 1 2 3 4
56789
25O
39794 39811 39829 39846 39863
39881 39898 39915 39933 39950
251
39967 39985 40002 40019 40037
40054 40071 40088 40106 40123
252
40140 40157 40175 40192 40209
40226 40243 40261 40278 40295
253
40312 40329 40346 40364 40381
40398 40415 40432 40449 40466
N 254
40483 40500 40518 40535 40552
40 569 40 586, 40 603 40 620 40 637
255
40654 40671 40688 40705 40722
40739 40756 40773 40790 40807
256
40824 408-11 40858 40875 40892
40909 40926 40943 .40960 40976
257
40993 41010 41027 41044 41061
41078 41095 41111 47128 41145
258
41162 41179 41196 41212 41229
41 246 41 263 41 280 41 296 41 313
259
41330 41347 41 363 41 380 41397
41414 41430 41447 41464 41481
26O
41 497 41 514 41 531 41 547 41 564
41581 41597 41614 41631 41647
261
41664 41681 41697 41714 41731
41747 41764 41780 41797 41814
262
41 830 41 847 41 863 41 880 41 896
41 913 41 929 41 946 41 963 41 979
263
41996 42012 42029 42045 42062
42078 42095 42111 42127 42144
264
42 160 42 177 42 193 42 210 42 226
42 243 42 259 42 275 42 292 42 308
265
42 7 25 42341 42357 42374 42390
42406 42423 42439 42455 42472
266
42488 42504 42521 42537 42553
42570 42586 42602 42619 42635
267
42651 42667 42684 42700 42716
42732 42749 42765 42781 42797
268
42813 42830 42846 42862 42878
42894 42911 42927 42943 42959
269 42975 42991 43008 43024 43040
43 056 43 072 43 088 43 104 43 120
27O
43 136 43 152 43 169 43 185 43 201
43217 43233 43249 43265 43281
271
43297 43313 43329 43345 43361
43377 43393 43409 43425 43441
272
43457 43473 43489 43505 43521
43537 43553 43569 43584 43600
273
43 616 43 632 43 648 43 664 43 680 43 696 43 712 43 727 43 743' 43 759
274
43775 43791 43807 43823 43838
43854 43870 43886 43902 43917
275
43933 43949 43965 43981 43996
44012 44028 44044 44059 44075
276
44 091 44 107 44 122 44 138 44 154
44170 44185 44201 44217 44232
277
44248 44264 44279 44295 44311
44326 44342 44358 44'373 44389
278
44404 44420 44436 44451 44467
44483 44498 44514 44529 44545
279
44560 44576 44592 44607 44623
44638 44654 44669 44685 44700
28O
44 716 44 731 44 747 44 762 44 778
44793 44809 44824 44840 44855
281
44871 44886 44902 44917 44932
44948 44963 44979 44994 45010
282
45025 45040 45056 45071 45086
45102 45117 45133 45148 45163
283 ! 45 179 45 194 45 209 45 225 45 240
45255 45271 45286 45301 45317
284! 45332 45347 45362 45378 45393
45408 45423 45439 45454 45469
285 1 45 484 45500 45515 45530 45545
45561 45576 45591 45606 45621
286
45637 45652 45667 45682 45697
45 712 45*28 45 743 45 758 45 773
287
45 788 45 803 45 818 45 834 45 849
45864 45879 45894 45909 45924
288
45 939 45 954 45 969 45 984 46 000
46015 46030 46045 46060 46075
289
46090 46105 46120 46135 46150
46165 46180 46195 46210 46225
29O
46240 46255 46270 46285 46300
46315 46330 46345 46359 46374
291
46389 46404 46419 46434 46449
46464 46479 46494 46509 46523
292
46538 46553 46568 46583 46598
46613 46627 46642 46657 46672
293
46687 46702 46716 46731 46746
46761 46776 46790 46805 46820
294
46 835 46 850 46 864 46 879 46 894
46909 46923 46938 46953 46967
295
296
46982 46997 47012 47026 47041
47129 47144 47159 47173 47188
47056 47070 47lfc 47100 47 !M
47202 47217 47232 47246 47 2B1
297
47276 47290 47305 47319 47334
47349 47363 47378 47392 47407
298
47422 47436 47451 47465 47480
47494 47509 47524 47538 47553
299
47567 47582 47596 47611 47625
47640 47654 47669 47683 47698
No.
01234
56789
250-299
14
300-349
No.
O
1
2
3
4
5
6
7
8
9
300
47712
47727
47741
47756
47770
47784
47799
47813
47828
47 842
301
47 857
47871
47885
47900
47914
47929
47 94.3
47958
47972
47986
302
48001
48 015
48029
48 044
48058
48073
48087
48101
48116
48130
303
48144
48159
48173
48187
48202
48216
48230
48244
48 259
48273
304
48287
48302
48316
48330
48344
48359
48373
48387
48401
48416
305
48430
48444
48458
48473
48487
48501
48515
48530
48544
48 558
306
48572
48586
48601
48615
48629
48643
48657
48671
48686
48700
307
48714
48728
48742
48756
48770
48785
48799
48813
48 827
48 841
308
48 855
48869
48 883
48897
48911
48926
48 940
48 954
48968
48982
309
48996
49010
49024
49038
49052
49066
49080
49094
49108
49122
31O
49136
49150
49164
49178
49192
49206
49220
49234
49248
49262
311
49 276
49290
49304
49318
49332
49346
49360
49374
49388
49402
312
49 415
49429
49443
49457
49471
49485
49499
49513
49527
49 541
313
49 554
49 568
49582
49 596
49610
49624
49638
49651
49665
49679
314
49693
49707
49721
49734
49748
49762
49776
49790
49803
49817
315
49831
49845
49859
49872
49886
49900
49914
49927
49941
49955
316
49969
49982
49996
50010
50024
50037
50051
50 065
50079
50092
317
50 106
50 120
50133
50 147
50161
50174
50188
50 202
50 215
50 229
318
50 243
50 256
50270
50284
50297
50311
50325
50 338
50352
50365
319
50379
50393
50406
50420
50433
50447
50461
50474
50488
50501
320
50515
50529
50542
50 556
50 569
50583
50 596
50610
50623
50637
321
50651
50664
50678
50691
50705
50718
50 732
50745
50759
50772
322
50 786
50799
50813
50 826
50 840
50853
50 866
50880
50893
50907
323
50 920
50934
50947
50961
50974
50987
51001
51014
51028
51041
324
51 055
51068
51081
51095
51108
51121
51135
51148
51162
51175
325
51188
51202
51 215
51228
51242
51 255
51268
51282
51295
51308
326
51322
-51335
51348
51 362
51 375
51388
51402
51415
51428
51441
327
^1415/51468
51481
51495
51508
51521
51534
51548
51561
51574
328
sStffefc
51601
51614
51627
51 640
51654
51667
51680
51693
51706
329
l '5l'20
51733
51746
51759
51772
51786
51799
51812
51825
51838
33O
51 851
51865
51 878
51891
51904
51917
51930
51943
51957
51970
331
51983
51996
52009
52022
52 035
52 048
52061
52075
52088
52101
332
52114
52127
52140
52 153
52 166
52179
52192
52205
52218
52231
333
52244
52257
52 270
52284
52 297
52310
52323
52336
52 349
52362
334
52375
52388
52401
52414
52427
52 440
52453
52466
52479
52492
335
52504
52 517
52530
52543
52556
52569
52582
52595
52608
52621
336
52634
52647
52 $60
52673
52686
52699
52711
52724
52737
52 750
337
52763
52776
52789
52802
52815
52827
52840
52853
52866
52879
338
52892
52905
52917
52 930
52943
52 956
52 969
52982
52994
53007
339
53020
53033
53046
53058
53071
53084
53097
53] 10
53122
53135
340
53148
53161
53 173
53186
53199
53 212
53224
53237
53250
53263
341
53275
53 288
53 301
53314
53326
53 339
53352
53364
53377
53390
342
53403
53415
53428
53441
53 453
53466
53479
53491
53504
53 517
343
53529
53542
53555
53567
53580
53 593
53 605
53 618
53631
53 643
344
53656
53668
53681
53694
53706
53719
53732
53744
53757
53769
345
53782
53794
53807
53820
53832
53845
53 857
53870
53882
53895
346
53908
53920
53 933
53945
53958
53 970
53 983
53 995
54008
54020
347
54033
54045
54058
54070
54083
54095
54108
54120
54133
54 145
348
54 158
54170
54183
54 195
54208
54220
54 233
54 245
54258
54270
349
54283
54295
54307
54320
54332
54345
54357
54370
54382
54394
No.
1
2
3
4
g
6
7
8
9
300-349
350-399
15
No.
O 1 2 3 4
56789
350
351
352
353
354
54407 54419 54 432 \5 4 444 54456
54531 54543 54555 54568 54580
54654 54667 54679 54691 54704
54 777 54 790 54 802 54 814 54 827
54900 54913 54925 54937 54949
54469 54481 54494 54506 54518
54593 54605 54617 54630 54642
54716 54728 54741 54753 54765
54839 54851 54864 54876 54888
54962 54974 54986 54998 55011
355
356
357
358
359
55023 55035 55047 55060 55072
55 145 55 157 55 169 55 182 55 194
55267 55279 55291 55303 55315
55388 55400 55413 55425 55437
55509 55522 55534 55546 55558
55084 55096 55108 55121 55133
55206 55218 55230 55242 55255
55328 55340 55352 55364 55376
55449 55461 55473 55485 55497
55570 55582 55594 55606 55618
36O
361
362
363
364
55630 55642 55654 55666 55678
55751 55763 55 775 55 787 55799
55871 55883 55895 55907 55919
55991 56003 56015 56027 56038
56110 56122 56134 56146 56158
55 691 55 703 55 715 55 727 55 739
55811 55823 55835 55*847 55859
55931 55943 55955 55967 55979
56050 56062 56074 56086 56098
56170 56182 56194 56205 56217
365
366.
367
368
369
56229 56241 56253 56265 56277
56348 56360 56372 56384 56396
56467 56478 56490 56502 56514
56585 56597 56608 56620 56632
56703 56714 56726 56738 56750
56289 56301 56312 56324 56336
56407 56419 56431 56443 56455
56526 56538 56549 56561 56573
56644 56656 56667 56679 56691
56761 56773 56785 56797 56808
37O
371
372
373
374
56820 56832 56844 56855 56867
56937 56949 56961 56972 56984
57054 57066 57078 57089 57101
57171 57183 57194 57206 57217
57287 57299 57310 57322 57334
56879 56891 56902 56914 56926
56996 57008 57019 57031 57043
57113 57124 57136 57148 57159
57229 57241 57252 57264 57276
57345 57357 57368 57380 57392
375
376
377
378
379
57403 57415 57426 57438 57449
57519 57530 57542 57553 57565
57634 57646 57657 57669 57680
57749 57761 57772 57784 57795
57864 57875 57887 -57898 57910
57461 57473 57484 57496 57507
57576 57588 57600 57611 57623
57692 57703 57715 57726 57738
57807 57818 57830 57841 57852
57921 57933 57944 57955 57967
38O
381
'382
383
384
57978 57990 58001 58013 58024
58092 58104 58115 58127 58138
58206 58218 58229 58240 58252
58320 58331 58343 58354 58365
58433 58444 58456 58467 58478
58035 58047 58058 58070 58081
58149 58161 58172 58184 58195
58263 58274 58286 58297 58309
58377 58388 58399 58410 58422
58490 58501 58512 58524 58535'
385
386
387
388
389
58546 58557 58569 58580 58591
58659 58670 58681 58692 58704
58771 58782 58794 58805 58816
58883 58894 58906 58917 58928
58995 59006 59017 59028 59040
58602 58614 58625 58636 58647
58715 5^726 58737 58749 58760
58827 58838 58850 58861 58872
58939 58950 58961 58973 58984
59051 59062 59073 59084 59095
39O
391
392
393
394
59106 59118 59129 59140 59151
59218 59229 59240 59251 59262
59329 59340 59351 59362 59373
59439 59450 59461 59472 59483
59550 59561 59572 59583 59594
59162 59173 59184^59195 59207
59273 59284 59295 59306 59318
59384 59395 59406 59417 59428
59494 59506 59517 59528 59539
59605 59616 59627 59638 59649
395
396
397
398
399
59660 59671 59682 59693 59704
59770 59780 59791 59802 59813
59879 59890 59901 59912 59923
59988 59999 60010 60021 60032
60097 60108 60119 60130 60141
59715 59726 59737 59 74S-.597Sa
59824 59835 59846 59857 5956$
59934 59945 59956 59966 59977
60043 60054 60065 60076 60016
60152 60163 60173 60184 60195
No.
O 1 2 3 4
56789
350-399
lli
400-449
]$0.
1
2
3
4
5
6
7
8
9
4OO
60206
60217
60228
60239
60249
60260
60271
60282
60293
60304
401
60314
60325
60 336
60347
60358
60369
60379
60390
60401
60412
402
60423
60 433
60 444
60455
60466
60477
60487
60498
60509
60520
403
60531
60541
60552
60 563
60574
60584
60595
60606
60617
60627
404
60638
60649
60660
60670
60681
60692
60703
60713
60724
60735
405
60746
60 756
60767
60778
60788
60799
60810
60821
60831
60842
406
60 853
60863
60874
60 885
60895
60906
60917
60927
60938
60949
407
60959
60970
60981
6Q991
61002
61013
61023
61034
61045
61 055
408
4 61 066
61077
61087
61 098
61 109
61 119
61 130
61140
61 151
61 162
409
61 172
61183
61194
61204
61215
61225
61236
61247
61257
61268
41O
61 278
61289
61300
61310
61321
61331
61342
61 352
61363
61374
411
61 384
61 395
61 405
61416
61426
61437
61448
61458
61469
61479
412
61490
61500
61511
61521
61532
61542
61553
61563
61 574
61584
413
61 595
61606
61616
61627
61637
61648
61 658
61 669
61679
61690
414
61700
61 711
61 721
61731
61742
61752
61763
61773
61784
61 794
415
61805
61815
61826
61836
61 847
61 857
61 868
61878
61888
61899
416
61909
61920
61930
61941
61951
61962
61972
61982
61 993
62003
^417
62014
62024
62034
62045
62 055
62066
62 076
62086
62097
62107
418
62118
62128
62138
62149
62 159
62170
62180
62190
62201
62211
419
62221
62232
62242
62252
62263
62273
62284
62294
62304
62315
420
62325
62335
62346
62356
62366
62377
62387
62397
62408
62418
421
62428
62439
62 449
62 459
62469
62 480
62490
62500
62511
62521
422
62 531
62 542
62552
62562
62 572
62 583
62593
62603
62613
62 624
423
62 634
62644
62655
62665
62675
62 685
62696
62 706
62716
62726
424
62 737
62747
62757
62 767
62778
62 788
62 798
62808
62818
62829
425
62 839
62849
62859
62870
62880
62890
62900
62910
62921
62931
-126
62941
62951
62961
62972
62 982
62992
63 002
63012
63022
63 033
427
63043
63053
63 063
63 073
63 083
63094
63 104
63114
63 124
63134
428 63 141
63 155
63165
63175
63185
63 195
63 205
63215
63 225
63236
429 63 246
63256
63266
63276
63286
63296
63306
63317
63327
63337
430
63347
63357
63367
63377
63 387
63397
63407
63417
63428
63438
431
63 448
63458
63468
63478
63 488
63 498
63 508
63 518
63 528
63538
432
63548
63 658
63568
63579
63 589
63 599
63 609
63619
63629
63639
433
63649
63 659
63669
63679
63 689
63 699
63709
63719
63729
63739
434
63749
63759
63769
63779
63789
63 799
63809
63819
63829
63 839
435
63 849
63 859
63869
63 879
63889
63899
63909
63919
63929
63939
436
63949
63 959
63 969
63979
63 988
63 998
64008
64018
64028
64 038
437
64048
64058
64068
64 078
64088
64098
64108
64118
64128
64137
438
64 147
64 157
64167
64 177-
64 187
64 197
64207
64217
64227
64237
439
64 246
64256
64266
64276
64286
64296
64306
64316
64326
64335
44O
64 345
64 355
64365
64375
64385
64395
64 404
64414
64424
64434
441
64 444
64 454
64464
64 473
64483
64493
64 503
64513
64 523
64532
442
64542
64 552
64562
64572
64582
64 591
64601
64611
64621
64631
443
64640
64650
64660
64670
64680
64689
64699
64709
64719
64729
444
64738
64748
64758
64768
64777
64787
64797
64807
64816
64826
445
64836
64846
64856
64 865
64 875
64885
64895
64904
64914
64 924
446
64933
64943
64953
64963
64972
64 982
64992
65 002
65011
65021
447
65031
65 040
65050
65 060
65 070
65 079
65 089
65 099
65108
65 118
448
65128
65 137
65147
65 157
65 167
65 176
65 186
65 196
65 205
65 215
449
65 225
65 234
65244
65254
65 263
65273
65283
65292
65302
65312
1
No. O
1
2
3
4
5
6
7
8
9
I
400-449
450-499
17
No.
1
2
3
4
5
6
7
8
9
450
65 321
65 331
65 341
65350
65360
65369
65 379
65 389
65 398
65 408
451
65418
65 427
65 437
65 447
65 456
65466
65 475
65485
65495
65 504
452
65 514
65 523
65 533
65 543
65 552
65 562
65571
65581
65 591
65600
453
65 6 10
65 619
65 629
65 639
65648
65 658
65667
65677
65686
65696
454
65 706
65715
65725
65734
65744
65753
65763
65772
65782
65792
\
455
65 801
65811
65 820
65 830
65839
65 849
65 858
65868
65877
65887
456
65 896
65 906
65916
65 925
65 935
65944
65 954
65963
65 973
65982
457
65 992
66001
66011
66020
66 030
66039
66049
66 058
66068
66 077
458
66087
66096
66 106
66115
66124
66134
66143
66153
66162
66172
459
66181
66191
66200
66 210
66219
66229
66238
66247
66257
66266
46O
66 276
66285
66295
66304
66314
66323
66332
66342
66351
66361
461
66 370
66380
66389
66398
66408
66417
66427
66436
66445
66 455
462
66 464
66 474
66483
66492
66502
66511
66521
66 530
66539
66549
463
66 558
66 567 "
66577
66 586
66 596
66605
66614
66624
66633
66642
464
66652
66 661
66671
66680
66689
66 699
66 708
66717
66727
66736
465
66 745
66 755
66764
66773
66783
66792
66801
66811
66820
66829
466
66839
66848
66857
66867
66876
66 885
66894
66904
66913
66922
467
66932
66 941
66950
66960
66969
66978
66987
66997
67006
67015
468
67025
67034
67043
67 052
67062
67071
67080
67089
67099
67108
469
67 1.17
67127
67136
67145
67154
67164
67173
67182
67191
67201
470
67210
67219
67228
67237
67247
67 256
67265
67274
67284
67293
471
67 302
67311
67321
67330
67339
67348
67357
67367
67376
67385
472
67394
67403
67413
67422
67431
67 440
67449
67 459
67468
67477
473
67486
67495
67 504
67514
67523
67 532
67541
67 550
67 560
67569
474
67578
67587
67596
67605
67614
67624
67633
67642
67651
67660
475
67669
67679
67688
67697
67706
67715
67724
67733
67742
67 752
476
67761
67770
67 779
67788
67797
67806
67815
67825
67834
67843
477
67852
67861
67870
67879
67888
67897
679J&6
67 916
67925
67934
478
67 943
67 952
67961
67 970
67979
67988
67997
68006
68015
68024
479
68 034
68043
68052
68061
68070
68079
68088
68097
68106
68115
480
68124
68133
68 142
68 151
68160
68169
68178
68187
68196
68 205
481
68215
68224
68 233
68242
68 251
68260
68269
68278
68287
68296
482
68 305
68314
68323
68332
68341
68350
68 359
68368
68377
68386
483
68395
68404
68 413
68422
68431
68440
68449
68 458
68467
68476
484
68485
68494
68502
68511
68520
68529
68538
68547
68556
68565
485
68574
68 583
68 592
68601
68610
68619
68628
68637
68646
68655
486
68664
68673
68681
68690
68 699
68708
68717
68726
68735
68744
487
68753
68762
687T1
68780
68789
68797
68806
68815
68824
68833
488
68 842
68851
68860
68869
68878
68.886
68 895
68904
68913
68922
489
68931
68940
68949
68 958
68 966
68 975
68984
68993
69002
69011
49O
69020
69028
69037
69046
69055
69064
69073
69082
69090
69 099
491
69 108
69117
69126
69135
69 144
69 152
69161
69170
69179
69188
492
69197
69 205
69214
69223
69232
69 241
69249
69 258
69267
69276
493
69 285
69294
69 302
69311
69320
69329
69338
69346
69 355
69364
494
69373
69381
69390
69 399
69408
69417
69425
69434
69443
69452
495
69461
69469
69478
69487
69496
69 504
69513
69522
69531
69 539
496
69 548
69 557
69 566
69574
69 583
69 592
69601
69 609
69618
69627
497
69636
69644
69 653
69662
69671
69679
69688
69697
69 705
69 714
498
69723
69732
69740
69749
69758
69 767
69775
69 784
69793
69801
499
69810
69819
69 827
69836
69845
69 854
69862
69871
69880
69888
No.
O
1
2
3
4
5
O
7
8
9
450-499
18
500-549
No.
O
1
2
3
4
5
6
7
8
9
5OO
69 897
69906
69914
69923
69932
69940
69949
69958
69966
69 975
501
69984
69992
70001
70-010
70018
70027
70036
70044
70053
70062
502
70070
70079
70088
70096
70105
70114
70122
70131
70140
70148
503
70 157
70165
70174
70183
70191
70200
70209
70217
70 226
70234
504
70243
70252
70260
70269
70278
70286
70295
70303
70312
70321
505
70329
70338
70346
70355
70364
70372
70381
70389
70398
70406
506
70415
70424
70432
70441
70449
70 458
70467
70475
70484
70492
507
70 501
70509
70518
70526
70535
70544
70 552
70561
70569
70578
508
70586
70 595
70603
70612
70621
70629
70 638
70 646
70655
70663
509
70672
70680
70689
70697
70706
70714
70723
70731
70740
70749
510
70757
70766
70774
70783
70791
70800
70808
70817
70825
70834
511
70842
70851
70859
70868
70 876
70885
70893
70 902
70910
70919
512
70927
70935
70944
70 952
70961
70969
70 978
70986
70995
71003
513
71012
71020
71029
71037
71046
71054
71 063
71071
71079
71088
514
71096
71105
71113
71122
71130
71 139
71 147
71155
71164
71172
515
71181
71189
71198
71206
71214
71223
71231
71240
71248
71 257
516
71265
71273
71282
71290
71 299
71 307
71315
71324
71332
71341
517
71 349
71357
71366
71374
71383
71391
71399
71408
71416
71 425
518
71433
71 441'
71450
71458
71466
71475
71483
71492
71 500
71508
519
71517
71525
71533
71 542
71 550
71559
71567
71575
71584
71592
520
71600
71609
71617
71625
71634
71642
71650
71659
71667
71675
521
71684
71692
71700
71709
71 717
71725
71734
71742
71750
71759
522
71767
71775
71784
71792
71800
71809
71817
71825
71834
71842
523
71850
71858
71867
71875
71 883
71892
71900
71908
71917
71925
524
71933
71941
71950
71958
71966
7197,5
71983
71991
71999
72008
525
72016
72024
72032
72041
72049
72057
72066
72074
72082
72090
526
72099
72107
72115
72123
72132
72140
72148
72156
72165
72173
527
72181
72189
72198
72206
72214
72222
72 230
72239
72247
72255
528
72263
72272
72280
72288
72296
72304
72313
72321
72329
72337
529
72346
72354
72362
72370
72378
72357
72 395
72403
72411
72419
53O
72428
72436
72444
72452
72460
72469
72477
72485
72493
72501
531
72 509
72518
72 526
72534
72 542
72 550
72 558
72567
72575
72583
532
72591
72599
72607
72616
72624
72 632
72 640
72648
72656
72665
533
72673
72681
72689
72697
72705
72713
72722
72 730
72738
72746
534
72754
72762
72770
72779
72787
72795
72803
72811
72819
72827
535
72 835
72843
72852
72860
72868
72876
72884
72892
72900
72908 1
536
72916
72925
72933
72941
72949
72957
72965
72973
72981
72989
537
72997
73006
73014
73022
73030
73038
73046
73054
73062
73070
538
73078
73086
73094
73 102
73 111
73119
73127
73135
73143
73151
539
73159
73167
73175
73183
73191
73199
73207
73215
73223
73231
54O
73239
73247
73 255
73263
73272
73280
73288
73296
73304
73312
541
73320
73328
73336
73344
73 352
73360
73368
73376
73384
73392
542
73 400
73408
73416
73424
73432
73440
73448
73456
73464
73472
543
73480
73488
73496
73 504
73512
73 520
73528
73536
73 544
73552
544
73560
73568
73576
73 584
73592
73600
73608
73616
73624
73632
545
73 640
73648
73 656
73664
73 672
73679
73687
73695
73703
73711
546
73719
73727
73 735
73743
73751
73759
73767
73775
73783
73791
547
73 799
73807
73815
73823
73830
73838
73846
73 854
73862
73870
548
73878
73 886
73894
73902
7391$
73918
73926
73933
73941
73 949
549
73957
73 965
73973
73981
73989
73997
74005
74013
74020
74028
No.
1
2
3
4
5
6
7
8
9
500-549
550-599
19
No.
O
1
2
3
4
5
6
7
8
9
550
74036
74044
74052
74060
74068
74076
74084
74092
74099
74107
551
74115
74123
74131
74139
74147
74155
74162
74170
74178
74186
552
74194
74202
74210
74218
74225
74233
74241
74249
74257
74265
553
74273
74280
74 288
74296
74304
74312
74320
74327
74335
74 343
554
74351
74359
74367
74374
74382
74390
74398
74406
74 414'
74421
\
555
74429
74437
74445
74453
74461
74468
74476
74484
74492
74500
556
74507
74515
74523
74531
74539
74547
74 554
74562
74570
74578
557
74 586
74593
74601
74609
74617
74624
74632
74640
74648
74656
558
74663
74671
74679
74 687
74695
74702
74710
74718
74726
74733
559
74741
74749
74757
74764
74772
74780
74788
74796
74803
74811
560
74819
74827
74834
74842
74850
74 858
74865
74873
74881
74889
561
74896
74904
74912
74920
74927
74935
74943
74950
74 958
74966
562
74974
74981
74 989
74997
75005
75 012
75020
75028
75035
75 043
563
75 051
75 059
75 066
75074
75082
75089
75097
75105
75 113
75120
564
75128
75 136
75 143
75 151
75159
75166
75174
75 182
75 189
75197
565
75205
75213
75220
75 228
75236
75243
75251
75259
75266
75274
566
75282
75289
75297
75305
75312
75320
75328
75335
75343
75351
567
75358
75366
75374
75 381
75389
75397
75404
75412
75420
75427
568
75435
75 442
75 450
75 458
75465
75473
75481
75488
75496
75504
569
75511
75519
75526
75534
75542
75549
75 557
75565
75572
75580
570
75 587
75595
75603
75610
75618
75626
75633
75641
75648
75656
571
75 664
75 671
75679
75686
75694
75702
75709
75717
75 724
75732
572
75740
75747
75 755
75762
75 770
75 778
75 785
75793
75800
75808
573
75815
75823
75831
75838
75846
75853
75861
75868
75876
75884
574
75891
75899
75906
75914
75921
75929
75937
75944
75952
75959
575
75967
75974
75982
75989
75997
76005
76012
76020
76027
76035
576
76042
76050
76057
76065
76072
76080
76087
76095
76103
76110
577
76118
76125
76133
76140
76148
76155
76163
76170
76178
76185
578
76193
76200
76208
76215
76223
76230
76238
76245
76 253
76260
579
76268
76275
76283
76290
76298
76305
76313
76320
76328
76 335
580
76343
76 350
76358
76365
76373
76380
76388
76395
76403
76410
581
76418
76 425
76433
76440
76448
76455
76462
76470
76477'
76485
582
76492
76500
76 507
76515
76522
76530
76537
76 545
76552
76559
583
76567
76574
76 582
76 589
76597
76604
76612
76619
76626
76634
584
76641
76649
76 656
76664
76671
76678
76686
76693
76701
76708
585
76716
76723
76730
76738
76 745
76753
76760
76768
76775
76782
586
76790
76797
76805
76812
76819
76827
76834
76842
76849
76856
587
76864
76871
76 879
76886
76 893
76901
76908
76916
76923
76930
588
76938
76945
76 953
76 960
76967 '
76975
76982
76989
76997
77004
589
77012
77019
77026
77034
77041
77048
77056
77063
77070
77078
59O
77085
77093
77100
77107
77115
77122
77129
77137
77144
77151
591
77 159
77166
77173
77181
77188
77195
77203
77210
77217
77225
592
77232
77240
77247
77 254
77262
77269
77276
77283
77291
77298
593
77 305
77313
77320
77327
77335
77342
77349
77357
77 364
77371
594
77379
77386
77393
77401
77408
77415
77422
77430
77437
77444
595
77 452
77 459
77 466
77 474
77481
77488
77495
77503
77510
77517
596
77525
77532
77539
77 546
77 554
77 561
77 568
77576
77583
77590
597
77597
77605
77612
77619
77627
77634
77641
77648
77 656
77663
598
77670
77677
77685
77692
77699
77706
77714
77721
77728
77 735
599
77743
77750
77757
77764
77772
77779
77786
77793
77801
77808
No.
1
2
3
4
5
6
7
8
9
550-599
20
600-649
No.
O
1
2
3
4
5
G
7
8
9
GOO
77815
77822
77830
77837
77 844
77 851
77 859
77866
77873
77 880
601
77887
77895
77902
77909
77916
77924
77931
77938
77 945
77 952
602
77960
77 967
77974
77981
77 988
77996
78003
78010
78017
78 025
603
78032
78 039
78046
78053
78 061
78068
78075
78082
78089
78 097
604
78104
78111
78118
78125
78132
78140
78147
78154
78161
78168
605
78176
78183
78190
78197
78204
78211
78219
78226
78233
78240
606
78247
78254
78262
78269
78276
78283
78290
78297
78305
78312
607
78319
78326
78333
78340
78347
78355
78362
78369
78376
78 383
608
78390
78398
78405
78412
78419
78426
78433
78440
78447
78455
609
78462
78469
78476
78483
78 490
78 497
78504
78512
78519
78 526
61O
78 533
78540
78 547
78554
78 561
78569
78576
78583
78590
78 597
611
78604
78611
78618
78 625
78 633
7S6HO
78647
78 654
78661
78668
612
78 675
78682
78 689
78696
78704
78711
78718
78725
78732
78739
613
78746
78 753
78760
78 767
78774
78 7S1
78789
78796
78 803
78810
614
78817
78824
78'S31
78838
78845
78852
78859
78866
78873
78880
615
78888
78895
78902
78909
78916
78923
78930
78937
78944
78951
616
78 958
78 965
78972
78979
78 986
78 993
79000
79007
79014
79021
617
79029
79036
79 043
79 050
79 057
79064
79071
79078
79085
79092
618
79099
79106
79 113
79120
79127
79134
79 141
79148
79 155
79162
619
79169
79176
79183
79190
79197
79204
79211
79218
79 225
79232
62O
79239
79246
79253
79260
79267
79274
79281
79288
79295
79302
621
79309
79316
79323
79330
79 337
79344
79351
79358
79365
79372
622
79379
79386
79393
79400
79 407
79414
79421
79428
79435
79442
623
79449
79 456
79463
79470
79 477
79 484
79 491
79498
79 505
79511
624
79518
79525
79532
79539
79546
79553
79 560
79567
79574
79581
625
79 588
79595
79602
79609
79616
79623
79630
79637
79644
79650
626
79657
79664
79671
79678
79 685
79692
79699
79706
79 713
79720
627
79727
79734
79741
79748
79 754
79761
79768
79 775
79782
79 789
628
79796
79803
79810
79817
79824
79831
79837
79 844
79851
79 858
629
79865
79872
79879
79886
79 893
79900
79 906
79913
79920
79927
63O
79934
79941
79948
79955
79962
79969
79975
79982
799S9
79996
631
80003
80010
80017
80024
80030
80037
80044
80051
80058
SO 065
632
80072
80079
80085
80092
80099
80106
80113
80120
80127
80134
633
80140
80147
80154
80161
80168
80175
80182
80188
80195
SO 202
634
80209
80216
80223
80229
80236
80243
80250
80257
SO 264
SO 271
635
80277
80284
80291
80298
80305
80312
80 318
80 325
80 332
80 339
636
80346
80 353
80359
80 366
80373
80380
80387
80393
80400
80407
637
80414
80421
80428
80 434
80441
80448
80 455
80462
SO 468
80 475
638
80482
80489
80496
80502
80509
80516
80 523
80 530
SO 536
80 543
639
80550
80557
80564
80570
80577
80 584
80591
80598
80604
SO 611
640
80618
80625
80632
80638
SO 645
80652
80 659
80665
SO 672
80679
641
80686
80693
80699
80706
80713
80720
80786
SO 733
80740
SO 747
642
80754
80760
80767
80774
80781
80787
80794
80801
80 SOS.
SO 814
643
80821
80828
80835
80841
80848
80855
80862
SOS6S
SO 875
80882
644
80889
80895
80902
80909
80916
80922
80929
80936
80943
80949
645
80956
80963
80969
80976
80983
80990
80996
SI 003
81010
81017
646
81023
81030
81037
81 043
81 050
81057
81064
81 070
81077
SI 084
647
81090
81097
81 104
81111
81 117
81 124
81131
81137
81144
SI 151
648
81 158
81 164
81171
81178
81 184
81191
81 198
81204
81211
81218
649
81224
81 231
81238
81245
81251
81258
81265
81271
8127S
81285
No.
O
1
2
3
4
5
G
7
8
9
600-649
650-699
21
No.
O 1 2 3 4
50789
65O
651
652
653
654
81291 81298 81305 81311 81318
81358 81365 81371 81378 81385
81425 81431 81438 81445 81451
81491 81498 81505 81511 81518
81 558 81 564 81 571 81 578 81 584
81 325 81 331 81 338 81 345 81 351
81391 81398 81405 81411 81418
81458 81465 81471 81478 81485
81525 81531 81538 81544 81551
81591 81598 81604 81611 81617
655
656
657
658
659
81624 81631 81637 81644 81651
81 690 81 697 81 704 81 710 81 717
81 757 81 763 81 770 81 776 81 783
81823 81829 81836 81842 81849
81889 81895 81902 81908 81915
81 657 81 664 81 671 81 677 81 684
81 723 81 730 81 737 81 743 81 750
81 790 81 796 81 803 81 809 81 816
81 856 81 862 81 869 81 875 81 882
81921 81928 81935 81941 81948
66O
661
662
663
664
81954 81961 81968 81974 81981
82020 82027 82033 82040 82046
82 086 82092 82099 82105 82112
82151 82-158 82164 82171 82178
82217 82223 82230 82236 82243
81987 81994 82000 82007 82014
82053 82060 82066 82073 82079
82119 82125 82132 82138 82145
82184 82191 82197 82204 82210
82249 82256 82263 82269 82276
665
666
667
668
669
82282 82289 82295 82302 82308
82347 82354 82360 82367 82373
82413 82419 82426 82432 82439
82478 82484 82491 82497 82504
82543 82549 82556 82562 82569
82315 82321 82328 82334 82341
82380 82387 82393 82400 82406
82445 82452 82458 82465 82471
82510 82517 82523 82530 82536
82575 82582 82588 82595 82601
67O
671
672
673
674
82607 82614 82620 82627 82633
82672 82679 82685 82692 82698
82737 82743 82750 82756 82763
82802 82808 82814 82821 82827
82866 82872 82879 82885 82892
82640 82646 82653 82659 82666
82705 82711 82718 82724 82730
82769 82776 82782 82789 82795
82834 82840 82847 82853 82860
82898 82905 82911 82918 82924
675
676
677
678
679
82930 82937 82943 82950 82956
82995 83001 83008 83014 83020
83059 83065 83072 83078 83085
83 123 83 129 83 136 83 142 83 149
83 187 83 193 83 200 83 206 83 213
82963 82969 82975 82982 82988
83027 83033 83040 83046 83052
83 091 83 097 83 104 83 110 83 117
83 155 83 161 83 168 83 174 83 181
83219 83225 83232 83238 83245
68O
681
682
683
684
83251 83257 83264 83270 83276
83315 83321 83327 83334 83340
83378 83385 83391 83398 83404
83442 83448 83455 83461 83467
83506 83512 83518 83525 83531
83283 83289 83296 83302 83308
83347 83353 83359 83366 83372
83410 83417 83423 83429 83436
83474 83480 83487 83493 83499
83537 83544 83550 83556 83563
685
686
687
688
689
83569 83575 83582 83588 83594
83632 83639 83645 83651 83658
S3 696 83 702 83 708 83 715 83 721
83 759 83 765 83 771 83 778 83 784
83822 83828 83835 83841 83847
83601 83607 83613 83620 83626
83664 83670 83677 83683 83689
83727 83734 83740 83746 83753
83790 83797 83803 83809 83816
83853 83860 83866 83872 83879
690
691
692
693
694
83885 83891 83897 83904 83910
83948 83954 83960 83967 83973
84011 84017 84023 84029 84036
84073 84080 84086 84092 84098
84 136 84 142 84 148 84 155 84 161
83916 83923 83929 83935 83942
83979 83985 83992 83998 84004
84042 84048 84055 84061 84067
84105 84111 84117 84123 84130
84167 84173 84180 84186 84 If 2
695
696
697
698
699
84198 84205 84211 84217 84223
84261 84267 842^ 84280 84286
84323 84330 84336 84342 84348
843S6 84392 84398 84404 84410
84448 84454 84460 84466 84473
84230 84236 84242 84248 A0d?
84292 84298 84305 84311 84317
84354 84361 84367 84373 84379
84417 84423 84429 84435 84442
84 479 84 485 84 491 84 497 84 504
No.
O 1 2 3 4
56789
650-699
9,9,
700-749
No.
O 1 2 3 4
5 6 7 89
700
701
702
703
704
84510 84516 84522 84528 84535
84572 84578 84584 84590 84597
84634 84640 84646 84652 84658
84696 84702 84708 84714 84720
84757 84763 84770 84776 84782
84541 84547 84553 84559 84566
84603 84609 84615 84621 84628
84665 84671 84677 84683 84689
84726 84733 84739 84745 84751
84788 84794 84800 84807 84813
705
706
707
708
709
84819 84825 84831 84837 84844
84880 84887 84893 84899 84905
84942 84948 84954 84960 84967
85003 85009 85016 85022 85028
85065 85071 85077 85083 85089
84850 84856 84862 84868 84874
84911 84917 84924 84930 84936
84973 84979 84985 84991 84997
85034 85040 85046 85052 85058
85 095 85 101 85 107 85 114 85 120
710
711
712
713
714
85 126 85 132 85 138 85 144 85 150
85 187 85 193 85 199 85 205 85 211
85 248 85 254 85 260 85 266 85 272
85309 85315 85321 85327 85333
85370 85376 85382 85388 85394
85156 85163 85169 85175 85181
85 217 85 224 85 230 85 236 85 242
85278 85285 85291 85297 85303
85339 85345 85352 85358 85364
85400 85406 85412 85418 85425
715
716
717
718
719
85431 85437 85443 85449 85455
85 491 85 497 85 503 85 509 85 516
85552 85558 85564 85570 85576
85612 85618 85625 85631 85637
85673 85679 85685 85691 85697
85461 85467 85473 85479 85485
85522 85528 85534 85540 85546
85582 85588 85594 85600 85606
85643 85649 85655 85661 85667
85703 85709 85715 85721 85727
72O
721
722
723
724
85 733 85 739 85 745 85 751 85 757
85794 85800 85806 85812 85818
85854 85860 85866 85872 85878
85914 85920 85926 85932 85938
85974 85980 85986 85992 85998
85 763 85 769 85 775 85 781 85 788
85824 85830 85836 85842 85848
85884 85890 85896 85902 85908
85944 85950 85956 85962 85968
86004 86010 86016 86022 86028
725
726
727
728
729
86034 86040 86046 86052 86058
86094 86100 86106 86112 86118
86153 86159 86165 86171 86177
86213 86219 86225 86231 86237
86273 86279 86285 86291 86297
86064 86070 86076 86082 86088
86124 86130 86136 86141 86147
86183 86189 86195 86201 86207
86243 86249 86255 86261 86267
86303 86308 86314 86320 86326
73O
731
732
733
734
86332 86338 86344 86350 86356
86392 86398 86404 86410 86415
864S1 86457 86463 86469 86475
86510 86516 86522 86528 86534
86570 86576 86581 86587 86593
86362 86368 86374 86380 86386
86421 86427 86433 86439 86445
86481 86487 86493 86499 86504
86540 86546 86552 86558 86564
86599 86605 86611 86617 86623
735
736
737
738
739
86629 86635 86641 86646 86652
86688 86694 86700 86705 86711
86747 86753 86759 86764 86770
86806 86812 86817 86823 86829
86864 86870 86876 86882 86888
86658 86664 86670 86676 86682
86717 86723 86729 86735 867-41
86 776 86 782 86 788 86 794 86 800
86835 86841 86847 86 853 86859
86894 86900 86906 86911 86917
74O
741
742
743
744
86923 86929 86935 86941 86947
86982 86988 86994 86999 87005
87040 87046 87052 87058 87064
87099 87105 87111 87116 87122
87157 87163 87169 87175 87181
86953 86958 86964 86970 86976
87011 87017 87023 87029 87035
87070 87075 87081 87087 87093
87128 87134 87140 87146 87151
87186 87192 87198 87204 87210
745
746
747
748
749
87216 87221 87227 87233 87239
87274 87280 87286 87291 87297
87332 87338 87344 87349 87355
87390 87396 87402 87408 87413
87448 87454 87460 87466 87471
87245 87251 87256 87262 87268
87303 87309 87315 87320 87326
87361 87367 87373 87379 87384
87419- 87425 87431 87437 87442
87477 87483 87489 87495 87500
No.
O 1 2 3 4
56789
700-749
750-799
23
No.
O 1 2 3 4
56789
750
751
752
753
754
87506 87512 87518 87523 87529
87564 87570 S7~576 87581 87587
87622 87628 87633 87639 87645
87679 87685 87691 87697 87703
87737 87743 87749 87754 87760
87535 87541 87547 87552 87558
87593 87599 87604 87610 87616
87651 87656 87662 87668 87674
87708 87714 87720 87726 87731
87766 87772 87777 87783 87789
755
756
757
758
759
87795 87800 87806 87812 87818
87 852 87858 87864 87869 87875
87910 87915 87921 87927 87933
87967 87973 87978 87984 87990
88024 88030 88036 88041 88047
87823 87829 87835 87841 87846
87881 87887 87892 87898 87904
87938 87944 87950 87955 87961
87996 88001 88007 88013 88018
88053 88058 88064 88070 88076
76O
761
762
763
764
SSOSl 88087 88093 88098 88104
88138 88144 88150 88156 88161
88195 88201 88207 88213 88218
88252 88258 88264 88270 88275
88 309 88315 88321 88326 88332
88110 88116 88121 88127 88133
88167 88173 88178 88184 88190
88224 88230 88235 88241 88247
88281 88287 88292 88298 88304
88338 88343 88349 88355 88360
765
766
767
768
769
88366 88372 88377 88383 88389
88423 88429 88434 88440 88446
88480 88485 88491 88497 88502
88536 88542 88547 88553 88559
88593 88598 88604 88610 88615
88395 88400 88406 88412 88417
88451 88457 88463 88468 88474
88508 88513 88519 88525 88530
88564 88570 88576 88581 88587
88621 88627 88632 88638 88643
770
771
772
773
774
88649 88655 88660 88666 88672
88705 88711 88717 88722 88728
88762 88767 88773 88779 88784
88818 88824 88829 88835 88840
88874 88880 88885 88891 88897
88677 88683 88689 88694 88700
88734 88739 88745 88750 88756
88790 88795 88801 88807 88812
88846 88852 88857 88863 88868
88902 88908 88913 88919 88925
775
776
777
778
779
88930 88936 88941 88947 88953
88986 88992 88997 89003 89009
-89042 89048 89053 89059 89064
89098 89104 89109 89115 89120
89154 89159 89165 89170 89176
88958 88964 88969 88975 88981
89014 89020 89025 89031 89037
89070 89076 89081 89087 89092
89126 89131 89137 89143 89148
89182 89187 89193 89198 89204
780
781
782
783
784
89209 89215 89221 89226 89232
89265 89271 89276 89282 89287
89321' 89326 89332 89337 89343
89376 89382 89387 89393 89398
89432 89437 89443 89448 89454
89237 89243 89248 89254 89260
89293 89298 89304 89310 89315
89348 89354 89360 89365 89371
89404 89409 89415 89421 89426
89459 89465 89470 89476 89481
785
786
787
788
789
89487 89492 89498 89504 89509
89542 89548 89553 89559 89564
89597 89603 89609 89614 89620
89653 89658 89664 89669 89675
89708 89713 89719 89724 89730
89515 89520 89526 89531 89537
89570 89575 89581 89586 89592
89625 89631 89636 89642 8^647
89680 89686 89691 89697 89702
89735 89741 89746 89752 89757
790
791
792
793
794
89763 89768 89774 89779 89785
89 818 89823 89829 89834 89840
89873 89878 89883 89889 89894
89927 89933 89938 89944 89949
89982 89988 89993 89998 90004
89790 89796 89801 89807 89812
89845 89851 89856 89862 89867
89900 89905 89911 89916 89922
89955 89960 89966 89971 89977
90009 90015 90020 90026 90031
795
796
797
798
799
90037 90042 90048 90053 90059
90091 90097 90102 90108 90113
90146 90151 90157 90162 90168
90200 90206 90211 90217 90222
90255 90260 90266 90271 90276
90064 90069 90075 90080 90086
90119 90124 90129 90135 90140
90173 90179 90184 90189 90195
90227 90233 90238 90244 90249
90282 90287 90293 90298 90304
No.
O 1 2 3 4
5 G 7 8 9
750-799
800-849
No.
O
1
2
3
4
5
6
7
8
9
80O
90309
90314
90320
90 325
90331
90 336
90342
90347
90 352
90 35S
801
90363
90369
90374
90380
90385
90390
90 396
90401
90 407
90412
802
90417
90423
90428
90 434
90 439
90445
90 450
90 455
90 461
90466
803
90 472
90 477
90 4 82
904SS
90493
90 499
90504
90 509
90515
90520
804
90526
90531
90536
90542
90547
90553
90558
90563
90569
90574
805
90 580
90585
90590
90596
90601
90607
90 612
90617
90623
90628
806
90 634
90 639
90 644
90650
90 655
90660
90 666
90671
90 677
90 682
807
90687
90693
90698
90703
90709
90714
90720
90 725
90 730
90 736
808
90741
90747
90752
90757
90763
90768
90773
90779
90 7S4
90 7 89
809
90795
90800
90806
90811
90 816
90822
90827
90 832
90 838
90843
810
90849
90854
90 859
90865
90S70
90875
90 SSI
90 886
90891
90 S97
811
90902
90 907
90 913
90918
90924
90 929
90934
90 940
90 945
90 950
812
90956
90961
90966
90 972
90 977
90 982
90 9SS
90 993
90 998
91 004
813
91009
91014
91020
91025
91030
91036
91041
91 046
91 052
91 057
814
91062
91068
91073
91 078
91084
91 OS9
91094
91100
91 105
91 110
815
91 116
91 121
91 126
91 132
91 137
91 142
91 148
91 153
91 158
91 164
816
91 169 .
91 174
91 ISO
91 185
91 190
91 196
91 201
91 206
91212
91217
817
91222
91228
91233
91238
91243
91 249
91 254
91 259
91 265
91270
818
91275
9128L
91 286
91 291
91297
91302
91307
91 312
91318
91323
819
91328
91334
91339
91344
91350
91355
91360
91365
91371
91376
82O
91381
91387
91 392
91397
91403
91 408
91413
91418
91424
91429
821
91434
91440
91445
91450
91 455
91461
91 466
91 471
91477
91 482
822
91487
91492
91 498
91 503
91 508
91514
91519
91 524
91 529
91535
823
91540
91 545
91 551
91556
91 561
91566
91 572
91 577
91 582
91 5S7
824
91 593
91 598
91603
91609
91614
91 619
91624
91 630
91 635
91 640
825
91645
91651
91656
91661
91666
91672
91677
91 682
91687
91693
826
91698
91 703
91 709
91714
91 719
91 724
91 730
91735
91 740
91 745
827
91751
91 756
91 761
91766
91772
91 777
91 782
91 787
91 793
91798
828
91803
91808
91814
91 819
91S24
91S29
91 834
91 840
91 845
91 850
829
91855
91 861
91866
91871
91876
91 882
91SS7
91892
91 S97
91 903
830
91 90S
91913
91918
91924
91929
91934
91939
91944
91 950
91 955
831
91960
91965
91971
91976
91 981
91 986
91 991
91 997
92002
92 007
832
92012
92018
92023
92028
92 033
92 038
92044
92049
92 054
92 059
833
92065
92070
92075
920SO
92 085
92 091
92 096
92101
92 106
92111
834
92117'
92122
92 127
92 132
92137
92143
92 148
92153
92158
92163
835
92169
92174
92179
92 184
921S9
92195
92200
92205
92210
92215
836
92221
92226
92231
92236
92 2-11
92 247
92 252
92 257
92 262
92267
837
92 273
92278
92 283
922SS
92293
92298
92 304
92 309
92314
92319
838
92324
92330
92335
92340
92345
92 350
92355
92361
92366
92371
839
92376
92381
92387
92392
92397
92402
92407
92412
92418
92423
840
92428
92433
92 438
92443
92449
92 454
92 459
92464
92469
92474
841
92480
92485
92 490
92495
92 500
92 505
92 511
92 516
92 521
92526
842
92531
92536
92542
92 547
92 552
92 557
92 562
92567
92572
92 5 78
843
92583
92588
92593
92598
92 603
92 609
92614
92619
92624
92629
844
92634
92639
92645
92650
92655
92660
92665
92670
92675
92681
845
92686
92691
92696
92701
92706
92711
92716
92722
92727
92 732
846
92 737
92742
92747
92 752,
92 758
92763
92 768
92773
92 778
92 783
847
92788
92 793
92799
92804
92809
92 814
92 819
92824
92 829
92 834
848
92840
92 845
92850
92855
92 860
92 865
92 870
92 875
92 SSI
92SS6
849
92891
92896
92901
92906
92911
92916
92921
92927
92 932
92 937
No.
O
1
2
3
4
5
6
7
8
9
800-849
850-899
25
No.
O
1
2
3
4
5
6
7
8
9
850
92942
92947
92952
92 957
92962
92967
92973
92978
92983
92988
851
92 993
92 Si98
93003
93008
93013
93018
93024
93029
93034
93039
852
93044
93049
93 054
93 059
93064
93069
93075
93080
93085
93090
853
93095
93 100
93105
93110
93115
93120
93125
93131
93136
93 141
854
93146
93151
93156
93161
93166
93171
93176
93181
93186
93192
855
93197
93202
93207
93212
93217
93222
93227
93232
93237
93242
856
93247
93252
93 258
93263
93268
93 273
93278
93283
93 288
93293
857
93 298
93303
93308
93313
93 318
93323
93328
93334
93339
93344
858
93 349
93 354
93 359
93364
93369
93374
93379
93384
93389
93394
859
93 399
93404
93409
93414
93420
93425
93430
93435
93440
93445
860
93450
93 455
93460
93 465
93470
93475
93480
93485
93490
93495
861
93 500
93 505
93510
93 515
93 520
93 526
93531
93536
93 541
93546
862
93551
93 556
93 561
93566
93571
93576
93 581
93 586
93591
93596
863
93601
93606
93611
93616
93621
93626
936.31
93636
93641
93646
864
93651
93656
93661
93666
93671
93676
93682
93687
93692
93697
865
93702
93707
93712
93717
93722
93727
93732
93737
93742
93747
866
93 752
93 757
93762
93767
93772
93777
93782
93787
93792
93797
867
93802
93807
93812
93817
93822
93827
93832
93837
93842
93847
868
93852
93857
93862
93867
93872
93877
93882
93887
93892
93897
869
93902
93907
93912
93917
93922
93927
93932
93937
93942
93947
870
93952
93957
93962
93967
93972
93977
93982
93987
93992
93997
871
94002
94007
94012
94017
94022
94027
94032
94037
94042
94047
872
94 052
94057
94 062
94067
94072
94077
94082
94086
94091
94096
873
94101
94106
94111
94116
94121
94126
94131
94136
94141
94146
874
94151
94156
94161
94166
94171
94176
94181
94186
94191
94196
875
94201
94206
94211
94216
94221
94226
94231
94236
94240
94245
876
94 250
94255
94260
94265
94270
94275
94280
94285
94290
94295
877
94 300
94305
94310
94315
94320
94325
94330
94335
94340
94345
878
94 349
94354
94359
94364
94369
94374
94379
94384
94389
94394
879
94399
94404
94409
94414
94419
94424
94429
94433
94438
94443
88O
94448
94453
94 458
94463
94468
94473
94478
94483
94488
94493
881
94498
94 503
94507
94512
94517
94522
94527
94532
94537
94542
882
94 547
94 552
94557
94 562
94 567
94571
94576
94581
94586
94591
883
94596
94601
94606
94611
94616
94621
94626
94630
94 635
94640
884
94645
94650
94655
94660
94665
94670
94675
94680
94685
94689
885
94694
94699
94 704
94709
94714
94719
94724
94729
94734
94738
886
94743
94748
94753
94 758
94763
94768
94773
94778
94783
94787
887
94792
94797
94802
94 807
94812
94817
94822
94827
94832
94836
888
94841
94846
94851
94856
94861
94866
94871
94876
94880
94 885
889
94890
94895
94900
94905
94910
94915
94919
94924
94929
94934
89O
94939
94944
94949
94954
94 959
94963
94968
94973
94978
94983
891
94988
94993
94998
95002
95007
95 012
95 017
95022
95 027
95 032
892
95036
95041
95046
95051
95 056
95061
95 066
95071
95075
95080
893
95085
95090
95095
95 100
95 105
95109
95 114
95119
95 124
95 129
894
95134
95139
95143
95148
95153
95 158
95163
95168
95173
95177
895
95 182
95187
95192
95197
95202
95 207
95211
95216
95221
95 226
896
95 231
95 236
95 240
95 245
95 250
95 255
95260
95265
95 270
95 274
897
95279
95284
95289
95294
95299
95 303
95308
95313
95318
95323
898
95328
95332
95337
95342
95 347
95 352
95357
95361
95 366
95 371
899
95376
95381
95386
95390
95395
95400
95405
95410
95415
95419
No.
O
1
2
3
4
5
6
7
8
9
850-899
26
900-949
No.
O
1
2
3
4
5
6
7
8
9
9OO
95424
95429
95434
95439
95444
95448
95 453
95458
95463
95468
901
95472
95477
95 482
95487
95492
95497
95501
95 406
95511
95516
902
95 521
95 525
95530
95535
95 540
95545
95 550
95 554
95559
95564
903
95569
95574
95578
95 583
95588
95 593
95 598
95602
95607
95612
904
95617
95622
95626
95631
95636
95641
95646
95650
95655
95660
905
95 665
95670
95674
95679
95684
95689
95694
95698
95703
95708
906
95713
95718
95 722
95727
95732
95737
95 742
95746
95751
95 756
907
95761
95 766
95770
95775
95 780
95785
95789
95794
95799
95804
908
95809
95813
95818
95823
95828
95832
95837
95 842
95 847
95 852
909
95856
95861
95866
95871
95875
95880
95885
95890
95895
95899
910
95904
95909
95 914
95918
95923
95928
95933
95 938
95942
95947
911
95952
95957
95961
95966
95 971
95976
95980
95985
95990
95995
912
95999
96004
96009
96014
96019
96023
96028
96033
96 038
96042
913
96 047
96052
96057
96061
96 066
96071
96076
96080
96085
96090
914
96095
96099
96104
96109
96114
96118
96123
96128
96133
96137
915
96142
96147
96152
96 156
96161
96166
96171
96175
96180
96185
916
96 190
96194
96 199
96204
96209
96213
96218
96223
96227
96232
917
96237
96242
96246
96251
96 256
96 261
96265
96270
96275
96280
918
96284
96289
96294
96298
96303
96308
96313
96317
96322
96327
919
96332
96336
96341
96346
96350
96355
96360
96365
96369
96374
92O
96379
96384
96388
96393
96398
96402
96407
96412
96417
96421
921
96426
96431
96435
96440
96445
96450
96454
96459
96464
96468
922
96473
96478
96483
96487
96492
96497
96501
96506
96511
96515
923
96520
96 525
96530
96 534
96539
96 544
96 548
96 553
96558
96562
924
96567
96572
96577
96581
96586
96591
96595
96600
96605
96 609
925
96614
96619
96624
96628
96633
96638
96642
96647
96652
96656
926
96661
96666
96670
96675
96680
96685
96689
96694
96699
96703
927
96708
96713
96717
96722
96727
96731
'96736
96741
96745
96750
928
96755
96759
96764
96 769
96774
96778
96783
96788
96792
96797
929
96802
96806
96811
96 816
96820
96825
96830
96834
96839
96844
93O
96848
96853
96858
96862
96867
96872
96876
96881
96886
96890
931
96895
96900
96904
96909
96914
96918
96923
96928
96 932
96937
932
96942
96946
96951
96956
96 960
96965
96 970
96974
96979
96984
933
96988
96993
96997
97002
97007
97011
97016
97021
97025
97030
934
97035
97039
97044
97049
97053
97058
97063
97067
97072
97077
935
97081
97086
97090
97 095
97100
97104
97109
97114
97118
97123
936
97128
97132
97137
97142
97146
97151
97 155
97160
97165
97169
937
97174
97179
97183
97188
97192
97197
97202
97206
97211
97216
938
97220
97225
97230
97234
97239
97243
97248
97253
97257
97262
939
97267
97271
97276
97280
97285
97290
97294
97299
97304
97308
940
97313
97317
97322
97327
97331
97336
97340
97345
97350
97354
941
97359
97364
97368
97373
97377
97382
97387
97391
97396
97400
942
97405
97410
97414
97419
97424
97428
97 433
97437
97442
97 447
943
97451
97 456
97 460
97465
97470
97474
97479
97483
97488
97493
944
97497
97 502
97506
97511
97516
97 520
97 525
97529
97534
97539
945
97543
97548
97 552
97557
97562
97566
9757,1
97575
97580
97585
946
97 589
97594
97598
97603
97607
97612
97617
97621
97626
97630
947
97 635
97640
97 644
97649
97653
97658
97663
97667
97672
97676
948
97681
97685
97690
97695
. 97 699
97704
97708
97713
97717
97722
949
97727
97731
97736
97740
97745
97749
97 754
97759
97763
97768
No.
O
1
2
3
4
5
6
7
8
9
900-949
950-1000
27
No.
o
1
2
3
4
5
6
7
8
9
950
97772
97777
97782
97 786
97791
97795
97800
97804
97 809
97813
951
97818
97823
97827
97 832
97836
97841
97 845
97850
97855
97859
952
97864
97868
97873
97877
97882
97886
97891
97896
97900
97905
953
97909
97 914
97918
97923
97928
97932
97937
97941
97946
97950
954
97955.
97959
97964
97968
97973
97978
97982
97987
97991
97996
955
98000
98005
98009
98014
98019
98023
98028
98032
98037
98041
956
98046
98050
98055
98059
98064
98068
98073
98078
98082
98087
957
98091
98096
98100
98105
98109
98114
98118
98123
98127
98132
958
98137
98141
98146
98150
98155
98 159
98164
98 168
98173
98177
959
98182
98186
98191
98195
98200
98204
98209
98214
98218
98223
96O
98227
98 232
98 236
98241
98 245
98250
98254
98259
98263
98268
961 98272
98277
98281
98286
98290
98295
98299
98304
98308
98313
962 98318
98322
98327
98331
98336
98340
98345
98349
98354
98358
963 98363
98367
98372
98376
98 381
98385
98390
98394
98399
98403
964
98408
98412
98417
98421
98426
98430
98435
98439
98444
98448
965
98453
98457
98462
98466
98471
98475
98480
98484
98489
98493
966
98498
98502
98507
98511
98516
98520
98525
98529
98534
98538
967
98 543
98547
98552
98556
98561
98565
98570
98574
98579
98 583
968
98588
98592
98597
98601
98605
98610
98614
98 619
98623
98628
969
98632
98637
98641
98646
98650
98 655
98659
98664
98668
98673
97O
98677
98682
98686
98691
98695
98700
98704
98709
98713
98717
971
98722
98726
98731
98735
98740
98744
98749
98753
98758
98762
972
98767
98771
98776
98780
98 784 .
98789
98793
98798
98802
98807
973
98811
98816
98820
98825
98829
98834
98838
98843
98847
98851
974
98856
98860
98865
98869
98874
98878
98883
98887
98892
98896
975
98900
98905
98909
98914
98918
98923
98927
98932
98936
98941
976
98945
98949
98954
98958
98963
98967
98972
98976
98981
98985
977
98989
98 994
98998
99003
99007
99012
99016
99021
99025
99029
978
99034
99038
99043
99047
99052
99056
99061
99065
99 069
99074
979
99078
99083
99087
99092
99096
99100
99105
99 109
99114
99118
980
99123
99127
99131
99136
99 140
99 145
99149
99154
99158
99162
981
99167
99171
99176
99 180
99185
99189
99193
99198
99202
99207
982
99211
99216
99220
99224
99 229
99233
99238
99242
99247
99251
983
99255
99260
99264
99269
99273
99277
99282
99286
99291
99 295
984
99300
99304
99308
99313
99317
99322
99326
99330
99335
99339
985
99 344
99348
99352
99357
99361
99366
99370
99374
99379
99383
986
99 388
99392
99396
99401
99405
99410
99414
99419
99423
99427
987
99432
99436
99441
99445
99449
99454
99458
99463
99467
99471
988 99476
99480
99 484
99489
99493
99498
99 502
99 506
99511
99515
989 99520
99524
99528
99533
99537
99542
99546
99550
99 555
99559
990
99 564
99568
99 572
99 577
99 581
99 585
99590
99594
99599
99603
991
99607
99612
99616
99621
99625
99629
99 634
99638
99642
99647
992
99651
99656
99660
99664
99 669
99673
99677
99682
99686
99691
993
99695
99699
99704
99708
99712
99717
99721
99726
99730
99734
994 99739
99743
99747
99752
99756
99760
99765
99769
99774
99778
995
99782
99787
99791
99795
99800
99804
99808
99813
99817
99822
996 99826
99 830
99835
99839
99843
99848
99 852
99 856
99861
99865
997 99870
99874
99878
99883
99887
99891
99896
99900
99904
99909
998 99913
99917
99922
99926
99930
99935
99939
99944
99948
99952
999
99957
99961
99965
99970
99974
99978
99983
99987
99991
99996
1OOO 00000
00004
00009
00013
00017
00022
00026
00030
00035
00039
No.
1
2
3
4
5
6
7
8
9
950-1000
TABLE II -USEFUL CONSTANTS AND THEIR LOGARITHMS
LOG
Circumference of the Circle in Degrees 360
2. 55 630 250
Circumference of the Circle in Minutes = 21 600
4.3344537.S
Circumference of the Circle in Seconds 1296000
6.11 260500
If the radius = 1, the semi-circumference is
TT 3. 14 159 265 358 979 323 846 764 338 378
0.49714987
ALSO
LOG
7r 2 = 9.86960440
0.99429975
27r= 6.28318531
0.79817987
= 0.10132118
9.00570025 - 10
47r = 12.56637061
1.09920986
7T-
= 1.57079633
0.19611988
VV = 1. 77 245 385
0.24857494
2
= 1.04719755
0.02002862
JL =0.56418958
9. 75 142 506 - 10
3
VTT
~= 4.18879020
0. 62 208 861
/I
3
\/- = 0.97 720 502
9.98998569- 10
= 0.78539816
9.89508988- 10
4
/4
\/-= 1.12837917
0. 05 245 506
= 0.52359878
9.71899862- 10
\7T
6
i
-v/TT = 1.46459189
0.16571662
-i.= 0.31830989
9. 50 285 013 - 10
7T
1
-^ = 0.68278406
9.83428338-10
= 0. 15 915 494
9.20182013-10
VTT
2-jr
= 0.95492966
9.97997138- 10
vV = 2. 14 502 940
0.33143325
7T
8/~5~"
^-= 1.27323954
0.10491012
V /A. = 0.62 035 049
\4w
9.79263713 - 10
7T
= 0. 23 873 241
9. 37 791 139 - 10
\ 8 / = 0.80 599 598
9.90633287- 10
4?r
\ 6
Angle 0, whose arc is equal to the radius r, is
180
in degrees, = =57.29577951
7T
1.75812263
10800
in minutes, 0' 3437 74677'
3.53627388
7T
648000
in seconds 0" 206264806"
5.31 442 513
Angle 2 0, whose arc is equal to twice the radius, 2 r, is
in degrees, 20= = 114. 59 155 903
2. 05 915 263
7T
in minutes, 2 0' - 21 60 -687549354'
3.837303*88
7T
in seconds 20" 12 96000 412529612"
5.61545513
7T
If the radius r = 1, the length of the arc is :
for 1 degree = -~- = = 0. 01 745 329 .
8.24187737-10
J 180
for 1 minute = -L = ?_ = 0. 00 029 089 ,
6.46372612- 10
0' 10 800
for 1 second = -L- = - = 0. 00 000 485
4. 68 557 487 ~ 10
0" 648000
for & degree = -^ = ^ =0.00872665 ....
7.94084737- 10
for 2 minute -
1 TT c
1.00014544 ....
6.16269612 10
2 0' 21 600
for .> second -
1 r -
.00000242 ....
4.38454487 10
20" 1296000
Sin 1", when the radius r - 1, is . . .=0.00000485 . . . .
4.68557487- 10
TABLE III
LOGARITHMS
OF THE
TRIGONOMETRIC FUNCTIONS
From 0' to 3', and from 89 57' to 90, for every second
From to 2, and from 88 to 90, for every ten seconds
From 1 to 89, for every minute
NOTE. The characteristic of every logarithm in the following table is too
large by 10. Therefore, 10 should be written after every logarithm.
L sin and L tan U I< sin and L tan
//.
1'
6.46 373
6.47 090
6.47 797
6.48 492
6.49 175
2'
6.76476
6.76 836
6.77 193
6.77 548
6.77 900
//
//
O'
6.16270
6.17694
6.19072
6.20 409
6.21 705
1'
6.63 982
6.64 462
6.64 936
6.65 406
6.65 870
2'
6.86 167
6.86455
6.86 742
6.87 027
6.87 310
//
O
1
2
3
4
6O
59
58
57
56
30
31
32
33
34
30
29
28
27
26
4.68 557
4.98 660
5.16270
5.28 763
5
6
7
8
9
5.38 454
5.46373
5.53 067
5.58 866
5.63 982
6.49 849
6.50512
6.51 165
6.51 808
6.52 442
6.78 248
6.78 595
6.78 938
6.79278
6.79616
55
54
53
52
51
35
36
37
38
39
6.22 964
6.24 188
6.25 378
6.26 536
6.27 664
6.66 330
6.66 785
6.67 235
6.67 680
6.68 121
6.87 591
6.87 870
6.88 147
6.88 423
6.88 697
25
24
23
22
21
10
11
12
13
14
5.68557
5.72 697
5.76476
5.79952
5.83 170
6.53 067
6 53 683
6.54 291
6.54 890
6.55 481
6.79952
6.80 285
6.80615
6.80 943
6.81 268
50
49
48
47
46
40
41
42
43
44
6.28 763
6.29 836
6.30 882
6.31 904
6.32 903
6.68 557
6.68 990
6.69 418
6.69 841
6.70 261
6.88 969
6.89 240
6.89 509
6.89 776
6.90 042
2O
19
18
17
16
15
16
17
18
19
5.86 167
5.88969
5.91 602
5.94085
5.96433
6.56064
6.56 639
6.57 207
6.57 767
6.58 320
6.81 591
6.81 911
6.82 230
6.82 545
6.82 859
45
44
43
42
41
45
46
47
48
49
6.33 879
6.34 833
6.35 767
6.36 682
6.37 577
6.70676
6.71 088
6.71 496
6.71 900
6.72 300
6.90306
6.90 568
6.90 829
6.91 088
6.91346
15
14
13
12
11
20
21
22
23
24
5.98660
6.00 779
6.02 800
6.04 730
6.06579
6.58 866
6.59 406
6.59 939
6.60 465
6.60 985
6.83 170
6.83479
6.83 786
6.84 091
6.84394
40
39
38
37
66
50
51
52
53
54
6.38 454
6.39315
6.40 158
6.40 985
6.41 797
6.72 697
6.73 090
6.73 479
6.73 865
6.74 248
6.91 602
6.91 857
6.92 110
6.92 362
6.92 612
1O
9
8
7
6
25
26
27
28
29
6.08351
6.10055
6.11694
6.13 273
6.14 797
6.61 499
6.62 007
6.62 509
6.63 006
6.63 496
6.84 694
6.84 993
6.85 289
6.85 584
6.85 876
35
34
33
32
31
55
56
57
58
59
6.42 594
6.43 376
6.44 145
6.44 900
6.45 643
6.74 627
6.75 003
6.75 376
6.75 746
6.76 112
6.92 861
6.93 109
6.93 355
6.93 599
6.93 843
5
4
3
2
1
30
6.16270
59'
6.63 982
58'
6.86 167
57'
30
6O
6.46373
59'
6.76 476
58'
6.94 085
57'
O
"
"
"
"
L cos and L cot
89'
L cos and L cot
29
30
/ //
L sin L tan L cos
// /
/ //
L sin L tan I cos
// /
On
10 00000
fiO
1O
746^7^ 7 46 37"? 1000000
fft
V
10
5.68557 5.68557 10.00000
V/ \J VJ
50
J_ Vr \J
10
/ . I O O t O / . i O O / O \J.\J\J \J\J\J
7.47090 7.47091 10.00000
U Ovr
50
20
5.98660 5.98660 10.00000
40
20
7.47797 7.47797 10.00000
40
30
6.16270 6.16270 10.00000
30
30
7.48491 7.48492 10.00000
30
40
6.28763 6.28763 10.00000
20
40
7.49175 7.49175 10.00000
20
50
6.38454 6.38454 10.00000
10
50
7.49 849 7.49849 10.00000
10
1
6.46373 6.46373 10.00000
59
11
7.50512 7.50512 10.00000
049
10
6.53067 6.53067 10.00000
50
10
7.51165 7.51165 10.00000
50
20
6.58866 6.58866 10.00000
40
20
7.51808 7.51809 10.00000
40
30
6.63982 6.63982 10.00000
30
30
7.52442 7.52443 10.00000
30
40
6.68557 6.68557 10.00000
20
40
7.53067 7.53067 10.00000
20
50
6.72697 6.72697 10.00000
10
50
7.53683 7.53683 10.00000
10
2
6.76476 6.76476 10.00000
58
12
7.54291 7.54291 10.00000
048
10
6.79952 6.79952 10.00000
50
10
7.54890 7.54890 10.00000
50
20
6.83170 6.83170 10.00000
40
20
7.55481 7.55481 10.00000
40
30
6.86167 6.86167 10.00000
30
30
7.56064 7.56064 10.00000
30
40
6.88969 6.88969 10.00000
20
40
7.56639 7.56639 10.00000
20
50
6.91602 6.91602 10.00000
10
50
7.57206 7.57206 10.00000
10
3
6.94085 6.94085 10.00000
57
13
7.57767 7.57767 10.00000
047
10
6.96433 6.96433 10.00000
50
10
7.58320 7.58320 10.00000
50
20
6.98660 6.98661 10.00000
40
20
7.58866 7.58867 10.00000
40
30
7.00779 7.00779 10.00000
30
30
7.59406 7.59406 10.00000
30
40
7.02800 7.02800 10.00000
20
40
7.59939 7.59939 10.00000
20
50
7.04730 7.04730 10.00000
10
50
7.60465 7.60466 10.00000
10
4
7.06579 7.06579 10.00000
56
14
7.60985 7.60986 10.00000
046
10
7.08351 7.08352 10.00000
50
10
7.61499 7.61500 10.00000
50
20
7.10055 7.10055 10.00000
40
20
7.62007 7.62008 10.00000
40
30
7.11694 7.11694 10.00000
30
30
7.62509 7.62510 10.00000
30
40
7.13273 7.13273 10.00000
20
40
7.63006 7.63006 10.00000
20
50
7.14797 7.14797 10.00000
10
50
7.63496 7.63497 10.00000
10
5
7.16270 7.16270 10.00000
55
15
7.63982 7.63982 10.00000
045
10
7.17694 7.17694 10.00000
50
10
7.64461 7.64462 10.00000
50
20
7.19072 7.19073 10.00000
40
20
7.64936 7.64937 10.00000
40
30
7.20409 7.20409 10.00000
30
30
7.65406 7.65406 10.00000
30
40
7.21705 7.21705 10.00000
20
40
7.65870 7.65871 10.00000
20
50
7.22964 7.22964 10.00000
10
50
7.66330 7.66330 10.00000
10
6
7.24188 7.24188 10.00000
54
16
7.66784 7.66785 10.00000
044
10
7.25378 7.25378 10.00000
50
10
7.67235 7.67235 10.00000
50
20
7.26536 7.26536 10.00000
40
20
7.67680 7.67680 10.00000
40
30
7.27664 7.27664 10.00000
30
30
7.68121 7.68121 10.00000
30
40
7.28763 7.28764 10.00000
20
40
7.68557 7.68558 9.99999
20
50
7.29836 7.29836 10.00000
10
50
7.68989 7.68990 9.99999
10
7
7.30882 7.30882 10.00000
53
17
7.69417 7.69418 9.99999
043
10
7.31904 7.31904 10.00000
50
10
7.69841 7.69842 9.99999
50
20
7.32903 7.32903 10.00000
40
20
7.70261 7.70261 9.99999
40
30
7.33879 7.33879 10.00000
30
30
7.70676 7.70677 9.99999
30
40
7.34833 7.34833 10.00000
20
40
7.71088 7.71088 9.99999
20
50
7.35767 7.35767 10.00000
10
50
7.71496 7.71496 9.99999
10
8
7.36682 7.36682 10.00000
52
18
7.71900 7.71900 9.99999
042
10
7.37577 7.37577 10.00000
50
10
7.72300 7.72301 9.99999
50
20
7.38454 7.38455 10.00000
40
20
7.72697 7.72697 9.99999
40
30
7.39314 7.39315 10.00000
30
30
7.73090 7.73090 9.99999
30
40
7.40158 7.40158 10.00000
20
40
7.73479 7.73480 9.99999
20
50
7.40985 7.40985 10.00000
10
50
7.73865 7.73866 9.99999
10
9
7.41797 7.41797 1000000
51
19
7.74248 7.74248 9.99999
041
10
7.42594 7.42594 10.00000
50
10
7.74627 7.74628 9.99999
50
20
7.43376 7.43376 10.00000
40
20
7.75003 7.75004 9.99999
40
30
7.44145 7.44145 10.00000
30
30
7.75376 7.75377 9.99999
30
40
7.44900 7.44900 10.00000
20
40
7.75745 7.75746 9.99999
20
50
7.45643 7.45643 10.00000
10
50
7.76112 7.76113 9.99999
10
10
7.46373 7.46373 10.00000
5O
20
7.76475 7.76476 9.99999
04O
/ //
L cos L cot L sin
// /
/ //
L cos L cot L sin
// /
89
81
/ //
L sin L tan L cos
// /
/ //
L sin L tan L cos
// /
2O
7.76475 7.76476 9.99999
4O
3O
7.94084 7.94086 9.99998
03O
10
7.76836 7.76837 9.99999
50
10
7.94325 7.94326 9.99998
50
20
7.77 193 7.77 194 9.99 999
40
20
7.94564 7.94566 9.99998
40
30
7.77548 7.77549 9.99999
30
30
7.94802 7.94804 9.99998
30
40
7.77899 7.77900 9.99999
20
40
7.95039 7.95040 9.99998
20
50
7.78248 7.78249 9.99999
10
50
7.95 274 7.95 276 9.99 998
10
21
7.78594 7.78595 9.99999
39
31
7.95508 7.95510 9.99998
029
10
7.78938 7.78938 9.99999
50
10
7.95 741 7.95 743 9.99 998
50
20
7.79278 7.79279 9.99999
40
20
7.95973 7.95974 9.99998
40
30
7.79616 7.79617 9.99999
30
30
7.96203 7.96205 9.99998
30
40
7.79952 7.79952 9.99999
20
40
7.96432 7.96434 9.99998
20
50
7.80284 7.80285 9.99999
10
50
/.96660 7.96662 9.99998
10
22
7.80615 7.80615 9.99999
38
32
7.96887 7.96889 9.99998
028
10
7.80942 7.80943 9.99999
50
10
7.97113 7.97114 9.99998
50
20
7.81 268 7.81 269 9.99 999
40
20
7.97337 7.97339 9.99998
40
30
7.81591 7.81591 9.99999
30
30
7.97560 7.97562 9.99998
30
40
7.81911 7.81912 9.99999
20
40
7.97782 7.97784 9.99998
20
50
7.82229 7.82230 9.99999
10
50
7.98003 7.98005 9.99998
10
23
7.82545 7.82546 9.99999
37
33
7.98223 7.98225 9.99998
027
10
7.82859 7.82860 9.99999
50
10
7.98442 7.98444 9.99998
50
20
7.83170 7.83171 9.99999
40
20
7.98660 7.98662 9.99998
40
30
7.83479 7.83480 9.99999
30
30
7.98876 7.98878 9.99998
30
40
7.83786 7.83787 9.99999
20
40
7.99092 7.99094 9.99998
20
50
7.84091 7.84092 9.99999
10
50
7.99306 7.99308 9.99998
10
24
7.84393 .7.84394 9.99999
36
34
7.99520 7.99522 9.99998
026
10
7.84694 7.84695 9.99999
50
10
7.99732 7.99734 9.99998
50
20
7.84992 7.84993 9.99999
40
20
7.99943 7.99946 9.99998
40
30
7.85289 7.85290 9.99999
30
30
8.00154 8.00156 9.99998
30
40
7.85583 7.85584 9.99999
20
40
8.00363 8.00365 9.99998
20
50
7.85876 7.85877 9.99999
10
50
8.00571 8.00574 9.99998
10
25
7.86166 7.86167 9.99999
35
35
8.00779 8.00781 9.99998
025
10
7.86455 7.86456 9.99999
50
10
8.00985 8.00987 9.99998
50
20
7.86741 7.86743 9.99999
40
20
8.01 190 8.01 193 9.99 998
40
30
7.87026 7.87027 9.99999
30
30
8.01395 8.01397 9.99998
30
40
7.87309 787310 9.99999
20
40
8.01 598 8.01 600 9.99 998
20
50
7.87590 7.87591 9.99999
10
50
8.01801 S.01'803 9.99998
10
26
7.87870 7.87871 9.99999
34
36
8.02002 8.02004 9.99998
024
10
7.88147 7.88148 9.99999
50
10
8.02203 8.02205 9.99998
50
20
7.88423 7.88424 9.99999
40
20
8.02402 8.02405 9.99998
40
30
7.88697 7.88698 9.99999
30
30
8.02601 8.02604 9.99998
30
40
7.88969 7.88970 9.99999
20
40
8.02799 8.02801 9.99998
20
50
7.89240 7.89241 9.99999
10
50
8.02996 8.02998 9.99998
10
27
7.89509 7.89510 9.99999
33
37
8.03 192 8.03 194 9.99 997
023
10
7.89776 7.89777 9.99999
50
10
8.03387 8.03390 9.99997
50
20
7.90041 7.90043 9.99999
40
20
8.03581 8.03584 9.99997
40
30
7.90305 7.90307 9.99999
30
30
8 03 775 8.03 777 9.99 997
30
40
7.90568 7.90569 9.99999
20
40
8.03967 8.03970 9.99997
20
50
7.90829 7.90830 9.99999
10
50
8.04 159 8.04 162 9.99 997
10
28
7.91088 7.91089 9.99999
32
38
8.04350 8.04353 9.99997
022
10
7.91346 7.91347 9.99999
50
10
804540 8.04543 9.99997
50
20
7.91602 7.91603 9.99999
40
20
8.04729 8.04732 9.99997
40
30
7.91857 7.91858 9.99999
30
30
8.04918 8.04921 9.99997
30
40
7.92110 7.92111 9.99998
20
40
8.05 105 8.05 108 9.99 997
20
50
7.92362 7.92363 9.99998
10
50
8.05292 8.05295 9.99997
1(5
29
7.92612 7.92613 9.99998
31
39
8.05 478 8.05 481 9.99 997
021
10
7.92861 7.92862 9.99998
50
10
8.05663 8.05666 9.99997
50
20
7.93108 7.93110 9.99998
40
20
8.05848 8.05851 9.99997
40
30
7.93354 7.93356 9.99998
30
30
8.06031 8.06034 9.99997
30
40
7.93599 7.93601 9.99998
20
40
8.06214 8.06217 9.99997
20
50
7.93842 7.93844 9.99998
10
50
8.06396 8.06399 9.99997
10
3O
7.94084 7.94086 9.99998
3O
4O
8.06578 8.06581 9.99997
02O
/ //
L cos L cot L sin
// /
/ //
L cos L cot L sin
// /
89
/ //
L sin L tan L cos
// /
/ //
L sin L tan L cos
// /
4O
8.06578 8.06581 9.99997
2O
5O
8.16268 8.16273 9.99995
01O
10
806758 8.06761 9.99997
50
10
8.16413 8.16417 9.99995
50
20
8.06938 8.06941 9.99997
40
20
8.16557 8.16561 9.99995
40
30
8.07117 8.07120 9.99997
30
30
8.16700 8.16705 9.99995
30
40
8.07295 8.07298 9.99997
20
40
8.16843 8.168-18 9.99995
20
50
8.07473 8.07476 9.99997
10
50
8.16986 8.16991 9.99995
10
41
8.07650 8.07653 9.99997
19
51
8.17128 8.17133 9.99995
9
10
8.07826 8.07829 9.99997
50
10
8.17270 8.17275 9.99995
50
20
8.08002 8.08005 9.99997
40
20
8.17411 8.17416 9.99995
40
30
8.08176 8.08180 9.99997
30
30
8.17552 8.17557 9.99995
30
40
8.08350 8.08354 9.99997
20
40
8.17692 8.17697 9.99995
20
50
8.08524 8.08527 9.99997
10
50
8.17832 8.17837 9.99995
10
42
8.08696 8.08700 9.99997
18
52
8.17971 8.17976 9.99995
8
10
8.08868 8.08872 9.99997
50
10
8.18110 8.18115 9.96995
50
20
8.09040 8.09043 9.99997
40
20
8.18249 8.18254 9.99995
40
30
8.09210 8.09214 9.99997
30
30
8.18387 8.18392 9.99995
30
40
8.09 380 8.09384 9.99997
20
40
8.18524 8.18530 999995
20
50
8.09550 8.09553 9.99997
10
50
8.18662 8.18667 9.99995
10
43
8.09718 8.09722 9.99997
17
53
8.18798 8.18804 9.99995
7
10
8.09886 8.09890 9.99997
50
10
8.18935 8.18940 9.99995
50
20
8.10054 8.10057 9.99997
40
20
8.19071 8.19076 9.99995
40
30
8.10220 8.10224 9.99997
30
30
8.19206 8.19211 9.99995
30
40
8.10386 8.10390 9.99996
20
40
8.19341 8.19347 9.99995
20
50
8.10552 8.10555 9.99996
10
50
8.19476 8.19481 9.99995
10
44
8.10717 8.10720 9.99996
16
54
8.19610 8.19616 9.99995
6
10
8.10881 8.10884 9.99996
50
10
8.19744 8.19749 9.99995
50
20
8.11044 8.11018 9.99996
40
20
8.19877 8.19883 9,99995
40
30
811207 8.11 211 9.99996
30
30
8.20010 8.20016 9-99995
30
40
8.11370 8.11373 9.99996
20
40
8.20143 8.20149 9.99995
20
50
8.11531 8.11535 9.99996
10
50
8.20275 8.20281 9.99994
10
45
8.11693 8.11696 9.99996
15
55
8.20407 8.20413 9.99994
5
10
8.11853 8.11857 9.99996
50
10
8.20 538 8.20 544 9.99 994
50
20
8.12013 8.12017 9.99996
40
20
8.20669 820675 9.99994
40
30
8.12172 8.12 176 9.99996
30
30
8.20800 8.20806 9.99994
30
40
8.12331 812335 9.99996
20
40
8.20930 8.20936 9.99994
20
50
8.12489 8.12*493 9.99996
10
50
8.21060 8.21066 9.99994
10
46
8.12647 8.12651 9.99996
14
56
8.21189 8.21 195 9.99994
4
10
8.12804 8.12808 9.99996
50
10
8.21319 8.21324 9.09994
50
20
8.12961 8.12965 9.99996
40
20
8.21 447 8.21 453 9.99 994
40
30
8.13117 8.13 121 9.99996
30
30
8.21 576 8.21581 9.99994
30
40
8.13272 8.13276 9.99996
20
40
8.21 703 8.21 709 9.99 994
20
50
8.13427 8.13431 9.99996
10
50
8.21831 8.21 837 9.99994
10
47
8.13581 8.13585 9.99996
13
57
8.21958 8.21964 9.99994
3
10
8.13735 8.13 739 9.99996
50
10
8.22085 8.22091 9.99994
50
20
8.13888 8.13892 9.99996
40
20
8.22211 8.22217 9.99994
40
30
8.14041 8.14045 9.99996
30
30
822337 8.22343 9.99994
30
40
8.14193 8.14197 9.99996
20
40
822463 8.22-169 9.99994
20
50
8.14344 8.143-18 9.99996
10
50
8.22588 8.22595 9.99994
10
48
8.14495 8.14500 9.99996
12
58
8.22713 8.22720 9.99994
2
10
8.14646 8.14650 9.99996
50
10
8.22838 8.22844 9.99994
50
20
8.14796 8.14800 9.99996
40
20
8.22962 8.22968 9.99994
40
30
814945 8.14950 9.99996
30
30
8.23086 8.23092 9.99994
30
40
8.15094 8.15099 9.99996
20
40
8.23210 8.23216 9.99994
20
50
8.15243 8.15247 9.99996
10
50
8.23333 8.23339 9.99994
10
49
8.15391 8.15395 9.99996
11
59
8.23456 8.23462 999994
1
10
8.15538 8.15543 9.99996
50
10
8.23578 8.23585 9.99994
50
20
8.15685 8.15690 9.99996
40
20
8.23 700 8.23 707 9.99 994
40
30
8.15832 8.15836 9.99995
30
30
8.23822 8.23829 9.99993
30
40
8.15978 8.15982 9.99995
20
40
8.23944 8.23950 9.99993
20
50
8.16123 8.16128 9.99995
10
50
8.24065 8.24071 9.99993
10
50
8.16268 8.16273 9.99995
1C
6O
8.24186 8.24192 9.99993
O
/ //
L cos L cot L sin
// /
/ //
L cos L cot L sin
// /
89'
33
/ //
L sin L tan L cos
// /
/ //
L sin L tan L cos
// /
O
8.24 186 8.24 192 9.99 993
6O
1O
8.30879 8.30888 9.99991
05O
10
8.24306 8.24313 9.99993
50
10
8.30983 8.30992 9.99991
50
20
8.24426 8.24433 9.99993
40
20
8.31086 8.31095 9.99991
40
30
824546 8.24553 9.99993
30
30
8.31188 8.31198 9.99991
30
40
8.24665 824672 9.99993
20
40
8.31291 8.31300 9.99991
20
50
8.24785 8.24791 9.99993
10
50
8.31393 8.31403 9.99991
10
1
8.24903 8.24910 9.99993
59
11
8.31495 8.31505 9.99991
049
10
8.25022 8.25029 9.99993
50
10
8.31597 8.31606 9.99991
50
20
8.25 140 8.25 147 9.99 993
40
20
8.31699 8.31 708 9.99991
40
30
8.25258 8.25265 9.99993
30
30
8.31800 8.31809 9.99991
30
40
.8.25375 8.25382 9.99993
20
40
8.31901 8.31911 9.99991
20
50
8.25493 8.25500 9.99993
10
50
8.32002 8.32012 9.99991
10
2
8.25609 8.25616 9.99993
58
12
8.32103 8.32112 9.99990
048
10
8.25 726 8.25 733 9.99 993
50
10
8.32203 8.32213 9.99990
50
20
8.25842 8.25849 9.99993
40
20
8.32303 8.32313 9.99990
40
30
8.25958 8.25965 9.99993
30
30
8.32403 8.32413 9.99990
30
40
8.26074 8.26081 9.99993
20
40
8.32503 8.32513 9.99990
20
50
8.26189 8.26196 9.99993
10
50
8.32602 8.32612 9.99990
10
3
8.26304 8.26312 9.99993
57
13
8.32702 8.32711 9.99990
047
10
8.26419 826426 9.99993
50
10
8.32801 8.32811 9.99990
50
20
8.26553 8.26541 9.99993
40
20
8.32899 8.32909 9.99990
40
30
8.26648 8.26655 9.99993
30
30
8.32998 8.33008 9.99990
30
40
8.26761 8.26769 9.99993
20
40
8.33096 8.33106 9.99990
20
50
8.26875 8.26882 9.99993
10
50
8.33195 8.33205 9.99990
10
4
8.26988 8.26996 9.99992
56
14
8.33292 8.33302 9.99990
046
10
8.27101 8.27109 9.99992
50
10
8.33390 8.33400 9.99990
50
20
8.27214 8.27221 9.99992
40
20
8.33488 8.33498 9.99990
40
30
8.27326 8.27334 9.99992
30
30
8.33585 8.33595 9.99990
30
40
8.27438 8.27446 9.99992
20
40
8.33682 8.33692 9.99990
20
50
8.27550 8.27558 9.99992
10
50
8.33779 8.33789 9.99990
10
5
8.27661 8.27669 9.99992
55
15
8.33875 8.33886 9.99990
045
10
8.27773 8.27780 9.99992
50
10
8.33972 8.33982 9.99990
50
20
8.27883 8.27891 9.99992
40
20
8.34086 8.34078 9.99990
40
30
8.27994 8.28002 9.99992
30
30
8.34164 8.34174 9.99990
30
40
8.28104 828112 9.99992
20
40
8.34260 8.34270 9.99989
20
50
8.28215 8.28223 9.99992
10
50
8.34355 8.34366 9.99989
10
6
8.28324 8.28332 9.99992
54
16
8.34450 8.34461 9.99989
044
10
8.28434 8.28442 9.99992
50
10
8.34546 8.34556 9.99989
50
20
8.28543 8.28551 9.99992
40
20
8.34640 8.34651 9.99989
40
30
8.28652 8.28660 9.99992
30
30
8.34735 8.34746 9.99989
30
40
8.28761 8.28769 9.99992
20
40
8.34830 8.34840 9.99989
20
50
8.2S869 8.28877 9.99992
10
50
8.34924 8.34935 9.99989
10
7
8.28977 8.28986 9.99992
53
17
8.35 018 8.35 029 9.99 989
043
10
8.29085 8.29094 9.99992
50
10
8.35112 8.35 123 9.99989
50
20
8.29093 8.29201 9.99992
40
20
8.35206 8.35217 9.99989
40
30
8.29300 8.29309 9.99992
30
30
8.35 299 8.35 310 9.99 989
30
40
8.29407 8.29416 9.99992
20
40
8.35 392 8.35 403 9.99 989
20
50
8.29514 8.29523 9.99992
10
50
8.35 485 8.35 497 9.99 989
10
8
8.29621 829629 9.99992
52
18
8.35578 8.35590 9.99989
042
10
8.29727 8.29736 9.99991
50
10
8.35671 8.35682 9.99989
50
20
8.29833 8.29842 9.99991
40
20
8.35764 8.35 775 9.99989
40
30
8.29939 8.29947 9.99991
30
30
8.35 856 8.35 867 9.99 989
30
40
8.30044 8.30053 9.99991
20
40
8.35948 8.35959 9.99989
20
-50
8.30150 8.30158 9.99991
10
50
8.36040 8.36051 9.99989
10
9
8.30255 8.30263 9.99991
51
19
8.36131 8.36143 9.99989
041
10
8.30359 8.30368 9.99991
50
10
8.36223 8.36235 9.99988
50
20
8.30464 8.30473 9.99991
40
20
8.36314 8.36326 9.99988
40
30
8.30568 8.30577 9.99991
30
30
8.36405 8.36417 9.99988
30
40
8.30672 8.30681 9.99991
20
40
836496 8.36508 9.99988
20
50
8.30776 8.30785 9.99991
50
8.36587 8.36599 9.99988
10
10
8.30879 8.30888 9.99991
*0 5O
2O
8.36678 8.36689 9.99988
04O
/ //
L cos L cot L sin
// /
/ //
L cos L cot L sin
// /
88 C
34
/ //
L sin L tan L cos
// /
/ //
L sin L tan L cos
// /
2O
8.36678 8.36689 9.99988
40
3O
8.41792 8.41807 9.99985
03O
10
8.36768 8.36780 9.99 988
50
10
8.41872 8.41887 9.99985
50
20
8.36858 8.36870 9.99988
40
20
8.41952 8.41967 9.99985
40
30
8.36948 8.36960 9.99988
30
30
8.42032 8.42048 9.99985
30
40
8.37038 8.37050 9.99988
20
40
8.42112 8.42127 9.99985
20
50
8.37128 8.37140 9.99988
10
50
8.42192 8.42207 9.99985
10
21
8.37217 8.37229 9.99988
39
31
8.42272 8.42287 9.99985
029
10
8.37306 8.37318 9.99988
50
10
8.42351 8.42366 9.99985
50
20
8.37395 8.37408 9.99988
40
20
8.42430 8.42446 9.99985
40
30
8.37484 8.37497 9.99988
30
30
8.42510 8.42525 9.99985
30
40
8.37573 8.37585 9.99988
20
40
8.42589 8.42406 9.99985.
20
50
8.37662 8.37674 9.99988
10
50
8.42667 8.42683 9.99985
10
22
8.37750 8.37762 9.99988
38
32
8.42746 8.42762 9.99984
028
10
8.37838 8.37850 9.99988
50
10
8.42825 8.42840 9.96984
50
20
8.37926 8.37938 9.99988
40
20
8.42903 8.42919 9.99984
40
30
8.38014 8.38026 9.99987
30
30
8.42 982 8.42997 9.99984
30
40
8.38101 8.38114 9.99987
20
40
8.43060 8.43075 9.99984
20
50
8.38189 8.33202 9.99987
10
50
8.43 138 8.43 154 9.99 984
10
23
8.38276 8.38289 9.99987
37
33
8.43 216 8.43 232 9.99 984
027
10
8.38363 8.38376 9.99987
50
10
8.43293 8.43309 9.99984
50
20
8.38450 8.38463 9.99987
40
20
8.43371 8.43387 9.99984
40
30
8.38537 8.38550 9.99987
30
30
8.43448 8.43464 9.99984
30
40
8.38624 838636 9.99987
20
40
8.43526 8.43542 9.99984
20
50
8.38710 8.38723 9.99987
10
50
8.43603 8.43619 9.99984
10
24
8.38796 8.38809 9.99987
36
34
8.43680 8.43696 9.99984
026
10
8.38882 8.38895 9.99987
50
10
8.43757 8.43773 9.99984
50
20
8.38968 8.38981 9.99987
40
20
8.43834 8.43850 9.99984
40
30
8.39054 8.39067 9.99987
30
30
8.43910 8.43927 9.99984
30
40
8.39139 8.39153 9.99987
20
40
8.43987 8.44003 9.99984
20
50
8.39225 8.39238 9.99987
10
50
8.44063 8.44080 9.99983
10
25
8.39310 8.39323 9.99987
35
35
8.44139 8.44156 9.99983
025
10
8.39395 8.39408 9.99987
50
10
8.44216 8.44232 9.99983
50
20
8.39480 8.39493 9.99987
40
20
8.44292 8.44308 9.99983
40
30
8.39565 8.39587 9.99987
30
30
8.44367 8.44384 9.99983
30
40
8.39649 8.39663 9.99987
20
40
8.44443 8.44460 9.99983
20
50
8.39734 8.39747 9.99986
10
50
8.44519 8.44536 9.99983
10
26
8.39818 8.39832 9.99986
34
36
8.44594 8.44611 9.99983
024
10
8.39902 8.39916 9.99986
50
10
8.44669 8.44686 9.99 983
50
20
8.39986 8.40000 9.99986
40
20
8.44745 8.44762 9.99983
40
30
8.40070 8.40083 9.99986
30
30
8.44820 8.44837 9.99983
30
40
8.40153 8.40167 9.99986
20
40
8.44895 8.44912 9.99983
20
50
8.40237 8.40251 9.99986
10
50
8.44969 8.44987 9.99983
10
27
8.40320 8.40334 9.99986
33
37
8.45044 8.45061 9.99983
023
10
8.40403 8.40417 9.99986
50
10
8.45119 8.45 136 9.99983
50
20
8.40486 8.40500 9.99986
40
20
8.45 193 8.45 210 9.99 983
40
30
8.40569 8.40583 9.99986
30
30
8.45267 8.45285 9.99983
30
40
8.40651 8.40665 9.99986
20
40
8.45341 8.45359 9.99982
20
50
8.40734 8.40748 9.99986
10
50
8.45415 8.45433 9.99982
10
28
8.40816 8.40830 9.99986
32
38
8.45489 8.45507 9.99982
022
10
8.40898 8.40913 9.99986
50
10
8.45563 8.45581 9.99982
50
20
8.40 980 8.40995 9.99986
40
20
8.45637 8.45655 9.99982
40
30
8.41062 8.41077 9.99986
30
30
8.45 710 8.45 728 9.99 982
30
40
841144 8.41158 9.99986
20
40
8.45 784 8.45 802 9.99 982
20
50
8.41225 8.41240 9.99986
10
50
8.45857 8.45875 9.99982
10
29
8.41307 8.41321 9.99985
31
39
8.45930 8.45948 9.99982
021
10
8.41388 8.41403 9.99985
50
10
8.46003 8.46021 9.99982
50
20
8.41469 8.41484 9.99985
40
20
8.46076 8.46094 9.99982
40
30
8.41550 8.41565 9.99985
30
30
8.46149 8.46167 9.99982
30
40
8.41631 8.41646 9.99985
20
40
8.46222 8.46240 9.99982
20
50
8.41711 8.41726 9.99985
10
50
8.46294 8.46312 9.99982
10
3O
8.41 792 8.41 807 9.99 985
30
4O
8.46366 8.46385 9.99932
020
/ //
L cos L cot L sin
// /
/ //
L cos L cot L sin
// /
1 i
88'
/ //
L sin L tan L cos
// /
/ //
L sin L tan L cos
// /
40
8.46366 8.46385 9.99982
2O
5O
8.50504 8.50527 9.99978
010
10
8.46439 8.46457 9.99982
50
10
8.50570 8.50593 9.99978
50
20
8.46511 8.46529 9.99982
40
20
8.50636 8.50658 9.99978
40
30
8.46583 8.46602 9.99981
30
30
8.50701 8.50724 9.99978
30
40
8.46655 8.46674 9.99981
20
40
8.50767 8.50789 9.99977
20
50
8.46727 8.46745 9.99981
10
50
8.50832 8.50855 9.99977
10
41
8.46799 8.46817 9.99981
19
51
8.50897 8.50920 9.99977
9
> 10
8.46870 8.46889 9.99981
50
10
8.50963 8.50985 9.99977
50
20
8.46942 8.46960 9.99981
40
20
8.51028 8.51050 9.99977
40
30
8.47013 8.47032 9.99981
30
30
8.51092 8.51015 9.99977
30
40
8.47084 8.47103 9.99981
20
40
8.51157 8.51180 9-99977
20
50
8.47155 8.47174 9.99981
10
50
8.51222 8.51245 9.99977
10
42
8.47226 8.47245 9.99981
18
52
8.51287 8.51310 9.99977
8
10
8.47297 8.47316 9.99981
50
10
8.51351 8.51374 9.99977
50
20
8.47368 847387 9.99981
40
20
8.51416 8.51439 9.99977
40
30
8.47439 8.47458 9.99981
30
30
8.51480 851 503 9.99977
30
40
8.47509 8.47528 9.99981
20
40
8.51544 8.51568 9.99977
20
50
8.47580 8.47599 9.99981
10
50
8.51609 8.51632 9.99-977
10
43
8.47650 8.47669 9.99981
17
53
8.51673 8.51696 9.99977
7
10
8.47720 8.47740 9.99980
50
10
8.51737 8.51760 9.99976
50
20
8.47790 8.47810 9.99980
40
20
8.51801 8.51824 9.99976
40
30
8.47860 8.47880 9.99980
30
30
8.51864 8.51888 9.99976
30
40
8.47930 8.47950 9.99980
20
40
8.51928 8.51952 9.99976
20
50
8.48000 8.48020 9.99980
10
50
8.51992 8.52015 9.99976
10
44
8.48096 8.48090 9.99980
16
54
8.52055 8.52079 9.99976
6
10
8.48139 8.48159 9.99980
50
10
8.52119 8.52143 9.99976
50
20
8.48208 8.48228 9.99980
40
20
8.52 182 8.52 206 9.99 976
40
30
8.48278 8.48298 9.99980
30
30
8.52245 8.52269 9.99976
30
40
8.48347 8.48367 9.99980
20
40
8.52308 8.52332 9.99976
20
50
8.48416 8.48436 9.99980
10
50
8.52371 8.52396 9.99976
10
45
8.48485 8.48505 9.99980
15
55
8.52434 8.52459 9.99976
5
10
8.48554 8.48574 9.99980
50
10
8.52497 8.52522 9.99976
50
20
8.48622 8.48643 9.99980
40
20
8.52560 8.52584 9.99976
40
30
8.48691 8.48711 9.99980
30
30
8.52623 8.52647 9.99975
30
40
8.48760 8.48780 9.99979
20
40
8.52685 8.52710 9.99975
20
50
8.48828 8.48849 9.99979
10
50
8.52748 8.52772 9.99975
10
46
8.48896 8.48917 9.99979
14
56
8.52810 8.52835 9.99975
4
10
8.48965 8.48985 9.99979
50
10
8.52872 8.52897 9.99975
50
20
8.49033 8.49053 9.99979
40
20
8.52935 8.52960 9.99975
40
30
8.49101 8.49121 9.99979
30
30
8.52997 8.53022 9.99975
30
40
8.49169 8.49189 9.99979
20
40
8.53059 8.53084 9.99975
20
50
8.49236 8.49257 9.99979
10
50
8.53121 8.53146 9.99975
10
47
8.49304 8.49325 9.99979
13
57
8.53183 8.53208 9.99975
3
10
8.49372 8.49393 9.99979
50
10
8.53245 8.53270 9.99975
50
20
8.49439 8.49460 9.99979
40
20
8.53306 8.53332 9.99975
40
30
8.49506 8.49528 9.99979
30
30
8.53368 8.53393 9.99975
30
40
8.49574 8.49595 9.99979
20
40
8.53 429 8.53 455 9.99 975
20
50
8.49641 8.49662 9.99979
10
50
8.53491 8.53516 9.99974
10
48
8.49708 8.49729 9.99979
12
58
8.53552 8.53578 9.99974
2
10
8.49775 8.49796 9.99979
50
10
8.53614 8.53639 9.99974
50
20
8.49842 8.49863 9.99978
40
20
8.53675 8.53700 9.99974
40
30
8.49908 8.49930 9.99978
30
30
8.53736 8.53762 9.99974
30
40
8.49975 8.49997 9.99978
20
40
8.53797 8.53823 9.99974
20
50
8.50042 8.50063 9.99978
10
50
8.53858 8.53884 9.99974
10
49
8.50108 850130 9.99978
11
59
8.53919 8.53945 9.99974
1
10
8.50174 8.50196 9.99978
50
10
8.53979 8.54005 9.99974
50
20
8.50241 8.50263 9.99978
40
20
8.54040 8.54066 9.99974
40
30
850307 8.50329 9.99978
30
30
8.54101 8.54127 9.99974
30
40
8.50373 8.50395 9.99978
20
40
8.54161 8.54187 9.99974
20
50
8.50439 8.50461 9.99978
10
50
8.54222 8.54248 9.99974
10
5O
8.50504 8.50527 9.99978
1O
6O
8.54282 8.54308 9.99974
O
/ //
L cos L cot L sin
// /
/ //
L cos L cot L sin
// /
36
/
SLsin SLtan llLcot 9Lcos
/
o
.24186 .24192 .75808 .99993
60
1
.24903 .24910 .75090 .99993
59
2
.25609 .25616 .74384 .99993
58
3
.26304 .26312 .73688 .99993
57
4
.26988 .26996 .73004 .99992
56
5
.27661 .27669 .72331 .99992
55
6
.28324 .28332 .71668 .99992
54
7
.28977 .28986 .71014 .99992
53
8
.29621 .29629 .70371 .99992
52
9
.30255 .30263 .69737 .99991
51
1C
.30879 .30888 .69112 .99991
50
11
.31495 .31505 .68495 .99991
49
12
.32103 .32112 .67888 .99990
48
13
.32702 .32711 .67289 .99990
47
14
.33292 .33302 .66698 .99990
46
15
.33875 .33886 .66114 .99990
45
16
.34450 .34461 .65539 .99989
44
17
.35018 .35029 .64971 .99989
43
18
.35578 .35590 .64410 .99989
42
19
.36131 .36143 .63857 .99989
41
20
.36678 .36689 .63311 .99988
40
21
.37217 .37229 .62771 .99988
39
22
.37750 .37762 .62238 .99988
38
23
.38276 .38289 .61711 .99987
37
24
.38796 .38809 .61191 .99987
36
25
.39310 .39323 .60677 .99987
35
26
.39818 .39832 .60168 .99986
34
27
.40320 .40334 .59666 .99986
33
28
.40816 .40830 .59170 .99986
32
29
.41307 .41321 .58679 .99985
31
3O
.41792 .41807 .58193 .99985
30
31
.42272 .42287 .57713 .99985
29
32
.42746 .42762 .57238 .99984
28
33
.43216 .43232 .56768 .99984
27
34
.43680 .43696 .56304 .99984
26
35
.44139 .44156 .55844 .99983
25
36
.44594 .44611 .55389 .99983
24
37
.45044 .45061 .54939 .99983
23
38
.45489 .45507 .54493 .99982
22
39
.45930 .45948 .54052 ,99982
21
40
.46366 .46385 .53615 .99982
2O
41
.46799 .46817 .53183 .99981
19
42
.47226 .47245 .52755 .99981
18
43
.47650 .47669 .52331 .99981
17
44
.48069 .48089 .51911 .99980
16
45
.48485 .48505 .51495 .99980
15
46
.48896 .48917 .51083 .99979
14
47
.49304 .49325 .50675 .99979
13
48
.49708 .49729 .50271 .99979
12
49
.50108 .50130 .49870 .99978
11
50
.50504 .50527 .49473 .99978
1O
51
.50897 .50920 .49080 .99977
9
52
.51287 .51310 .48690 .99977
8
53
.51673 .51696 .48304 .99977
7
54
.52055 .52079 .47921 .99976
6
55
.52434 .52459 .47541 .99976
5
56
.52810 .52835 .47165 .99975
4
57
.53183 .53208 .46792 .99975
3
58
.53552 .53578 .46422 .99974
2
59
.53919 .53945 .46055 .99974
1
60
.54282 .54308 .45692 .99974
O
/
SLcos SLcot ULtan 9Lsin
/
/
SLsin SLtan 11 L cot 9Lcos
/
O
.54282 .54308 .45692 .99974
60
1
.54642 .54669 .45331 .99973
59
2
.54999 .55027 .44973 .99973
58
3
.55354 .55382 .44618 .99972
57
4
.55705 .55734 .44266 .99972
56
5
.56054 .56083 .43917 .99971
55
6
.56400 .56429 .43571 .99971
54
7
.56743 .56773 .43227 .99970
53
8
.57084 .57114 .42886 .99970
52
9
.57421 .57452 .42548 .99969
51
10
.57757 .57788 .42212 .99969
r>o
11
.58089 .58121 .41879 .99968
49
12
.58419 .58451 .41549 .99968
48
13
.58747 .58779 .41221 .99967
47
14
.59072 .59105 .40895 .99967
46
15
.59395 .59428 .40572 .99967
45
16
.59715 .59749 .40251 .99966
44
17
.60033 .60068 .39932 .99966
43
18
.60349 .60384 .39616 .99965
42
19
.60662 .60698 .39302 .99964
41
20
.60973 .61009 .38991 .99964
4O
21
.61282 .61319 .38681 .99963
39
22
.61589 .61626 .38374 .99963
38
23
.61894 .61931 .38069 .99962
37
24
.62196 .62234 .37766 .99962
36
25
.62497 .62535 .37465 .99961
35
26
.62795 .62834 .37166 .99961
34
27
.63091 .63131 .36869 .99960
33
28
.63385 .63426 .36574 .99960
32
29
.63678 .63718 .36282 .99959
31
30
.63968 .64009 .35991 .99959
30
31
.64256 .64298 .35702 .99958
29
32
.64543 .64585 .35415 .99958
28
33
.64827 .64870 .35130 .99957
27
34
.65110 .65154 .34846 .99956
26
35
.65391 .65435 .34565 .99956
25
36
.65670 .65715 .34285 .99955
24
37
.65947 .65993 .34007 .99955
23
38
.66223 .66269 .33731 .99954
22
39
.66497 .66543 .33457 .99954
21
4O
.66769 .66816 .33184 .99953
2O
41
.67039 .67087 .32913 .99952
19
42
.67308 .67356 .32644 .99952
18
43
.67575 .67624 .32376 .99951
17
44
.67841 .67890 .32110 .99951
16
45
.68104 .68154 .31846 .99950
15
46
.68367 .68417 .31583 .99949
14
47
.68627 .68678 .31322 .99949
13
48
.68886 .68938 .31062 .99948
12
49
.69144 .69196 .30804 .99948
11
50
.69400 .69453 .30547 .99947
10
51
.69654 .69708 .30292 .99946
9
52
.69907 .69962 .30038 .99946
8
53
.70159 .70214 .29786 .99945
7
54
.70409 .70465 .29535 .99944
6
55
.70658 .70714 .29286 .99944
5
56
.70905 .70962 .29038 .99943
4
57
.71151 .71208 .28792 .99942
3
58
.71395 .71453 .28547 .99942
2
59
.71638 .71697 .28303 .99941
1
60
.71880 .71940 .28060 .99940
O
/
SLcos SLcot ULtan 9Lsin
/
88 C
87'
37
/
SLsin SLtan llLcot 9Lcos
/
o
.71880 .71940 .28060 .99940
60
1
.72120 .72181 .27819 .99940
59
2
.72359 .72420 .27580 .99939
58
3
.72597 .72659 .27341 .99938
57
4
.72834 .72896 .27104 .99938
56
5
.73069 .73132 .26868 .99937
55
* 6
.73303 .73366 .26634 .99936
54
7
.73535 .73600 .26400 .99936
53
8
.73767 .73832 .26168 .99935
52
9
.73997 .74063 .25937 .99934
51
10
.74226 .74292 .25708 .99934
50
11
.74454 .74521 .25479 .99933
49
12
.74680 .74748 .25252 .99932
48
13
.74906 .74974 .25026 .99932
47
14
.75130 .75199 .24801 .99931
46
IS
.75353 .75423 .24577 .99930
45
16
.75575 .75645 .24355 .99929
44
17
.75795 .75867 .24133 .99929
43
18
.76015 .76087 .23913 .99928
42
19
.76234 .76306 .23694 .99927
41
20
.76451 .76525 .23475 .99926
4O
21
.76667 .76742 .23258 .99926
39
22
.76883 .76958 .23042 .99925
38
23
.77097 .77173 .22827 .99924
37
24
.77310 .77387 .22613 .99923
36
25
.77522 .77600 .22400 .99923
35
26
.77733 .77811 .22189 .99922
34
27
.77943 .78022 .21978 .99921
33
28
.78152 .78232 .21768 .99920
32
29
.78360 .78441 .21559 .99920
31
30
.78568 .78649 .21351 .99919
30
31
.78774 .78855 .21145 .99918
29
32
.78979 .79061 .20939. .99917
28
33
.79183 .79266 .20734 .99917
27
34
.79386 .79470 .20*530 .99916
26
35
.79588 .79673 .20327 .99915
25
36
.79789 .79875 .20125 .99914
24
37
.79990 .80076 .19924 .99913
23
38
.80189 .80277 .19723 .99913
22
39
.80388 .80476 .19524 .99912
21
4O
.80585 .80674 .19326 .99911
2O
41
.80782 .80872 .19128 .99910
19
[ 42
.80978 .81068 .18932 .99909
18
43
.81173 .81264 .18736 .99909
17
44
.81367 .81459 .18541 .99908
16
45
.81560 .81653 .18347 .99907
15
46
.81752 .81846 .18154 .99906
14
47
.81944 .82038 .17962 .99905
13
48
.82134 .82230 .17770 .99904
12
49
.82324 .82420 .17580 .99904
11
50
.82513 .82610 .17390 .99903
1O
51
.82701 .82799 .17201 .99902
9
52
.82888 .82987 .17013 .'99901
8
53
.83075 .83175 .16825 .99900
7
54
.83261 .83361 .16639 .99899
6
55
.83446 .83547 .16453 .99898
5
56
.83630 .83732 .16268 .99898
4
57
.83813 .83916 .16084 .99897
3
58
.83996 .84100 .15900 .99896
2
59
.84177 .84282 .15718 .99895
1
52.
.84358 .84464 .15536 .99894
/
SLcos SLcot llLtan 9Lsin
/
/
SLsin
SLtan
11 Loot
DLcos
/
O
.84 358
.84 464
.15 536
.99894
6O
1
.84 539
.84646
.15354
.99893
59
2
.84718
.84 826
.15 174
.99 892
58
3
.84 897
.85006
.14994
.99891
57
4
.85 075
.85 185
.14815
.99 891
56
5
.85 252
.85 363
.14637
.99 890
55
6
.85 429
.85 540
.14460
.99889
54
7
.85 605
.85 717
.14283
.99888
53
8
.85 780
.85 893
.14 107
.99887
52
9
.85 955
.86069
.13931
.99886
51
1O
.86 128
.86243
.13757
.99885
50
11
.86301
.86417
.13583
.99884
49
12
.86474
.86591
.13409
.99883
48
13
.86645
.86 763
.13237
.99882
47
14
.86816
.86935
.13065
.99881
46
15
.86987
.87 106
.12 894
.99880
45
16
.87156
.87277
.12 723
.99 879
44
17
.87325
.87447
.12 553
.99 879
43
18
.87494
.87616
.12384
.99 878
42
19
.87661
.87 785
.12215
.99 877
41
2O
.87 829
.87953
.12047
.99876
40
21
.87995
.88 120
.11880
.99875
39
22
.88 161
.88287
.11713
.99874
38
23
.88326
.88453
.11547
.99 873
37
24
.88490
.88618
.11382
.99872
36
25
.88654
.88 783
.11217
.99871
35
26
.88817
.88948
.11052
.99 870
34
27
.88980
.89111
.10889
.99869
33
28
.89 142
89274
.10726
.99868
32
29
.89304
.89437
.10563
.99867
31
3O
.89464
.89 598
.10402
.99866
30
31
.89625
.89 760
.10240
.99865
29
32
.89 784
.89920
.10080
.99864
28
33
.89943
.90080
.09920
.99863
27
34
.90 102
.90 240
.09 760
.99862
26
35
.90260
.90399
.09601
.99861
25
36
.90417
.90557
.09443
.99860
24
37
.90574
.90715
.09 285
.99859
23
38
.90 730
.90872
.09 128
.99858
22
39
.90885
.91 029
.08971
.99857
21
4O
.91 040
.91 185
.08815
.99856
2O
41
.91 195
.91 340
.08660
.99855
19
42
.91 349
.91 495
.08 505
.99854
18
43
.91 502
.91 650
.08350
.99853
17
44
.91655
.91 803
.08 197
.99852
16
45
.91 807
.91957
.08043
.99851
15
46
.91 959
.92110
.07 890
.99850
14
47
.92110
.92 262
.07 738
.99848
13
48
.92 261
.92414
.07 586
.99847
12
49
.92411
.92 565
.07 435
.99846
11
5O
.92 561
.92716
.07 284
.99845
10
51
.92710
.92 866
.07 134
.99844
9
52
.92 859
.93 016
.06984
.99843
8
53
.93 007
.93 165
.06835
.99842
7
54
.93 154
.93313
.06687
.99841
6
55
.93301
.93 462
.06538
.99840
5
56
.93 448
.93 609
.06391
.99839
4
57
.93 594
.93 756
.06244
.99838
3
58
.93 740
.93 903
.06097
.99837
2
59
.93 885
.94 049
.05951
.99836
1
60
.94030
.94 195
.05 805
.99834
O
/
SLcos
SLcot
11 L tan
9Lsin
/
86
85 C
38
/
SLsin SLtan llLcot 9Lcos
/
o
.94030 .94195 .05805 .99834
6O
1
.94174 .94340 .05660 .99833
59
2
.94317 .94485 .05515 .99832
58
3
.94461 .94630 .05370 .99831
57
4
.94603 .94773 .05227 .99830
56
5
.94746 .94917 .05083 .99829
55
6
.94887 .95060 .049-10 .99828
54
7
.95029 .95202 .04798 .99827
53
8
.95170 .95344 .04656 .99825
52
9
.95310 .95486 .04514 .99824
51
ID
.95450 .95627 .04373 .99823
50
11
.95589 .95767 .04233 .99822
49
12
.95728 .95908 .04092 .99821
48
13
.95867 .96047 .03953 .99820
47
14
.96005 .96187 .03813 .99819
46
15
.96143 .96325 .03675 .99817
45
16
.96280 .96464 .03536 .99816
44
17
.96417 .96602 .03398 .99815
43
18
.96553 .96739 .03261 .99814
42
19
.96689 .96877 .03123 .99813
41
20
.96825 .97013 .02987 .99812
40
21
.96960 .97150 .02850 .99810
39
22
.97095 .97285 .02715 .99809
38
23
.97229 .97421 .02579 .99808
37
24
.97363 .97556 .02444 .99807
36
25
.97496 .97691 .02309 .99806
35
26
.97629 .97825 .02175 .99804
34
27
.97762 .97959 .02041 .99803
33
28
.97894 .98092 .01908 .99802
32
29
.98026 .98225 .01775 .99801
31
30
.98157 .98358 .01642 .99800
30
31
.98288 .98490 .01510 .99798
29
32
.98419 .98622 .01378 .99797
28
33
.98549 .98753 .01247 .99796
27
34
.98679 .98884 .01116 .99795
26
35
.98808 .99015 .00985 .99793
25
36
.98937 .99145 .00855 .99792
24
37
.99066 .99275 .00725 .99791
23
38
.99194 .99405 .00595 .99790
22
39
.99322 .99534 .00466 .99788
21
40
.99450 .99662 .00338 .99787
20
41
.99577 .99791 .00209 .99786
19
42
.99704 .99919 .00081 .99785
18
43
.99830 .00046 .99954 .99783
17
44
.99956 .00174 .99826 .99782
16
45
.00082 .00301 .99699 .99781
15
46
.00207 .00427 .99573 .99780
14
47
.00332 .00553 .99447 .99778
13
48
.00456 .00679 .99321 .99777
12
49
.00581 .00805 .99195 .99776
11
5O
.00704 .00930 .99070 .99775
1O
51
.00828 .01055 .98945 .99773
9
52
.00951 .01179 .98821 .99772
8
53
.01074 .01303 .98697 .99771
7
54
.01196 .01427 .98573 .99769
6
55
.01318 .01550 .98450 .99768
5
56
.01440 .01673 .98327 .99767
4
57
.01561 .01796 .98204 .99765
3
58
.01682 .01918 .98082 .99764
2
59
.01803 .02040 .97960 .99763
1
60
.01923 .02162 .97838 .99761
O
/
9 L cos 9 L cot 1O L tan 9 L sin
/
/
9Lsin 9LtanlOLcot 9Lcos
/
.01923 .02162 .97838 .99761
6O
1
.02043 .02283 .97717 .99760
59
2
.02163 .02404 .97596 .99759
58
3
.02283 .02525 .97475 .99757
57
4
.02402 .02645 .97355 .99756
56
5
.02520 .02766 .97234 .99755
55
6
.02639 .02885 .97115 .99753
54
7
.02757 .03005 .96995 .99752
53
8
.02874 .03124 .96876 .99751
52
9
.02992 .03242 .96758 .99749
51
1O
.03109 .03361 .96639 .99748
5O
11
.03226 .03479 .96521 .99747
49
12
.03342 .03597 .96403 .99745
48
13
.03458 .03714 .96286 .99744
47
14
.03574 .03832 .96168 .99742
46
15
.03690 .03948 .96052 .99741
45
16
.03805 .04065 .95935 .99740
44
17
.03920 .04181 .95819 .99738
43
18
.04034 .04297 .95703 .99737
42
19
.04149 .04413 .95587 .99736
41
2O
.04262 .04528 .95472 .99734
40
21
.04376 .04643 .95357 .99733
39
22
.04490 .04758 .95242 .99731
38
23
.04603 .04873 .95127 .99730
37
24
.04715 .04987 .95013 .99728
36
25
.04828 .05101 .94899 .99727
35
26
.04940 .05214 .94786 .99726
34
27
.05052 .05328 .94672 .99724
33
28
.05164 .05441 .94559 .99723
32
29
.05275 .05553 .94447 .99721
31
3O
.05386 .05666 .94334 .99720
3O
31
.05497 .05778 .94222 .99718
29
32
.05607 .05890 .94110 .99717
28
33
.05717 .06002 .93998 .99716
27
34
.05827 .06113 .93887 .99714
26
35
.05937 .06224 .93776 .99713
25
36
.06046 .06335 .93665 .99711
24
37
.06155 .06445 .93555 .99710
23
38
.06264 .06556 .93444 .99708
22
39
.06372 .06666 .93334 .99707
21
4O
.06481 .06775 .93225 .99705
2O
41
.06589 .06885 .93115 .99704
19
42
.06696 .06994 .93006 .99702
18
43
.06804 .07103 .92897 .99701
17
44
.06911 .07211 .92789 .99699
16
45
.07018 .07320 .92680 .99698
15
46
.07124 .07428 .92572 .99696
14
47
.07231 .07536 .92464 .99695
13
48
.07337 .07643 .92357 .99693
12
49
.07442 .07751 .92249 .99692
11
5O
.07548 .07858 .92142 .99690
1O
51
.07653 .07964 .92036 .99689
9
52
.07758 .08071 .91929 .99687
8
53
.07863 .08177 .91823 .99686
7
54
.07968 .08283 .91717 .99684
6
55
.08072 .08389 .91611 .99683
5
56
.08176 .08495 .91505 .99681
4
57
.08280 .08600 .91400 .99680
3
58
.08383 .08705 .91295 .99678
2
59
.08486 .08810 .91190 .99677
1
6O
.08589 .08914 .91086 .99675
O
7
9 L cos 9 L cot 1O L tan 9 L sin
/
84 C
83<
7
8 C
39
/
9Lsin 9Ltan 10 L cot 9Lcos
/
.08589 .08914 .91086 .99675
6O
1
.08692 .09019 .90981 .99674
59
2
.08795 .09123 .90877 .99672
58
3
.08897 .09227 .90773 .99670
57
4
.08999 .09330 .90670 .99669
56
5
.09101 .09434 .90566 .99667
55
6
.09202 .09537 .90463 .99666
54
7
.09304 .09640 .90360 .99664
53
8
.09405 .09742 .90258 .99663
52
9
.09506 .09845 .90155 .99661
51
1C
.09606 .09947 .90053 .99659
50
11
.09707 .10049 .89951 .99658
49
12
.09807 .10150 .89850 .99656
48
13
.09907 .10252 .89748 .99655
47
14
.10006 .10353 .89647 .99653
46
15
.10106 .10454 .89546 .99651
45
16
.10205 .10555 .89445 .99650
44
17
.10304 .10656 .89344 .99648
43
18
.10402 .10756 .89244 .99647
42
19
.10501 .10856 .89144 .99645
41
2O
.10599 .10956 .89044 .99643
4O
21
.10697 .11056 .88944 .99642
39
22
.10795 .11155 .88845 .99640
38
23
.10893 ..11254 .88746 .99638
37
24
.10990 .11353 .88647 .99637
36
25
.11087 .11452 .88548 .99635
35
26
-.11184 .11551 .88449 .99633
34
27
.11281 .11649 .88351 .99632
33
28
.11377 .11747 .88253 .99630
32
29
.11474 .11845 .88155 .99629
31
30
.11570 .11943 .88057 .99627
30
31
.11666 .12040 .87960 .99625
29
32
.11761 .12138 .87862 .99624
28
33
.11857 .12235 .87765 ,.99622
27
34
.11952 .12332 .87668 .99620
26
35
.12047 .12428 .87572 .99618
25
36
.12142 .12525 .87475 .99617
24
37
.12236 .12621 .87379 .99615
23
38
.12331 .12717 .87283 .99613
22
39
.12425 .12813 .87187 .99612
21
40
.12519 .12909 .87091 .99610
2O
41
.12612 .13004 .86996 .99608
19
42
.12706 .13099 .86901 .99607
18
43
.12799 .13194 .86806 .99605
17
44
.12892 .13289 .86711 .99603
16
45
.12985 .13384 .86616 .99601
15
46
.13078 .13478 .86522 .99600
14
47
.13171 .13573 .86427 .99598
13
48
.13263 .13667 .86333 .99596
12
49
.13355 .13761 .86239 .99595
11
50
.13447 .13854 .86146 .99593
10
51
.13539 .13948 .86052 .99591
9
52
.13630 .14041 .85959 .99589
8
53
.13722 .14134 .85866 .99588
7
54
.13813 .14227 .85773 .99586
6
55
.13904 .14320 .85680 .99584
5
56
.13994 .14412 .85588 .99582
4
57
.14085 .14504 .85496 .99581
3
58
.14175 .14597 .85403 .99579
2
59
.14266 .14688 .85312 .99577
1
6O
.14356 .14780 .85220 .99575
/
9 L cos 9 L cot 10 L tan 9 L sin
/
/
9Lsin 9Ltan lOLcot 9Lcos
/
o
.14356 .14780 .85220 .99575
60
1
.14445 .14872 .85128 .99574
59.
2
.14535 .14963 .85037 .99572
58
3
.14624 .15054 .84946 .99570
57
4
.14714 .15145 .84855 .99568
56
5
.14803 .15236 .84764 .99566
55
6
.14891 .15327 .84673 .99565
54
7
.14980 .15417 .84583 .99563
53
8
.15069 .15508 .84492 .99561
52
9
.15157 .15598 .84402 .99559
51
1O
.15245 .15688 .84312 .99557
50
11
.15333 .15777 .84223 .99556
49
12
.15421 .15867 .84133 .99554
48
13
.15508 .15956 .84044 .99552
47
14
.15596 .16046 .83954 .99550
46
15
.15683 .16135 .83865 .99548
45
16
.15770 .16224 .83776 .99546
44
17
.15857 .16312 .83688 .99545
43
18
.15944 .16401 .83599 .99543
42
19
.16030 .16489 .83511 .99541
41
2O
.16116 .16577 .83423 .99539
40
21
.16203 .16665 .83335 .99537
39
22
.16289 .16753 .83247 .99535
38
23
.16374 .16841 .83159 .99533
37
24
.16460 .16928 .83072 .99532
36
25
.16545 .17016 .82984 .99530
35
26
.16631 .17103 .82897 .99528
34
27
.16716 .17190 .82810 .99526
33
28
.16801 .17277 .82723 .99524
32
29
.16886 .17363 .82637 .99522
31
3O
.16970 .17450 .82550 .99520
30
31
.17055 .17536 .82464 .99518
29
32
.17139 .17622 .82378 .99517
28
33
.17223 .17708 .82292 .99515
27
34
.17307 .17794 .82206 .99513
26
35
.17391 .17880 .82120 .99511
25
36
.17474 .17965 .82035 .99509
24
37
.17558 .18051 .81949 .99507
23
38
.17641 .18136 .81864 .99505
22
39
.17724 .18221 .81779 .99503
21
40
.17807 .18306 .81694 .99501
2O
41
.17800 .18391 .81609 .99499
19
42
.17973 .18475 .81525 .99497
18
43
.18055 .18560 .81440 .99495
17
44
.28137 .18644 .81356 .99494
16
45
.18220 .18728 .81272 .99492
15
46
.18302 .18812 .81188 .99490
14
47
.18383 .18896 .81104 .99488
13
48
.18465 .18979 .81021 .99486
12
49
.18547 .19063 .80937 .99484
11
50
.18628 .19146 .80854 .99482
10
51
.18709 .19229 .80771 .99480
9
52
.18790 .19312 .80688 .99478
8
53
.18871 .19395 .80605 .99476
7
54
.18952 .19478 .80522 .99474
6
55
.19033 .19561 .80439 .99472
5
56
.19113 .19643 .80357 .99470
4
57
.19193 .19725 .80275 .99468
3
58
.19273 .19807 .80193 .99466
2
59
.19353 .19889 .80111 .99464
1
60
.19433 .19971 .80029 .99462
O
/
9 L cos 9 L cot 1O L tan 9 L sin
/
81
40
10
/
9Lsin 9Ltan lOLcot 9Lcos
/
o
.19433 .19971 .80029 .99462
6O
I
.19513 .20053 .79947 .99460
59
2
.19592 .20134 .79866 .99458
58
3
.19672 .20216 .79784 .99456
57
4
.19751 .20297 .79703 .99454
56
5
.19830 .20378 .79622 .99452
55
6
.19909 .20459 .79541 .99450
54
7
.19988 .20540 .79460 .99448
53
8
.20067 .20621 .79379 .99446
52
9
.20145 .20701 .79299 .99444
51
1O
.20223 .20782 .79218 .99442
50
11
.20302 .20862 .79138 .99440
49
12
.20380 .20942 .79058 .99438
48
13
.20458 .21022 .78978 .99436
47
14
.20535 .21102 .78898 .99434
46
15
.20613 .21182 .78818 .99432
45
16
.20691 .21 261 .78739 .99429
44
17
.20768 .21341 .78659 .99427
43
18
.20845 .21420 .78580 .99425
42
19
.20922 .21499 .78501 .99423
41
20
.20999 .21578 .78422 .99421
4O
21
.21076 .21657 .78343 .99419
39
22
.21153 .21736 .78264 .99417
38
23
.21229 .21814 .78186 .99415
37
24
.21306 .21893 .78107 .99413
36
25
.21382 .21971 .78029 .99411
35
26
.21458 .22049 .77951 .99409
34
27
.21534 .22127 .77873 .99407
33
28
.21610 .22205 .77795 .99404
32
29
.21685 .22283 .77717 .99402
31
3O
.21761 .22361 .77639 .99400
3O
31
.21836 .22438 .77562 .99398
29
32
.21912 .22516 .77484 .993.96
28
33
.21987 .22593 .77407 .99394
27
34
.22062 .22670 .77330 .99392
26
35
.22137 .22747 .77253 .99390
25
36
.22211 .22824 .77176 .99388
24
37
.22286 .22901 .77099 .99385
23
38
.22361 .22977 .77023 .99383
22
39
.22435 .23054 .76946 .99381
21
40
.22509 .23130 .76870 .99379
20
41
.22583 .23206 .76794 .99377
19
42
.22657 .23283 .76717 .99375
18
43
.22731 .23359 .76641 .99372
17
44
.22805 .23435 .76565 .99370
16
45
.22878 .23510 .76490 .99368
15
46
.22952 .23586 .76414 .99366
14
47
.23025 .23661 .76339 .99364
13
48
.23098 .23737 .76263 .99362
12
49
.23171 .23812 .76188 .99359
11
5O
.23244 .23887 .76113 .99357
10
51
.23317 .23962 .76038 .99355
9
52
.23390 .24037 -.75963 .99353
8
53
.23462 .24112 .75888 .99351
7
54
.23535 .24186 .75814 .99348
6
55
.23607 .24261 .75739 .99346
5
56
.23679 .24335 .75665 .99344
4
57
.23752 .24410 .75590 .99342
3
58
.23823 .24484 .75516 .99340
2
59
.23895 .24558 .75442 .99337
1
60
.23967 .24632 .75368 .99335
O
/
9Lcos 9Lcot lOLtan 9Lsin
/
/
9Lsin
9Ltan
1O L cot
9 Lcos
/
O
.23 967
.24 632
.75 368
.99335
750
1
.24 039
.24 706
.75 294
.99 333
59
2
.24110
.24 779
.75 221
.99331
58
3
.24181
.24 853
.75 147
.99328
57
4
.24253
.24926
.75074
.99 326
56
5
.24324
.25000
.75000
.99324
55
6
.24 395
.25 073
.74 927
.99322
54
7
.24466
.25 146
.74 854
.99319
53
8
.24 536
.25 219
.74 781
.99317
52
9
.24607
.25 292
.74 708
.99315
51
10
.24677
.25 365
.74 635
.99313
5O
11
.24 748
.25 437
.74563
.99310
49
12
.24818
.25510
.74490
.99308
48
13
.24 888
.25 582
.74418
.99306
47
14
.24958
.25 655
.74345
.99304
46
15
.25 028
.25 727
.74273
.99301
45
16
.25 098
.25 799
.74 201
.99299
44
17
.25 168
.25 871
.74129
.99 297
43
18
.25 237
.25 943
.74057
.99 294 i . 42
19
.25307
.26015
.73985
.99 292 41
2O
.25 376
.26086
.73914
.99290 4O
21
.25 445
.26158
.73842
.99288 39
22
.25514
.26 229
.73 771
.99 285 38
23
.25 583
.26301
.73699
.99 283
37
24
.25 652
.26372
.73 628
.99281
36
25
.25 721
.26443
.73557
.99278
35
26
.25 790
.26514
.73486
.99 276
34
27
.25 858
.26 585
.73415
.99274
33
28
.25927
.26655
.73 345
.99271
32
29
.25 995
.26 726
.73 274
.99269
31
3O
.26063
.26 797
.73 203
.99 267
30
31
.26131
.26867
.73133
.99 264
29
32
.26 199
.26937
.73 063
.99262
28
33
.26267
..27008
.72992
.99260
27
34
.26335
.27078
.72 922
.99257
26
35
.26403
.27148
.72852
.99255
25
36
.26470
.27218
.72 782
.99252
24
37
.26 538
.27 288
.72712
.99 250
23
38
.26605
.27357
.72643
.99 248
22
39
.26672
.27427
.72573
.99 245
21
40
.26 739
.27496
.72 504
.99243
2O
41
.26806
.27 566
.72434
.99 241
19
42
.26 873
.27635
.72365
.99 238
18
43
.26940
.27704
.72 296
.99 236
17
44
.27007
.27773
.72227
.99 233
16
45
.27073
.27 842
.72155
.99231
15
46
.27140
.27911
.72 089
.99 229
14
47
.27 206
.27 980
.72020
.99 226
13
48
.27273
.28049
.71951
.99 224
12
49
.27339
.28117
.71883
.99221
11
50
.27405
.28 186
.71814
.99219
10
51
.27471
.28254
.71 746
.99217
9
52
.27537
.28323
.71677
.99214
8
53
.27 602
.28391
.71 609
.99212
7
54
.27668
.28459
.71 541
.99 209
6
55
.27 734
.28 527
.71473
.99207
5
56
.27 799
.28 595
.71405
.99 204
4
57
.27 864
.28 662
.71338
.99 202
3
58
.27930
.28 730
.71270
.99 200
2
59
.27995
.28 798
.71 202
.99 197
1
60
.28060
.28 865
.71 135
.99 195
O
/
9 Lcos
9Lcot
lOLtan
9Lsin
/
80 C
79
ir
41
/
9 L sin 9 L tan 1O L cot 9 L cos
/
o
.28060 .28865 .71135 .99195
6O
I
.28125 .28933 .71067 .99192
59
2
.28190 .29000 .71000 .99190
58
3
.28254 .29067 .70933 .99187
57
4
.28319 .29134 .70866 .99185
56
5
.28384 .29201 .70799 .99182
55
6
.28448 .29268 .70732 .99180
54
7
.28512 .29335 .70665 .99177
53
8
.28577 .29402 .70598 .99175
52
9
.28641 .29468 .70532 .99172
51
1C
.28705 .29535 .70465 .99170
50
11
.28769 .29601 .70399 .99167
49
12
.28833 .29668 .70332 .99165
48
13
.28896 .29734 .70266 .99162
47
. 14
.28960 .29800 .70200 .99160
46
15
.29024 .29866 .70134 .99157
45
16
.29087 .29932 .70068 .99155
44
17
.29150 .29998 .70002 .99152
43
18
.29214 .30064 .69936 .99150
42
19
.29277 .30130 .69870 .99147
41
2O
.29340 .30195 .69805 .99145
40
21
.29403 .30261 .69739 .99142
39
22
.29466 .30326 .69674 .99140
38
23
.29529 .30391 .69609 .99137
37
I 24
.29591 .30457 .69543 .99135
36
25
.29654 .30522 .69478 .99132
35
26
.29716 .30587 .69413 .99130
34
27
.29779 .30652 .69348 .99127
33
28
.29841 .30717 .69283 .99124
32
29
.29903 .30782 .69218 .99122
31
30
.29966 .30 846 '.69 154 .99119
30
31
.30028 .30911 .69089 .99117
29
32
.30090 .30975 .69025 .99114
28
33
.30151 .31040 .68960 .99112
27
34
.30213 .31104 .68896 .99109
26
35
.30275 .31168 .68832 .99106
25
36
.30336 .31233 .68767 .99104
24
37
.30398 .31297 .68703 .99101
23
38
.30459 .31361 .68639 .99099
22
39
.30521 .31425 .68575 .99096
21
40
.30582 .31489 .68511 .99093
2O
41
.30643 .31552 .68448 .99091
19
42
.30704 .31616 .68384 .99088
18
43
.30765 .31679 .68321 .99086
17
44
.30826 .31743 .68257 .99083
16
45
.30887 .31806 .68194 .99080
15
46
.30947 .31870 .68130 .99078
14
47
.31008 .31933 .68067 .99075
13
48
.31068 .31996 .68004 .99072
12
49
.31129 .32059 .67941 .99070
11
50
.31189 .32122 .67878 .99067
1O
51
.31250 .32185 .67815 .99064
9
52
.31310 .32248 .67752 .99062
8
53
.31370 .32311 .67689 .99059
7
54
.31430 .32373 .67627 .99056
6
55
.31490 .32436 .67564 .99054
5
56
.31549 .32498 .67502 .99051
4
57
.31609 .32561 .67439 .99048
3
58
.31669 .32623 .67377 .99046
2
59
.31728 .32685 .67315 .99043
1
6O
.31788 .32747 .67253 .99040
O
/
9 L cos 9 L cot 1O L tan 9 L sin
/
/
9Lsin
9Ltan
1O L cot
9Lcos -L-
O
.31 788
.32 747
.67 253
.99040
ou
CO
1
.31847
.32810
.67 190
.99038
V
2
.31907
.32872
.67 128
.99035
58
3
.31 966
.32933
.67 067
.99032
57
4
.32025
.32995
.67005
.99030
56
5
.32084
.33057
.66943
.99027
55
6
.32 143
.33 119
.66881
.99024
54
7
.32 202
.33 180
.66 820
.99022
53
CO
8
.32 261
.33 242
.66 758
.99019
OL
9
.32319
.33303
.66697
.99016
51
1.O
.32378
.33 365
.66635
.99013
50
zin
11
.32437
.33 426
.66574
.99011
*ty
12
.32495
.33487
.66513
.99008
48
13
.32553
.33548
.66452
.99005
47
14
.32612
.33609
.66391
.99002
46
15
.32670
.33 670
.66330
.99000
45
16
.32 728
.33 731
.66 269
.98997
44
17
.32 786
.33 792
.66208
.98994
43
18
.32 844
.33 853
.66 147
.98991
42
19
.32902
.33913
.66087
.98989
41
20
.32960
.33974
.66026
.98986
4O
21
.33 018
.34 034
.65 966
.98983
39
'22
.33 075
.34095
.65 905
.98980
38
1 r-
23
.33 133
.34 155
.65 845
.98978
o/
ox;
24
.33 190
.34215
.65 785
.98975
36
25
.33 248
.34 276
.65 724
.98972
35
26
.33 305
.34336
.65 664
.98969
34
27
.33 362
.34 396
.65 604
.98967
33
28
.33 420
.34456
.65 544
.98964
32
29
.33 477
.34516
.65 484
.98961
31
30
.33 534
.34576
.65 424
.98958
30
31
.33 591
.34 635
.65 365
.98955
29
32
.33647
.34695
.65 305
.98953
28
33
.33 704
.34 755
.65 245
.98950
27
34
.33 761
.34814
.65 186
.98947
26
35
.33 818
.34874
.65 126
.98944
25
36
.33 874
.34 933
.65 067
.98941
24
37
.33931
.34992
.65 008
.98938
23
38
.33 987
.35051
.649-19
.98 936
22
39
.34043
.35 111
.64 889
.98933
21
40
.34 100
.35 170
.64830
.98930
2O
41
.34 156
.35 229
,64771
.98927
19
42
.34212
.35 288
.64712
.98924
18
43
.34 268
.35 347
.64653
.98921
17
44
.34324
.35 405
.64595
.98919
16
45
.34 380
.35 464
.64 536
.98916
15
46
.34 436
.35 523
.64477
.98913
14
47
.34491
.35 581
.64419
.98910
13
48
.34 547
.35 640
.64360
.9890?
12
49
.34602
.35 698
.64302
.98904
11
50
.34658
.35 757
.64 243
.98901
1O
51
.34 713
.35815
.64 185
.98898
9
52
.34 769
.35 873
.64127
.98896
8
53
.34 824
.35931
.64069
.98893
7
54
.34 879
.35989
.64011
.98890
6
55
.34934
.36047
.63 953
.98887
5
56
.34989
.36 105
.63 895
.98884
4
57
.35 044
.36 163
.63 837
.98 881
3
58
.35 099
.36221
.63 779
.98878
2
59
.35 154
.36279
.63 721
.98 875
1
60
.35 209
.36336
.63 664
.98 872
/
9Lccs
9Lcot
1O L tan
9Lsin
/
78 C
77 C
42
13
14 C
/
9 L sin 9 L tan 1O L cot 9 L cos
/
o
.35209 .36336 .63664 .98872
60
1
.35263 .36394 .63606 .98869
59
2
.35318 .36452 .63548 .98867
58
3
.35373 .36509 .63491 .98864
57
4
.35427 .36566 .63434 .98861
56
5
.35481 .36624 .63376 .98858
55
6
.35536 .36681 .63319 .98855
54
7
.35590 .36738 .63262 .98852
53
8
.35644 .36795 .63205 .98849
52
9
.35698 .36852 .63148 .98846
51
1C
.35752 .36909 .63091 .98843
5O
11
.35806 .36966 .63034 .98840
49
12
.35860 .37023 .62977 .98837
48
13
.35914 .37080 .62920 .98834
47
14
.35968 .37137 .62863 .98831
46
15
.36022 .37193 .62807 .98828
45
16
.36075 .37250 .62750 .98825
44
17
.36129 .37306 .62694 .98822
43
18
.36182 .37363 .62637 .98819
42
19
.36236 .37419 .62581 .98816
41
20
.36289 .37476 .62524 .98813
40
21
.36342 .37532 .62468 .98810
39
22
.36395 .37588 .62412 .98807
38
23
.36449 .37644 .62356 .98804
37
24
.36502 .37700 .62300 .98801
36
25
.36555 .37756 .62244 .98798
35
26
.36608 .37812 .62188 .98795
34
27
.36660 .37868 .62132 .98792
33
28
.36713 .37924 .62076 .98789
32
29
.36766 .37980 .62020 .98786
31
30
.36819 .38035 .61965 .98783
30
31
.36871 .38091 .61909 .98780
29
32
.36924 .38147 .61853 .98777
28
33
.36976 .38202 .61798 .98774
27
34
.37028 .38257 .61743 .98771
26
35
.37081 .38313 .61687 .98768
25
36
.37133 .38368 .61632 .98765
24
37
.37185 .38423 .61577 .98762
23
38
.37237 .38479 .61521. .98759
22
39
.37289 .38534 .61466 .98756
21
4O
.37341 .38589 .61411 .98753
2O
41
.37393 .38644 .61356 .98750
19
42
.37445 .38699 .61301 .98746
18
43
.37497 .38754 .61246 .98743
17
44
.37549 .38808 .61192 .98740
16
45
.37600 .38863 .61137 .98737
15
46
.37652. .38918 .61082 .98734
14
47
.37703 .38972 .61028 .98731
13
48
.37755 .39027 .60973 .98728
12
49
.37806 .39082 .60918 .98725
11
50
.37858 .39136 .60864 .98722
10
51
.37909 .39190 .60810 .98719
9
52
.37960 .39245 .60755 .98715
8
53
.38011 .39299 .60701 .98712
7
54
.38062 .39353 .60647 .98709
6
55
.38113 .39407 .60593 .98706
5
56
.38164 .39461 .60539 '.98703
4
57
.38215 .39515 .60485 .98700
3
58
.38266 .39569 .60431 .98697
2
59
.38317 .39623 .60377 .98694
1
60
.38368 .39677 .60323 .98690
O
/
9Lcos 9LcotlOLtan9Lsin
/
/
9 L sin 9 L tan 1O L cot 9 L cos
/
~o
.38368 .39677 .60323 .98690
6O
1
.38418 .39731 .60269 .98687
59
2
.38469 .39785 .60215 .98684
58
3
.38519 .39838 .60162 .98681
57
4
.38570 .39892 .60108 .98678
56
5
.38620 .39945 .60055 .98675
55
6
.38670 .39999 .60001 .98671
54
7
.38721 .40052 .59948 .98668
53
8
.38771 .40106 .59894 .98665
52
9
.38821 .40159 .59841 .98662
51
10
.38871 .40212 .59788 .98659
5O
11
.38921 .40266 .59734 .98656
49
12
.38971 .40319 .59681 .98652
48
13
.39021 .40372 .59628 .98649
47
14
.39071 .40425 .59575 .98646
46
15
.39121 .40478 .59522 .98643
45
16
.39170 .40531 .59469 .98640
44
17
.39220 .40584 .59416 .98636
43
18
.39270 .40636 .59364 .98633
42
19
.39319 .40689 .59311 .98630
41
20
.39369 .40742 .59258 .98627
40
21
.39418 .40795 .59205 .98623
39
22
.39467 .40847 .59153 .98620
38
23
.39517 .40900 .59100 .98617
37
24
.39566 .40952 .59048 .98614
36
25
.39615 .41005 .58995 .98610
35
26
.39.664 .41057 .58943 .98607
34
27
.39713 .41109 .58891 .98604
33
28
.39762 .41161 .58839 .98601
32
29
.39811 .41214 .58786 .98597
31
3O
.39860 .41266 .58734 .98594
30
31
.39909 .41318 .58682 .98591
29
32
.39958 .41370 .58630 .98588
28
33
.40006 .41422 .58578 .98584
27
34
.40055 .41474 .58526 .98581
26
35
.40103 .41526 .58474 .98578
25
36
.40152 .41578 .58422 .98574
24
37
.40200 .41629 .58371 .98571
23
38
.40249 .41681 .58319 .98568
22
39
.40297 .41733 .58267 .98565
21
4O
.40346 .41784 .58216 .98561
20
41
.40394 .41836 .58164 .98558
19
42
.40442 .41887 .58113 .98555
18
43
.40490 .41939 .58061 .98551
17
44
.40538 .41990 .58010 .98548
16
45
.40586 .42041 .57959 .98545
15
46
.40634 .42093 .57907 .98541
14
47
.40682 .42144 .57856 .98538
13
48
.40730 .42195 .57805 .98535
12
49
.40778 .42246 .57754 .98531
11
50
.40825 .42297 .57703 .98528
1O
51
.40873 .42348 .57652 .98525
9
52
.40921 .42399 .57601 .98521
8
53
.40968 .42450 .57550 .98518
7
54
.41016 .42501 .57499 .98515
6
55
.41063 .42552 .57448 .98511
5
56
.41111 .42603 .57397 .98508
4
57
.41158 .42653 .57347 .98505
3
58
.41205 .42704 .57296 .98501
2
59
.41252 .42755 .57245 .98498
1
60
.41300 .42805 .57195 .98494
O
/
9Lcos 9LcotlOLtan 9Lsin
/
76'
75 C
15'
16
43
/
9Lsin 9Ltan lOLcot 9Lcos
/
o
.41300 .42805 .57195 .98494
60
I
.41347 .42856 .57144 .98491
59
2
.41394 .42906 .57094 .98488
58
3
.41441 .42957 .57043 .98484
57
4
.41488 .43007 .56993 .98481
56
5
.41535 .43057 .56943 .98477
55
6
.41582 .43108 .56892 .98474
54
7
.41628 .43158 .56842 .98471
53
8
.41675 .43208 .56792 .98467
52
9
.41722 .43258 .56742 .98464
51
10
.41768 .43308 .56692 .98460
50
11
ATSIS .43358 .56642 .98457
49
12
.41861 .43408 .56592 .98453
48
13
.41908 .43458 .56542 .98450
47
14
.41954 .43508 .56492 .98447
46
15
.42001 .43558 .56442 .98443
45
16
.42047 .43607 .56393 .98440
44
17
.42093 .43657 .56343 .98436
43
18
.42140 .43707 .56293 .98433
42
19
.42186 .43756 .56244 .98429
41
2O
.42232 .43806 .56194 .98426
4O
21
.42278 .43855 .56145 .98422
39
22
.42324 .43905 .56095 .98419
38
23
.42370 .43954 .56046 .98415
37
24
.42416 .44004 .55996 .98412
36
25
.42461 .44053 .55947 .98409
35
26
.42507 .44102 .55898 .98405
34
27
.42553 .44151 .55849 .98402
33
28
.42599 .44201 .55799 .98398
32
29
.42644 .44250 .55750 .98395
31
3O
.42690 .44299 .55701 .98391
30
31
.42735 .44348 .55652 .98388
29
32
.42781 .44397 .55603 .98384
28
33
.42-826 .44446 .55554 .98381
27
34
.42872 .44495 .55505 .98377
26
35
.42917 .44544 .55456 .98373
25
36
.42962 .44592 .55408 .98370
24
37
.43008 .44641 .55359 .98366
23
38
.43053 .44690 .55310 .98363
22
39
.43098 .44738 .55262 .98359
21
4O
.43143 .44787 .55213 .98356
2O
41
.43188 .44836 .55164 .98352
19
42
.43233 .44884 .55116 .98349
18
43
.43278 .44933 .55067 .98345
17
44
.43323 .44981 .55019 .98342
16
45
.43367 .45029 .54971 .98338
15
46
.43412 .45078 .54922 .98334
14
47
.43457 .45126 .54874 .98331
13
48
.43502 .45174 .54826 .98327
12
49
.43546 .45222 .54778 .98324
11
50
.43591 .45271 .54729 .98320
10
51
.43635 .45319 .54681 .98317
9
52
.43680 .45367 .54633 .98313
8
53
.43724 .45415 .54585 .98309
7
54
.43769 .45463 .54537 .98306
6
55
.43813 .45511 .54489 .98302
5
56
.43857 .45559 .54441 .98299
4
57
.43901 .45606 .54394 .98295
3
58
.43946 .45654 .54346 .98291
2
59
.43990 .45702 .54298 .98288
1
6O
.44034 .45750 .54250 .98284
O
/
9 L cos 9 L cot 10 L tan 9 L sin
/
/
9 Lain
9Ltan
10 L cot
9Lcos
/
.44034
.45 750
.54250
.98284
60
1
.44 078
.45 797
.54203
.98 281
59
2
.44 122
.45 845
.54 155
.98277
58
3
.44 166
.45 892
.54 108
.98 273
57
4
.44 210
.45 940
.54060
.98 270
56
5
.44 253
.45 987
.54013
.98266
55
6
.44 297
.46035
.53 965
.98 262
54
7
.44341
.46082
.53918
.98 259
53
8
.44385
.46 130
.53870
.98255
52
9
.44428
.46 177
.53823
.98251
51
1O
.44472
.46224
.53 776
.98248
50
11
.44516
.46271
.53 729
.98244
49
12
.44559
.46319
.53681
.98240
48
13
.44602
.46366
.53 634
.98237
47
14
.44646
.46413
.53587
.98233
46
15
.44 689
.46 460
.53 540
.98 229
45
16
.44 733
.46 507
.53 493
.98 226
44
17
.44 776
.46554
.53 446
.98 222
43
18
.44819
.46601
.53 399
.98 218
42
19
.44862
.46648
.53 352
.98215
41
2O
.44905
.46694
.53 306
.98211
40
21
.44 948
.46741
.53259
.98 207
39
22
.44992
.46 788
.53212
.98 204
38
23
.45 035
.46 835
.53165
.98 200
37
24
.45 077
.46881
.53119
.98 196
36
25
.45 120
.46928
.53072
.98 192
35
26
.45 163
.46975
.53 025
.98 189
34
27
.45 206
.47021
.52979
.98 185
33
28
.45 249
.47068
.52932
.98181
32
29
.45 292
.47 114
.52886
.98177
31
30
.45 334
.47 160
.52840
.98174
30
31
.45377
.47 207
.52 793
.98170
29
32
.45 419
.47 253
.52 747
.98 166
28
33
.45 462
.47 299
.52701
.98 162
27
34
.45 504
.47346
.52654
.98159
26
35
.45 547
.47 392
.52608
.98 155
25
36
.45 589
.47438
.52562
.98151
24
37
.45 632
.47 484
.52516
.98 147
23
38
.45 674
.47530
.52470
.98 144
22
39
.45 716
.47576
.52424
.98 140
21
40
.45 758
.47 622
.52378
.98 136
2O
41
.45 801
.47 668
.52332
.98 132
19
42
.45 843
.47714
.52 286
.98 129
18
43
.45 885
.47 760
.52 240
.98 125
17
44
.45 927
.47806
.52 194
.98 121
16
45
.45969
.47 852
.52 148
.98117
15
46
.46011
.47 897
.52 103
.98113
14
47
.46053
.47 943
.52057
.98110
13
48
.46095
.47 989
.52011
.98 106
12
49
.46 136
.48035
.51965
.98 102
11
50
.46178
.48080
.51920
.98098
1O
51
.46 220
.48 126
.51874
.98094
9
52
.46262
.48 171
.51829
.98090
8
53
.46303
.48217
.51783
.98087
7
54
.46345
.48 262
.51 738
.98083
6
55
.46386
.48307
.51693
.98079
5
56
.46428
.48353
.51647
.98075
4
57
.46469
.48398
.51 602
.98071
3
58
.46511
.48443
.51557
.98067
2
59
.46552
.48489
.51511
.98063
1
60
.46594
.48 534
.51466
.98060
O
/
9Lcos
9Lcot
1O L tan
91 sin
/
73
44
17
18 C
/
9Lsin 9Ltan lOLcot 9Lcos
/
o
.46594 .48534 .51466 .98060
6O
1
.46635 .48579 .51421 .98056
59
2
.46676 .48624 .51376 .98052
58
3
.46717 .48669 .51331 .98048
57
4
.46758 .48714 .51286 .98044
56
5
.46800 .48759 .51241 .98040
55
6
.46841 .48804 .51196 .98036
54
7
.46882 .48849 .51151 .98032
53
8
.46923 .48894 .51106 .98029
52
9
.46964 .48939 .51061 .98025
51
10
.47005 .48984 .51016 .98021
5O
11
.47045 .49029 .50971 .98017
49
12
.47086 .49073 .50927 .98013
48
13
.47127 .49118 .50882 .98009
47
14
.47168 .49163 .50837 .98005
46
15
.47209 .49207 .50793 .98001
45
16
.47249 .49252 .50748 .97997
44
17
.47290 .49296 .50704 .97993
43
18
.47330 .49341 .50659 .97989
42
19
.47371 .49385 .50615 .97986
41
20
.47411 .49430 .50570 .97982
40
21
.47452 .49474 .50526 .97978
39
22
.47492 .49519 .50481 .97974
38
23
.47533 .49563 .50437 .97970
37
24
.47573 .49607 .50393 .97966
36
25
.47613 .49652 .50348 .97962
35
26
.47654 .49696 .50304 .97958
34
27
.47694 .49740 .50260 .97954
33
28
.47734 .49784 .50216 .97950
32
29
.47774 .49828 .50172 .97946
31
30
.47814 .49872 .50128 .97942
30
31
.47854 .49916 .50084 .97938
29
32
.47894 .49960 .50040 .97934
28
33
.47934 .50004 .49996 .97930
27
34
.47974 .50048 .49952 .97926
26
35
.48014 .50092 .49908 .97922
25
36
.48054 .50136 .49864 .97918
24
37
.48094 .50180 .49820 .97914
23
38
.48133 .50223 .49777 .97910
22
39
.48173 .50267 .49733 .97906
21
4O
.48213 .50311 .49689 .97902
2O
41
.48252 .50355 .49645 .97898
19
42
.48292 .50398 .49602 .97894
18
43
.48332 .50442 .49558 .97890
17
44
.48371 .50485 .49515 .97886
16
45
.48411 .50529 .49471 .97882
15
46
.48450 .50572 .49428 .97878
14
47
.48490 .50616 .49384 .97874
13
48
.48529 .50659 .49341 .97870
12
49
.48568 .50703 .49297 .97866
11
50
.48607 .50746 .49254 .97861
1O
51
.48647 .50789 .49211 .97857
9
52
.48686 .50833 .49167 .97853
8
53
.48725 .50876 .49124 .97849
7
54
.48764 .50919 .49081 .97845
6
55
.48803 .50962 .49038 .97841
5
56
.4S8H2 .51005 .48995 .97837
4
57
.48881 .51048 .48952 .97833
3
58
.48920 .51092 .48908 .97829
2
59
.48959 .51135 .48865 .97825
1
60
.48998 .51 178 .48822 .97821
/
9Lcos 9Lcot lOLtan 9Lsin
/
/
9Lsin 9Ltan 10 L cot 9Lcos
/
O
.48998 .51178 .48822 .97821
6O
1
.49037 .51221 .48779 .97817
59
2
.49076 .51264 .48736 .97812
58
3
.49115 .51306 .48694 .97808
57
4
.49153 .51349 .48651 .97804
56
5
.49192 .51392 .48608 .97800
55
6
.49231 .51435 .48565 .97796
54
7
.49269 .51478 .48522 .97792
53
8
.49308 .51520 .48480 .97788
52
9
.49347 .51563 .48437 .97784
51
1O
.49385 .51606 .48394 .97779 5O
11
.49424 .51648 .48352 .97775
49
12
.49462 .51691 .48309 .97771
48
13
.49500 .51734 .48266 .97767
47
14
.49539 .51776 .48224 .97763
46
15
.49577 .51819 .48181 .97759
45
16
.49615 .51861 .48139 .97754
44
17
.49654 .51903 .48097 .97750
43
18
.49692 .51946 .48054 .97746
42
19
.49730 .51988 .48012 .97742
41
2O
.49768 .52031 .47969 .97738
40
21
.49806 .52073 .47927 .97734
39
22
.49844 .52115 .47885 .97729
38
- 23
.49882 .52157 .47843 .97725
37
24
.49920 .52200 .47800 .97721
36
25
.49958 .52242 .47758 .97717
35
26
.49996 .52284 .47716 .97713
34
27
.50034 .52326 .47674 .97708
33
28
.50072 .52368 .47632 .97704
32
29
.50110 .52410 .47590 .97700
31
30
.50148 .52452 .47548 .97696
3O
31
.50185, .52494 .47506 .97691
29
32
.50223 .52536 .47464 .97687
28
33
.50261 .52578 .47422 .97683
27
34
.50298 .52620 .47380 .97679
26
35
.50336 .52661 .47339 .97674
25
36
.50374 .52703 .47297 .97670
24
37
.50411 .52745 .47255 .97666
23
38
.50449 .52787 .47213 .97662
22
39
.50486 .52829 .47171 .97657
21
4O
.50523 .52870 .47130 .97653
20
41
.50561 .52912 .47088 .97649
19
42
.50598 .52953 .47047 .97645
18
43
.50635 .52995 .47005 .97540
17
44
.50673 .53037 .46963 .97636
16
45
.50710 .53078 .46922 .97632
15
46
.50747 .53120 .46880 .97628
14
47
.50784 .53161 .46839 .97623
13
48
.50821 .53202 .46798 .97619
12
49
.50858 .53244 .46756 .97615
11
50
.50896 .53285 .46715 .97610
10
51
.50933 .53327 .46673 .97606
9
52
.50970 .53368 .46632 .97602
8
53
.51007 .53409 .46591 .97597
7
54
.51043 .53450 .46550 .97593
6
55
.51080 .53492 .46508 .97589
5
56
.51117 .53533 .46467 .97584
4
57
.51154 .53574 .46426 .97580
3
58
.51191 .53615 .46385 .97576
2
59
.51227 .53656 .46344 .97571
1
60
.51264 .53697 .46303 .97567
O
/
9 L cos 9 L cot 1O L tan 9 L sin
/
72 C
71
19
20 C
45
/
9 L sin 9 L tan 1O L cot 9 L cos
/
o
.51264 .53697^.46303 .97567
6O
1
.51301 .53738 .46262 .97563
59
2
.51338 .53779 .46221 .97558
58
3
.51374 .53820 .46180 .97554
57
4
.51411 .53861 .46139 .97550
56
5
.51447 .53902 .46098 .97545
55
6
.51484 .53943 .46057 .97541
54
7
.51520 .53984 .46016 .97536
53
8
.51557 .54025 .45975 .97532
52
9
.51593 .54065 .45935 .97528
51
1O
.51629 .54106 .45894 .97523
5O
11
.51666 .54147 .45853 .97519
49
12
.51702 .54187 .45813 .97515
48
13
.51738 .54228 .45772 .97510
47
14
.51774 .54269 .45731 .97506
46
15
.51811 .54309 .45691 97501
45
16
.51847 .54350 .45650 .97497
44
17
.51883 .54390 .45610 .97492
43
18
.51919 .54431 .45569 .97488
42
19
.51955 .54471 .45529 .97484
41
2O
.51991 .54512 .45488 .97479
4O
21
.52027 .54552 .45448 .97475,
39
22
.52063 .54593 .45407 .97470
38
23
.52099 .54633 .45367 .97466
37
24
.52135 .54673 .45327 .97461
36
25
.52171 .54714 .45286 .97457
35
26
.52207 .54754 .45246 .97453
34
27
.52242 .54794 .45206 .97448
33
28
.52 278 .54 835 .45 165 .97 444
32
29
.52314 .54875 .45125 .97439
31
3O
.52350 .54915 .45085 .97435
3Q
31
.52385 .54955 .45045 .97430
29
32
.52421 .54995 .45005 .97426
28
33
.52456 .55035 .44965 .97421
27
34
.52492 .55075 .44925 .97417
26
35
.52527 .55115 .44885 .97412
25
36 .52563 .55155 .44845 .97408
24
37 .52598 .55195 .44805 .97403
23
*3S
.52634 .55235 .44765 .97399
22
39
.52669 .55275 .44725 .97394
21
4O .52705 .55315 .44685 .97390
2O
41 .52740 .55355 .44645 .97385
19
42
.52775 .55395 .44605 .97381
18
43
.52811 .55434 .44566 .97376
17
44
.52846 .55474 .44526 .97372
16
45
.52881 .55514 .44486 .97367
15
46
.52916 .55554 .44446 .97363
14
47
.52951 .55593 .44407 .97358
13
48
.52986 .55633 .44367 .97353
12
49
.53021 .55673 .44327 .97349
11
5O
.53056 .55712 .44288 .97344
1O
51
.53092 .55752 .44248 .97340
9
52
.53126 .55791 .44209 .97335
8
53
.53161 .55831 .44169 .97331
7
54
.53196 .55870 .44130 .97326
6
55
.53231 .55910 .44090 .97322
5
56
.53266 .55949 .44051 .97317
4
57
.53301 .55989 .44011 .97312
3
58
.53336 .56028 .43972 .97308
2
59
.53370 .56067 .43933 .97303
1
60
.53405 .56107 .43893 .97299
O
/
9Lcos 9Lcot lOLtan 9Lsin
/
/
9Lsin 9Ltan lOLcot 9Lcos
/
O
.53405 .56107 .43893 .97299
6O
1
.53440 .56146 .43854 .97294
59
2
.53475 .56185 .43815 .97289
58
3
.53509 .56224 .43776 .97285
57
4
.53544 .56264 .43736 .97280
56
5
.53578 .56303 .43697 .97276
55
6
.53613 .56342 .43658 .97271
54
7
.53647 .56381 .43619 .97266
53
8
.53682 .56420 .43580 .97262
52
9
.53716 .56459 .43541 '97257
51
1O
.53751 .56498 .43502 .97252
5O
11
.53785 .56537 .43463 .97248
49
12
.53819 .56576 .43424 .97243
48
13
.53854 .56615 .43385 .97238
47
14
.53888 .56654 .43346 .97234
46
15
.53922 .56693 .43307 .97.229
45
16
.53957 .56732 .43268 .97224
44
17
.53991 .56771 .43229 .97220
43
18
.54025 .56810 .43190 .97215
42
19
.54059 .56849 .43151 .97210
41
20
.54093 .56887 .43113 .97206
4O
21
.54127 .56926 .43074 .97201
39
22
.54161 .56965 .43035 .97196
38
23
.54195 .57004 .42996 .97192
37
24
.54229 .57042 .42958 .97187
36
25
.54263 .57081 .42919 .97182
35
26
.54297 .57120 .42880 .97178
34
27
.54331 .57158 .42842 .97173
33
28
.54365 .57197 .42803 .97168
32 1
29
.54399 .$7235 .42765 .97163
31
30
.54433 .57274 .42726 .97159
30
31
.54466 .57312 .42688 .97154
29
32
.54500 .57351 .42649 .97149
28
33
.54534 .57389 .42611 .97145
27
34
.54567 .57428 .42572 .97140
26
35
.54601 .57466 .42534 .97135
25
36
.54635 .57504 .42496 .97130
24
37
.54668 .57543 .42457 .97126
23
38
.54702 .57581 .42419 .97121
22
39
.54735 .57619 .42381 .97116
21
4O
.54769 .57658 .42342 .97111
2O
41
.54802 .57696 .42304 .97107
19
42
.54836 .57734 .42266 .97102
18
43
.54869 .57772 .42228 .97097
17
44
.54903 .57810 .42190 .97092
16
45
.54936 .57849 .42151 .97087
15
46
.54969 .57887 .42113 .97083
14
47
.55003 .57925 .42075 .97078
13
48
.55036 .57963 .42037 .97073
12
49
.55069 .58001 .41999 .97068
11
50
.55102 .58039 .41961 .97063
10
51
.55136 .58077 .41923 .97059
9
52
.55169 .58115 .41885 .97054
8
53
.55202 .58153 .41847 .97049
7
54
.55235 .58191 .41809 .97044
6
55
.55268 .58229 .41771 .97039
5
56
.55301 .58267 .41733 .97035
4
57
.55334 .58304 .41696 .97030
3
58
.55367 .58342 .41658 .97025
2
59
.55400 .58380 .41620 .97020
1
60
.55433 .58418 .41582 .97015
O
/
9Lcos 9 Loot lOLtan 9Lsin
/
70 C
69
22'
/
9Lsin
9Ltan
1O L cot
9 Lcos
/
o
.55433
.58418
.41 582
.97015
6O
1
.55 466
.58455
.41545
.97010
59
2
.55 499
.58493
.41507
.97 005
58
3
.55 532
.58531
.41 469
.97 001
57
4
.55 564
.58 569
.41431
.96996
56
5
.55 597
.58606
.41 394
.96991
55
6
.55 630
.58644
.41 356
.96986
54
7
.55 663
.58681
.41319
.96981
53
8
.55 695
.58719
.41 281
.96976
52
9
.55 728
.58757
.41 243
.96971
51
10
.55 761
.58 794
.41 206
.96966
5O
11
.55 793
.58832
.41 168
.96962
49
12
.55 826
.58869
.41 131
.96957
48
13
.55 858
.58907
.41 093
.96952
47
14
.55 891
,58944
.41 056
.96947
46
15
.55.923
.58981
.41 019
.96942
45
16
.55 956
.59019
.40981
.96937
44
17
.55 988
.59056
.40944
.96932
43
18
.56021
.59094
.40906
.96927
42
19
.56053
.59131
.40869
.96922
41
20
.56085
.59168
.40832
.96917
40
21
.56118
.59205
.40 795
.96912
39
22
.56150
.59243
.40757
.96907
38
23
.56182
.59280
.40 720
.96903
37
24
.56215
.59317
.40683
.96898
36
25
.56247
.59354
.40646
.96893
35
26
.56279
.59391
.40 609
.96888
34
27
.56311
.59429
.40571
.96883
33
28
.56343
.59466
.40534
.96878
32
29
.56375
.59503
.40497
.96873
31
30
.56408
.59540
.40460
.96868
30
31
.56440
.59577
.40423
.96 863
29
32
.56472
.59614
.40386
.96 858
28
33
.56504
.59651
.40349
.96853
27
34
.56536
.59688
.40312
.96848
26
35
.56568
.59725
.40275
.96843
25
36
.56 599
.59 762
.40 238
.96838
24
37
.56631
.59 799
.40 201
.96833
23
38
.56663
.59835
.40 165
.96828
22
39
.56695
.59872
.40 128
.96823
21
40
.56727
.59909
.40091
.96818
20
41
.56759
.59946
.40054
.96813
19
42
.56 790
.59983
.40017
.96808
18
43
.56822
.60019
.39981
.96803
17
44
.56854
.60056
.39944
.96 798
16
45
.56886
.60093
.39907
.96793
15
46
.56917
.60 130
.39870
.96 788
14
47
.56949
.60 166
.39834
.96 783
13
48
.56980
.60 203
.39 797
.96778
12
49
.57012
.60240
.39 760
.96772
11
50
.57044
.60276
.39 724
.96 767
10
51
.57075
.60313
.39687
.96 762
9
52
.57107
.60349
.39651
.96757
8
53
.57138
.60386
.39614
.96752
7
54
.57169
.60422
.39578
.96 747
. 6
55
.57201
.60459
.39 541
.96 742
5
56
.57232
.60495
.39505
.96737
4
"57
.57264
.60532
.39468
.96732
3
58
.57295
.60568
.39432
.96 727
2
59
.57326
.60605
.39395
.96 722
1
60
.57358
.60641
.39359
.96 717
O
/
9Lcos
9LcotlOLtan9Lsin
/
/
9Lsin
9Ltan
1O L cot
9 Lcos
/
O
.57358
.60641
.39359
.96717
6O
1
.57389
.60677
.39323
.96711
59
2
.57420
.60 714
.39286
.96 706
58
3
.57451
.60 750
.39250
.96 701
57
4
.57482
.60786
.39 214
.96696
56
5
.57514
.60823
.39177
.96691
55
6
.57545
.60859
.39 141
.96686
54
7
.57576
.60895
.39 105
.96681
53
8
.57607
.60931
.39069
.96676
52
9
.57638
.60967
.39033
.96670
51
1O
.57669
.61 004
.38996
.96665
50
11
.57700
.61 040
.38960
.96660
49
12
.57731
.61 076
.38924
.96655
48
13
.57762
.61] 12
.38 888
.96650
47
14
.57793
.61 148
.38852
.96645
46
15
.57824
.61 184
.38816
.96640
45
16
.57855
.61 220
.38 780
.96634
44
17
.57885
.61 256
.38 744
.96629
43
18
.57916
.61 292
.38 708
.96624
42
19
.57947
.61 328
.38672
.96619
41
20
.57978
.61 364
.38636
.96614
40
21
.58008
.61 400
.38 600
.96608
39
22
.58039
.61436
.38 564
.96603
38
23
.58070
.61 472
.38528
.96 598
37
24
.58101
.61 508
.38492
.96593
36
25
.58131
.61 544
.38456
.96 588
35
26
.58 162
.61 579
.38421
.96 582
34
27
.58 192
.61 615
.38385
.96577
33
28
.58223
.61651
.38349
.96572
32
29
.58253
.61 687
.38313
.96567
31
.30
.58284
.61 722
.38278
.96562
30
31
.58314
.61 758
.38 242
.96556
29
32
,58345
.61 794
.38 206
.96551
28
33
.58375
.61 830
.38 170
.96546
27
34
.58406
.61 865
.38 135
.96541
26
35
.58436
.61901
.38099
.96535
25
36
.58467
.61936
.38064
.96530
24
37
.58497
.61 972
.38028
.96 525
23
38
.58527
.62008
.37992
.96 520
22
39
.58557
.62043
.37957
.96514
21
4O
.58588
.62079
.37921
.96509
2O
41
.58618
.62 114
.37886
.96504
19
42
.58648
.62 150
.37850
.96498
18
43
.58678
.62185
.37815
.96493
17
44
.58 709
.62221
.37779
.96488
16
45
.58739
.62256
.37 744
.96483
15
46
.58769
.62 292
.37 708
.96477
14
47
.58 799
.62327
.37673
.96472
13
48
.58829
.62 362
.37638
.96467
12
49
.58859
.62398
.37602
.96461
11
5O
.58889
.62433
.37567
.96456
10
51
.58919
.62 468
.37532
.96451
9
52
.58949
.62 504
.37496
.96445
8
53
.58979
.62 539
.37461
.96440
7
54
.59009
.62 574
.37426
.96435
6
55
.59039
.62 609
.37391
.96429
5
56
.59069
.62645
.37355
.96424
4
57
.59098
.62680
.37320
.96419
3
58
.59128
.62715
.37 285
.96413
2
59
.59158
.62 750
.37 250
.96408
1
60
.59188
.62 785
.37215
.96403
/
9 Lcos
9 L cot 1O L tan
9 L sin /
68'
67
23'
47
/
9 L sin 9 L tan 1O L cot 9 L cos
/
.59188 .62785 .37215 .96403
6O
1
.59218 .62820 .37180 .96397
59
2
.59247 .62855 .37145 .96392
58
3
.59277 .62890 .37110 .96387
57
4
.59307 .62926 .37074 .96381
56
5
.59336 .62961 .37039 .96376
55
6
.59366 .62996 .37004 .96370
54
7
.59396 .63031 .36969 .96365
53
8
.59425 .63066 .36934 .96360
52
9
.59455 .63101 -36899 .96354
51
1C
.59484 .63135 .36865 .96349
50
11
.59514 .63170 .36830 .96343
49
12
.59543 .63205 .36795 .96338
48
13
.59573 .63240 .36760 .96333
47
14
.59602 .63275 .36725 .96327
46
15
.59632 .63310 .36690 .96322
45
16
.59661 .63345 .36655 .96316
44
17
.59690 .63379 .36621 .96311
43
18
.59720 .63414 .36586 .96305
42
19
.59749 .63449 .36551 .96300
41
2O
.59778 .63484 .36516 .96294
40
21
.59808 .63519 .36481 .96289
39
22
.59837 .63553 .36447 .96284
38
23
.59866 .63588 .36412 .96278
37
24
.59895 .63623 .36377 .96273
36
25
.59924 .63657 .36343 .96267
35
26
.59954 .63692 .36308 .96262
34
27
.59983 .63726 .36274 .96256
33
28
.60012 .63761 .36239 .96251
32
29
.60041 .63796 .36204 .96245
31
30
.60070 .63830 .36170 .96240
30
31
.60099 .63865 .36135 .96234
29
32
.60128 .63899 .36101 .96229
28
33
.60157 .63934 .36066 .96223
27
34
.60186 .63968 .36032 .96218
26
35
.60215 .64003 .35997 .96212
25
36
.60244 .64037 .35963 .96207
24
37
.60273 .64072 .35928 .96201
23
38
.60302 .64106 .35894 .96196
22
39
.60331 .64140 .35860 .96190
21
4O
.60359 .64175 .35825 .96185
2O
41
.60388 .64209 .35791 .96179
19
42
.60417 .64243 .35757 .96174
18
43
.60446 .64278 .35722 .96168
17
44
.60474 .64312 .35688 .96162
16
45
.60503 .64346 .35654 .96157
15
46
.60532 .64381 .35619 .96151
14
47
.60561 .64415 .35585 .96146
13
48
.60589 .64449 .35551 .96140
12
49
.60618 .64483 .35517 .96135
11
50
.60646 .64517 .35483 .96129
1O
51
.60675 .64552 .35448 .96123
9
52
.60704 .64586 .35414 .96118
8
53
.60732 .64620 .35380 .96112
7
54
.60761 .64654 .35346 .96107
6
55
.60789 .64688 .35312 .96101
5
56
.60818 .64722 .35278 .96095
4
57
.60846 .64756 .35244 .96090
3
58
.60875 .64790 .35210 .96084
2
59
.60903 .64824 .35176 .96079
1
60
.60931 .64858 .35142 .96073
O
/
9Lcos 91 cot lOLtan 9Lsin
/
/
9Lsin
9Ltan
lOLcot
9Lcos
/
O
.60931
.64858
.35 142
.96073
6O
1
.60960
.64892
.35 108
.96067
59
2
.60988
.64 926
.35 074
.96062
58
3
.61016
.64960
.35 040
.96056
57
4
.61 045
.64994
.35 006
.96050
56
5
.61 073
.65 028
.34972
.96045
55
6
.61 101
.65 062
.34938
.96039
54
7
.61 129
.65096
.34904
96034
53
8
.61 158
.65 130
.3^870
.96028
52
9
.61 186
.65 164
.34836
.96022
51
1O
.61 214
.65 197
.34803
.96017
5O
11
.61 242
.65 231
.34 769
.96011
49
12
.61 270
.65 265
.34735
.96005
48
13
.61 298
.65 299
.34 701
.96000
47
14
.61 326
.65333
.34667
.95 994
46
15
.61 354
.65 366
.34634
.95988
45
16
.61 382
.65 400
.34600
.95 982
44
17
.61411
.65 434
.34 566
.95 977
43
18
.61 438
.65 467
.34 533
.95971
42
19
.61 466
.65 501
.34499
.95 965
41
2O
.61 494
.65 535
.34465
.95 960
40
21
.61 522
.65 568
.34 432
.95 954
39
22
.61 550
.65 602
.34398
.95 948
38
23
.61 578
.65 636
.34364
.95 942
37
24
.61 606
.6T669
.34331
.95937
36
25
.61 634
.65 703
.34297
.95931
35
26
.61 662
.65 736
.34 264
.95 925
34
27
.61 689
.65 770
.34 230
.95920
33
28
.61 717
.65 803
.34 197
.95 914
32
29
.61 745
.65 837
.34163
.95 908
31
30
.61 773
.65 870
.34 130
.95 902
30
31
.61 800
.65 904
.34096
.95 897
29
32
.61 828
.65937
.34063
.95 891
28
33
.61 856
.65 971
.34 029
.95 885
27
34
.61 883
.66004
.33 996
.95 879
26
35
.61911
.66038
.33 962
.95 873
25
35
.61 939
.66071
.33 929
.95 868
24
37
.61 966
.66104
.33 896
.95 862
23
38
.61 994
.66138
.33862
.95 856
22
39
.62021
.66171
.33 829
.95 850
21
40
.62049
.66204
.33 796
.95 844
2O
41
.62 076
.66238
.33 762
.95 839
19
42
.62 104
.66271
.33 729
.95 833
18
43,
.62 131
.66304
.33 696
.95 827
17
^
44
.62 159
.66337
.33663
.95 821
16
45
.62 186
.66371
.33 629
.95 815
15
46
.62 214
.66404
.33 596
.95 810
14
47
.62 241
.66437
.33 563
.95 804
13
48
.62 268
.66470
.33 530
.95 798
12
49
.62 296
.66503
.33497
.95 792
11
5O
.62323
.66537
.33 463
.95 786
1O
51
.62 350
.66570
.33 430
.95 780
9
52
.62377
.66603
.33397
.95 775
8
53
.62 405
.66636
.33 364
.95 769
7
54
.62432
.66669
.33331
.95 763
6
55
.62 459
.66 702
.33 298
.95 757
5
56
.62 486
.66 735
.33 265
.95 751
4
57
.62513
.66 768
.33 232
.95 745
3
58
.62 541
.66801
.33 199
.95 739
2
59
.62 568
.66834
.33 166
.95 733
' 1
60
.62595
.66867
.33 133
.95 728
O
/
9Lcos
9 Loot 10 L tan
9 L sin /
66
65 C
48
25
26
/ DLsin 9Ltan lOLcot 9Lcos
/
O
.62595 .66867 .33133 .95728
6O
I
.62622 .66900 .33100 .95722
59
2
.62649 .66933 .33067 .95716
58
3
.62676 .66966 .33034 .95710
57
4
.62703 .66999 .33001 .95704
56
5
.62730 .67032 .32968 .95698
55
6
.62757 .67065 .32935 .95692
54
7
.62784 .67098 .32902 .95686
53
8
.62811 .67131 .32869 .95680
52
9
.62838 .67163 .32837 .95674
51
10
.62865 .67196 .32804 .95668
50
11
.62892 .67229 .32771 .95663
49
12
.62918 .67262 .32738 .95657
48
13
.62945 .67295 .32705 .95651
47
14
.62972 .67327 .32673 .95645
46
15
.62999 .67360 .32640 .95639
45
16
.63026 .67393 .32607 .95633
44
17
.63052 .67426 .32574 .95627
43
18
.63079 .67458 .32542 .95621
42
19
.63106 .67491 .32509 .95615
41
2O
.63133 .67524 .32476 .95609
40
21
.63159 .67556 .32444 .95603
39
22
.63186 .67589 .32411 .95597
38
23
.63213 .67622 .32378 .95591
37
24
.63239 .67654 .32346 .95585
36
25
.63266 .67687 .32313 .95579
35
26
.63292 .67719 .32281 .95573
34
27
.63319 .67752 .32248 .95567
33
28
.63345 .67785 .32215 .95561
32
29
.63372 .67817 .32183 .95555
31
30
.63398 .67850 .32150 .95549
30
31
.63425 .67882 .32118 .95543
29
32
.63451 .67915 .32085 .95537
28
33
.63478 .67947 .32053 .95531
27
34
.63504 .67980 .32020 .95525
26
35
.63531 .68012 .31988 .95519
25
36
.63557 .68044 .31956 .95513
'24
37
.63583 .68077 .31923 .95507
23
38
.63610 .68109 .31891 .95500
22
39
.63636 .68142 .31858 .95494
21
40
.63662 .68174 .31826 .95488
2O
41
.63689 .68206 .31794 .95482
19
42
.63715 .68239 .31761 .95476
18
43
.63741 .68271 .31729 .95470
17
44
.63767 .68303 .31697 .95464
16
45
.63794 .68336 .31664 .95458
15
46
.63820 .68368 .31632 .95452
14
47
.63846 .68400 .31600 .95446
13
48
.63872 .68432 .31568 .95440
12
49
.63898 .68465 .31535 .95434
11
5O
.63924 .68497 .31503 .95427
10
51
.63950 .68529 .31471 .95421
9
52
.63976 .68561 .31439 .95415
8
53
.64002 .68593 .31407 .95409
7
54
.64028 .68626 .31374 .95403
6
55
.64054 .68658 .31342 .95397
5
56
.64080 .68690 .31310 .95391
4
57
.64106 .68722 .31278 .95384
3
58
.64132 .68754 .31246 .95378
2
59
.64158 .68786 .31214 .95372
1
6O
.64184 .68818 .31182 .95366
O
/
9 L cos 9 L cot 10 L tan 9 L sin
/
/
9Lsin
9Ltan
1O L cot
9 L cos
/
O
.64 184
.68818
.31 182
.95 366
6O
1
.64 210
.68850
.31150
.95 360
59
2
.64 236
.68882
.31118
.95 354
58
3
.64 262
.68914
.31086
.95 348
57
4
.64 288
.68946
.31 054
.95 341
56
5
.64313
.68978
.31022
.95 335
55
6
.64339
.69010
.30990
.95 329
54
7
.64365
.69042
.30958
.95 323
53
8
.64391
.69074
.30926
.95317
52
9
.64417
.69 106
.30894
.95310
51
10
.64442
.69 138
.30862
.95 304
5O
11
.64 468
.69 170
.30830
.95 298
49
12
.64494
.69 202
.30 798
.95 292
48
13
.64519
.69 234
.30 766
.95 286
47
14
.64545
.69266
.30 734
.95 279
46
15
.64571
.69 298
.30 702
.95 273
45
16
.64 596
.69329
.30671
.95 267
44
17
.64622
.69361
.30639
.95 261
43
18
.64647
.69393
.30607
.95 254
42
19
.64673
.69425
.30575
.95 248
41
2O
.64698
.69457
.30 543
.95 242
4O
21
.64 724
.69488
.30512
.95 236
39
22
.64 749
.69520
.30480
.95 229
38
23
.64 775
.69552
.30448
.95 223
37
24
.64800
.69 584
.30416
.95217
36
25
.64826
.69615
.30385
.95211
35
26
.64851
.69647
.30353
.95 204
34
27
.64877
.69 679
.30321
.95 198
33
28
.64902
69710
.30290
.95 192
32
29
.64927
.69 742
.30258
.95 185
31
3O
.64953
.69 774
.30226
.95 179
3O
31
.64978
.69805
.30 195
.95 173
29
32
.65 003
.69837
.30163
.95 167
28
33
.65 029
.69868
.30 132
.95 160
27
34
.65 054
.69900
.30 100
.95 154
26
35
.65 079
.69932
.30068
.95 148
25
36
.65 104
.69963
.30037
.95141
24
37
.65 130
.69995
.30005
.95 135
23
38
.65 155
.70026
.29974
.95 129
22
39
.65 180
.70058
.29942
.95 122
21
40
.65 205
.70089
.29911
.95116
20
41
.65 230
.70121
.29879
.95 110
19
42
.65 255
.70152
.29 848
.95 103
18
43
.65 281
.70184
.29816
.95 097
17
44
.65 306
.70215
.29 785
.95 090
16
45
.65331
.70247
.29 753
.95 084
15
46
.65 356
.70278
.29 722
.95 078
14
47
.65 381
.70309
.29691
.95 071
13
48
.65 406
.70341
.29 659
.95 065
12
49
.65431
.70372
.29628
.95 059
11
50
.65 456
.70404
.29596
.95 052
1O
51
.65 481
.70435
.29565
.95 046
9
52
.65 506
.70466
.29 534
.95 039
8
53
.65 531
.70498
.29 502
.95 033
7
54
.65 556
.70 529
.29471
.95 027
6
55
.65 580
.70560
.29440
.95 020
5
56
.65 605
.70 592
.29408
.95 014
4
57
.65 630
.70623
.29377
.95 007
3
58
.65 655
.70654
.29346
.95 001
2
59
.65 680
.70685
.29315
.94995
1
60
.65 705
.70717
.29283
.94988
O
/
9Lcos
9Lcot
10 L tan
9Lsin
/
64 C
63 C
27'-
28 C
49
/
9 L sin 9 L tan 10 L cot 9 L cos
/
o
.65705 .70717 .29283 .94988
00
1
.65729 .70748 .29252 .94982
59
2
.65754 .70779 .29221 .94975
58
3
.65779 .70810 .29190 .94969
57
4
.65804 .70841 .29159 .94962
56
5
.65828 .70873 .29127 .94956
55
6
.65853 .70904 .29096 .94949
54
7
.65878 .70935 .29065 .94943
53
8
.65902 .70966 .29034 .94936
52
9
.65927 .70997 .29003 .94930
51
1C
.65952 .71028 .28972 .94923
50
11
.65976 .71059 .28941 .94917
49
12
.66001 .71090 .28910 .94911
48
13
.66025 .71121 .28879 .94904
47
14
.66050 .71153 .28847 .94898
46
15
.66075 .71184 .28816 .94891
45
16
.66099 .71215 .28785 .94885
44
17
.66124 .71246 .28754 .94878
43
18
.66148 .71277 .28723 .94871
42
19
.66173 .71308 .28692 .94865
41
20
.66197 .71339 .28661 .94858
4O
21
.66221 .71370 .28630 .94852
39
22
.66246 .71401 .28599 .94845
38
23
.66270 .71431 .28569 .94839
37
24
.66295 .71462 .28538 .94832
36
25
.66319 .71493 .28507 .94826
35
26
.66343 .71524 .28476 .94819
34
27
.66368 .71555 .28445 .94813
33
28
.66392 .71586 .28414 .94806
32
29
.66416 .71617 .28383 .94799
31
SO
.66441 .71648 .28352 .94793
3O
3H.66465 .71679 .28321 .94786
29
32
.66489 .71709 .28291 .94780
28
33
.66513 .71740 .28260 .94773
27
34
.66537 .71771 .28229 .94767
26
35
.66562 .71802 .28198 .94760
25
36
.66586 .71833 .28167 .94753
24
37
.66610 .71863 .28137 .94747
23
38
.66634 .71894 .28106 .94740
22
39
.66658 .71925 .28075 .94734
21
4O
.66682 .71955 .28045 .94727
2O
41
.66706 .71986 .28014 .94720
19
42
.66731 .72017 .27983 .94714
18
43
.66755 .72048 .27952 .94707
17
44
.66779 .72078 .27922 .94700
16
45
.66803 .72109 .27891 .94694
15
46
.66827 .72140 .27860 .94687
14
47
.66851 .72170 .27830 .94680
13
48
.66875 .72201 .27799 .94674
12
49
.66899 .72231 .27769 .94667
11
50
.66922 .72262 .27738 .94660
1O
51
.66946 .72293 .27707 .94654
9
52
.66970 .72323 .27677 .94647
8
53
.66994 .72354 .27646 .94640
7
54
.67018 .72384 .27616 .94634
6
55
.67042 .72415 .27585 .94627
5
56
.67066 .72445 .27555 .94620
4
57
.67090 .72476 .27524 .94614
3
58
.67113 .72506 .27494 .94607
2
59
.67137 .72537 .27463 .94600
1
6O
.67161 .72567 .27433 .94593
O
/
9Lcos 9Lcot 10 L tan 9Lsin
/
/
9Lsin 9Ltan lOLcot 9Lcos
/
.67161 .72567 .27433 .94593
60
1
.67185 .72598 .27402 .94587
59
2
.67208 .72628 .27372 .94580
58
3
.67232 .72659 .27341 .94573
57
4
.67256 .72689 .27311 .94567
56
5
.67280 .72720 .27280 .'94560
55
6
.67303 .72750 .27250 .94553
54
7
.67327 .72780 .27220 .94546
53
8
.67350 .72811 .27189 .94540
52
9
.67374 .72841 .27159 .94533
51
1O
.67398 .72872 .27128 .94526
5O
11
.67421 .72902 .27098 .94519
49
12
.67445 .72932 .27068 .94513
48
13
.67468 .72963 .27037 .94506
47
14
.67492 .72993 .27007 .94499
46
15
.67515 .73023 .26977 .94492
45
16
.67539 .73054 .26946 .94485
44
17
.67562 .73084 .26916 .94479
43
18
.67586 .73114 .26886 .94472
42
19
.67609 .73144 .26856 .94465
41
2O
.67633 .73175 .26825 .94458
4O
21
.67656 .73205 .26795 .94451
39
22
.67680 .73235 .26765 .94445
38
23
.67703 .73265 .26735 .94438
37
24
.67726 .73295 .26705 .94431
36
25
.67750 .73326 .26674 .94424
35
26
.67773 .73356 .26644 .94417
34
27
.67796 .73386 .26614 .94410
33
28
.67820 .73416 .26584 .94404
32
29
.67843 .73446 .26554 .94397
31
3O
.67866 .73476 .26524 .94390
30
31
.67890 .73507 .26493 .94383
29
32
.67913 .73537 .26463 .94376
28
33
.67936 .73567 .26433 .94369
27
34
-.67959 .73597 .26403 .94362
26
35
.67982 .73627 .26373 .94355
25
36
.68006 .73657 .26343 .94349
24
37
.68029 .73687 .26313 .94342
23
38
.68052 .73717 .26283 .94335
22
39
.68075 .73747 .26253 .94328
21
40
.68098 .73777 .26223 .94321
2O
41
.68121 .73807 .26193 .94314
19
42
.68144 .73837 .26163 .94307
18
43
.68167 .73867 .26133 .94300
17
44
.68190 .73897 .26103 .94293
16
45
.68213 .73927 .26073 .94286
15
46
.63237 .73957 .26043 .94279
14
47
.68260 .73987 .26013 .94273
13
48
.68283 .74017 .25983 .94266
12
49
.68305 .74047 .25953 .94259
11
5O
.68328 .74077 .25923 .94252
10
51
.68351 .74107 .25893 .94245
9
52
.68374 .74137 .25863 .94238
8
53
.68397 .74166 .25834 .94231
7
54
.68420 .74196 .25804 .94224
6
55
.68443 .74226 .25774 .94217
5
56
.68466 .74256 .25744 .94210
4
57
.68489 .74286 .25714 .94203
3
58
.68512 .74316 .25684 .94196
2
59
.68534 .74345 .25655 .94189
1
6O
.68557 .74375 .25625 .94182
O
/
9Lcos 9Lcot lOLtan 9Lsin
/
62'
61
50
29
30<
/
9 L sin 9 L tan 1O L cot 9 L cos
/
.68557 .74375 .25625 .94182
60
1
.68580 .74405 .25595 .94175
59
2
.68603 .74435 .25565 .94168
58
3
.68625 .74465 .25535 .94161
57
4
.68648 .74494 .25506 .94154
56
5
.68671 .74524 .25476 .94147
55
6
.68694 .74554 .25446 .94140
54
7
.68716 .74583 .25417 .94133
53
8
.68739 .74613 .25387 .94126
52
9
.68762 .74643 .25357 .94119
51
10
.68784 .74673 .25327 .94112
50
11
.68807 .74702 .25298 .94105
49
12
.68829 .74732 .25268 .94098
48
13
.68852 .74762 .25238 .94090
47
14
.68875 .74791 .25209 .94083
46
15
.68897 .74821 .25179 .94076
45
16
.68920 .74851 .25149 .94069
44
17
.68942 .74880 .25120 .94062
43
18
.68965 .74910 .25090 .94055
42
19
.68987 .74939 .25061 .94048
41
20
.69010 .74969 .25031 .94041
4O
21
.69032 .74998 .25002 .94034
39
22
.69055 .75028 .24972 .94027
38
23
.69077 .75058 .24942 .94020
37
24
.69100 .75087 .24913 .94012
36
25
.69122 .75117 .24883 .94005
35
26
.69144 .75146 .24854 .93998
34
27
.69167 .75176 .24824 .93991
33
28
.69189 .75205 .24795 .93984
32
29
.69212 .75235 .24765 .93977
31
30
.69234 .75264 .24736 .93970
30
31
.69256 .75294 .24706 .93963
29
32
.69279 .75323 .24677 .93955
28
33
.69301 .75353 .24647 .93948
27
34
.69323 .75382 .24618 .93941
26
35
.69345 .75411 .24589 .93934
25
36
.69368 .75441 .24559 .93927
24
37
.69390 .75470 .24530 .93920
23
38
.69412 .75500 .24500 .93912
22
39
.69434 .75529 .24471 .93905
21
40
.69456 .75558 .24442 .93898
20
41
.69479 .75588 .24412 .93891
19
42
.69501 .75617 .24383 .93884
18
43
.69523 .75647 .24353 .93876
17
44
.69545 .75676 .24324 .93869
16
45
.69567 .75705 .24295 .93862
15
46
.69589 .75735 .24265 .93855
14
47
.69611 .75764 .24236 .93847
13
48
.69633 .75793 .24207 .93840
12
49
.69655 .75822 .24178 .93833
11
5O
.69677 .75852 .24148 .93826
1O
51
.69699 .75881 .24119 .93819
9
52
.69721 .75910 .24090 .93811
8
53
.69743 .75939 .24061 .93804
7
54
.69765 .75969 .24031 .93797
6
55
.69787 .75998 .24002 .93789
5
56
.69809 .76027 .23973 .93782
4
57
.69831 .76056 .23944 .93775
3
58
.69853 .76086 .23914 .93768
2
59
.69875 .76115 .23885 .93760
1
6O
'.69897 .76144 .23856 .93753
O
/
9 L cos 9 L cot 1O L tan 9 L sin
/
/
9 L sin 9 L tan 1O L cot 9 L cos
/
O
.69897 .76144 .23856 .93753
6O
1
.69919 .76173 .23827 .93746
59
2
.69941 .76202 .23798 .93738
58
3
.69963 .76231 .23769 .93731
57
4
.69984 .76261 .23739 .93724
56
5
.70006 .76290 .23710 .93717
55
6
.70028 .76319 .23681 .93709
54
7
.70050 .76348 .23652 .93702
53
8
.70072 .76377 .23623 .93695
52
9
.70093 .76406 .23594 .93687
51
1O
.70115 .76435 .23565 .93680
5O
11
.70137 .76464 .23536 .93673
49
12
.70159 .76493 .23507 .93665
48
13
.70180 .76522 .23478 .93658
47
14
.70202 .76551 .23449 .93650
46
15
.70224 .76580 .23420 .93643
45
16
.70245 .76609 .23391 .93636
44
17
.70267 .76639 .23361 .93628
43
18
.70288 .76668 .23332 .93621
42
19
.70310 .76697 .23303 .93614
41
20
.70332 .76725 .23275 .93606
40
21
.70353 .76754 .23246 .93599
39
22
.70375 .76783 .23217 .93591
36
23
.70396 .76812 .23188 .93584
37
24
.70418 .76841 .23159 .93577
36
25
.70439 .76870 .23130 .93569
35
26
.70461 .76899 .23101 .93562
34
27
.70482 .76928 .23072 .93554
33
28
.70504 .76957 .23043 .93547
32
29
.70525 .76986 .23014 .93539
31
30
.70547 .77015 .22985 .93532
30
31
.70568 .77044 .22956 .93525
29
32
.70590 .77073 .22927 .93517
28
33
.70611 .77101 .22899 .93510
27
34
.70633 .77130 .22870 .93502
26
35
.70654 .77159 .22841 .93495
25
36
.70675 .77188 .22812 .93487
24
37
.70697 .77217 .22783 .93480
23
38
.70718 .77246 .22754 .93472
22
39
.70739 .77274 .22726 .93465
21
4O
.70761 .77303 .22697 .93457
2O
41
.70782 .77332 .22668 .93450
19
42
.70803 .77361 .22639 .93442
18
43
.70824 .77390 .22610 .93435
17
44
.70846 .77418 .22582 .93427
16
45
.70867 .77447 .22553 .93420
15
46
.70888 .77476 .22524 .93412
14
47
.70909 .77505 .22495 .93405
13
48
.70931 .77533 .22467 .93397
12
49
.70952 .77562 .22438 .93390
11
50
.70973 .77591 .22409 .93382
1O
51
.70994 .77619 .22381 .93375
9
52
.71015 .77648 .22352 .93367
8
53
.71036 .77677 .22323 .93360
7
54
.71058 .77706 .22294 .93352
6
55
.71079 .77734 .22266 .93344
5
56
.71100 .77763 .22237 .93337
4
57
.71121 .77791 .22209 .93329
3
58
.71142 .77820 .22180 .93322
2
59
.71163 .77849 .22151 .93314
1
60
.71184 .77877 .22123 .93307
O
/
9 L cos 9 L cot 1O L tan 9 L sin
/
60
59
32'
51
/
9Lsin 9Ltan lOLcot9Lcos
/
o
.71184 .77877 .22123 .93307
7*0^
1
.71205 .77906 .22094 .93299
59
2
.71226 .77935 .22065 .93291
58
3
.71247 .77963 .22037 .93284
57
4
.71268 .77992 .22008 .93276
56
5
.71289 .78020 .21980 .93269
55
6
.71310 .78049 .21951 .93261
54
7
.71331 .78077 .21923 .93253
53
8
.71352 .78106 .21894 .93246
52
9
.71373 .78135 .21865 .93238
51
1O
.71393 .78163 .21837 .93230
5O
11
.71414 .78192 .21808 .93223
49
12
.71435 .78220 .21780 .93215
48
13
.71456 .78249 .21751 .93207
47
14
.71477 .78277 .21723 .93200
46
15
.71498 .78306 .21694 .93192
45
16
.71519 .78334 .21666 .93184
44
17
.71539 .78363 .21637 .93177
43
18
.71560 .78391 .21609 .93169
42
19
.71581 .78419 .21581 .93161
41
20
.71602 .78448 .21552 .93154
40
21
.71622 .78476 .21524 .93146
39
22
.71643 .78505 .21495 .93138
38
23
.71664 .78533 .21467 .93131
37
24
.71685 .78562 .21438 .93123
36
25
.71705 .78590 .21410 .93115
35
26
.71726 .78618 .21382 .93108
34
27
.71747 .78647 .21353 .93100
33
28
.71767 .78675 .21325 .93092
32
29
.71788 .78704 .21296 .93084
31
3O
.71809 .78732 .21268 .93077
3O
31
.71829 .78760 .21240 .93069
29
32
.71850 .78789 .21211 .93061
28
33
.71870 .78817 .21183 .93053
27
34
.71891 .78845 .21155 .93046
26
35
.71911 .78874 .21126 .93038
25
36
.71932 .78902 .21098 .93030
24
37
.71952 .78930 .21070 .93022
23
38
.71973 .78959 .21041 .93014
22
39
.71994 .78987 .21013 .93007
21
40
.72014 .79015 .20985 .92999
2O
41
.72034 .79043 .20957 .92991
19
42
.72055 .79072 .20928 .92983
18
43
.72075 .79100 .20900 .92976
17
44
.72096 .79128 .20872 .92968
16
45
.72116 .79156 .20844 .92960
15
46
.72137 .79185 .20815 .92952
14
47
.72157 .79213 .20787 .92944
13
48
.72177 .79241 .20759 .92936
12
49
.72198 .79269 .20731 .92929
11
50
.72218 .79297 .20703 .92921
1O
51
.72238 .79326 .20674 .92913
9
52
.72259 .79354 .20646 .92905
8
53
.72279 .79382 .20618 .92897
7
54
.72299 .79410 .20590 .92889
6
55
.72320 .79438 .20562 .92881
5
56
.72340 .79466 .20534 .92874
4
57
.72360 .79495 .20505 .92866
3
58
.72381 .79523 .20477 .92858
2
59
.72401 .79551 .20449 .92850
1
60
.72421 .79579 .20421 .92842
/
9Lcos 9Lcot lOLtan 9Lsin
/
/
9Lsin 9Ltan lOLcot 9Lcos
/
O
.72421 .79579 .20421 .92842
6O
1
.72441 .79607 .20393 .92834
59
2
.72461 .79635 .20365 .92826
58
3
.72482 .79663 .20337 .92818
57
4
.72502 .79691 .20309 .92810
56
5
.72522 .79719 .20281 .92803
55
6
.72542 .79747 .20253 .92795
54
7
.72562 .79776 .20224 .92787
53
8
.72582 .79804 .20196 .92779
52
9
.72602 .79832 .20168 .92771
51
10
.72 622 .79 860 .20 140 ,.92 763
5O
11
.72643 .79888 .20112 .92755
49
12
.72663 .79916 .20084 .92747
48
13
.72683 .79944 .20056 .92739
47
14
.72703 .79972 .20028 .92731
46
15
.72723 .80000 .20000 .92723
45
16
.72743 .80028 .19972 .92715
44
17
.72763 .80056 .19944 .92707
43
18
.72783 .80084 .19916 .92699
42
19
.72803 .80112 .19888 .92691
41
2O
.72823 .80140 .19860 .92683
40
21
.72843 .80168 .19832 .92675
39
22
.72863 .80195 .19805 .92667
38
23
.72883 .80223 .19777 .92659
37
24
.72902 .80251 .19749 .92651
36
25
.72922 .80279 .19721 .92643
35
26
.72942 .80307 .19693 .92635
34
27
.72962 .80335 .19665 .92627
33
28
.72982 .80363 .19637 .92619
32
29
.73002 .80391 .19609 .92611
31
3O
.73022 .80419 .19581 .92603
3O
31
.73041 .80447 .19553 .92595
29
32
.73061 .80474 .19526 .92587
28
33
.73081 .80502 .19498 .92579
27
34
.73101 .80530 .19470 .92571
26
35
.73121 .80558 .19442 .92563
25
36
.73140 .80586 .19414 .92555
24
37
.73160 .80614 .19386 .92546
23
38
.73180 .80642 .19358 .92538
22
39
.73200 .80669 .19331 .92530
21
4O
.73219 .80697 .19303 .92522
2O
41
.73239 .80725 .19275 .92514
19
42
.73259 .80753 .19247 .92506
18
43
.73278 .80781 .19219 .92498
17
44
.73298 .80808 .19192 .92490
16
45
.73318 .80836 .19164 .92482
15
46
.73337 .80864 .19136 .92473
14
47
.73357 .80892 .19108 .92465
13
48
.73377 .80919 .19081 .92457
12
49
.73396 .80947 .19053 .92449
11
5O
.73416 .80975 .19025 .92441
1O
51
.73435 .81003 .18997 .92433
9
52
.73455 .81030 .18970 .92425
8
53
.73474 .81058 .18942 .92416
7
54
.73494 .81086 .18914 .92408
6
55
.73513 .81113 .18887 .92400
5
56
.73533 .81141 .18859 .92392
4
57
.73552 .81169 .18831 .92384
3
58
.73572 .81196 .18804 .92376
2
59
.73591 .81224 .18776 .92367
1
60
.73611 .81252 .18748 .92359
O
/
9 L cos 9 L cot 1O L tan 9 L sin
/
58 C
57 C
52
33
34'
/
9Lsin 9Ltan lOLcot 9Lcos
/
o
.73611 .81252 .18748 .92359
60
1
.73630 .81279 .18721 .92351
59
2
.73650 .81307 .18693 .92313
58
3
.73669 .81335 .18665 .92335
57
4
.73689 .81362 .18638 .92326
56
5
.73708 .81390 .18610 .92318
55
6
.73727 .81418 .18582 .92310
54
7
.73747 .81445 .18555 .92302
53
8
.73766 .81473 .18527 .92293
52
9
.73785 .81500 .18500 .92285
51
10
.73805 .81528 .18472 .92277
5O
11
.73824 .81556 .18444 .92269
49
12
.73843 .81583 .18417 .92260
48
13
.73863 .81611 .18389 .92252
47
14
.73882 .81638 .18362 .92244
46
15
.73901 .81666 .18334 .92235
45
16
.73921 .81693 .18307 .92227
44
17
.73940 .81721 .18279 .92219
43
18
.73959 .81748 .18252 .92211
42
19
.73978 .81776 .18224 .92202
41
20
.73997 .81803 .18197 .92194
4O
21
.74017 .81831 .18169 .92186
39
22
.74036 .81858 .18142 .92177
38
23
.74055 .81886 .18114 .92169
37
24
.74074 .81913 .18087 .92161
36
25
.74093 .81941 .18059 .92152
35
26
.74113 .81968 .18032 .92144
34
27
.74132 .81996 .18004 .92136
33
28
.74151 .82023 .17977 .92127
32
29
.74170 .82051 .17949 .92119
31
3O
.74189 .82078 .17922 .92111
30
31
.74208 .82106 .17894 .92102
29
32
.74227 .82133 .17867 .92094
28
33
.74246 .82161 .17839 .92086
27
34
.74265 .82188 .17812 .92077
26
35
.74284 .82215 .17785 .92069
25
36
.74303 .82243 .17757 .92060
24
37
.74322 .82270 .17730 .92052
23
38
.74341 .82298 .17702 .92044
22
39
.74360 .82325 .17675 .92035
21
4O
.74379 .82352 .17648 .92027
20
41
.74398 .82380 .17620 .92018
19
42
.74417 .82407 .17593 .92010
18
43
.74436 .82435 .17565 .92002
17
44
.74455 .82462 .17538 .91993
16
45
.74474 .82489 .17511 .91985
15
46
.74493 .82517 .17483 .91976
14
47
.74512 .82544 .17456 .91968
13
48
.74531 .82571 .17429 .91959
12
49
.74549 .82599 .17401 .91951
11
5O
.74568 .82626 .17374 .91942
1O
51
.74587 .82653 .17347 .91934
9
52
.74606 .82681 .17319 .91925
8
53
.74625 .82708 .17292 .91917
7
54
.74644 .82735 .17265 .91908
6|
55
.74662 .82762 .17238 .91900
5
56
.74681 .82790 .17210 .91891
4
57
.74700 .82817 .17183 .91883
3
58
.74719 .82844 .17156 .91874
2
59
.74737 .82871 .17129 .91866
1
60
.74756 .82899 .17101 .91857
O
/
9Lcos 9LcotlOLtan9Lsin
/
/
9 L sin 9 L tan 1O L cot 9 L cos
/
O
.74756 .82899 .17101 .91857'
60
1
.74775 .82926 .17074 .91849
59
2
.74794 .82953 .17047 .91840
58
3
.74812 .82980 .17020 .91832
57
4
.74831 .83008 .16992 .91823
56
5
.74850 .83035 .16965 .91815
55
6
.74868 .83062 .16938 .91806
54
7
.74887 .83089 .16911 .91798
53
8
.74906 .83117 .16883 .91789
52
9
.74924 .83144 .16856 .91781
51
1O
.749-13 .83171 .16829 .91772
50
11
.74961 .83198 .16802 .91763
49
12
.74980 .83225 .16775 .91755
48
13
.74999 .83252 .16748 .91746
47
14
.75017 .83280 .16720 .91738
46
15
.75036 .83307 .16693 .91729
45
16
.75054 .83334 .16666 .91720
44
17
.75073 .83361 .16639 .91712
43
18
.75091 .83388 .16612 .91703
42
19
.75110 .83415 .16585 .91695
41
2O
.75128 .83442 .16558 .91686
4O
21
.75 147 .83470 .16530 .91677
39
22
.75165 .83497 .16503 .91669
38
23
.75184 .83524.16476 .91660
37
24
.75202 .83551 .16449 .91651
36
25
.75221 .83578 .16422 .91643
35
26
.75239 .83605 .16395 .91634
34
27
.75258 .83632 .16368 .91625
33
28
.75276 .83659 .16341 .91617
32
29
.7-5294 .83686 .16314 .91608
31
30
.75313 .83713 .16287 .91599
30
31
.75331 .83740 .16260 .91591
29
32
.75350 .83768 .16232 .91582
28
33
.75368 .83795 .16205 .91573
27
34
.75386 .83822 .16178 .91565
26
35
.75405 .83849 .16151 .91556
25
36
.75423 .83876 .16124 .91547
24
37
.75441 .83903 .16097 .91538
23
38
.75459 .83930 .16070 .91530
22
39
.75478 .83957 .16043 .91521
21
4O
.75496 .83984 .16016 .91512
2O
41
.75514 .84011 .15989 .91504
19
42
.75533 .84038 .15962 .91495
18
43
.75551 .84065 .15935 .91486
17
44
.75569 .84092 .15908 .91477
16
45
.75587 .84119 .15881 .91469
15
46
.75605 .84146 .15854 .91460
14
47
.75624 .84173 .15827 .91451
13
48
.75642 .84200 .15800 .91442
12
49
.75660 .84227 .15773 .91433
11
50
.75678 .84254 .15746 .91425
1OI
51
.75696 .84280 .15720 .91416
9
52
.75714 .84307 .15693 .91407
8
53
.75733 .84334 .15666 .91398
7
54
.75751 .84361 .15639 .91389
6
55
.75769 .84388 .15612 .91381
5
56
.75787 .84415 .15585 .91372
4
57
.75805 .84442 .15558 .91363
3
58
.75823 .84469 .15531 .91354
2
59
.75841 .84496 .15504 .91345
1
60
.75859 .84523 .15477 .91336
O
~7
9 L cos 9 L cot 1O L tan 9 L sin
/
56
55
35
36<
53
/
9Lsin
9Ltan
1O L cot
9 Lcos
/
.75 859
.84 523
.15477
.91 336
6O
1
.75 877
.84550
.15450
.91 328
59
2
.75 895
.84 576
.15 424
.91319
58
3
.75913
.84 603
.15397
.91 310
57
4
.75931
.84630
.15370
.91 301
56
5
.75949
.84657
.15343
.91 292
55
6
.75 967
.84684
.15316
.91 283
54
7
.75985
.84711
.15 289
.91 274
53
8
.76003
.84 738
.15262
.91 266
52
9
.76021
.84764
.15 236
.91 257
51
10
.76039
.84 791
.15 209
.91 248
50
11
.76057
.84818
.15 182
.91 239
49
12
.76075
.84 845
.15 155
.91 230
48
13
.76093
.84 872
.15 128
.91 221
47
14
.76111
.84 899
.15101
.91212
46
15
.76 129
.84925
.15075
.91 203
45
16
.76 146
.84952
.15048
.91 194
44
17
.76 164
.84979
.15021
.91 185
43
18
.76 182
.85 006
.14994
.91 176
42
19
.76 200
.85033
.14967
.91 167
41
2O
.76218
.85 059
.14941
.91 158
40
21
.76 236
.85 086
.14914
.91 149
39
22
.76253
.85 113
.14887
.91 141
38
23
.76271
.85 140
.14860
.91 132
37
24
.76 289
.85 166
.14834
.91 123
36
25
.76307
.85 193
.14807
.91114
35
26
.76324
.85 220
.14780
.91 105
34
27
.76342
.85 247
.14753
.91 096
33
28
.76360
.85 273
.14 727
.91087
32
29
.76378
.85 300
.14 700
.91 078
31
30
.76395
.85 327
.14673
.91 069
30
31
.76413
.85 354
.14646
.91 060
29
32
.76431
.85 380
.14620
.91051
28
33
.76448
.85 407
.14593
.91 042
27
34
.76466
.85 434
.14 566
.91033
26
35
.76484
.85 460
.14540
.91 023
25
36
.76501
.85 487
.14513
.91014
24
37
.76519
.85 514
.14486
.91 005
23
38
.76537
.85 540
.14460
.90996
22
39
.76554
.85 567
.14433
.90987
21
4O
.76572
.85 594
.14406
.90978
2O
41
.76 590
.85 620
.14380
.90969
19
42
.76607
.85 647
.14353
.90960
18
43
.76625
.85 674
.14326
.90951
17
44
.76642
.85 700
.14300
.90942
16
45
.76660
.85 727
.14273
.90933
15
46
.76677
.85 754
.14246
.90924
14
47
.76695
.85 780
.14220
.90915
13
48
.76712
.85 807
.14 193
.90906
12
49
.76 730
.85 834
.14 166
.90896
11
50
.76 747
.85 860
.14 140
.90887
1O
51
.76 765
.85 887
.14113
.90878
9
52
.76 782
.85 913
.14087
.90869
8
53
.76800
.85 940
.14060
.90860
7
54
.76817
.85 967
.14033
.90851
6
55
.76835
.85 993
.14007
.90842
5
56
.76852
.86020
.13 980
.90832
4
57
.76870
.86046
.13954
.90823
3
58
.76887
.86073
.13927
.90814
2
59
.76904
.86 103
.13900
.90805
1
60
.76922
.86 126
.13874
.90 796
O
/
9 Lcos
9Lcot
10 L tan
9Lsin
/
/
91 sin
9Ltan
lOLcot
9 Lcos
t
O
.76922
.86126
.13874
.90 796
60
1
.76939
.86153
.13847
.90787
59
2
.76957
.86179
.13821
.90777
58
3
.76974
.86 206
.13 794
.90 768
57
4
.76991
.86232
.13 768
.90759
56
5
.77009
.86259
.13 741
.90 750
55
6
.77026
.86285
.13 715
.90 741
54
7
.77043
.86312
.13688
.90 731
53
8
.77061
.86338
.13662
.90 722
52
9
.77078
.86365
.13635
.90713
51
1O
.77 095
.86392
.13608
.90 704
5O
11
.77112
.86418
.13 582
.90694
49
12
.77 130
.86445
.13 555
.90685
48
13
.77 147
.86471
.13 529
.90676
47
14
.77 164
.86498
.13 502
.90667
46
15
.77181
.86524
.13476
.90657
45
16
.77 199
.86551
.13449
.90648
44
17
.77216
.86577
.13 423
.90639
43
18
.77233
.86603
.13397
.90630
42
19
.77250
.86630
.13370
.90620
41
2O
.77 268
.86656
.13 344
.90611
40
21
.77285
.86683
.13317
.90602
39
22
.77302
.86 709
.13291
.90592
38
23
.77319
.86 736
.13 264
.90 583
37
24
.77336
.86762
.13 238
.90574
36
25
.77353
.86 789
.13211
.90565
35
26
.77370
.86815
.13 185
.90555
34
27
.77387
.86842
.13158
.90 546
33
28
.77405
.86868
.13 132
.90537
32
29
.77422
.86894
.13 106
.90527
31
30
.77439
.86921
.13079
.90518
30
31
.77456
.86947
.13053
.90 509
29
32
.77473
.86974
.13026
.90499
28
33
.77 490
.87000
.13000
.90490
27
34
.77507
.87027
.12973
.90480
26
35
.77524
.87053
.12947
.90471
25
36
.77541
.87079
.12921
.90462
24
37
.77558
.87 106
.12 894
.90452
23
38
.77575
.87 132
.12868
.90443
22
39
.77 592
.87 158
.12842
.90434
21
4O
.77609
.87 185
.12815
.90424
2O
41
.77 626
.87211
.12 789
.90415
19
42
.77 643
.87 238
.12 762
.90405
18
43
.77660
.87264
.12 736
.90396
17
44
.77677
.87 290
.12710
.90386
16
45
.77694
.87317
.12683
.90377
15
46
.77711
.87343
.12657
.90368
14
47
.77 728
.87 369
.12631
.90358
13
48
.77 744
.87 396
.12604
.90349
12
49
.77761
.87422
.12578
.90339
11
5O
.77 778
.87448
.12552
.90330
10
51
.77 795
.87475
.12525
.90320
9
52
.77812
.87501
.12499
.90311
8
53
.77829
.87 527
.12473
.90301
7
54
.77 846
.87554
.12 446
.90292
6
55
.77 862
.87 580
.12420
.90282
5
56
.77879
.87 606
.12394
.90273
4
57
.77896
.87 633
.12367
.90263
3
58
.77913
.87659
.12341
.90254
2
59
.77930
.87685
.12315
.90244
1
60
.77946
.87711
.12289
.90235
O
/
9 Lcos
9Lcot
10 L tan
9Lsin
/
54 C
53'
54
37
38
/
9Lsin
9Ltan
lOLcot
9 Lcos
/
o
.77946
.87711
.12289
.90 235
60
1
.77963
.87 738
.12262
.90 225
59
2
.77980
.87 764
.12236
.90216
58
3
.77997
.87 790
.12210
.90 206
57
4
.78013
.87817
.12183
.90 197
56
5
.78030
.87 843
.12157
.90187
55
6
.78047
.87 869
.12131
.90178
54
7
.78063
.87895
.12105
.90 168
53
8
.78080
.87922
.12078
.90 159
52
9
.78097
.87948
.12052
.90 149
51
1O
.78113
.87974
.12026
.90 139
50
11
.78 130
.88000
.12000
.90 130
49
12
.78 147
.88027
.11973
.90 120
48
13
.78 163
.88053
.11947
.90111
47
14
.78 180
.88079
.11921
.90 101
46
15
.78 197
.88 105
.11895
.90091
45
16
.78213
.88131
.11869
.90082
44
17
.78 230
.88 158
.11 842
.90072
43
18
.78 246
.88 184
.11816
.90063
42
19
.78 263
.88210
.11790
.90053
41
20
.78280
.88236
.11764
.90043
40
21
.78 296
.88 262
.11738
.90034
39
22
.78313
.88 289
.11711
.90024
38
23
.78329
.88315
.11685
.90014
37
24
.78346
.88341
.11659
.90005
36
25
.78362
.88367
.11633
.89995
35
26
.78379
.88393
.11607
.89985
34
27
.78 395
.88420
.11580
.89976
33
28
.78412
.88446
.11554
.89966
32
29
.78428
.88472
.11528
.89956
31
30
.78445
.88498
.11502
.89947
3O
31
.78461
.88 524
.11476
.89937
29
32
.78478
.88550
.11450
.89927
28
33
.78494
.88577
.11423
.89918
27
34
.78510
.88603
.11397
.89908
26
35
.78527
.88629
.11371
.89898
25
36
.78543
.88655
.11345
.89888
24
37
.78560
.88681
.11319
.89879
23
38
.78576
.88 707
.11293
.89869
22
39
.78 592
.88 733
.11267
.89859
21
40
.78609
.88759
.11241
.89849
2O
41
.78625
.88 786
.11214
.89840
19
42
.78642
.88812
.11188
.89830
18
43
.78658
.88838
.11162
.89 820
17
44
.78674
.88864
.11136
.89810
16
45
.78691
.88890
.11110
.89801
15
46
.78 707
.88916
.11084
.89791
14
47
.78 723
.88942
.11058
.89781
13
48
.78739
.88968
.11032
.89771
12
49
.78756
.88994
.11 006
.89 761
11
5O
.78 772
.89020
.10980
.89 752
10
51
.78 788
.89046
.10954
.89 742
9
52
.78805
.89073
.10927
.89 732
8
53
.78821
.89099
.10901
.89 722
7
54
.78837
.89 125
.10875
.89712
6
55
.78853
.89151
.10849
.89 702
5
56
.78869
.89177
.10823
.88693
4
57
.78 886
.89 203
.10797
.89683
3
58
.78902
.89 229
.10771
.89673
2
59
.78918
.89255
.10745
.89663
1
60
.78934
.89281
.10719
.89653
/
9 Lcos
9Lcot
1O L tan 9 L sin
/
/
9 L sin 9 L tan 1O L cot 9 L cos
/
.78934 .89281 .10719 .89653
60
1
.78950 .89307 .10693 .89643
59
2
.78967 .89333 .10667 .89633
58
3
.78983 .89359 .10641 .89624
57
4
.78999 .89385 .10615 .89614
56
5
.79015 .89411 .10589 .89604
55
6
.79031 .89437 .10563 .89594
54
7
.79047 .89463 .10537 .89584
53
8
.79063 .89489 .10511 .89574
52
9
.79079 .89515 .10485 .89564
51
1O
.79095 .89541 .10459 .89554
5O
11
.79111 .89567 .10433 .89544
49
12
.79128 .89593 .10407 .89534
48
13
.79144 .89619 .10381 .89524
47
14
.79160 .89645 .10355 .89514
46
15
.79176 .89671 .10329 .89504
45
16
.79192 .89697 .10303 .89495
44
17
.79208 .89723 .10277 .89485
43
18
.79224 .89749 .10251 .89475
42
19
.79240 .89775 .10225 .89465
41
20
.79256 .89801 .10199 .89455
4O
21
.79272 .89827 .10173 .89445
39
22
.79288 .89853 .10147 .89435
38
23
.79304 .89879 .10121 .89425
37
24
.79319 .89905 .10095 .89415
36
25
.79335 .89931 .10069 .89405
35
26
.79351 .89957 .10043 .89395
34
27
.79367 .89983 .10017 .89385
33
28
.79383 .90009 .09991 .89375
32
29
.79399 .90035 .09965 .89364
31
3O
.79415 .90061 .09939 .89354
3O
31
.79431 .90086 .09914 .89344
29
32
.79447 .90112 .09888 .89334
28
33
.79463 .90138 .09862 .89324
27
34
.79478 .90164 .09836 .89314
26
35
.79494 .90190 .09810 .89304
25
36
.79510 .90216 .09784 .89294
24
37
.79526 .90242 .09758 .89284
23
38
.79542 .90268 .09732 .89274
22
39
.79558 .90294 .09706 .89264
21
4O
.79573 .90320 .09680 .89254
2O
41
.79589 .90346 .09654 .89244
19
42
.79605 .90371 .09629 .89233
18
43
.79621 .90397 .09603 .89223
17
44
.79636 .90423 .09577 .89213
16
45
.79652 .90449 .09551 .89203
15
46
.79668 .90475 .09525 .89193
14
47
.79684 .90501 .09499 .89183
13
48
.79699 .90527 .09473 .89173
12
49
.79715 .90553 .09447 .89162
11
50
.79731 .90578 .09422 .89152
1O
51
.79746 .90604 .09396 .89142
9
52
.79762 .90630 .09370 .89132
8
53
.79778 .90656 .09344 .89122
7
54
.79793 .90682 .09318 .89112
6
55
.79809 .90708 .09292 .89101
5
56
.79825 .90734 .09266 .89091
4
57
.79840 .90759 .09241 .89081
3
58
.79856 .90785 .09215 .89071
2
59
.79872 .90811 .09189 .89060
1
60
.79887 .90837 .09163 .89050
/
9 Lcos 9Lcot lOLtan 9Lsin
/
52 C
51 C
39
40'
55
/
9 L sin 9 L tan 1O L cot 9 L cos
/
o
.79887 .90837 .09163 .89050
BO
1
.79903 .90863 .09137 .89040
59
2
.79918 .90889 .09111 .89030
58
3
.79934 .90914 .09086 .89020
57
4
.79950 .90940 .09060 .89009
56
5
.79965 .90966 .09034 .88999
55
6
.79981 .90992 .09008 .88989
54
7
.79996 .91018 .08982 .88978
53
8
.80012 .91043 .08957 .88968
52
9
.80027 .91069 .08931 .88958
51
10
.80043 .91095 .08905 .88948
50
11
.80058 .91121 .08879 .88937
49
12
.80074 .91147 .08853 .88927
48
13
.80089 .91172 .08828 .88917
47
14
.80105 .91198 .08802 .88906
46
15
.80120 .91224 .08776 .88896
45
16
.80136 .91250 .08750 .88886
44
17
.80151 .91276 .08724 .88875
43
18
.80166 .91301 .08699 .88865
42
19
.80182 .91327 .08673 .88855
41
2O
.80197 .91353 .08647 .88844
4O
21
.80213 .91379 .08621 .88834
39
22
.80228 .91404 .08596 .88824
38
23
.80244 .91430 .08570 .88813
37
24
.80259 .91456 .08544 .88803
36
25
.80274 .91482 .08518 .88793
35
26
.80290 .91507 .08493 .88782
34
27
.80305 .91533 .08467 .88772
33
28
.80320 .91559 .08441 .88761
32
29
.80336 .91585 .08415 .88751
31
30
.80351 .91610 .08390 .88741
3O
31
.80366 .91636 .08364 .88730
29
32
.80382 .91662 .08338 .88720
28
33
.80397 .91688 .08312 .88709
27
34
.80412 .91713 .08287 .88699
26
35
.80428 .91739 .08261 .88688
25
36
.80443 .91765 .08235 .88678
24
37
.80458 .91791 .08209 .88668
23
38
.80473 .91816 .08184 .88657
22
39
.80489 .91842 .08158 .88647
21
4O
.80504 .91868 .08132 .88636
20
41
.80519 .91893 .08107 .88626
19
42
.80534 .91919 .08081 .88615
18
43
.80550 .91945 .08055 .88605
17
44
.80565 .91971 .08029 .88594
16
45
.80580 .91996 .08004 .88584
15
46
.80595 .92022 .07978 .88573
14
47
.80610 .92048 .07952 .88563
13
48
.80625 .92073 .07927 .88552
12
49
.80641 .92099 .07901 .88542
11
50
.80656 .92125 .07875 .88531
10
51
.80671 .92150 .07850 .88521
9
52
.80686 .92176 .07824 .88510
8
53
.80701 .92202 .07798 .88499
7
54
.80716 .92227 .07773 .88489
6
55
.80731 .92253 .07747 .88478
5
56
.80746 .92279 .07721 .88468
4
57
.80762 .92304 .07696 .88457
3
58
.80777 .92330 .07670 .88447
2
59
.80792 .92356 .07644 .88436
1
60
.80807 .92381 .07619 .88425
O
/
9Lcos 9Lcot lOLtan 9Lsin
/
/
9Lsin
9Ltan
10 L cot
9Lcos
/
O
.80807
.92381
.07 619
.88425
60
1
.80822
.92407
.07 593
.88415
59
2
.80837
.92433
.07 567
.88404
58
3
.80852
.92458
.07 542
.88394
57
4
.80867
.92484
.07516
.88383
56
5
.80882
.92510
.07 490
.88372
55
6
.80897
.92 535
.07 465
.88362
54
7
.80912
.92561
.07 439
.88351
53
8
.80927
.92 587
.07413
.88340
52
9
.80942
.92612
.07388
.88330
51
10
.80957
.92638
.07 362
.88319
50
11
.80972
.92663
.07 337
.88308
49
12
.80987
.92 689
.07311
.88 298
48
13
.81 002
.92 715
.07 285
.88 287
47
14
.81017
.92 740
.07 260
.88276
46
15
.81032
.92 766
.07 234
.88 266
45
16
.81 047
.92 792
.07 208
.88255
44
17
.81061
.92817
.07 183
.88 244
43
18
.81076
.92 843
.07157
.88 234
42
19
.81 091
.92 868
.07 132
.88223
41
2O
.81 106
.92894
.07 106
.88212
40
21
.81 121
.92920
.07080
.88 201
39
22
.81 136
.92 945
.07055
.88 191
38
23
.81151
.92971
.07029
.88 180
37
24
.81 166
.92996
.07004
.88 169
36
25
.81 180
.93022
.06978
.88 158
35
26
.81 195
.93048
.06952
.88 148
34
27
.81210
.93073
.06927
.88137
33
28
.81 225
.93099
.06901
.88126
32
29
.81 240
.93 124
.06876
.88115
31
3O
.81 254
.93 150
.06850
.88 105
3O
31
.81269
.93175
.06825
.88094
29
32
.81 284
.93 201
.06 799
.88083
28
33
.81 299
.93 227
.06 773
.88072
27
34
.81314
.93252
.06748
.88061
26
35
.81328
.93 278
.06 722
.88051
25
36
.81 343
.93 303
.06697
.88040
24
37
.81358
.93329
.06671
.88029
23
38
.81372
.93 354
.06646
.88018
22
39
.81387
.93380
.06620
.88007
21
4O
.81 402
.93 406
.06594
.87996
20
41
.81417
.93431
.06569
.87985
19
42
.81431
.93457
.06543
.87975
18
43
.81 446
.93 482
.06518
.87964
17
44
.81461
.93 508
.06492
.87953
16
45
.81 475
.93 533
.06467
.87942
15
46
.81 490
.93 559
.06441
.87931
14
47
.81 505
.93 584
.06416
.87 920
13
48
.81519
.93610
.06390
.87 909
12
49
.81 534
.93 636
.06364
.87 898
11
50
.81 549
.93 661
.06339
.87 887
1O
51
.81 563
.93 687
.06313
.87877
9
52
.81578
.93 712
.06288
.87 866
8
53
.81 592
.93 738
.06262
.87855
7
54
.81 607
.93 763
.06237
.87844
6
55
.81622
.93 789
.06211
.87 833
5
56
.81636
.93 814
.06 186
.87822
4
57
.81651
.93 840
.06 160
.87811
3
58
.81 665
.93 865
.06 135
.87 800
2
59
.81680
.93 891
.06 109
.87 789
1
60
.81 694
.93 916
.06084
.87 778
r
9Lcos
9Lcot
lOLtan 9Lsin
'
50 C
49'
41 C
42 C
/
9Lsin 9Ltan lOLcot 9Lcos /
o
.81694 .93916 .06084 .87778
GO
1
.81709 .93942 .06058 .87767
59
2
.81723 .93967 .06033 .87756
58
3
.81738 .93993 .06007 .87745
57
4
.81752 .94018 .05982 .87734
56
5
.81767 .94044 .05956 .87723
55
6
.81781 .94069 .05931 .87712
54
*7
.81796 .94095 .05905 .87701
53
8
.81810 .94120 .05880 .87690
52
9
.81825 .94146 .05854 .87679
51
10
.81839 .94171 .05829 .87668
50
11
.81854 .94197 .05803 .87657
49
12
.81868 .94222 .05778 .87646
48
13
.81882 .94248 .05752 .87635
47
14
.81897 .94273 .05727 .87624
46
15
.81911 .94299 .05701 .87613
45
16
.81926 .94324 .05676 .87601
44
17
.81940 .94350 .05650 .87590
43
18
.81955 .94375 .05625 .87579
42
19
.81969 .94401 .05599 .87568
41
20
.81983 .94426 .05574 .87557
40
21
.81998 .94452 .05548 .87546
39
22
.82012 .94477 .05523 .87535
38
23
.82026 .94503 .05497 .87524
37
24
.82041 .94528 .05472 .87513
36
25
.82055 .94554 .05446 .87501
35
26
.82069 .94579 .05421 .87490
34
27
.82084 .94604 .05396 .87479
33
28
.82098 .94630 .05370 .87468
32
29
.82112 .94655 .05345 .87457
31
3O
.82126 .94681 .05319 .87446
3O
31
.82141 .94706 .05294 .87434
29
32
.82155 .94732 .05268 .87423
28
33
.82169 .94757 .05243 .87412
27
34
.82184 .94783 .05217 .87401
26
35
.82198 .94808 .05192 .87390
25
36
.82212 .94834 .05166 .87378
24
37
.82226 .94859 .05141 .87367
23
38
.82240 .94884 .05116 .87356
22
39
.82255 .94910 .05090 .87345
21
4O
.82269 .94935 .05065 .87334
20
41
.82283 .94961 .05039 .87322
19
42
.82297 .94986 .05014 .87311
18
43
.82311 .95012 .04988 .87300
17
44
.82326 .95037 .04963 .87288
16
45
.82340 .95062 .04938 .87277
15
46
.82354 .95088 .04912 .87266
14
47
.82368 .95113 .04887 .87255
13
48
.82382 .95139 .04861 .87243
12
49
.82396 .95164 .04836 .87232
11
5O
.82410 .95190 .04810 .87221
1O
51
.82424 .95215 .04785 .87209
9
52
.82439 .95240 .04760 .87198
8
53
.82453 .95266 .04734 .87187
7
54
.82467 .95291 .04709 .87175
6
55
.82481 .95317 .04683 .87164
5
56
.82495 .95342 .04658 .87153
4
57
.82509 .95368 .04632 .87141
3
58
.82523 .95393 .04607 .87130
2
59
.82537 .95418 .04582 .87119
1
60
.82551 .95444 .04556 .87107
/
9Lcos 9LcotlOLtan9Lsin
/
/
9Lsin
9Ltan
lOLcot 9 Lcos |_/_
O
.82551
.95 444
.04 556
.87 107
GO
1
.82 565
.95 469
.04531
.87 096
59
2
.82579
.95 495
.04 505
.87085
58
3
.82 593
.95 520
.04480
.87073
57
4
.82607
.95 545
.04 455
.87062
56
5
.82621
.95 571
.04 429
.87050
55
6
.82635
.95 596
.04 404
.87 039
54
7
.82649
.95 622
.04378
.87028
53
8
.82 663
.95647
.04353
.87016
52
9
.82677
.95 672
.04328
.87 005
51
1O
.82691
.95 698
.04 302
.86993
50
11
.82 705
.95 723
.04 277
.86982 i 49
12
.82719
.95 748
.04 252
.86970 48
13
.82 733
.95 774
.04 226
.86959 47
14
.82 747
.95 799
.04 201
.86947
46
15
.82 761
.95 825
.04 175
.86936
45
16
.82775
.95 850
.04 150
.86924
44
17
.82 788
.95 875
.04 125
.86913
43
18
.82 802
.95 901
.04 099
.86902
42
19
.82816
.95 926
.04074
.86890
41
20
.82830
.95952
.04048
.86879
4O
21
.82 844
.95 977
.04023
.86867
39
22
.82858
.96002
.03 998
.86855
38
23
.82872
.96028
.03 972
.86844
37
24
.82885
.96 053
.03947
.86832
36
25
.82 899
.96078
.03 922
.86821
35
26
.82913
.96 104
.03 896
.86809
34
27
.82927
.96129
.03871
.86 798
33
28
.82941
.96 155
.03 845
.86 786
32
29
.82955
.96 180
.03 820
.86 775
31
30
.82968
.96205
.03 795
.86 763
30
31
.82982
.96231
.03 769
.86752
29
32
.82996
.96256
.03 744
.86 740
28
33
.83 010
.96 281
.03 719
.86 728
27
34
.83 023
.96307
.03 693
.86717
26
35
.83 037
.96332
.03 668
.86705
25
36
.83051
.96357
.03 643
.86694
24
37
.83 065
.96383
.03617
.86682
23
38
.83 078
.96408
.03 592
.86670
22
39
.83 092
.96433
.03 567
.86659
21
40
.83 106
.96459
.03 541
.86647
20
41
.83120
.96484
.03516
.86635
19
42
.83 133
.96510
.03 490
.86624
18
43
.83 147
.96535
.03465
.86612
17
44
.83 161
.96560
.03 440
.86600
16
45
.83 174
.96 586
.03 414
.86589
15
46
.83 188
.96611
.03389
.86577
14
47
.83 202
.96636
.03 364
.86 565
13
48
.83215
.96662
.03 338
.86554
12
49
.83 229
.96687
.03313
.86542
11
50
.83 242
.96712
.03 288
.86530
1O
51
.83 256
.96 738
.03 262
.86518
9
52
.83 270
.96 763
.03 237
.86 507
8
53
.83 283
.96 788
.03 212
.86495
7
54
.83 297
.96814
.03 186
.86483
6
55
.83310
.96839
.03 161
.86472
5
56
.83 324
.96864
.03 136
.86460
4
57
.83338
.96890
.03 110
.86448
3
58
.83351
.96915
.03085
.86436
2
59
.83365
.96940
.03 060
.86425
1
6O
.83 378
.96966
.03034
.86413
/
9Lcos
9LcotlOLtan
9 L sin /
48'
47
43'
44 C
57
/
9 L sin 9 L tan 1O L cot 9 L cos
/
o
.83378 .96966 .03034 .86413
W
1
.83392 .96991 .03009 .86401
59
2
.83405 .97016 .02984 .86389
58
3
.83419 .97042 .02958 .86377
57
4
.83.432 .97067 .02933 .86366
56
5
.83446 .97092 .02908 .86354
55
6
.83459 .97118 .02882 .86342
54
7
.83473 .97143 .02857 .86330
53
8
.83486 .97168 .02832 .86318
52
9
.83500 .97193 .02807 .86306
51
1C
.83513 .97219 .02781 .86295
50
11
.83527 .97244 .02756 .86283
49
12
.83540 .97269 .02731 .86771
48
13
.83554 .97295 .02705 .86259
47
14
.83567 .97320 .02680 .86247
46
IS
.83581 .97345 .02655 .86235
45
16
.83594 .97371 .02629 .86223
44
17
.83608 .97396 .02604 .86211
43
18
.83621 .97421 .02579 .86200
42
19
.83634 .97447 .02553 .86188
41
2O
.83648 .97472 .02528 .86176
40
21
.83661 .97497 .02503 .86164
39
22
.83674 .97523 .02477 .86152
38
23
.83688 .97548 .02452 .86140
37
24
.83701 .97573 .02427 .86128
36
25
.83715 .97598 .02402 .86116
35
26
.83728 .97624 .02376 .86104
34
27
.83741 .97649 .02351 .86092
33
28
.83755 .97674 .02326 .86080
32
29
.83768 .97700 .02300 .86068
31
30
.83781 .97725 .02275 .86056
30
31
.83795 .97750 .02250 .86044
29
32
.83808 .97776 .02224 .86032
28
33
.83821 .97801 .02199 .86020
27
34
.83834 .97826 .02174 .86008
26
35
.83848 .97851 .02149 .85996
25
36
.83861 .97877 .02123 .85984
24
37
.83874 .97902 .02098 .85972
23
38
.83887 .97927 .02073 .85960
22
39
.83901 .97953 .02047 .85948
21
4O
.83914 .97978 .02022 .85936
2O
41
.83927 .98003 .01997 .85924
19
42
.83940 .98029 .01971 .85912
18
43
.88954 .98054 .01946 .85900
17
44
.83967 .98079 .01921 .85888
16
45
.83980 .98104 .01896 .85876
15
46
.83993 .98130 .01870 .85864
14
47
.84006 .98155 .01845 .85851
13
48
.84020 .98180 .01820 .85839
12
49
.84033 .98206 .01794 .85827
11
50
.84046 .98231 .01769 .85815
1O
51
.84059 .98256 .01744 .85803
9
52
.84072 .98281 .01719 .85791
8
53
.84085 .98307 .01693 .85779
7
54
.84098 .98332 .01668 .85766
6
55
.84112 .98357 .01643 .85754
5
56
.84125 .98383 .01617 .85742
4
57
.84138 .98408 .01592 .85730
3
58
.84151 .98433 .01567 .85718
2
59
.84164 .98458 .01542 .85706
1
60
.84177 .98484 .01516 .85693
/
9Lcos 9Lcot 10 L tan 9Lsin
/
/
9 L sin 9 L tan 1O L cot 9 L cos
/
~cT
.84177 .98484 .01516 .85693
6O
i
.84190 .98509 .01491 .85681
59
2
.84203 .98534 .01466 .85669
58
3
.84216 .98560 .01440 .85657
57
4
.84229 .98585 .01415 .85645
56
5
.84242 .98610 .01390 .85632
55
6
.84255 .98635 .01365 .85620
54
.7
.84269 .98661 .01339 .85608
53
8
.84282 .98686 .01314 .85596
52
9
.84295 .98711 .01289 .85583
51
1O
.84308 .98737 .01263 .85571
50
11
.84321 .98762 .01238 .85559
49
12
.84334 .98787 .01213 .85547
48
13
.84347 .98812 .01188 .85534
47
14
.84360 .98838 .01162 .85522
46
15
.84373 .98863 .01137 .85510
45
16
.84385 .98888 .01112 .85497
44
17
.84398 .98913 .01087 .85485
43
18
.84411 .98939 .01061 .85473
42
19
.84424 .98964 .01036 .85460
41
2O
.84437 .98989 .01011 .85448
4O
21
.84450 .99015 .00985 .85436
39
22
.84463 .99040 .00960 .85423
38
23
.84476 .99065 .00935 .85411
37
24
.84489 .99090 .00910 .85399
36
25
.84502 .99116 .00884 .85386
35
26
.84515 .99141 .00859 .85374
34
27
.84528 .99166 .00834 .85361
33
28
.84540 .99191 .00809 .85349
32
29
.84553 .99217 .00783 .85337
31
30
.84566 .99242 .00758 .85324
3O
31
.84579 .99267 .00733 .85312
29
32
.84592 .99293 .00707 .85299
28
33
.84605 .99318 .00682 .85287
27
34
.84618 .99343 .00657 .85274
26
35
.84630 .99368 .00632 .85262
25
36
.84643 .99394 .00606 .85250
24
37
.84656 .99419 .00581 .85237
23
38
.84669 .99444 .00556 .85225
22
39
.84682 .99469 .00531 .85212
21
40
.84694 .99495 .00505 .85200
20
41
.84707 .99520 .00480 .85187
19
42
.84720 .99545 .00455 .85175
18
43
.84733 .99570 .00430 .85162
17
44
.84745 .99596 .00404 .85150
16
45
.84758 .99621 .00379 .85137
15
46
.84771 .99646 .00354 .85125
14
47
.84784 .99672 .00328 .85112
13
48
.84796 .99697 .00303 .85100
12
49
.84809 .99722 .00278 .85087
11
50
.84822 .99747 .00253 .85074
1O
51
.84835 .99773 .00227 .85062
9
52
.84847 .99798 .00202 .85049
8
53
.84860 .99823 .00177 .85037
7
54
.84873 .99848 .00152 .85024
6
55
.84885 .99874 .00126 .85012
5
56
.84898 .99899 .00101 .84999
4
57
.84911 .99924 .00076 .84986
3
58
.84923 .99949 .00051 .84974
2
59
.84936 .99975 .00025 .84961
1
6O
.84949 .00000 .00000 .S49J2_.
O
/
9 L cos 1O L cot 1O L tan 9~lTsiii
/
46'
45<
58
TABLE IV NATURAL FUNCTIONS
/
sin tan cot cos
/
o
.00000 .00000 co 1.0000
60
1
.00029 .00029 3437.7 1.0000
59
2
.00058 .00058 1718.9 1.0000
58
3
.00087 .00087 1145.9 1.0000
57
4
.00116 .00116 859.44 1.0000
56
5
.00 145 .00 145 687.55 1.0000
55
6
.00 175 .00175 572.96 1.0000
54
7
.00204 .00204 491.11 1.0000
53
8
.00233 .00233 429.72 1.0000
52
9
.00262 .00262 381.97 1.0000
51
1C
.00291 .00291 343.77 1.0000
5O
11
.00320 .00320 312.52 .99999
49
12
.00349 .00349 286.48 .99999
48
13
.00378 .00378 264.44 .99999
47
14
.00407 .00407 245.55 .99999
46
15
.00436 .00436 229.18 .99999
45
16
.00465 .00465 214.86 .99999
44
17
.00495 .00495 202.22 .99999
43
18
.00524 .00524 190.98 .99999
42
19
.00553 .00553 180.93 .99998
41
20
.00 582 -.00 582 171.89 .99 998
4O
21
.00611 .00611 163.70 .99998
39
22
.00640 .00640 156.26 .99998
38
23
.00669 .00669 149.47 .99998
37
24
.00698 .00698 143.24 .99998
36
25
.00727 .00727 137.51 .99997
35
26
.00756 .00756 132.22 .99997
34
27
.00785 .00785 127.32 .99997
33
28
.00814 .00815 122.77 .99997
32
29
.00844 .00844 118.54 .99996
31
30
.00873 .00873 114.59 .99996
3O
31
.00902 .00902 110.89 .99996
29
32
.00931 .00931 107.43 .99996
28
33
.00960 .00960 104.17 .99995
27
34
.00989 .00989 101.11 .99995
26
35
.01018 .01018 98.218 .99995
25
36
.01047 .01047 95.489 .99995
24
37
.01 076 .01 076 92.908 .99 994
23
38
.01105 .01 105 90.463 .99994
22
39
.01 134 .01 135 88.144 .99994
21
40
.01 164 .01 164 85.940 .99993
20
41
.01 193 .01 193 83.844 .99 993
19
42
.01222 .01222 81.847 .99993
18
43
.01251 .01251 79.943 .99992
17
44
.01280 .01280 78.126 .99992
16
45
.01 309 .01 309 76.390 .99 991
15
46
.01338 .01338 74.729 .99991
14
47
.01367 .01367 73.139 .99991
13
48
.01396 .01396 71.615 .99990
12
49
.01425 .01425 70.153 .99990
11
50
.01 454 .01 455 68.750 .99 989
10
51
.01483 .01484 67.402 .99989
9
52
.01513 .01 513 66.105 .99989
8
53
.01542 .01 542 64.858 .99988
7
54
.01571 .01571 63.657 .99988
6
55
.01 600 .01 600 62.499 .99987
5
56
.01629 .01629 61.383 .99987
4
57
.01 658 .01 658 60.306 .99 986
3
58
.01 687 .01 687 59.266 .99 986
2
59
.01716 .01 716 58.261 .99985
1
6O
.01 745 .01 746 57.290 .99985
O
/
cos cot tan sin
/
89
1
/
sin tan cot cos
/
O
.01 745 .01 746 57.290 .99 985
00
1
.01774 .01775 56.351 .99984
59
2
.01803 .01804 55.442 .99984
58
3
.01832 .01833 54.561 .99983
57
4
.01862 .01862 53.709 .99983
56
5
.01891 .01891 52.882 .99982
55
6
.01920 .01 920 52.081 .99982
54
7
.01949 .01949 51.303 .99981
53
8
.01978 .01978 50.549 .99980
52
9
.02007 .02007 49.816 .99980
51
1O
.02036 .02036 49.104 .99979
50
11
.02065 .02066 48.412 .99979
49
12
.02094 .02095 47.740 .99978
48
13
.02123 .02124 47.085 .99977
47
14
.02152 .02153 46.449 .99977
46
15
.02181 .02182 45.829 .99976
45
16
.02211 .02211 45.226 .99976
44
17
.02 240 .02 240 44.639 .99 975
43
18
.02 269 .02 269 44.066 .99 974
42
19
.02298 .02298 43.508 .99974
41
2O
.02327 .02328 42.964 .99973
40
21
.02356 .02357 42.433 .99972
39
22
.02385 .02386 41.916 .99972
38
23
.02414 .02415 41.411 .99971
37
24
.02443 .02444 40.917 .99970
36
25
.02 472 .02 473 40.436 .99 969
35
26
.02501 .02502 39.965 .99969
34
27
.02530 .02531 39.506 .99968
33
28
.02560 .02560 39.057 .99967
32
29
.02589 .02589 38.618 .99966
31
3O
.02618 .02619 38.188 .99966
30
31
.02647 .02648 37.769 .99965
29
32
.02676 .02677 37.358 .99964
28
33
.02 705 .02 706 36.956 .99 963
27
34
.02 734 .02 735 36.563 .99 963
26
35
.02763 .02764 36.178 .99962
25
36
.02792 .02793 35.801 .99961
24
37
.02821 .02822 35.431 .99960
23
38
.02850 .02851 35.070 .99959
22
39
.02879 .02881 34.715 .99959
21
40
.02908 .02910 34.368 .99958
20
41
.02938 .02939 34.027 .99957
19
42
.02967 .02968 33.694 .99956
18
43
.02996 .02997 33.366 .99955
17
44
.03 025 .03 026 33.045 .99 954
16
45
.03054 .03055 32.730 .99953
15
46
.03083 .03084 32.421 .99952
14
47
.03112 .03114 32.118 .99952
13
48
.03141 .03143 31.821 .99951
12
49
.03 170 .03 172 31.528 .99950
11
50
.03199 .03201 31.242 .99949
10
51
.03 228 .03 230 30.960 .99 948
9
52
.03257 .03259 30.683 .99947
8
53
.03 286 .03 288 30.412 .99 946
7
54
.03316 .03317 30.145 .99945
6
55
.03345 .03346 29.882 .99944
5
56
.03374 .03376 29.624 .99943
4
57
.03403 .03405 29.371 .99942
3
58
.03432 .03434 29.122 .99941
2
59
.03461 .03463 28.877 .99940
1
GO
.03 490 .03 492 28.636 .99 939
O
/
cos cot tan sin
/
88
NATURAL FUNCTIONS
59
2
/
sin tan cot cos
/
.03490 .03492 28.636 .99939
60
1
.03519 .03521 28.399 .99938
59
2
.03548 .03550 28.166 .99937
58
3
.03577 .03579 27.937 .99936
57
4
.03 606 .03 609 27.712 .99 935
56
5
.03 635 .03 638 27.490 .99 934
55
6
.03664 .03667 27.271 .99933
54
7
.03693 .03696 27.057 .99932
53
8
.03723 .03725 26.845 .99931
52
9
.03 752 .03 754 26.637 .99 930
51
1C
.03 781 .03 783 26.432 .99 929
5O
11
.03810 .03812 26.230 .99927
49
12
.03839 .03842 26.031 .99926
48
13
.03868 .03871 25.835 .99925
47
14
.03897 .03900 25.642 .99924
46
15
.03 926 .03 929 25.452 .99 923
45
16
.03955 .03958 25.264 .99922
44
17
.03984 .03987 25.080 .99921
43
18
.04013 .04016 24.898 .99919
42
19
.04042 .04046 24.719 .99918
41
2O
.04071 .04075 24.542 .99917
4O
21
.04 100 .04 104 24.368 .99 916
39
22
.04129 .04133 24.196 .99915
38
23
.04 159 .04 162 24.026 .99913
37
24
.04188 .04191 23.859 .99912
36
25
.04217 .04220 23.695 .99911
35
26
.04 246 .04 250 23.532 .99 910
34
27
.04 275 .04 279 23.372 .99 909
33
28
.04304 .04308 23.214 .99907
32
29
.04333 .04337 23.058 .99906
31
3O
.04362 .04366 22.904 .99905
30
31
.04391 .04395 22.752 .99904
29
32
.04420 .04424 22.602 .99902
28
33
.04449 .04454 22.454 .99901
27
34
.04478 .04483 22.308 .99900
26
35
.04507 .04512 22.164 .99898
25
36
.04536 .04541 22.022 .99897
24
37
.04565 .04570 21.881 .99896
23
38
.04 594 .04 599 21.743 .99 894
22
39
.04623 .04628 21.606 .99893
21
40
.04653 .04658 21.470 .99892
2O
41
.04682 .04687 21.337 .99890
19
42
.04711 .04716 21.205 .99889
18
43
.04 740 .04 745 21.075 .99 888
17
44
.04 769 .04 774 20.946 .99 886
16
45
.04 798 .04 803 20.819 .99 885
15
46
.04827 .04833 20.693 .99883
14
47
.04856 .04862 20.569 .99882
13
48
.04885 .04891 20.446 .99881
12
49
.04914 .04920 20.325 .99879
11
5O
.04943 .04949 20.206 .99878
1O
51
.04 972 .04 978 20.087 .99 876
9
52
.05 001 .05 007 19.970 .99 875
8
53
.05 030 .05 037 19.855 .99 873
7
54
.05 059 .05 066 19.740 .99 872
6
55 i .05 OSS .05 095 19.627 .99 870
5
56
.05117 .05124 19.516 .99869
4
57
.05 146 .05 153 19.405 .99867
3
58
.05 175 .05 182 19.296 .99866
2
59
.05 205 .05 212 19.188 .99 864
1
60
.05 234 .05 241 19.081 .99 863
/
cos cot tan sin
/
87
3
/
sin tan cot cos
/
.05 234 .05 241 19.081 .99 863
6O
1
.05 263 .05 270 18.976 .99 861
59
2
.05 292 .05 299 18.871 .99 860
58
3
.05 321 .05 328 18.768 .99 858
57
4
.05350 .05357 18.666 .99857
56
5
.05 379 .05 387 18.564 .99 855
55
6
.05408 .05416 18.464 .99854
54
7
.05437 .05445 18.366 .99852
53
8
.05466 .05474 18.268 .99851
52
9
.05495 .05503 18.171 .99849
51
1O
.05 524 .05 533 18.075 .99 847
50
11
.05 553 .05 562 17.980 .99 846
49
12
.05582 .05591 17.886 .99844
48
13
.05611 .05620 17.793 .99842
47
14
.05 640 .05 649 17.702 .99 841
46
15
.05669 .05678 17.611 .99839
45
16
.05.698 ,05 708 17.521 a83&
44
17
.05727 .05737 17.431 .99836
43
18
.05 756 .05 766 17.343 .99834
42
19
.05785 .05795 17.256 .99833
41
20
.05 814 .05 824 17.169 .99 831
4O
21
.05 844 .05 854 17.084 .99 829
39
22
.05 873 .05 883 16.999 .99 827
38
23
.05902 .05912 16.915 .99826
37
24
.05931 .05941 16.832 .99824
36
25
.05 960 .05 970 16.750 .99 822
35
26
.05989 .05999 16.668 .99821
34
27
.06018 .06029 16.587 .99819
33
28
.06047 .06058 16.507 .99817
32
29
.06076 .06087 16.428 .99815
31
30
.06105 .06116 16.350 .99813
3O
31
.06 134 .06 145 16.272 .99 812
29
32
.06163 .06175 16.195 .99810
28
33
.06192 .06204 16.119 .99808
27
34
.06221 .06233 16.043 .99806
26
35
.06250 .06262 15.969 .99804
25
36
.06279 .06291 15.895 .99803
24
37
.06308 .06321 15.821 .99801
23
38
.06337 .06350 15.748 .99799
22
39
.06 366 .06 379 15.676 .99 797
21
4O
.06395 .06408 15.605 .99795
20
41
.06424 .06438 15.534 .99793
19
42
.06453 .06467 15.464 .99792
18
43
.06482 .06496 15.394 .99790
17
44
.06511 .06525 15.325 .99788
16
45
.06 540 .06 554 15.257 .99 786
15
46
.06569 .06584 15.189 .99784
14
47
.06 59S .06 613 15.122 .99 782
13
48
.06627 .06642 15.056 .99780
12
49
.06 656 .06 671 14.990 .99 778
11
50
.06685 .06700 14.924 .99776
10
51
.06714 .06730 14.860 .99774
9
52
.06743 .06759 14.795 .99772
8
53
.06 773 .06 788 14.732 .99 770
7
54
.06802 .06817 14.669 .99768
6
55
.06831 .06847 14.606 .99766
5
56
.06 860 .06 876 14.544 .99 764
4
57
.06889 .06905 14.482 .99762
3
58
.06918 .06934 14.421 .99760
2
59
.06947 .06963 14.361 .99758
1
6O
.06 976 .06 993 14.301 .99 756
O
/
cos cot tan sin
/
86
60
NATURAL FUNCTIONS
4
/
sin tan cot cos
/
o
.06976 .06993 14.301 .99756
6O
1
.07005 .07022 14.241 .99754
59
2
.07034 .07051 14.182 .99752
58
3
.07063 .07080 14.124 .99750
57
4
.07 092 .07 110 14.065 .99 748
56
5
.07 121 .07 139 14.008 .99 746
55
6
.07 150 .07 168 13.951 .99 744
54
7
.07 179 .07 197 13.894 .99 742
53
8
.07208 .07227 13.838 .99740
52
9
.07 237 .07 256 13.782 .99 738
51
10
.07 266 .07 285 13.727 .99 736
50
11
.07295 .07314 13.672 .99734
49
12
.07324 .07344 13.617 .99731
48
13
.07353 .07373 13.563 .99729
47
14
.07382 .07402 13.510 .99727
46
15
.07411 .07431 13.457 .99725
45
16
.07 440 .07 461 13.404 .99 723
44
17
.07 469 .07 490 13.352 .99 721
43
18
.07498 .07519 13.300 .99719
42
19
.07527 .07548 13.248 .99716
41
20
.07556 .07578 13.197 .99714
4O
21
.07585 .07607 13.146 .99712
39
22
.07 614 .07 636 13.096 .99 710
38
23
.07 643 .07 665 13.046 .99 708
37
24
.07 672 .07 695 12.996 .99 705
36
25
.07 701 .07 724 12.947 .99 703
35
26
.07 730 .07 753 12.898 .99 701
34
27
.07 759 .07 782 12.850 .99 699
33
28
.07 788 .07 812 12 801 .99 696
32
29
.07817 .07841 12.754 .99694
31
30
.07846 .07870 12.706 .99692
3O
31
.07 875 .07 899 12.659 .99 689
29
32
.07904 .07929 12.612 .99687
28
33
.07933 .07958 12.566 .99685
27
34
.07962 .07987 12.520 .99683
26
35
.07991 .08017 12.474 .99680
25
36
.08020 .08046 12.429 .99678
24
37
.080*9 .08075 12.384 .99676
23
38
.08 078 .08 104 12.339 .99 673
22
39
.08107 .08134 12.295 .99671
21
40
.08136 .08163 12.251 .99668
20
41
.08 165 .08 192 12.207 .99 666
19
42
.08 194 .08 221 12.163 .99 664
18
43
.08223 .08251 12.120 .99661
17
44
.08252 .08280 12.077 .99659
16
45
.08 281 .08 309 12.035 .99 657
15
46
.08310 .08339 11.992 .99654
14
47
.08339 .08368 11.950 .99652
13
48
.08368 .08397 11.909 .99649
12
49
.08397 .08427 11.867 .99647
11
50
.08426 .08456 11.826 .99644
1O
51
.08455 .08485 11.785 .99642
9
52
.08484 .08514 11.745 .99639
8
53
.08513 .08544 11.705 .99637
7
54
.08542 .08573 11.664 .99635
6
55
.08571 .08602 11.625 .99632
5
56
.08600 .08632 11.585 .99630
4
57
.08629 .08661 11.546 .99627
3
58
.08658 .08690 11.507 .99625
2
59
.08687 .08720 11.468 .99622
1
60
.08716 .08749 11.430 .99619
/
cos cot tan sin
/
85
5
/
sin tan cot cos
/
O
.08 716 .08 749 11.430 .99 619
TK>
1
.08745 .08778 11.392 .99' 617
59
2
.08774 .08807 11.354 .99614
58
3
.08803 .08837 11.316 .99612
57
4
.08831 .08866 11.279 .99609
56
5
.08860 .08895 11.242 .99607
55
6
.08889 .08925 11.205 .99604
54
7
.08918 .08954 11.168 .99602
.53
8
.08947 .08983 11.132 .99599
52
9
.08976 .09013 11.095 .99596
51
10
.09005 .09042 11.059 .99594
5O
11
.09034 .09071 11.024 .99591
49
12
.09 063 .09 101 10.988 .99 588
48
13
.09092 .09130 10.953 .99586
47
14
.09 121 .09 159 10.918 .99 583
46
15
.09150 .09189 10.883 .99580
45
16
.09179 .09218 10.848 .99578
44
17
.09208 .09247 10.814 .99575
43
18
.09237 .09277 10.780 .99572
42
19
.09 266 .09 306 10.746 .99 570
41
20
.09295 .09335 10.712 .99567
40
21
.09324 .09365 10.678 .99564
39
22
.09353 .09394 10.645 .99562
38
23
.09382 .09423 10.612 .99559
37
24
.09411 .09453 10.579 .99556
36
25
.09440 .09482 10.546 .99553
35
26
.09469 .09511 10.514 .99551
34
27
.09498 .09541 10.481 .99548
33
28
.09527 .09570 10.449 .99545
32
29
.09556 .09600 10.417 .99542
31
30
.09585 .09629 10.385 .99540
30
31
.09614 .09658 10.354 .99537
29
32
.09642 .09688 10.322 .99534
28
33
.09671 .09717 10.291 .99531
27
34
.09 700 .09 746 10.260 .99 528
26
35
.09 729 .09 776 10.229 .99 526
25
36
.09758 .09805 10.199 .99523
24
37
.09 787 .09 834 10.168 .99 520
23
38
.09816 .09864 10.138 .99517
22
39
.09845 .09893 10.108 .99514
21
40
.09874 .09923 10.078 .99511
20
41
.09903 .09952 10.048 .99508
19
42
.09932 .09981 10.019 .99506
18
43
.09961 .10011 9.9893 .99503
17
44
.09990 .10040 9.9601 .99500
16
45
.10019 .10069 9.9310 .99497
15
46
.10048 .10099 9.9021 .99494
14
47
.10077 .10128 9.8734 .99491
13
48
.10106 .10158 9.8448 .99488
12
49
.10 135 .10 187 9.8164 .99 485
11
50
.10164 .10216 9.7882 .99482
1O
51
.10 192 .10 246 9.7601 .99 479
9
52
.10221 .10275 9.7322 .99476
8
53
.10250 .10305 9.7044 .99473
7
54
.10279 .10334 9.6768 .99470
6
55
.10308 .10363 9.6493 .99467
5
56
.10337 .10393 9.6220 .99464
4
57
.10366 .10422 9.5949 .99461
3
58
.10395 .10452 9.5679 .99458
2
59
.10424 .10481 9.5411 .99455
1
6O
.10453 .10510 9.5144 .99452
O
/
cos cot tan sin
/
84
NATURAL FUNCTIONS
61
6 D
/
sin tan cot cos
/
o
.10453 .10510 9.5144 .99452
GO
1
.10482 JO 540 9.4878 .99449
59
2
.10511 .10569 9.46L4 .99446
58
3
.10540 .10599 9.4352 .99443
57
4
.10569 .10628 9.4090 .99440
56
5
.10597 .10657 9.3S31 .99437
55
6
.10626 .10687 9.3572 .99434
54
7
.106S5 .10716 93315 .99431
53
8
.10681 .10746 9.3060 .99428
52
9 .10713 .10775 9.2806 .99-124
51
10
.10742 .10805 9.2553 .99421
50
11
.10771 .10834 9.2302 .99418
49
12
.10800 .10863 9.2052 .99415
48
13
.10829 .10893 9.1803 .99412
47
14
.10858 .10922 9.1555 .99409
46
15
.10837 .10952 9.1309 .99406
45
16
.10916 .10981 9.1065 .99402
44
17
.10945 .11011 9.0821 .99399
43
18
.10973 .11040 9.0579 .99396
42
19
.11002 .11070 9.0338 .99393
41
20
.11031 .11099 9.0098 .99390
4O
21
.11050 .11 128 8.9860 .99386
39
22
.11089 .11 158 8.9623 .99383
38
23
.11 118 .11 187 8.9387 .99380
37
24
.11 147 .11217 8.9152 .99377
36
25
.11176 .11246 8.8919 .99374
35
26
.11205 .11276 8.8686 .99370
34
27
.11234 .11305 8.8455 .99367
33
28
.11263 .11335 8.8225 .99364
32
29
.11291 .11364 8.7996 .99360
31
30
.11 320 .11394 8.7769 .99357
30
31
.11349 .11423 8.7542 .99354
29
32
.11378 .11452 8.7317 .99351
28
33
.11407 .11482 8.7093 .99347
27
34
.11436 .11511 8.6870 .99344
26
35
.11465 .11541 8.6648 .99341
25
36
.11494 .11570 8.6427 .99337
24
37
.11523 .11600 8.6208 .99334
23
38
.11552 .11629 8.5989 .99331
22
39
.11580 .11659 8.5772 .99327
21
4O
.11609 .11688 8.5555 .99324
2O
41
.11 638 .11 718 8.5340 .99320
19
42
.11667 .11 747 8.5126 .99317
18
43
.11696 .11 777 8.4913 .99314
17
44
.11725 .11806 8.4701 .99310
16
45
.11 754 .11 836 8.4490 .99307
15
46
.11 783 .11865 8.4280 .99303
14
47
'.11812 .11895 8.4071 .99300
13
48
.11840 .11924 8.3863 .99297
12
49
.11869 .11954 8.3656 .99293
11
50
.11898 .11983 8.3450 .99290
1O
51
.11927 .12013 8.3245 .99286
9
52
.11956 .12042 8.3041 .99283
8
53
.11985 .12072 8.2838 .99279
7
54
.12014 .12101 8.2636 .99276
6
55
.12043 .12131 8.2434 .99272
5
56
.12071 .12160 8.2234 .99269
4
57
.12 100 .12 190 8.2035 .99265
3
58
.12 129 .12 219 8.1837 .99 262
2
59
.12 158 .12249 8.1640 .99258
1
6O
.12187 .12278 8.1443 .99255
O
/
cos cot tan sin
/
83
7
/
sin tan cot cos
/
.12187 .12278 8.1443 .99255
60
1
.12216 .12308 8.1248 .99251
59
2
.12245 .12338 8.1054 .99248
58
3
.12274 .12367 8.0860 .99244
57
4
.12302 .12397 8.0667 .99240
56
5
.12331 .12426 8.0476 .99237
55
6
.12360 .12456 80285 .99233
54
7
.12389 .12485 8.C095 .99230
53
8
.12418 .12515 7.9906 .99226
52
9
.12447 .12544 7.9718 .99222
51
1O
.12476 .12574 7.9530 .99219
50
11
.12501 .12603 7.9344 .99215
49
12
.12533 .12633 7.9158 .99211
48
13
.12562 .12662 7.8973 .99208
47
14
.12591 .12692 7.8789 .99204
46
15
.12 620 .12 722 7.8606 .99 200
45
16
.12649 .12751 7.8424 .99197
44
17
.12678 .12781 7.8243 .99193
43
18
.12 706 .12 810 7.8062 .99 189
42
19
.12735 .12840 7.7882 .99186
41
2O
.12764 .12869 7.7704 .99182
4O
21
.12793 .12899 7.7525 .99178
39
22
.12822 .12929 7.7348 .99175
38
23
.12851 .12958 7.7171 .99171
37
24
.12880 .12988 7.6996 .99167
36
25
.12908 .13017 7.6821 .99163
35
26
.12937 .13047 7.6647 .99160
34
27
.12966 .13076 7.6473 .99156
33
28
.12 995 .13 106 7.6301 .99 152
32
29
.13024 .13136 7.6129 .99148
31
30
.13053 .13165 7.5958 .99144
30
31
.13081 .13 195 7.5787 .99141
29
32
.13110 .13224 7.5618 .99137
28
33
.13 139 .13 254 7.5449 .99 133
27
34
.13 168 .13 284 7.5281 .99 129
26
35
.13197 .13313 7.5113 .99125
25
36
.13226 .13343 7.4947 .99122
24
37
.13254 .13372 7.4781 .99118
23
38
.13283 .13402 7.4615 .99114
22
39
.13 312 .13 432 7.4451 .99 110
21
4O
.13341 .13461 7.4287 .99106
20
41
.13370 .13491 7.4124 .99102
19
42
.13 399 .13 521 7.3962 .99 098
18
43
.13427 .13550 7.3300 .99094
17
44
.13456 .13580 7.3639 .99091
16
45
.13485 .13609 7.3479 .99087
15
46
.13514 .13639 7.3319 .99083
14
47
.13543 .13669 7.3160 .99079
13
48
.13 572 .13 698 7.3002 .99 075
12
49
.13 600 .13 728 7.2844 .99 071
11
50
.13629 .13758 7.2687 .99067
1O
51
.13658 .13 787 7.2531 .99063
9
52
.13687 .13817 7.2375 .99059
8
53
.13716 .13846 7.2220 .99055
7
54
.13744 .13876 7.2066 .99051
6
55
.13773 .13906 7.1912 .99047
5
56
.13802 .13935 7.1759 .99043
4
57
.13831 .13965 7.1607 .99039
3
58
.13860 .13995 7.1455 .99035
2
59
.13 889 .14 024 7.1304 .99 031
1
60
.13917 .14054 7.1154 .99027
O
/
cos cot tan sin
/
82
62
NATURAL FUNCTIONS
8
/
sin tan cot cos
/
o
.13917 .14054 7.1154 .99027
6O
1
.13946 .14084 7.1004 .99023
59
2
13975 .14113 7.0855 .99019
58
3
.14004 .14143 7.0706 .99015
57
4
.14033 .14173 7.0558 .99011
56
5
.14061 .14202 7.0410 .99006
55
6
.14 090 .14 232 7.0264 .99 002
54
7
.14119 .14262 7.0117 .98998
53
8
.14 148 .14291 6.9972 .98994
52
9
.14177 .14321 6.9827 .98990
51
1O
.14205 .14351 6.9682 .98986
5O
11
.14234 .14381 6.9538 .98982
49
12
.14263 .14410 6.9395 .98978
48
13
.14 292 .14 440 6.9252 .98 973
47
14
.14320 .14470 6.9110 .98969
46
15
.14349 .14499 6.8969 .98965
45
16
.14 378 .14 529 6.8828 .98 961
44
17
.14407 .14559 6.8687 .98957
43
18
.14 436 .14 588 6.8548 .98 953
42
19
.14464 .14618 6.8408 .98948
41
2O
.14493 .14648 6.8269 .98944
4O
21
.14522 .14678 6.8131 .98940
39
22
.14551 .14707 6.7994 .98936
38
23
.14580 .14737 6.7856 .98931
37
24
.14 608 .14 767 6.7720 .98 927
36
25
.14637 .14796 6.7584 .98923
35
26
.14666 .14826 6.7448 .98919
34
27
.14695 .14856 6.7313 .98914
33
28
.14723 .14886 6.7179 .98910
32
29
.14752 .14915 6.7045 .98906
31
30
.14781 .14945 6.6912 .98902
30
31
.14810 .14975 6.6779 .98897
29
32
.14 838 .15 005 6.6646 .98 893
28
33
.14867 .15034 6.6514 .98889
27
34
.14 896 .15 064 6.6383 .98 884
26
35
.14925 .15094 6.6252 .98880
25
36
.14954 .15 124 6.6122 .98876
24
37
.14 982 .15 153 6.5992 .98 871
23
38
.15011 .15 183 6.5863 .98867
22
39
.15 040 .15 213 6.5734 .98 863
21
40
.15 069 .15 243 6.5606 .98 858
20
41
.15097 .15272 6.5478 .98854
19
42
.15 126 .15 302 6.5350 .98 849
18
43
.15 155 .15 332 6.5223 .98 845
17
44
.15 184 .15 362 6.5097 .98 841
16
45
.15212 .15391 6.4971 .98836
15
46
.15 241 .15 421 6.4846 .98 832
14
47
.15270 .15451 6.4721 .98827
13
48
.15 299 .15 481 6.4596 .98 823
12
49
.15327 .15511 6.4472 .98818
11
50
.15 356 .15 540 6.4348 .98 814
10
51
.15385 .15570 6.4225 .98809
9
52
.15 414 .15 600 6.4103 .98 805
8
53
.15442 .15630 6.3980 .98800
7
54
.15471 .]5660 6.3859 .98796
6
55
.15 500 .15 689 6.3737 .98 791
5
56
.15 529 .15 719 6.3617 .98 787
4
57
.15557 .15749 6.3496 .98782
3
58
.15 586 .15 779 6.3376 .98 778
2
59
.15 615 .15 809 6.3257 .98 773
1
6O
.15 643 .15 838 6.3138 .98 769
O
/
cos cot tan sin
/
81
9
/
sin tan cot cos
/
O
.15643 .15838 6.3138 .98769
60^
1
.15 672 .15 868 6.3019 .98 764
59
2
.15 701 .15898 6.2901 .98760
58
3
.15 730 .15928 6.2783 .98755
57
4
.15 758 .15958 6.2666 .98751
56
5
.15 787 .15 988 6.2549 .98 746
55
6
.15816 .16017 6.2432 .98741
54
7
.15845 .16047 6.2316 .98737
53
8
.15873 .16077 6.2200 .98732
52
9
.15 902 .16 107 6.2085 .98 728
51
10
.15931 .16137 6.1970 .98723
50
11
.15 959 .16 167 6.1856 .98 718
49
12
.15988 .16196 6.1742 .98714
48
13
.16017 .16226 6.1628 .98709
47
14
.16046 .16256 6.1515 .98704
46
15
.16074 .16286 6.1402 .98700
45
16
.16103 .16316 6.1290 .98695
44
17
.16132 .16346 6.1178 .98690
43
18
.16 160 .16 376 6.1066 .98 686
42
19
.16189 .16405 6.0955 .98681
41
20
.16218 .16435 6.0844 .98676
40
21
.16246 .16465 6.0734 .98671
39
22
.16275 .16495 6.0624 .98667
38
23
.16304 .16525 6.0514 .98662
37
24
.16333 .16555 6.0405 .98657
36
25
.16361 .16585-6.0296 .98652
35
26
.16390 .16615 6.0188 .98648
34
27
.16419 .16645 6.0080 .98643
33
28
.16447 .16674 5.9972 .98638
32
29
.16476 .16704 5.9865 .98633
31
30
.16505 .16734 5.9758 .98629
30
31
.16533 .16764 5.9651 .98624
29
32
.16562 .16794 5.9545 .98619
28
33
.16591 .16824 5.9439 .98614
27
34
.16620 .16854 5.9333 .98609
26
35
.16648 .16884 5.9228 .98604
25
36
.16677 .16914 5.9124 .98600
24
37
.16706 .16944 5.9019 .98595
23
38
.16734 .16974 5.8915 .98590
22
39
.16763 .17004 5.8811 .98585
21
40
.16792 .17033 5.8708 .98580
2O
41
..16820 .17063 5.8605 .98575
19
42
.16849 .17093 5.8502 .98570
18
43
.16878 .17123 5.8400 .98565
17
44
.16906 .17153 5.8298 .98561
16
45
.16935 .17183 5.8197 .98556
15
46
.16964 .17213 5.8095 .98551
14
47
.16992 .17243 5.7994 .98546
13
48
.17021 .17273 5.7894 .98541
12
49
.17050 .17303 5.7794 .98536
11
5O
.17078 .17333 5.7694 .98531
10
51
.17107 .17363 5.7594 .98526
9
52
.17136 .17393 5.7495 .98521
8
53
.17164 .17423 5.7396 .98516
7
54
.17193 .17453 5.7297 .98511
6
55
.17222 .17483 5.7199 .98506
5
56
.17250 .17513 5.7101 .98501
4
57
.17279 .17543 5.7004 .98496
3
58
.17308 .17573 5.6906 .98491
2
59
.17336 .17603 5.6809 .98486
1
60
.17365 .17633 5.6713 .98481
O
/
cos cot tan sin
/
80
NATURAL FUNCTIONS
63
10
/
sin tan cot cos
/
o
.17365 .17633 5.6713 .98481
6O
1
.17393 .17663 5.6617 .98476
59
2
.17422 .17693 5.6521 .98471
58
3
.17451 .17723 5.6425 .98466
57
4
.17479 .17753 5.6329 .98461
56
5
.17 508 .17 783 5.6234 .98 455
55
6
.17537 .17813 5.6140 .98450
54
7
.17565 .17843 5.6045 .98445
53
8
.17594 .17873 5.5951 .98440
52
9
.17623 .17903 5.5857 .98435
51
1O
.17651 .17933 5.5764 .98430
50
11
.17680 .17963 5.5671 .98425
49
12
.17 708 .17 993 5.5578 .98 420
48
13
.17737 .18023 5.5485 .98414
47
14
.17766 .18053 5.5393 .98409
46
15
.17794 .18083 5.5301 .98404
45
16
.17823 .18113 5.5209 .98399
44
17
.17852 .18143 5.5118 .98394
43
18
.17880 .18173 5.5026 .98389
42
19
.17909 .18203 5.4936 .98383
41
20
.17937 .18233 5.4845 .98378
40
21
.17 966 .18 263 5.4755 .98 373
39
22
.17 995 .18 293 5.4665 .98 368
38
23
.18023 .18323 5.4575 .98362
37
24
.18052 .18353 5.4486 .98357
36
25
.18081 .18384 5.4397 .98352
35
26
.18 109 .18 414 5.4308 .98 347
34
27
.18 138 .18 444 5.4219 .98 341
33
28
.18166 .18474 5.4131 .98336
32
29
.18195 .18504 5.4043 .98331
31
'30
.18 224 .18 534 5.3955 .98 325
3O
31
.18 252 .18 564 5.3868 .98 320
29
32
.18281 .18594 5.3781 .98315
28
33
.18309 .18624 5.3694 .98310
27
34
.18 338 .18 654 5.3607 .98 304
26
35
.18367 .18684 5.3521 .98299
25
36
.18395 .18714 5.3435 .98294
24
37
.18 424 .18 745 5.3349 .98 288
23
38
.18452 .18775 5.3263 .98283
22
39
.18481 .18805 5.3178 .98277
21
40
.18 509- .18 835 5.3093 .98272
2O
41
.18 538 .18 865 5.3008 .98 267
19
42
.18 567 .18 895 5.2924 .98 261
18
43
.18 595 .18 925 5.2839 .98 256
17
44
.18624 .18955 5.2755 .98250
16
45
.18652 .18986 5.2672 .98245
15
46
.18681 .19016 5.2588 .98240
14
47
.13710 .19046 5.2505 .98234
13
48
.18738 .19076 5.2422 .98229
12
49
.18 767 .19 106 5.2339 .98 223
11
50
.18 795 .19 136 5.2257 .98 218
1O
51
.18 824 .19 166 5.2174 .98 212
9
52
.18 852 .19 197 5.2092 .98 207
8
53
.18881 .19227 5.2011 .98201
7
54
.18910 .19257 5.1929 .98196
6
55
.18938 .19287 5.1848 .98190
5
56
.18967 .19317 5.1767 .98185
4
57
.18995 .19347 5.1686 .98179
3
58
.19024 .19378 5.1606 .98174
2
59
.19052 .19408 5.1526 .98168
1
60
.19 081 .19 438 5.1446 .98 163
/
cos cot tan sin
/
79
11
/
sin tan cot cos
/
.19081 .19438 5.1446 .98163
60
1
.19109 .19468 5.1366 .98157
59
2
.19138 .19498 5.1286 .98152
58
3
.19 167 .19 529 5.1207 .98 146
57
4
.19195 .19559 5.1128 .98140
56
5
.19 224 .19 589 5.1049 .98 135
55
6
.19 252 .19 619 5.0970 .98 129
54
7
.19281 .19649 5.0892 .98124
53
8
.19309 .19680 5.0814 .98118
52
9
.19 338 .19 710 5.0736 .98 112
51
1O
.19 366 .19 740 5.0658 .98 107
5O
11
.19395 .19770 5.0581 .98101
49
12
.19423 .19801 5.0504 .98096
48
13
.19452 .19831 5.0427 .98090
47
14
.19481 .19861 5.0350 .98084
46
15
.19 509 .19 891 5.0273 .98 079
45
16
.19538 .19921 5.0197 .98073
44
17
.19566 .19952 5.0121 .98067
43
18
.19595 .19982 5.0045 .98061
42
19
.19 623 .20 012 4.9969 .98 056
41
2O
.19652 .20042 4.9894 .98050
4O
21
.19680 .20073 4.9819 .98044
39
22
.19709 .20103 4.9744 .98039
38
23
.19 737 .20 133 4.9669 .98 033
37
24
.19766 .20164 4.9594 .98027
36
25
.19794 .20194 4.9520 .98021
35
26
.19 823 .20 224 4.9446 .98 016
34
27
.19851 .20254 4.9372 .98010
33
28
.19 880 .20 285 4.9298 .98 004
32
29
.19908 .20315 4.9225 .97998
31
30
.19937 .20345 4.9152 .97992
3O
31
.19 965 .20 376 4.9078 .97 987
29
32
.19994 .20406 4.9006 .97981
28
33
.20022 .20436 4.8933 .97975
27
34
.20051 .20466 4.8860 .97969
26
35
.20079 .20497 4.8788 .97963
25
36
.20 108 .20 527 4.8716 .97 958
24
37
.20 136 .20 557 4.8644 .97 952
23
38
.20 165 .20 588 4.8573 .97 946
22
39
.20 193 .20 618 4.8501 .97 940
21
4O
.2022? .20648 4.8430 .97934
2O
41
.20 250 .20 679 4.8359 .97 928
19
42
.20 279 .20 709 4.8288 .97 922
18
43
.20307 .20739 4.8218 .97916
17
44
.20 336 .20 770 4.8147 .97 910
16
45
.20364 .20800 4.8077 .97905
15
46
.20393 .20830 4.8007 .97899
14
47
.20421 .20861 4.7937 .97893
13
48
.20450 .20891 4.7867 .97887
12
49
.20478 .20921 4.7798 .97881
11
5O
.20 507 .20 952 4.7729 .97 875
1O
51
.20 535 .20 982 4.7659 .97 869
9
52
.20 563 .21 013 4.7591 .97 863
8
53
.20 592 .21 043 4.7522 .97 857
7
54
.20 620 .21 073 4.7453 .97 851
6
55
.20649 .21 104 4.7385 .97845
5
56
.20677 .21 134 4.7317 .97839
4
57
.20706 .21 164 4.7249 .97833
3
58
.20 734 .21 195 4.7181 .97 827
2
59
.20763 .21225 4.7114 .97821
1
60
.20 791 .21 256 4.7046 .97 815
O
/
cos cot tan sin
/
78
64
NATURAL FUNCTIONS
12
/
sin tan cot cos
/
o
.20 791 .21 256 4.7046 .97 815
60^
1
.20 820 .21 286 4.6979 .97 809
59
2
.20848 .21316 4.6912 .97803
58
3
.20 877 .21 347 4.6845 .97 797
57
4
.20905 .21377 4.6779 .97791
56
5
.20 933 .21 408 4.6712 .97 784
55
6
.20 962 .21 438 4.6646 .97 778
54
7
.20 990 .21 469 4.6580 .97 772
53
8
.21 019 .21 499 4.6514 .97 766
52
9
.21 047 .21 529 4.6448 .97 760
51
1O
.21 076 .21 560 4.6382 .97 754
5O
11
.21 104 .21 590 4.6317 .97 748
49
.12
.21 132 .21 621 4.6252 .97 742
48
13
.21 161 .21 651 4.6187 .97 735
47
14
.21 189 .21 682 4.6122 .97 729
46
15
.21 218 .21 712 4.6057 .97 723
45
16
.21 246 .21 743 4.5993 .97 717
44
17
.21 275 .21 773 4.5928 .97 711
43
18
.21 303 .21 804 4.5864 .97 705
42
19
.21331 .21834 4.5800 .97698
41
20
.21 360 .21 864 4.5736 .97 692
4O
21
.21 388 .21 895 4.5673 .97 686
39
22
.21 417 .21 925 4.5609 .97 680
38
23
.21445 .21956 4.5546 .97673
37
24
.21 474 .21 986 4.5483 .97 667
36
25
.21502 .22017 4.5420 .97661
35
26
.21 530 .22 047 4.5357 .97 655
34
27
.21 559 .22 078 4.5294 .97 648
33
28
.21 587 .22 108 4.5232 .97 642
32
29
.21616 .22139 4.5169 .97636
31
30
.21644 .22 169 4.5107 .97630
30
31
.21 "57^.22 200 4.5045 .97623
29
32
.21701 .22231 4.4983 .97617
28
33
.21 729 .22261 4.4922 .97611
27
34
.21 758 .22 292 4.4860 .97 604
26
35
.21 786 .22 322 4.4799 .97 598
25
36
.21 814 .22 353 4.4737 .97 592
24
37
.21 843 .22 383 4.4676 .97 585
23
38
.21871 .22414 4.4615 .97579
22
39
.21 899 .22 444 4.4555 .97 573
21
4O
.21928 .22475 4.4494 .97566
20
41
.21 956 .22 505 4.4434 .97 560
19
42
.21 985 .22 536 4.4373 .97 553
18
43
.22013 .22567 4.4313 .97547
17
44
.22 041 .22 597 4.4253 .97 541
16
45
.22 070 .22 628 4.4194 .97 534
15
46
.22 098 .22 658 4.4134 .97 528
14
47
.22 126 .22 689 4.4075 .97 521
13
48
.22155 .22719 4.4015 .97515
12
49
.22 183 .22 750 4.3956 .97 508
11
50
.22 212 .22 781 4.3897 .97 502
1O
51
.22240 .22811 4.3838 .97496
9
52
.22 268 .22 842 4.3779 .97 489
8
53
.22297 .22872 4.3721 .97483
7
54
.22' 325 .22903 4.3662 .97476
6
55
.22 353 .22 934 4.3604 .97 470
5
56
.22 382 .22 964 4.3546 .97 463
4
57
.22410 .22995 4.3488 .97457
3
58
.22 438 .23 026 4.3430 .97 450
2
59
.22 467 .23 056 4.3372 .97 444
1
6O
.22 495 .23 087 4.3315 .97 437
/
cos cot tan sin
/
77
13
/
sin tan cot cos
/
O
.22495 .23087 4.3315 .97437
60
1
.22523 .23117 4.3257 .97430
59
2
.22 552 .23 148 4.3200 .97 424
58
3
.22580 .23179 4.3143 .97417
57
4
.22 608 .23 209 4.3086 .97 411
56
5
.22 637 .23 240 4.3029 .97 404
55
6
.22 665 .23 271 4.2972 .97 398
54
7
.22 693 .23 301 4.2916 .97 391
53
8
.22 722 .23 332 4.2859 .97 384
52
9
.22 750 .23 363 4.2803 .97 378
51
10
.22 778 .23 393 4.2747 .97 371
5O
11
.22 807 .23 424 4.2691 .97 365
49
12
.22 835 .23 455 4.2635. .97 358
48
13
.22863 .23485 4.2580 .97351
47
14
.22892 .23 516 4.2524 .97345
46
- 15
.22 920 .23 547 4.2468 .97 338
45
16
.22948 .23578 4.2413 .97331
44
17
.22 977 .23 608 4.2358 .97 325
43
18
.23 005 .23 639 4.2303 .97 318
42
19
.23033 .23670 4.2248 .97311
41
2O
.23 062 .23 700 4.2193 .97 304
4O
21
.23 090 .23 731 4.2139 .97 298
39
22
.23 118 .23 762 4.2084 .97291
38
23
.23 146 .23 793 4.2030 .97 284
37
21
.23 175 .23 823 4.1976 .97 278
36
25
.23 203 .23 854 4.1922 .97 271
35
26
.23 231 .23 885 4.1868 .97 264
34
27
.23 260 .23 916 4.1814 .97 257
33
28
.23 288 .23 946 4.1760 .97 251
32
29
.23 316 .23 977 4.1706 .97 244
31
3O
.23345 .24008 4.1653 .97237
3O
31
.23373 .24039 4.1600 .97230
29
32
.23 401 .24 069 4.1547 .97 223
28
33
.23 429 .24 100 4.1493 .97 217
27
34
.23 458 .24 131 4.1441 .97 210
26
35
.23 486 .24 162 4.1388 .97 203
25
36
.23 514 .24 193 4.1335 .97 196
24
37
.23 542 .24 223 4.1282 .97 189
23
38
.23 571 .24 254 4.1230 .97 182
22
39
.23 599 .24 285 4.1178 .97 176
21
4O
.23627 .24316 4.1126 .97169
2O
41
.23 656 .24 347 4.1074 .97 162
19
42
.23 684 .24 377 4.1022 .97 155
18
43
.23 712 .24 408 4.0970 .97 148
17
44
.23 740 .24 439 4.0918 .97 141
16
45
.23 769 .24 470 4.0867 .97 134
15
46
.23797 .24 501 4.0815 .97127
14
47
.23 825 .24 532 4.0764 .97 120
13
48
.23 853 .24 562 4.0713 .97 113
12
49
.23 882 .24 593 4.0662 .97 106
11
50
.23910 .24624 4.0611 .97100
10
51
.23 938 .24 655 4.0560 .97 093
9
52
.23 966 .24 686 4.0509 .97 086
8
53
.23 995 .24 717 4.0459 .97 079
7
54
.24 023 .24 747 4.0408 .97 072
6
55
.24051 .24778 4.0358 .97065
5
56
.24 079 .24 809 4.0308 .97 058
4
57
.24 108 .24 840 4.0257 .97 051
3
58
.24 136 .24 871 4.0207 .97 044
2
59
.24 164 .24 902 4.0158 .97 037
1
60
.24192 .24933 4.0108 .97030
O
/
cos cot tan sin
/
76
NATURAL FUNCTIONS
65
14
/
sin tan cot cos
/
o
.24 192 .24 933 4.0108 .97 030
6O
1
.24 220 .24 964 4.0058 .97 023
59
2
.24249 .24995 4.0009 .97015
58
3
.24 277 .25 026 3.9959 .97 008
57
4
.24 305 .25 056 3.9910 .97 001
56
5
.24 333 .25 087 3.9861 .96 994
55
6
.24362 .25 118 3.9812 .96987
54
7
.24 390 .25 149 3.9763 .96 980
53
8
.24 418 .25 180 3.9714 .96973
52
9
.24446 .25211 3.9665 .96966
51
10
.24 474 .25 242 3.9617 .96 959
5O
11
.24 503 .25 273 3.9568 .96 952
49
12
.24 531 .25 304 3 9520 .96 945
48
13
.24559 .25335 3.9471 .96937
47
14
.24587 .25*366 3.9423 .96930
46
15
.24 615 .25 397 3.9375 .96 923
45
16
.24644 .25428 3.9327 .96916
44
17.
.24 672 .25 459 3.9279 .96 909
43
18
.24 700 .25 490 3.9232 .96 902
42
19
.24 728 .25 521 3.9184 .96 894
41
2O
.24756 .25552 3.9136 .96887
4O
21
.24 784 .25 583 3.9089 .96 880
39
22
.24 813 .25 614 3.9042 .96 873
38
23
.24841 .25645 3.8995 .96866
37
24
.24 869 .25 676 3.8947 .96 858
36
25
.24897 .25707 3.8900 .96851
35
26
.24 925 .25 738 3.8854 .96 844
34
27
.24 954 .25 769 3.8807 .96 837
33
| 28
.24982 .25800 3.8760 .96829
32
29
.25010 .25831 3.8714 .96822
31
3D
.25038 .25862 3.8667 .96815
30
31
.25066 .25893 3.8621 .96807
29
32
.25094 .25924 3.8575 .96800
28
33
.25 122 .25 955 3.8528 .96 793
27
34
.25 151 .25 986 3.8482 .96 786
26
35
.25179 .26017 3.8436 .96778
25
36
.25 207 .26 048 3.8391 .96 771
24
37
.25 235 .26 079 3.8345 .96 764
23
38
.25 263 .26 110 3.8299 .96 756
22
39
.25 291 .26 141 3.8254 .96 749
21
4O
.25 320 .26 172 3.8208 .96 742
2O
41
.25348 .26203 3.8163 .96734
19
42
.25376 .26235 3.8118 .96727
18
43
.25 404 .26 266 3.8073 .96 719
17
44
.25432 .26297 3.8028 .96712
16
45
.25 460 .26 328 3.7983 .96 705
15
46
.25 488 .26 359 3.7938 .96 697
14
47
.25516 .26390 3.7893 .96690
13
48
.25 545 .26 421 3.7848 .96 682
12
49
.25573 .26452 3.7804 .96675
11
50
.25 601 .26 483 3.7760 .96 667
10
51
.25629 .26515 3.7715 .96660
9
52
.25657 .26546 3.7671 .96653
8
53
.25685 .26577 3.7627 .96645
7
54
.25713 .26608 3.7583 .96638
6
55
.25741 .26639 3.7539 .96630
5
56
.25 769 .26 670 3.7495 .96 623
4
57
.25798 .26701 3.7451 .96615
3
58
.25 826 .26 733 3.7408 .96 608
2
59
.25 854 .26 764 3.7364 .96 600
1
60
.25882 .26795 3.7321 .96593
O
/
cos cot tan sin
/
75
15
/
sin tan cot cos
/
O
.25 882 .26 795 3.7321 .96 593
60
1
.25 910 .26 826 3.7277 .96 585
59
2
.25 938 .26 857 3.7234 .96 578
58
3
.25 966 .26 888 3.7191 .96 570
57
4
.25 994 .26 920 3.7148 .96 562
56
5
.26022 .26951 3.7105 .96555
55
6
.26050 .26982 3.7062 .96547
54
7
.26079 .27013 3.7019 .96540
53
8
.26 107 .27 044 3.6976 .96 532
52
9
.26 135 .27 076 3.6933 .96 524
51
1O
.26163 .27107 3.6891 .96517
5O
11
.26 191 .27 138 3.6848 .96 509
49
12
.26 219 .27 169 3.6806 .96 502
48
13
.26247 .27201 3.6764 .96494
47
14
.26 275 .27 232 3.6722 .96 486
46
15
.26303 .27263 3.6680 .96479
45
16
.26331 .27294 3.6638 .96471
44
17
.26 359 .27 326 3.6596 .96 463
43
18
.26387 .27357 3.6554 .96456
42
19
.26415 .27388 3.6512 .96448
41
2O
.26443 .27419 3.6470 .96440
4O
21
.26471 .27451 3.6429 .96433
39
22
.26500 .27482 3.6387 .96425
38
23
.26528 .27513 3.6346 .96417
37
24
.26556 .27545 3.6305 .96410
36
25
.26584 .27576 3.6264 .96402
35
26
.26612 .27607 3.6222 .96394
34
27
.26 640 .27 638 3.6181 .96 386
33
28
.26 668 .27 670 3.6140 .96 379
32
29
.26 696 .27 701 3.6100 .96 371
31
30
.26 724 .27 732 3.6059 .96 363
3O
31
.26 752 .27 764 3.6018 .96 355
29
32
.26 780 .27 795 3.5978 .96 347
28
33
.26 808 .27 826 3.5937 .96 340
27
34
.26 836 .27 858 3.5897 .96 332
26
35
.26 864 .27 889 3.5856 .96 324
25
36
.26892 .27921 3.5816 .96316
24
37
.26920 .27952 3.5776 .96308
23
38
.26948 .27983 3.5736 .96301
22
39
.26976 .28015 3.5696 .96293
21
4O
.27004 .28046 3.5656 .96285
2O
41
.27032 .28077 3.5616 .96277
19
42
.27 060 .28 109 3.5576 .96 269
18
43
.27 088 .28 140 3.5536 .96 261
17
44
.27 116 .28 172 3.5497 .96 253
16
45
.27 144 .28 203 3.5457 .96 246
15
46
.27 172 .28 234 3.5418 .96 238
14
47
.27 200 .28 266 3.5379 .96 230
13
48
.27 228 .28 297 3.5339 .96 222
12
49
.27256 .28329 3.5300 .96214
11
50
.27284 .28360 3.5261 .96206
10
51
.27312 .28391 3.5222 .96198
9
52
.27340 .28423 3.5183 .96190
8
53
.27 368 .28 454 3.5144 .96 182
7
54
.27396 .28486 3.5105 .96174
6
55
.27424 .28517 3.5067 .96166
5
56
.27 452 .28 549 3.5028 .96 158
4
57
.27480 .28580 3.4989 .96150
3
58
.27508 .28612 3.4951 .96142
2
59
.27536 .28643 3.4912 .96134
1
6O
.27564 .28675 3.4874 .96126
O
/
cos cot tan sin
/
74
66
NATURAL FUNCTIONS
16
/
sin tan cot cos
/
|O
.27 564 .28 675 3.4874 .96 126
6O
1
.27592 .28706 3.4836 .96118
59
2
.27620 .28738 3.4798 .96110
58
3
.27 648 .28 769 3.4760 .96 102
57
4
.27676'.28801 3.4722 .96094
56
5
.27 704 .28 832 3.4684 .96 086
55
6
.27731 .28864 3.4646 .96078
54
7
.27759 .28895 3.4608 .96970
53
8
.27787 .28927 3.4570 .96062
52
9
.27815 .28958 3.4533 .96954
51
10
.27843 .28990 3.4495 .96046
5O
11
.27871 .29021 3.4458 .96037
49
12
.27899 .29053 3.4420 .96029
48
13
.27927 .29084 3.4383 .96021
47
14
.27955 .29116 3.4346 .96013
46
15
.27 983 .29 147 3.4308 .96 005
45
16
.28 Oil .29 179 3.4271 .95 997
44
17
.28 039 .29 210 3.4234 .95 989
43
18
.28 067 .29 242 3.4197 .95 981
42
19
.28095 .29274 3.4160 .95972
41
20
.28 123 .29 305 3.4124 .95 964
40
21
.28150 .29337 3.4087 .95956
39
22
.28178 .29368 3.4050 .95948
38
23
.28206 .29400 3.4014 .95940
37
24
.28234 .29432 3.3977 .95931
36
25
.28 262 .29 463 3.3941 .95 923
35
26
.28 290 .29 495 3.3904 .95 915
34
27
.28 318 .29 526 3.3868 .95 907
33
28
.28346 .29558 3.3832 .95898
32
29
.28 374 .29 590 3.3796 .95 890
31
30
.28402 .29621 3.3759 .95882
30
31
.28429 .29653 3.3723 .95874
29
32
.28457 .29685 3.3687 .95865
28
33
.28485 .29716 3.3652 .95857
27
34
.28513 .29748 3.3616 .95849
26
35
.28 541 .29 780 3.3580 .95 841
25
36
.28569 .29811 3.3544 .95832
24
37
.28 597 .29 843 3.3509 .95 824
23
38
.28625 .29875 3.3473 .95816
22
39
.28 652 .29 906 3.3438 .95 807
21
4O
.28 680 .29 938 3.3402 .95 799
20
41
.28708 .29970 3.3367 .95791
19
42
.28 736 .30 001 3.3332 .95 782
18
43
.28 764 .30 033 3.3297 .95 774
17
44
.28 792 .30 065 3.3261 .95 766
16
45
.28820 .30097 3.3226 .95757
15
46
.28 847 .30 128 3.3191 .95 749
14
47
.28875 .30160 3.3156 .95740
13
48
.28903 .30192 3.3122 .95732
12
49
.28931 .30224 3.3087 .95724
11
50
.28 959 .30 255 3.3052 .95 715
1O
51
.28 987 .30 287 3.3017 .95 707
9
52
.29015 .30319 3.2983 .95698
8
53
.29042 .30351 3.2948 .95690
7
54
.29 070 .30 382 3.2914 .95 681
6
55
.29 098 .30 414 3.2879 .95 673
5
56
.29 126 .30 446 3.2845 .95 664
4
1 57
.29154 .30478 3.2811 .95656
3
58
.29 182 .30 509 3.2777 .95 647
2
59
.29 209 .30 541 3.2743 .95 639
1
6O
.29237 .30573 3.2709 .95630
O
/
cos cot tan sin
/
73
17
/
sin tan cot cos
/
O
.29237 .30573 3.27Q9 .95630
GO
1
.29265 .30605 3.2675 .95622
59
2
.29293 .30637 3.2641 .95613
58
3
.29321 .30669 3.2607 .95605
57
4
.29348 .30700 3.2573 .95596
56
5
.29376 .30732 3.2539 .95588
55
6
.29404 .30764 3.2506 .95579
54
7
.29432 .30796 3.2472 .95571
53
8
.29460 .30828 3.2438 .95562
52
9
.29487 .30860 3.2405 .95554
51
1O
.29515 .30891 3.2371 .95545
50
11
.29543 .30923 3.2338 .95536
49
12
.29571 .30955 3.2305 .95528
48
13
.29599 .30987 3.2272 .95519
47
14
.29626 .31019 3.2238 .95511
46
15
.29654 .31051 3.2205 .95502
45
16
.29682 .31083 3.2172 .95493
44
17
.29710 .31115 3.2139 .95485
43
18
.29 737 .31 147 3.2106 .95 476
42
19
.29765 .31 178 3.2073 .95467
41
20
.29 793 .31 210 3.2041 .95 459
40
21
.29 821 .31 242 3.2008 .95 450
39
22
.29849 .31274 3.1975 .95441
38
23
.29876 .31306 3.1943 .95433
37
24
.29904 .31338 3.1910 .95424
36
25
.29932 .31370 3.1878 .95415
35
26
.29960 .31402 3.1845 .95407
34
27
.29987 .31434 3.1813 .95398
33
28
.30015 .31466 3.1780 .95389
32
29
.30043 .31498 3.1748 .95380
31
30
.30071 .31530 3.1716 .95372
30
31
.30098 .31562 3.1684 .95363
29
32
.30 126 .31 594 3.1652 .95 354
28
33
.30 154 .31 626 3.1620 .95 345
27
34
.30 182 .31 658 3.1588 .95 337
26
35
.30 209 .31 690 3.1556 .95 328
25
36
.30 237 .31 722 3.1524 .95 319
24
37
.30 265 .31 754 3.1492 .95 310
23
38
.30292 .31786 3.1460 .95301
22
39
.30 320 .31 818 3.1429 .95 293
21
40
.30348 .31850 3.1397 .95284
2O
41
.30376 .31882 3.1366 .95275
19
42
.30403 .31914 3.1334 .95266
18
43
.30431 .31946 3.1303 .95257
17
44
.30459 .31978 3.1271 .95248
16
45
.30486 .32010 3.1240 .95240
15
46
.30514 .32042 3.1209 .95231
14
47
.30542 .32074 3.1178 .95222
13
48
.30570 .32106 3.1146 .95213
12
49
.30597 .32139 3.1115 .95204
11
50
-.30625 .32171 3.1084 .95 195
10
51
.30653 .32203 3.1053 .95 186
9
52
.30680 .32235 3.1022 .95 177
8
53
.30 708 .32 267 3.0991 .95 168
7
54
.30 736 .32 299 3.0961 .95 159
6
55
.30763 .32331 3.0930 .95150
5
56
.30 791 .32 363 3.0S99 .95 142
4
57
.30 819 .32 396 3.0868 .95 133
3
58
.30 846 .32 428 3.0838 .95 124
2
59
.30874 .32460 3.0807 .95 115
1
6O
.30902 .32492 3.0777 .95 106
O
/
cos cot tan sin
/
72
NATURAL FUNCTIONS
6T
18
/
sin tan cot cos
/
O
.30 902 .32 492 3.0777 .95 106
6O
1
.30 929 .32 524 3.0746 .95 097
59
2
.30957 .32556 3.0716 .95088
58
3
.30985 .32588 3.0686 .95079
57
4
.31012 .32621 3.0655 .95070
56
5
.31 040 .32 653 3.0625 .95 061
55
6
.31068 .32685 3.0595 .95052
54
7
.31 095 .32 717 3.0565 .95 043
53
8
.31 123 .32 749 3.0535 .95 033
52
9
.31151 .32782 3.0505 .95024
51
10
.31 178 .32 814 3.0475 .95 015
50
11
.31 206 .32 846 3.0445 .95 006
49
12
.31233 .32878 3.0415 .94997
48
13
.31261 .32911 3.0385 .94988
47
14
.31289 .32943 3.0356 .94979
46
15
.31316 .32975 3.0326 .94970
45
>16
.31 344 .33 007 3.0296 .94 961
44
17
.31 372 .33 040 3.0267 .94 952
43
18
.31399 .33072 3.0237 .94943
42
19
.31 427 .33 104 3.0208 .94 933
41
2O
.31 454 .33 136 3.0178 .94924
40
21
.31482 .33169 3.0149 .94915
39
22
.31510 .33201 3.0120 .94906
38
23
.31 537 .33 233 3.0090 .94 897
37
24
.31565 .33266 3.0061 .94888
36
25
.31593 .33298 3.0032 .94878
35
26
.31620 .33330 3.0003 .94869
34
27
.31648 .33363 2.9974 .94860
33
28
.31675 .33395 2.9945 .94851
32
29
r. 31 703 .33427 2.9916 .94842
31
30
\31730 .33460 2.9887 .94832
30
31
.31 758 .33 492 2.9858 .94 823
29
32
.31 786 .33 524 2.9829 .94 814
28
33
.31 813 .33 557 2.9800 .94 805
27
34
.31 841 .33 589 2.9772 .94 795
26
35
.31 868 .33 621 2.9743 .94 786
25
36
.31 896 .33 654 2.9714 .94 777
24
37
.31923 .33686 2.9686 .94768
23
38
.31951 .33718 2.9657 .94758
22
39
.31 979 .33 751 2.9629 .94 749
21
4O
.32 006 .33 783 2.9600 .94 740
2O
41
.32 034 .33 816 2.9572 .94 730
19
42
.32 061 .33 848 2.9544 .94 721
18
43
.32089 .33881 2.9515 .94712
17
44
.32116 .33913 2.9487 .94702
16
45
.32144 .33945 2.9459 .94693
15
46
.32 171 .33 978 2.9431 .94 684
14
47
.32 199 .34 010 2.9403 .94 674
13
48
.32227 .34043 2.9375 .94665
12
49
.32 254 .34 075 2.9347 .94 656
11
50
.32282 .34108 2.9319 .94646
1O
51
.32 309 .34 140 2.9291 .94 637
9
52
.32 337 .34 173 2.9263 .94 627
8
53
.32364 .34205 2.9235 .94618
7
54
.32392 .34238 2.9208 .94609
6
55
.32 419 .34 270 2.9180 .94 599
5
56
.32447 .34303 2.9152 .94590
4
57
.32474 .34335 2.9125 .94580
3
58
.32502 .34368 2.9097 .94571
2
59
.32 529 .34 400 2.9070 .94 561
1
6O
.32557 .34433 2.9042 .94552
/
cos cot tan sin
/
71
19
/
sin tan cot cos
/
o
.32557 .34433 2.9042 .94552
6O
1
.32 584 .34 465 2.9015 .94 542
59
2
.32612 .34498 2.8987 .94533
58
3
.32 639 .34 530 2.8960 .94 523
57
4
.32667 .34563 2.8933 .94514
56
5
.32 694 .34 596 2.8905 .94 504
55
6
.32 722 .34 628 2.8878 .94 495
54
7
.32749 .34661 2.8851 .94485
53
8
.32 777 .34 693 2.8824 .94 476
52
9
.32 804 .34 726 2.8797 .94 466
51
10
.32832 .34758 2.8770 .94457
5O
11
.32859 .34791 2.8743 .94447
49
12
.32887 .34824 2.8716 .94438
48
13
.32914 .34856 2.8689 .94428
47
14
.32942 .34889 2.8662 .94418
46
15
.32969 .34922 2.8636 .94409
45
16
.32997 .34954 2.8609 .94399
44
17
.33 024 .34 987 2.8582 .94 390
43
18
.33051 .35020 2.8556 .94380
42
19
.33 079 .35 052 2.8529 .94 370
41
20
.33 106 .35 085 2.8502 .94 361
4O
21
.33134 .35118 2.8476 .94351
39
22
.33 161 .35 150 2.8449 .94 342
38
23
.33 189 .35 183 2.8423 .94 332
37
24
.33216 .35216 2.8397 .94322
36
25
.33 244 .35 248 2.8370 .94 313
35
26
.33 271 .35 281 2.8344 .94 303
34
27
.33298 .35314 2.8318 .94293
33
28
.33 326 .35 346 2.8291 .94 284
32
29
.33 353 .35 379 2.8265 .94 274
31
3O
.33 381 .35 412 2.8239 .94 264
30
31
.33 408 .35 445 2.8213 .94 254
29
32
.33436 .35477 2.8187 .94245
28
33
.33463 .35510 2.8161 .94235
27
34
.33 490 .35 543 2.8135 .94 225
26
35
.33 518 .35 576 2.8109 .94 215
25
36
.33 545 .35 608 2.8083 .94 206
24
37
.33 573 .35 641 2.8057 .94 196
23
38
.33 600 .35 674 2.8032 .94 186
22
39
.33 627 .35 707 2.8006 .94 176
21
4O
.33 655 .35 740 2.7980 .94 167
2O
41
.33 682 .35 772 2.7955 .94 157
19
42
.33 710 .35 805 2.7929 .94 147
18
43
.33 737 .35 838 2.7903 .94 137
17
44
.33 764 .35 871 2.7878 .94 127
16
45
.33 792 .35 904 2.7852 .94 118
15
46
.33 819 .35 937 2.7827 .94 108
14
47
.33846 .35969 2.7801 .94098
13
48
.33 874 .36 002 2.7776 .94 088
12
49
.33901 .36035 2.7751 .94078
11
50
.33 929 .36 068 2.7725 .94 068
1O
51
.33 956 .36 101 2.7700 .94 058
9
52
.33 983 .36 134 2.7675 .94 049
8
53
.34011 .36167 2.7650 .94039
7
54
.34 038 .36 199 2.7625 .94 029
6
55
.34065 .36232 2.7600 .94019
5
56
.34093 .36265 2.7575 .94009
4
57
.34120 .36298 2.7550 .93999
3
58
.34147 .36331 2.7525 .93989
2
59
.34 175 .36 364 2.7500 .93 979
1
6O
.34 202 .36 397 2.7475 .93 969
/
cos cot tan sin
/
70
68
NATURAL FUNCTIONS
20
/
sin tan cot cos
/
o
.34202 .36397 2.7475 .93969
6O
1
.34 229 .36 430 2.7450 .93 959
59
2
.34 257 .36 463' 2.7425 .93 949
58
3
.34 284 .36 496 2.7400 .93 939
57
4
.34311 .36529 2.7376 .93929
56
5
.34339 .36562 2.7351 .93919
55
6
.34 366 .36 595 2.7326 .93 909
54
7
.34 393 .36 628 2.7302 .93 899
53
8
.34 421 .36 661 2.7277 .93 889
52
9
.34 448 .36 694 2.7253 .93 879
51
10
.34475 .36727 2.7228 .93869
50
11
.34 503 .36 760 2.7204 .93 859
49
12
.34 530 .36 793 2.7179 .93 849
48
13
.34 557 .36 826 2.7155 .93 839
47
14
.34584 .36859 2.7130 .93829
46
15
.34612 .36892 2.7106 .93819
45
16
.34 639 .36 925 2.7082 .93 809
44
17
.34666 .36958 2.7058 .93799
43
18
.34 694 .36 991 2.7034 .93 789
42
19
.34 721 .37 024 2.7009 .93 779
41
20
.34748 .37057 2.6985 .93769
4O
21
.34.775 .37090 2.6961 .93759
39
22
.34 803 .37 123 2.6937 .93 748
38
23
.34830 .37157 2.6913 .93738
37
24
.34 857 .37 190 2.6889 .93 728
36
25
.34 884 .37 223 2.6865 .93 718
35
26
.34 912 .37 256 2.6841 .93 708
34
27
.34 939 .37 289 2.6818 .93 698
33
28
.34 966 .37 322 2.6794 .93 688
32
29
.34 993 .37 355 2.6770 .93 677
31
30
.35 021 .37 388 2.6746 .93 667
30
31
.35048 .37422 2.6723 .93657
29
32
.35 075 '.37 455 2.6699 .93 647
28
33
.35 102 .37 488 2.6675 .93 637
27
34
.35 130 .37 521 2.6652 .93 626
26
35
.35157 .37554 2.6628 .93616
25
36
.35 184 .37 588 2.6605 .93 606
24
37
.35211 .37621 2.6581 .93596
23
38
.35 239 .37 654 2.6558 .93 585
22
39
.35 266 .37 687 2.6534 .93 575
21
40
.35 293 .37 720 2.6511 .93 565
20
41
.35 320 .37 754 2.6488 .93 555
19
42
;35 347 .37 787 2.6464 .93 544
18
43
.35 375 .37 820 2.6441 .93 534
17
44
.35402 .37853 2.6418 .93524
16
45
.35429 .37887 2.6395 .93514
15
46
.35 456 .37 920 2.6371 .93 503
14
47
.35 484 .37 953 2.6348 .93 493
13
48
.35 511 .37 986 2.6325 .93 483
12
49
.35 538 .38 020 2.6302 .93 472
11
50
.35565 .38053 2.6279 ;93 462
1O
51
.35 592 .38 086 2.6256 .93 452
9
52
.35 619 .38 120 2.6233 .93 441
8
53
35647 .38153 2.6210 .93431
7
54
.35 674 .38 186 2.6187 .93 420
6
55
.35 701 .38 220 2.6165 .93 410
5
56
.35 728 .38 253 2.6142 .93 400
4
57
.35 755 .38 286 2.6119 .93 389
3
58
.35 782 .38320 2.6096 .93379
2
59
.35 810 .38 353 2.6074 .93 368
1
6O
.35837 .38386 2.6051 .93358
O
/
cos cot tan sin
/
69
21
/
sin tan cot cos
/
O
.35837 .38386 2.6051 .93358
6O
1
.35 864 .38 420 2.6028 .93 348
59
2
.35891 .38453 2.6006 .93337
58
3
.35 918 .38 487 2.5983 .93 327
57
4
.35 945 .38 520 2.5961 .93 316
56
5
.35 973 .38 553 2.5938 .93 306
55
6
.36 000 .38 587 2.5916 .93 295
54
7
.36 027 .38 620 2.5893 .93 285
53
8
.36054 .38654 2.5871 .93274
52
9
.36 081 .38 687 2.5848 .93 264
51
1O
.36 108 .38 721 2.5826 .93 253
5O
11
.36 135 .38 754 2.5804 .93 243
49
12
.36 162 .38 787 2.5782 .93 232
48
13
.36 190 .38 821 2.5759 .93 222
47
14
.36217 .38854 2.5737 .93211
46
15
.36 244 .38 888 2.5715 .93 201
45
16
.36 271 .38 921 2.5693 .93 190
44
17
.36 298 .38 955 2.5671 .93 180
43
18
.36 325 .38 988 2.5649 .93 169
42
19
.36 352 .39 022 2.5627 .93 159
41
20
.36 379 .39 055 2.5605 .93 148
4O
21
.36406 .39089 2.5583 .93137
39
22
.36 434 .39 122 2.5561 .93 127
38
23
.36461 .39156 2.5539 .93 116
37
24
.36488 .39190 2.5517 .93106
36
25
.36515 .39223 2.5495 .93095
35
26
.36542 .39257 2.5473 .93084
34
27
.36 569 .39 290 2.5452 .93 074
33
28
.36 596 .39 324 2.5430 .93 063
32
29
.36623 .39357 2.5408 .93052
31
30
.36 650 .39 391 2.5386 .93 042
30
31
.36677 .39425 2.5365 .93031
29
32
.36 704 .39 458 2.5343 .93 020
28
33
.36731 .39492 2.5322 .93010
27
34
.36 758 .39 526 2.5300 .92 999
26
35
.36785 .39559 2.5279 .92988
25
36
.36812 .39593 2.5257 .92978
24
37
.36 839 .39 626 2.5236 .92 967
23
38
.36867 .39660 2.5214 .92956
22
39
.36894 .39694 2.5193 .92945
21
4O
.36921 .39727 2.5172 .92935
2O
41
.36948 .39761 2.5150 .92924
19
42
.36975 .39795 2.5129 .92913
18
43
.37002 .39829 2.5108 .92902
17
44
.37029 .39862 2.5086 .92892
16
45
.37 056 .39 896 2.5065 .92 881
15
46
.37 083 .39 930 2.5044 .92 870
14
47
.37110 .39963 2.50Z3 .92859
13
48
.37 137 .39 997 2.5002 .92 849
12
49
.37 164 .40 031 2.4981 .92 838
11
5O
.37191 .40065 2.4960 .92827
10
51
.37218 .40098 2.4939 .92816
9
52
.37 245 .40 132 2.4918 .92 805
8
53
.37 272 .40 166 2.4897 .92 794
7
54
.37 299 .40 200 2.4876 .92 784
6
55
.37 326 .40 234 2.4855 .92 773
5
56
.37 353 .40 267 2.4834 .92 762
4
57
.37380 .40301 2.4813 .92751
3
58
.37407 .40335 2.4792 .92740
2
59
.37 434 .40 369 2.4772 .92 729
1
6O
.37461 .40403 2.4751 .92718
/
cos cot tan sin
/
68
NATURAL FUNCTIONS
69
22
/
sin tan cot cos
/
o
.37461 .40403 2.4751 .92718
60
1
.37 488 .40 436 2.4730 .92 707
59
2
.37515 .40470 2.4709 .92697
58
3
.37 542 .40 504 2.4689 .92 686
57
4
.37 569 .40 538 2.4668 .92 675
56
5
.37595 .40572 2.4648 .92664
55
6
.37622 .40606 2.4627 .92653
54
7
.37649 .40640 2.4606 .92642
53
8
.37676 .40674 2.4586 .92631
52
9
.37 703 .40 707 2.4566 .92 620
51
1C
.37 730 .40 741 2.4545 .92 609
50
11
.37757 .40775 2.4525 .92598
49
12
.37 784 .40 809 2.4504 .92 587
48
13
.37811 .40843 2.4484 .92576
47
14
.37 838 .40 877 2.4464 .92 565
46
15
.37865 .40911 2.4443 .92554
45
16
.37892 .40945 2.4423 .92543
44
17
.37 919 .40 979 2.4403 .92 532
43
18
.37 946 .41 013 2.4383 .92 521
42
19
.37973 .41047 2.4362 .92510
41
20
.37 999 .41 081 2.4342 .92 499
4O
21
.38 026 .41 115 2.4322 .92 488
39
22
.38 053 .41 149 2.4302 .92 477
38
23
.38 080 .41 183 2.4282 .92 466
37
24
.38 107 .41 217 2.4262 .92 455
36
25
.38134 .41251 2.4242 .92444
35
26
.38161 .41285 2.4222 .92432
34
27
.38 188 .41 319 2.4202 .92 421
33
28
.38215 .41353 2.4182 .92410
32
29
.38 241 .41 387 2.4162 .92 399
31
30
.38268 .41421 2.4142 .92388
30
31
.38295 .41455 2.4122 .92377
29
32
.38 322 .41 490 2.4102 .92 366
28
33
.38349 .41524 2.4083 .92355
27
34
.38 376 .41 558 2.4063 .92 343
26
35
.38 403 .41 592 2.4043 .92 332
25
36
.38430 .41626 2.4023 .92321
24
37
.38 456 .41 660 2.4004 .92 310
23
38
.38 483 .41 694 2.3984 .92 299
22
39
.38 510 .41 728 2.3964 .92 287
21
40
.38 537 .41 763 2.3945 .92 276
2O
41
.38 564 .41 797 2.3925 .92 265
19
42
.38 591 .41 831 2.3906 .92 254
18
43
.38 617 .41 865 2.3886 .92 243
17
44
.38644 .41899 2.3867 .92231
16
45
.38671 .41933 2.3847 .92220
15
46
.38 698 .41 968 2.3828 .92 209
14
47
.38 725 .42 002 2.3808 .92 198
13
48
.38 752 .42 036 2.3789 .92 186
12
49
.38 778 .42 070 2.3770 .92 175
11
5O
.38 805 .42 105 2.3750 .92 164
1O
51
.38 832 .42 139 2.3731 .92 152
9
52
.38 859 .42 173 2.3712 .92 141
8
53
.38 886 .42 207 2.3693 .92 130
7
54
.38912 .42242 2.3673 .92119
6
55
.38 939 .42 276 2.3654 .92 107
5
56
.38966 .42310 2.3635 .92096
4
57
.38993 .42345 2.3616 .92085
3
58
.39020 .42379 2.3597 .92073
2
59
.39046 .42413 2.3578 .92062
1
6O
.39073 .42447 2.3559 .92050
O
/
cos cot tan sin
/
67
23
/
sin tan cot cos
/
.39073 .42447 2.3559 .92050
60
1
.39 100 .42 482 2.3539 .92 039
59
2
.39 127 .42 516 2.3520 .92 028
58
3
.39153 .42551 2.3501 .92016
57
4
.39 180 .42 585 2.3483 .92 005
56
5
.39207 .42619 2.3464 .91994
55
6
.39 234 .42 654 2.3445 .91 982
54
7
.39 260 .42 688 2.3426 .91 971
53
8
.39 287 .42 722 2.3407 .91 959
52
9
.39 314 .42 757 2.3388 .91 948
51
10
.39 341 .42 791 2.3369 .91 936
5O
11
.39 367 .42 826 2.3351 .91 925
49
12
.39 394 .42 860 2.3332 .91 914
48
13
.39421 .42894 2.3313 .91902
47
14
.39 448 .42 929 2.3294 .91 891
46
15
.39474 .42963 2.3276 .91879
45
16
.39 501 .42 998 2.3257 .91 868
44
17
.39 528 .43 032 2.3238 .91 856
43
18
.39 555 .43 067 2.3220 .91 845
42
19
.39 581 .43 101 2.3201 .91 833
41
2O
.39 608 .43 136 2.3183 .91 822
4O
21
.39635 .43170 2.3164 .91*810
39
22
.39 661 .43 205 2.3146 .91 799
38
23
.39 688 .43 239 2.3127 .91 787
37
24
.39 715 .43 274 2.3109 .91 775
36
25
.39 741 .43 308 2.3090 .91 764
35
26
.39 768 .43 343 2.3072 .91 752
34
27
.39 795 .43 378 2.3053 .91 741
33
28
.39 822 .43 412 2.3035 .91 729
32
29
.39 848 .43 447 2.3017 .91 718
31
30
.39 875 .43 481 2.2998 .91 706
30
31
.39902 .43516 2.2980 .91694
29
32
.39 928 .43 550 2.2962 .91 683
28
33
.39 955 .43 585 2.2944 .91 671
27
34
.39 982 .43 620 2.2925 .91 660
26
35
.40 008 .43 654 2.2907 .91 648
25
36
.40 035 .43 689 2.2889 .91 636
24
37
.40 062 .43 724 2.2871 .91 625
23
38
.40 088 .43 758 2.2853 .91 613
22
39
.40 115 .43 793 2.2835 .91 601
21
4O
.40 141 .43 828 2.2817 .91 590
2O
41
.40 168 .43 862 2.2799 .91 578
19
42
.40 195 .43 897 2.2781 .91 566
18
43
.40 221 .43 932 2.2763 .91 555
17
44
.40 2H8 .43 966 2.2745 .91 543
16
45
.40 275 .44 001 2.2727 .91 531
15
46
.40 301 .44 036 2.2709 .91 519
14
47
.40 328 .44 071 2.2691 .91 508
13
48
.40355 .44" 105 2.2673 .91496
12
49
.40381 .44140 2.2655 .91484
11
50
.40408 .44175 2.2637 .91472
1O
51
.40434 .44210 2.2620 .91461
9
52
.40461 .44244 2.2602 .91449
8
53
.40 488 .44 279 2.2584 .91 437
7
54
.40514 .44314 2.2566 .91425
6
55
.40 541 .44 349 2.2549 .91 414
5
56
.40567 .44384 2.2531 .91402
4
57
.40594 .44418 2.2513 .91390
3
58
.40621 .44453 2.2496 .91378
2
59
.40 647 .44 488 2.2478 .91 366
1
60
.406.74 .44 523 2.2460 .91 355
/
cos cot tan sin
/
66
70
NATURAL FUNCTIONS
24
/
sin tan cot cos
/
o
.40674 .44523 2.2460 .91355
6O
1
.40 700 .44 558 2.2443 .91 343
59
2
.40727 .44593 2.2425 .91331
58
3
.40753 .44627 2.2408 .91319
57
4
.40 780 .44 662 2.2390 .91 307
56
5
.40 806 .44 697 2.2373 .91 295
55
6
.40 833 .44 732 2.2355 .91 283
54
7
.40 860 .44 767 2.2338 .91 272
53
8
.40 886 .44 802 2.2320 .91 260
52
9
.40913 .44837 2.2303 .91248
51
1C
.40 939 .44 872 2.2286 .91 236
5O
11
.40 966 .44 907 2.2268 .91 224
49
12
.40992 .44942 2.2251- .91212
48
13
.41019 .44977 2.2234 .91200
47
14
.41 045 .45 012 2.2216 .91 1S8
46
15
.41 072 .45 047 2.2199 .91 176
45
16
.41 098 .45 082 2.2182 .91 164
44
17
.41 125 .45 117 2.2165 .91 152
43
18
.41 151 .45 152 2.2148 .91 140
42
19
.41 178 .45 187 2.2130 .91 128
41
2O
.41204 .45222 2.2113 .91116
40
21
.41 231 .45 257 2.2096 .91 104
39
22
.41 257 .45 292 2.2079 .91 092
38
23
.41 284 .45 327 2.2062 .91 080
37
24
.41 310 .45 362 2.2045 .91 068
36
25
.41 337 .45 397 2.2028 .91 056
35
26
.41 363 .45 432 2.2011 .91 044
34
27
.41 390 .45 467 2.1994 .91 032
33
28
.41 416 .45 502 2.1977 .91 020
32
29
.41 443 .45 538 2.1960 .91 008
31
30
.41469 .45573 2.1943 .90996
30
31
.41 496 .45 608 2.1926 .90984
29
32
.41 522 .45 643 2.1909 .90 972
28
33
.41549 .45678 2.1892 .90960
27
34
.41575 .45 713 2.1876 .90948
26
35
.41602 .45748 2.1859 .90936
25
36
.41 628 .45 784 2.1842 .90 924
24
37
.41655 .45819 2.1825 .90911
23
38
.41 681 .45 854 2.1808 .90 899
22
39
.41 707 .45 889 2.1792 .90 887
21
40
.41 734 .45 924 2.1775 .90 875
20
41
.41 760 .45 960 2.1758 .90 863
19
42
.41787 .45995 2.1742 .90851
18
43
.41813 .46030 2.1725 .90839
17
44
.41 840 .46 065 2.1708 .90 826
16
45
.41 866 .46 101 2.1692 .90 814
15
46
.41 892 .46 136 2.1675 .90 802
14
47
.41 919 .46 171 2.1659 .90 790
13
48
.41 945 .46 206 2.1642 .90 778
12
49
.41 972 .46 242 2.1625 .90 766
11
5O
.41998 .46277 2.1609 .90753
10
51
.42 024 .46 312 2.1592 .90 741
9
52
.42051 .46348 2.1576 .90729
8
53
.42077 .46383 2.1560 .90717
7
54
.42104 .46418 2.1543 .90704
6
55
.42 130 .46 454 2.1527 .90 692
5
56
.42156 .46489 2.1510 .90680
4
57
.42 183 .46 525 2.1494 .90 668
3
58
.42 209 .46 560 2.1478 .90 655
2
59
.42235 .46595 2.1461 .90643
1
60
.42262 .46631 2.1445 .90631
/
cos cot tan sin
/
65
25
/
sin tan cot cos
/
.42262 .46631 2.1445 .90631
60
1
.42 288 .46 666 2.1429 .90 618
59
2
.42315 .46702 2.1413 .90606
58
3
.42341 .46737 2.1396 .90594^
57
4
.42367 .46772 2.1380 .90582
56
5
.42 394 .46 808 2.1364 .90 569
55
6
.42420 .46843 2.1348 .90557
54
7
.42 446 .46 879 2.1332 .90 545
53
* 8
.42473 .46914 2.1315 .90532
52
% 9
.42499 .46950 2.1299 .90520
51
10
.42525 .46985 2.1283 .90507
50
11
.42552 .47021 2.1267 .90495
49
12
.42578 .47056 2.1251 .90483
48
13
.42604 .47092 2.1235 .90470
47
14
.42631 .47128 2.1219 .90458
46
15
.42657 .47163 2.1203 .90446
45
16
.42683 .47199 2.1187 .90433
44
17
.42709 .47234 2.1171 .90421
43
18
.42736 .47270 2.1155 .90408
42
19
.42762 .47305 2.1139 .90396
41
2O
.42788 .47341 2.1123 .90383
40
21
.42815 .47377 2.1107 .90371
39
22
.42841 .47412 2.1092 .90358
38
23
.42867 .47448 2.1076 .90346
37
24
.42 894 .47 483 2.1060 .90 334
36
25
.42920 .47519 2.1044 .90321
35
26
.42946 .47555 2.1028 .90309
34
27
.42 972 .47 590 2.1013 .90 296
33
28
.42999 .47626 2.0997 .90284
32
29
.43025 .47662 2.0981 .90271
31
3O
.43051 .47698 2.0965 .90259
3O
31
.43 077 .47 733 2.0950 .90 246
29
32
.43 104 .47 769 2.0934 .90 233
28
33
.43 130 .47 805 2.091S .90 221
27
34
.43 156 .47 840 2.0903 .90 208
26
35
.43 182 .47 876 2.0887 .90 196
25
36
.43 209 .47 912 2.0872 .90 183
24
37
.43235 .47948 2.0856 .90171
23
38
.43261 .47984 2.0840 .90158
22
39
.43 287 .48 019 2.0825 .90 146
21
40
.43313 .48055 2.0809 .90133
2O
41
.43 340 .48 091 2.0794 .90 120
19
42
.43 366 .48 127 2.0778 .90 108
18
43
.43 392 .48 163 2.0763 .90 095
17
44
.43 418 .48 198 2.0748 .90 082
16
45
.43445 .48234 2.0732 .90070
15
46
.43471 .48270 2.0717 .90057
14
47
.43497 .48306 2.0701 .90045
13
48
.43 523 .48 342 2.0686 .90 032
12
49
.43549 .48378 2.0671 .90019
11
50
.43575 .48414 2.0655 .90007
1O
51
.43 602 .48 450 2.0640 .89 994
9
52
.43628 .48486 2.0625 .89981
8
53
.43 654 .48 521 2.0609 .89 968
7
54
.43680 .48557 2.0594 .89956
6
55
.43 706 .48 593 2.0579 .89 943
5
56
.43 733 .48 629 2.0564 .89 930
4
57
.43 759 .48 665 2.0549 .89 918
3
58
.43 785 .48 701 2.0533 .89 905
2
59
.43 811 .48 737 2.0518 .89 892
1
6O
.43 837 .48 773 2.0503 .89 879
/
cos cot tan sin
/
64
NATURAL FUNCTIONS
71
26
/
sin tan cot cos
/
.43837 .48773 2.0503 .89879
60
1
.43863 .48809 2.0488 .89867
59
2
.43 889 .48 845 2.0473 .89 854
58
3
.43916 .48881 2.0458 .89841
57
4
.43942 .48917 2.0443 .89828
56
5
.43968 .48953 2.0428 .89816
55
6
.43 994 .48 989 2.0413 .89 803
54
7
.44020 .49026 2.0398 .89790
53
8
.44 046 .49 062 2.0383 .89 777
52
9
.44 072 .49 098 2.0368 .89 764
51
1O
.44098 .49134 2.0353 .89752
5O
11
.44 124 .49 170 2.0338 .89 739
49
12
.44151 .49206 2.0323 .89726
48
13
.44177 .49242 2.0308 .89713
47
14
.44203 .49278 2.0293 .89700
46
15
.44229 .49315 2.0278 .89687
45
16
.44255 .49351 2.0263 .89674
44
17
.44281 .49387 2.0248 .89662
43
18
.44307 .49423 2.0233 .89649
42
19
.44333 .49459 2.0219 .89636
41
2O
.44359 .49495 2.0204 .89623
4O
21
.44385 .49532 2.0189 .89610
39
22
.44411 .49568 2.0174 .89597
38
23
.44437 .49604 2.0160 .89584
37
24
.44 464 .49 640 2.0145 .89 571
36
25
.44 490 .49 677 2.0130 .89 558
35
26
.44 516 .49 713 2.0115 .89 545
34
27
.44 542 .49 749 2.0101 .89 532
33
28
.44568 .49786 2.0086 .89519
32
29
.44594 .49822 2.0072 .89506
31
'3O
.44620 .49858 2.0057 .89493
30
31
.44646 .49894 2.0042 .89480
29
32
.44672 .49931 2.0028 .89467
28
33
.44698 .49967 2.0013 .89454
27
34
.44 724 .50 004 1.9999 .89 441
26
35
.44750 .50040 1.9981- .89428
25
36
.44 776 .50 076 1.9970 .89 415
24
37
.44802 .50113 1.9955 .89402
23
38
.44 828 .50 149 1.9941 .89 389
22
39
.44 854 .50 185 1.9926 .89 376
21
40
.44880 .50222 1.9912 .89363
20
41
.44906 .50258 1.9897 .89350
19
42
.44932 .50295 1.9883 .89337
18
43
.44958 .50331 1.9868 .89324
17
44
.44984 .50368 1.9854 .89311
16
45
.45010 .50404 1.9840 .89298
15
46
.45036 .50441 1.9825 .89285
14
47
.45062 .50477 1.9811 .89272
13
48
.45 088 .50 514 1.9797 .89 259
12
49
.45114 .50550 1.9782 .89245
11
5O
.45 140 .50 587 1.9768 .89 232
10
51
.45 166 .50 623 1.9754 .89 219
9
52
.45192 .50660 1.9740 .89206
8
53
.45218 .50696 1.9725 .89193
7
54
.45243 .50733 1.9711 .89180
6
55
.45269 .50769 1.9697 .89167
5
56
.45 295 .50 806 1.9683 .89 153
4
57
.45321 .50843 1.9669 .89140
3
58
.45347 .50879 1.9654 .89127
2
59
.45 373 .50 916 1.9640 .89 114
1
60
.45399 .50953 1.9626 .89101
O
/
cos cot tan sin
/
63
27
/
sin tan cot cos
f
.45 399 .50 953 1.9626 .89 101
60
1
.45425 .50989 1.9612 .89087
59
2
.45451 .51026 1.9598 .89074
58
3
.45477 .51063 1.9584 .89061
57
4
.45503 .51099 1.9570 .89048
56
5
.45529 .51136 1.9556 .89035
55
6
.45554 .51173 1.9542 .89021
54
'7
.45580 .51209 1.9528 .89008
53
8
.45606 .51246 1.9514 .88995
52
9
.45 632 .51 283 1.9500 .88 981
51
1O
.45658 .51319 1.9486 .88968
5O
11
.45 684 .51 356 1.9472 .88 955
49
12
.45 7JO .51393 1.9458 .88942
48
13
.45736 .51430 1.9444 .88928
47
14
.45762 .51467 1.9430 .88915
46
15
.45 787 .51 503 1.9416 .88902
45
16
.45813 .51 540 1.9402 .88888
44
17
.45839 .51577 1.9388 .88875
43
18
.45865 .51 614 1.9375 .88862
42
19
.45891 .51651 1.9361 .88848
41
20
.45917 .51688 1.9347 .88835
4O
21
.45942 .51 724 1.9333 .88822
39
22
.45968 .51 761 1.9319 .88808
38
23
.45 994 .51 798 1.9306 .88 795
37
24
.46020 .51 835 1.9292 .88782
36
25
.46046 .51872 1.9278 .88768
35
26
.46072 .51909 1.9265 .88755
34
27
.46097 .51946 1.9251 .88741
33
28
.46123 .51983 1.9237 .88728
32
29
.46149 .52020 1.9223 .88715
31
30
.46 175 .52 057 1.9210 .88 701
3O
31
.46201 .52094 1.9196 .88688
29
32
.46226 .52131 1.9183 .88674
28
33
.46252 .52168 1.9169 .88661
27
34
.46278 .52205 1.9155 .88647
26
35
.46304 .52242 1.9142 .88634
25
36
.46330 .52279 1.9128 .88620
24
37
.46355 .52316 1.9115 .88607
23
38
.46381 .52353 1.9101 .88593
22
39
.46407 .52390 1.9088 .88580
21
40
.46433 .52427 1.9074 .88566
2O
41
.46458 .52464 1.9061 .88553
19
42
.46484 .52501 1.9047 .88539
18
43
.46 510 .52 538 1.9034 .88 526
17
44
.46536 .52575 1.9020 .88512
16
45
.46 561 .52 613 1.9007 .88 499
15
46
.46 587 .52 650 1.8993 .88 485
14
47
.46 613 .52 687 1.89SO .88 472
13
48
.46 639 .52 724 1.8967 .88 458
12
49
.46664 .52761 1.8953 .88445
11
50
.46690 .52798 1.8940 .88431
1O
51
.46716 .52836 1.8927 .88417
9
52
.46742 .52873 1.8913 .88404
8
53
.46767 .52910 1.8900 .88390
71
54
.46793 .52947 1.8887 .88377
6
55
.46819 .52985 1.8873 .88363
5
56
.46 844 .53 022 1.8860 .88 349
4
57
.46 870 .53 059 1.8847 .88 336
3
58
.46 896 .53 096 1.8834 .88 322
2
59
.46921 .53 134 1.8820 .88308
1
60
.46947 .53171 1.SS07 .88 295
/
cos cot tan sin
/
62
72
NATURAL FUNCTIONS
28
/
sin tan cot cos
/
o
.46947 .53171 1.8S07 .88295
60
1
.46973 .53208 1.8794 .88281
59
2
.46999 .53246 1.8781 .88267
58
3
.47 024 .53 283 1.8768 .88 254
57
4
.47050 .53320 1.8755 .88240
56
5
.47 076 .53 358 1.8741 .88 226
55
6
.47 101 .53 395 1.8728 .88 213
54
7
.47 127 .53 432 1.8715 .88 199
53
8
.47 153 .53470 1.8702 .88 185
52
9
.47 178 .53 507 1.8689 .88 172
51
10
.47 204 .53 545 1.8676 .88 158
5O
11
.47 229 .53 582 1.8663 .88 144
49
12
.47 255 .53 620 1.8650 .88 130
48
13
.47 281 .53 657 1.8637 .88 117
47
14
.47 306 .53 694 1.8624 .88 103
46
15
.47332 .53732 1.8611 .88089
45
16
.47 358 .53 769 1.8598 .88 075
44
17
.47 383 .53 807 1.8585 .88 062
43
18
.47409 .53844 1.8572 .88048
42
19
.47434 .53882 1.8559 .88034
41
2O
.47 460 .53 920 1.8546 .88 020
40
21
.47486 .53957 1.8533 .88006
39
22
.47511 .53995 1.8520 .87993
38
23
.47 537 .54 032 1.8507 .87 979
37
24
.47562 .54070 1.8495 .87965
36
25
.47588 .54307 1.8482 .87951
35
26
.47 614 .54 145 1.8469 .87 937
34
27
.47 639 .54 183 1.8456 .87 923
33
28
.47665 .54220 18443 .87909
32
29
.47 690 .54 258 1.8430 .87 896
31
3O
.47 716 .54 296 1.8418 .87 882
3O
31
.47741 .54333 1.8405 .87868
29
32
.47 767 .54 371 1.8392 .87 854
28
33
.47 793 .54 409 1.8379 .87 840
27
34
.47 818 .54 446 1.8367 .87 826
26
35
.47844 .54484 1.8354 .87812
25
36
.47 869 .54 522 1.8341 .87 798
24
37
.47 895 .54 560 1.8329 .87 784
23
38
.47 920 .54 597 1.8316 .87 770
22
39
.47 946 .54 635 1.8303 .87 756
21
4O
.47971 .54673 1.8291 .87743
2O
41
.47 997 .54 711 1.8278 .87 729
19
42
.48022 .54748 1.8265 .87715
18
43
.48048 .54786 1.8253 .87701
17
44
.48073 .54824 1.8240 .87687
16
45
.48099 .54862 1.8228 .87673
15
46
.48124 .54900 1.8215 .87659
14
47
.48150 .54938 1.8202 .87645
13
48
.48175 .54975 1.8190 .87631
12
49
.48201 .55013 1.8177 .87617
11
50
.48226 .55051 1.8165 .87603
1O
51
.48 252 .55 089 1.8152 .87 589
9
52
.48 277 .55 127 1.8140 .87 575
8
53
.48 303 .55 165 1.8127 .87 561
7
54
.48328 .55203 1.8115 .87546
6
55
.48 354 .55 241 1.8103 .87 532
5
56
.48379 .55279 1.8090 .87518
4
57
.48405 .55317 1.8078 .87504
3
58
.48430 .55355 1.8065 .87490
2
59
.48 456 .55 393 1.8053 .87 476
1
GO
.48481 .55431 1.8040 .87462
/
cos cot tan sin
/
61
29
/
sin tan cot cos
/
.48481 .55431 1.8040 .87462
6O
1
.48506 .55469 1.8028 .87448
59
2
.48 532 .55 507 1.8016 .87 434
58
3
.48 557 .55 545 1.8003 .87 420
57
4
.48 583 .55 583 1.7991 .87 406
56
5
.48608 .55621 1.7979 .87391
55
6
.48634 .55659 1.7966 .87377
54
7
.48 659 .55 697 1.7954 .87 363
53
8
.48684 .55 736 1.7942 .87349
52
9
.48710 .55774 1.7930 .87335
51
1O
.48735 .55812 1.7917 .87321
50
11
.48 761 .55 850 1.7905 .87 306
49
12
.48786 .55888 1.7893 .87292
48
13
.48811 .55926 1.7881 .87278
47
14
.48837 .55964 1.7868 .87264
46
15
.48862 .56003 1.7856 .87250
45
16
.48888 .56041 1.7844 .87235
44
17
.48913 .56079 .7832 .87221
43
18
.48938 .56117 .7820 .87207
42
19
.48 964 .56 156 .7808 .87 193
41
2O
.48 989 .56 194 .7796 .87 178
4O
21
.49 014 .56 232 .7783 .87 164
" 39
22
.49 040 .56 270 .7771 .87 150
38
23
.49065 .56309 .7759 .87136
37
24
.49 090 .56 347 .7747 .87 121
36
25
.49 116 .56 385 .7735 .87 107
35
26
.49141 .56424 .7723 .87093
34
27
.49166 .56462 .7711 .87079
33
28
.49192 .56 501 .7699 .87 064
32.
29
.49 217 .56 539 .7687 .87 05.0
31
30
.49242 .56577 .7675 .87036
3O
31
.49268 .56616 .7663 .87021
29
32
.49293 .56654 .7651 .87007
28
33
.49318 .56693 .7639 .86993
27
34
.49344 .56731 .7627 .86978
26
35
.49369 .56769 1.7615 .86964
25
36
.49394 .56808 1.7603 .86949
24
37
.49419 .56846 1.7591 .86935
23
38
.49445 .56885 1.7579 .86921
22
39
.49470 .56923 1.7567 .86906
21
40
.49495 .56962 1.7556 .86892
2O
41
.49521 .57000 1.7544 .86878
19
42
.49546 .57039 1.7532 .86863
18
43
.49 571 .57 078 1.7520 .86 849
17
44
.49596 .57116 1.7508 .86834
16
45
.49 622 .57 155 .7496 .86 820
15
46
.49647 .57 193 .7485 .86805
14
47
.49 672 .57 232 .7473 .86 791
13
48
.49 697 .57 271 .7461 .86 777
12
49
.49 723 .57 309 .7449 .86 762
11
50
.49 748 .57 348 .7437 .86 748
1O
51
.49 773 .57 386 .7426 .86 733
9
52
.49 798 .57 425 1.7414 .86 719
8
53
.49824 .57464 1.7402 .86704
7
54
.49849 .57503 1.7391 .86690
6
55
.49874 .57541 1.7379 .86675
5
56
.49899 .57580 1.7367 .86661
4
57
.49 924 .57 619 1.7355 .86 646
3
58
.49950 .57657 1.7344 .86632
2
59
.49975 .57696 1.7332 .86617
1
60
.50000 .57735 1.7321 .86603
O
/
cos cot tan sin
/
60
NATURAL FUNCTIONS
73
30
/
sin tan cot cos
/
o
.50000 .57735 1.7321 .86603
6O
1
.50025 .57774 1.7309 .86588
59
2
.50050 .57813 1.7297 .86573
58
3
.50076 .57851 1.7286 .86559
57
4
.50 101 .57 890 1.7274 .86 544
56
5
.50 126 .57 929 1.7262 .86 530
55
6
.50151 .57968 1.7251 .86515
54
7
.50176 .58007 1.7239 .86501
53
8
.50201 .58046 1.7228 .86486
52
9
.50227 .58085 1.7216 .86471
51
10
.50252 .58124 1.7205 .86457
50
11
.50277 .58162 1.7193 .86442
49
12
.50302 .58201 1.7182 .86427
48
13
.50327 .58240 1.7170 .86413
47
14
.50352 .58279 1.7159 .86398
46
15
.50377 .58318 1.7147 .86384
45
16
.50403 .58357 1.7136 .86369
44
17
.50428 .58396 1.7124 .86354
43
18
.50453 .58435 1.7113 .86340
42
! 19
.50478 .58474 1.7102 .86325
41
2O
.50503 .58513 1.7090 .86310
40
21
.50 528 .58 552 1.7079 .86 295
39
22
.50553 .58591 1.7067 .86281
38
23
.50578 .58631 1.7056 .86266
37
24
.50603 .58670 1.7045 .86251
36
25
.50628 .58709 1.7033 .86237
35
26
.50654 .58748 1.7022 .86222
34
27
.50679 .58787 1.7011 .86207
33
28
.50704 .58826 1.6999 .86192
32
29
.50729 .58865 1.6988 .86178
31
3O
.50 754 .58 905 1.6977 .86 163
30
31
.50779 .58944 1.6965 .86148
29
32
.50804 .58983 1.6954 .86133
28
33
.50829 .59022 1.6943 .86119
27
34
.50854 .59061 1.6932 .86104
26
35
.50879 .59101 1.6920 .86089
25
36
.50904 .59140 1.6909 .86074
24
37
.50929 .59179 1.6898 .86059
23
38
.50954 .59218 1.6887 .86045
22
39
.50979 .59258 1.6875 .86030
21
4O
.51004 .59297 1.6864 .86015
2O
41
.51029 .59336 1.6853 .86000
19
42
.51054 .59376 1.6S42 .85985
18
43
.51079 .59415 1.6831 .85970
17
44
.51104 .59454 1.6820 .85956
16
45
.51 129 .59494 1.6808 .85941
15
46
.51 154 .59533 1.6797 .85926
14
47
.51 179 .59573 1.67S6 .85911
13
48
.51204 .59612 1.6775 .85896
12
49
.51229 .59651 1.6764 .85881
11
50
.51254 .59691 1.6753 .85866
10
51
.51279 .59730 1.6742 .85851
9
52
.51304 .59770 1.6731 .85836
8
53
.51329 .59809 1.6720 .85821
7
54
.51354 .59849 1.6709 .85806
6
55
.51379 .59888 1.6698 .85792
5
56
.51404 .59928 1.6687 .85777
4
57
.51429 .59967 1.6676 .85762
3
58
.51454 .60007 1.6665 .85747
2
59
.51479 .60046 1.6654 .85732
1
60
.51504 .60086 1.6643 .85717
O
/
cos cot tan sin
/
59
31
/
sin tan cot cos
/
.51504 .60086 1.6643 .85717
60
1
.51 529 .60 126 1.6632 .85 702
59
2
.51 554 .60 165 1.6621 .85 687
58
3
.51579 .60205 1.6610 .85672
57
4
51604 .60245 1.6599 .85657
56
5
.51 628 .60 284 1.6588 .85 642
55
6
.51653 .60324 1.6577 .85627
54
'7
.51678 .60364 1.6566 .85612
53
8
.51703 .60403 1.6555 .85597
52
9
.51728 .60443 1.6545 .85582
51
1O
.51753 .60483 1.6534 .85567
50
11
.51 778 .60522 1.6523 .85551
49
12
.51803 .60562 1.6512 .85536
48
13
.51828 .60602 1.6501 .85521
47
14
.51852 .60642 1.6490 .85506
46
15
.51877 .60681 1.6479 .85491
45
16
.51902 .60721 1.6469 .85476
44
17
.51927 .60761 1.6458 .85461
43
18
.51952 .60801 1.6447 .85446
42
19
.51977 .60841 1.6436 .85431
41
2O
.52 002 .60 881 1.6426 .85 416
40
21
.52026 .60921 1.6415 .85401
39
22
.52051 .60960 1.6404 .85385
38
23
.52076 .61000 1.6393 .85370
37
24
.52 101 .61 040 1.6383 .85 355
36
25
.52 126 .61 080 1.6372 .85 340
35
26
.52151 .61 120 1.6361 .85325
34
27
.52175 .61160 1.6351 .85310
33
28
.52 200 .61 200 1.6340 .85 294
32
29
.52 225 .61 240 1.6329 .85 279
31
30
.52 250 .61 280 1.6319 .85 264
30
31
.52 275 .61 320 1.6308 .85 249
29
32
.52 299 .61 360 1.6297 .85 234
28
33
.52 324 .61 400 1.6287 .85 218
27
31
.52 349 .61 440 1.6276 .85 203
26
35
.52374 .61480 1.6265 .85188
25
36
.52 399 .61 520 1.6255 .85 173
24
37
.52 423 .61 561 1.6244 .85 157
23
38
.52 448 .61 601 1.6234 .85 142
22
39
.52473 .61641 1.6223 .85 127
21
4O
.52498 .61681 1.6212 .85112
2O
41
.52 522 .61 721 1.6202 .85 096
19
42
.57 547 .61 761 1.6191 .85 081
18
43
.52572 .61801 1.6181 .85066
17
44
.52 597 .61 842 1.6170 .85 051
16
45
.52 621 .61 882 1.6160 .85 035
15
46
.52 646 .61 922 1.6149 .85 020
14
47
.52671 .61962 1.6139 .85005
13
48
.52696 .62003 1.6128 .84989
12
49
.52720 .62043 1.6118 .84974
11
5O
.52745 .62083 1.6107 .84959
1O
51
.52770 .62 124 1.6097 .84943
9
52
.52 794 .62 164 1.6087 .84 928
8
53
.52 819 .62 204 1.6076 .84 913
7
54
.52 844 .62 245 1.6066 .84 897
6
55
.52 869 .62 285 1.6055 .84 882
5
56
.52893 .62325 1.6045 .84866
4
57
.52918 .62366 1.6034 .84851
3
58
.52 943 .62 406 1.6024 .84 836
2
59
.52967 .62446 1.6014 .84820
1
60
.52992 .62487 1.6003 .84805
/
cos cot tan sin
/
58
74
NATURAL FUNCTIONS
32
/
sin tan cot cos
/
~0
.52992 .62487 1.6003 .84805
6O
1
.53017 .62527 1.5993 .84789
59
2
.53041 .62568 1.5983 .84774
58
3
.53 066 .62 608 1.5972 .84 759
57
4
.53 091 .62 649 1.5962 .84 743
56
5
.53 115 .62 689 1.5952 .84 728
55
6
.53 140 .62 730 1.5941 .84 712
54
7
.53164 .62770 1.5931 .84697
53
8
.53189 .62811 1.5921 .84681
52
9
.53214 .62852 1.5911 .84666
51
10
.53 238 .62 892 1.5900 .84 650
50
11
.53 263 .62 933 1.5890 .84 635
49
12
.53288 .62973 1.5880 .84619
48
13
.53312 .63014 1.5869 .84604
47
14
.53337 .63055 1.5859 .84588
46
15
.53 361 .63 095 1.5849 .84 573
45
16
.53 386 .63 136 1.5839 .84 557
44
17
.53411 .63177 1.5829 .84542
43
18
.53435 .63217 1.5818 .84526
42
19
53460 .63258 1.5808 .84511
41
2O
.53 484 .63 299 1.5798 .84 495
40
21
.53 509 .63 340 1.5788 .84 480
39
22
.53534 .63380 1.5778 .84464
38
23
.53558 .63421 1.5768 .84448
37
24
.53583 .63462 1.5757 .84433
36
25
.53607 .63503 1.5747 .84417
35
26
.53632 .63544 1.5737 .84402
34
27
.53 656 .63 584 1.5727 .84 386
33
28
.53681 .63625 1.5717 .84370
32
29
.53705 .63666 1.5707 .84355
31
3D
.53 730 .63 707 1.5697 .84 339
30
31
.53 754 .63 748 1.5687 .84 324
29
32
.53 779 .63 789 1.5677 .84 308
28
33
.53 804 .63 830 1.5667 .84 292
27
34
.53 828 .63 871 1.5657 .84 277
26
35
.53853 .63912 1.5647 .84261
25
36
.53 877 .63 953 L5637 .84 245
24
37
.53902 .63994 1.5627 .84230
23
38
.53926 .64035 1.5617 .84214
22
39
.53951 .64076 1.5607 .84198
21
4O
.53975 .64117 1.5597 .84182
20
41
.54 000 .64 158 1.5587 .84 167
19
42
.54024 .64199 1.5577 .84151
18
43
.54 049 .64 240 1.5567 .84 135
17
44
.54073 .64281 1.5557 .84120
16
45
.54097 .64322 1.5547 .84104
15
46
.54122 .64363 1.5537 .84088
14
47
.54146 .64404 1.5527 .84072
13
48
.54171 .64446 1.5517 .84057
12
49
.54 195 .64 487 1.5507 .84 041
11
50
.54220 .64528 1.5497 .84025
10
51
.54244 .64569 1.5487 .84009
9
52
.54269 .64610 1.5477 .83994
8
53
.54293 .64652 1.5468 .83978
7
54
.54317 .64693 1.5458 .83962
6
55
.54342 .64734 1.5448 .83946
5
56
.54 366 .64 775 1.5438 .83 930
4
57
.54 391 .64 817 1.5428 .83 915
3
58
.54415 .64858 1.5418 .83899
2
59
.54 440 .64 899 1.5408 .83 883
1
6O
.54 464 .64 941 1.5399 .83 867
O
/
cos cot tan sin
/
57
33
/
sin tan cot cos
/
.54464 .64941 1.5399 .83867
6O
1
.54488 .64982 1.5389 .83851
59
2
.54 513 .65 024 1.5379 .83 835
58
3
.54 537 .65 065 1.5369 .83 819
57
4
.54 561 .65 106 1.5359 .83 804
56
5
.54 586 .65 148 1.5350 .83 788
55
6
.54610 .65 189 1.5340 .83772
54
7
.54 635 .65 231 1.5330 .83 756
53
8
.54 659 .65 272 1.5320 .83 740
52
9
.54683 .65314 1.5311 .83724
51
10
.54 708 .65 355 1.5301 .83 708
50
11
.54 732 .65 397 1.5291 .83 692
49
12
.54 756 .65 438 1.5282 .83 676
48
13
.54781 .65480 1.5272 .83660
47
14
.54805 .65521 1.5262 .83645
46
15
.54 829 .65 563 1.5253 .83 629
45
16
.54 854 .65 604 1.5243 .83 613
44
17
.54 878 .65 646 1.5233 .83 597
43
18
.54 902 .65 688 1.5224 .83 581
42
19
.54 927 .65 729 1.5214 .83 565
41
2O
.54951 .65771 1.5204 .83549
40
21
.54975 .65813 1.5195 .83533
39
22
.54999 .65854 1.5185 .83517
38
23
.55 024 .65 896 1.5175 .83 501
37
24
.55 048 .65 938 1.5166 .83 485
36
25
.55072 .65980 1.5156 .83469
35
26
.55097 .66021 1.5147 .83453
34
27
.55121 .66063 1.5137 .83437
33
28
.55-145 .66105 1.5127 .83421
32
29
.55 169 .66 147 1.5118 .83 405
31
30
.55 194 .66 189 1.5108 .83 389
30
31
.55 218 .66 230 1.5099 .83 373
29
32
.55 242 .66 272 1.5089 .83 356
28
33
.55266 .66314 1.5080 .83340
27
34
.55291 .66356 1.5070 .83324
26
35
.55315 .66398 1.5061 .83308
25
36
.55339 .66440 1.5051 .83292
24
37
.55 363 .66 482 1.5042 .83 276
23
38
.55 388 .66 524 1.5032 .83 260
22
39
.55412 .66566 1.5023 .83244
21
40
.55 436 .66 608 1.5013 .83 228
20
41
.55460 .66650 1.5004 .83212
19
42
.55484 .66692 1.4994 .83195
18
43
.55 509 .66 734 1.4985 .83 179
17
44
.55 533 .66 776 1.4975 .83 163
16
45
.55557 .66818 1.4966 .83147
15
46
.55581 .66860 1.4957 .83131
14
47
.55605 .66902 1.4947 .83115
13
48
.55630 .66944 1.4938 .83098
12
49
.55654 .66986 1.4928 .83082
11
50
.55 678 .67 028 1.4919 .83 066
10
51
.55 702 .67 071 1.4910 .83 050
9
52
.55726 .67113 1.4900 .83034
8
53
.55750 .67155 1.4891 .83017
7
54
.55 775 .67 197 1.4882 .83 001
6
55
.55 799 .67 239 1.4872 .82 985
5
56
.55823 .67282 1.4863 .82969
4
57
.55847 .67324 1.4854 .82953
3
58
.55871 .67366 1.4844 .82936
2
59
.55 895 .67 409 1.4835 .82 920
1
60
.55919 .67451 1.4826 .82904
/
cos cot tan sin
/
56
NATURAL FUNCTIONS
75
34
/
sin tan cot cos
/
o
.55919 .67451 1.4826 .82904
6O
1
.55 943 .67 493 1.4816 .82 887
59
2
.55968 .67536 1.4807 .82871
58
3
.55992 .67578 1.4798 .82855
57
4
.56016 .67620 1.4788 .82839
56
5
.56 040 .67 663 1.4779 .82 822
55
6
.56064 .67705 1.4770 .82806
54
7
.56088 .67748 1.4761 .82790
53
8
.56112 .67790 1.4751 .82773
52
9
.56 136 .67 832 1.4742 .82 757
51
10
.56 160 .67 875 1.4733 .82 741
50
11
.56 184 .67 917 1.4724 .82 724
49
12
.56 208 .67 960 1.4715 .82 708
48
13
.56232 .68002 1.4705 .82692
47
14
.56 256 .68 045 1.4696 .82 675
46
15
.56280 .68088 1.4687 .82659
45
16
.56 305 .68 130 1.4678 .82 643
44
17
.56 329 .68 173 1.4669 .82 626
43
18
.56353 .68215 1.4659 .82610
42
19
.56377 .68258 1.4650 .82593
41
2O
.56401 .68301 1.4641 .82577
40
21
.56425 .68343 1.4632 .82561
39
22
.56 449 .68 386 1.4623 .82 544
38
23
.56473 .68429 1.4614 .82528
37
24
.56497 .68471 1.4605 .82511
36
25
.56521 .68514 1.4596 .82495
35
26
.56545 .68557 1.4586 .82478
34
27
.56569 .68600 1.4577 .82462
33
28
.56593 .68642 1.4568 .82446
32
29
.56617 .68685 1.4559 .82429
31
30
.56641 .68728 1.4550 .82413
3O
31
.56665 .68771 1.4541 .82396
29
32
.56689 .68814 1.4532 .82380
28
33
.56713 .68857 1.4523 .82363
27
34
.56736 .68900 1.4514 .82347
26
35
.56760 .68942 1.4505 .82330
25
36
.56784 .68985 1.4496 .82314
24
37
.56808 .69028 1.4487 .82297
23
38
.56832 .69071 1.4478 .82281
22
39
.56856 .69114 1.4469 .82264
21
4O
.56880 .69157 1.4460 .82248
2O
41
.56904 .69200 1.4451 .82231
19
42
.56928 .69243 1.4442 .82214
18
43
.56952 .69286 1.4433 .82198
17
44
.56 976 .69 329 1.4424 .82 181
16
45
.57 000 .69 372 1.4415 .82 165
15
46
.57024 .69416 1.4406 .82148
14
47
.57047 .69459 1.4397 .82132
13
48
.57071 .69502 1.4388 .82115
12
49
.57095 .69545 1.4379 .82098
11
50
.57119 .69588 1.4370 .82082
1O
51
.57143 .69631 1.4361 .82065
9
52
.57167 .69675 1.4352 .82048
8
53
.57191 .69718 1.4344 .82032
7
54
.57215 .69761 1.4335 .82015
6
55
.57238 .69804 1.4326 .81999
5
56
.57262 .69847 1.4317 .81982
4
57
.57286 .69891 1.4308 .81965
3
58
.57310 .69934 1.4299 .81949
2
59
.57334 .69977 1.4290 .81932
1
6O
.57358 .70021 1.4281 .81915
O
/
cos cot tan sin
/
55
35
/
sin tan cot cos
/
O
.57358 .70021 1.4281 .81915
60
1
.57381 .70064 1.4273 .81899
59
2
.57405 .70107 1.4264 .81882
58
3
.57429 .70151 1.4255 .81865
57
4
.57 453 .70 194 1.4246 .81 848
56
5
.57 477 .70 238 1.4237 .81 832
55
6
.57 501 .70 281 1.4229 .81 815
54
7
.57 524 .70 325 1.4220 .81 798
53
8
.57548 .70368 1.4211 .81782
52
9
.57572 .70412 1.4202 .81765
51
10
.57596 .70455 1.4193 .81748
50
11
.57619 .70499 1.4185 .81731
49
12
.57643 .70542 1.4176 .81714
48
13
.57667 .70586 1.4167 .81698
47
14
.57691 .70629 1.4158 .81681
46
15
.57715 .70673 1.4150 .81664
45
16
.57738 .70717 1.4141 .81647
44
17
.57 762 .70 760 1.4132 .81 631
43
18
.57786 .70804 1.4124 .81614
42
19
.57810 .70848 1.4115 .81597
41
20
.57833 .70891 1.4106 .81580
40
21
.57857 .70935 1.4097 .81563
39
22
.57881 .70979 1.4089 .81546
38
23
.57904 .71023 1.4080 .81530
37
24
.57 928 .71 066 1.4071 .81 513
36
25
.57952 .71110 1.4063 .81496
35
26
.57976 .71154 1.4054 .81479
34
27
.57999 .71198 1.4045 .81462
33
28
.58 023 .71 242 1.4037 .81 445
32
29
.58 047 .71 285 1.4028 .81 428
31
3O
.58070 .71329 1.4019 .81412
30
31
.58 094 .71 373 1.4011 .81 395
29
32
.58 118 .71 417 1.4002 .81 378
28
33
.58 141 .71 461 1.3994 .81 361
27
34
.58 165 .71 505 1.3985 .81 344
26
35
.58189 .71549 1.3976 .81327
25
36
.58212 .71593 1.3968 .81310
24
37
.58236 .71637 1.3959 .81293
23
38
.58260 .71681 1.3951 .81 276
22
39
.58 283 .71 725 1.3942 .81 259
21
4O
.58307 .71769 1.3934 .81242
20
41
.58330 .71813 1.3925 .81225
19
42
.58354 .71857 1.3916 .81208
18
43
.58378 .71901 1.3908 .81191
17
44
.58401 .71946 1.3899 .81174
16
45
.58425 .71990 1.3891 .81157
15
46
.58 449 .72 034 1.3882 .81 140
14
47
.58472 .72078 1.3874 .81 123
13
48
.58 496 .72 122 1.3865 .81 106
12
49
.58 519 .72 167 1.3857 .81 089
11
5O
.58543 .72211 1.3848 .81072
1O
51
.58567 .72255 1.3840 .81055
9
52
.58590 .72299 1.3831 .81038
8
53
.58614 .72344 1.3823 .81021
7
54
.58637 .72388 1.3814 .81004
6
55
.58661 .72432 1.3806 .80987
5
56
.58684 .72477 1.3798 .80970
4
57
.58708 .72521 1.3789 .80953
3
58
.58731 .72565 1.3781 .80936
2
59
.58755 .72610 1.3772 .80919
1
60
.58 779 .72 654 1.3764 .80 902
/
cos cot tan sin
/
54
76
NATURAL FUNCTIONS
36
/
sin tan cot cos
/
~o
.58779 .72654 1.3764 .80902
6O
i
.58802 .72699 1.3755 .80885
59
2
.58 826 .72 743 1.3747 .80 867
58
3
.58 849 .72 788 1.3739 .80 850
57
4
.58 873 .72 832 1.3730 .80 833
56
5
.58896 .72877 1.3722 .80816
55
6
.58920 .72921 1.3713 .80799
54
7
.58943 .72966 1.3705 .80782
53
8
.58967 .73010 1.3697 .80765
52
9
.58 990 .73 055 1.3688 .80 748
51
10
.59014 .73100 1.3680 .80730
50
11
.59037 .73144 1.3672 .80713
49
12
.59 061 .73 189 1.3663 .80 696
48
13
.59084 .73234 1.3655 .80679
47
14
.59108 .73.278 1.3647 .80662
46
IS
.59131 .73323 1.3638 .80644
45
16
.59154 .73368 1.3630 .80627
44
17
.59178 .73413 1.3622 .80610
43
18
.59201 .73457 1.3613 .80593
42
19
.59225 .73502 1.3605 .80576
41
2O
.59248 .73' 547 1.3597 .80558
40
21
.59 272 .73 592 1.3588 .80 541
39
22
.59 295 .73 637 1.3580 .80 524
38
23
.59318 .73681 1.3572 .80507
37
24
.59 342 .73 726 1.3564 .80 489
36
25
.59365 .73771 1.3555 .80472
35
26
.59389 .73816 1.3547 .80455
34
27
.59412 .73861 1.3539 .80438
33
28
.59436 .73906 1.3531 .80420
32
29
.59459 .73951 1.3522 .80403
31
3O
.59482 .73996 1.3514 .80386
30
31
.59506 .74041 1.3506 .80368
29
32
.59529 .74086 1.3498 .80351
28
33
.59552 .74131 1.3490 .80334
27
34
.59576 .74176 1.3481 .80316
26
35
.59599 .74221 1.3473 .80299
25
36
.59622 .74267 1.3465 .80282
24
37
.59646 .74312 1.3457 .80264
23
38
.59669 .74357 1.3449 .80247
22
39
.59693 .74402 1.3440 .80230
21
40
.59716 .74447 1.3432 .80212
2O
41
.59739 .74492 1.3424 .80195
19
42
.59763 .74538 1.3416 .80178
18
43
.59 786 .74 583 1.3408 .80 160
17
44
.59809 .74628 1.3400 .80143
16
45
.59832 .74674 1.3392 .80125
15
46
.59 856 .74 719 1.3384 .80 108
14
47
.59 879 .74 764 1.3375 .80 091
13
48
.59902 .74810 1.3367 .80073
12
49
.59926 .74855 1.3359 .80056
11
50
.59949 .74900 1.3351 .80038
1O
51
.59972 .749-16 1.3343 .80021
9
52
.59995 .74991 1.3335 .80003
8
53
.60019 .75037 1.3327 .79986
7
54
.60042 .75082 1.3319 .79968
6
55
.60065 .75128 1.3311 .79951
5
56
.60089 .75173 1.3303 .79934
4
57
.60112 .75219 1.3295 .79916
3
58
.60135 .75264 1.3287 .79899
2
59
.60158 .75310 1.3278 .79881
1
60
.60 182 .75 355 1.3270 .79 864
O
/
cos cot tan sin
/
53
37
/
sin tan cot cos
/
O
.60182 .75355 1.3270 .79864
60
1
.60205 .75401 1.3262 .79846
59
2
.60228 .75447 1.3254 .79829
58
3
.60251 .75492 1.3246 .79811
57
4
.60 274 .75 538 1.3238 .79 793
.56
5
.60 298 .75 584 1.3230 .79 776
55
6
.60321 .75629 1.3222 .79758
54
7
.60344 .75675 1.3214 .79741
53
8
.60 367 .75 721 1.3206 .79 723
52
9
.60390 .75767 1.3198 .79706
51
1O
.60414 .75812 1.3190 .79688
50
11
.60437 .75858 1.3182 .79671
49
12
.60460 .75904 1.3175 .79653
48
13
.60483 .75950 1.3167 .79635
47
14
.60506 .75996 1.3159 .79618
46
15
.60529 .76042 1.3151 .79600
45
16
.60553 .76088 1.3143 .79583
44
17
.60576 .76134 1.3135 .79565
43
18
.60599 .76180 1.3127 .79547
42
19
.60622 .76226 1.3119 .79530
41
2O
.60645 .76272 1.3111 .79512
4O
21
.60668 .76318 1.3103 .79494
39
22
.60691 .76364 1.3095 .79477
38
23
.60714 .76410 1.3087 .79459
37
24
.60738 .76456 1.3079 .79441
36
25
.60761 .76502 1.3072 .79424
35
26
.60784 .76548 1.3064 .79406
34
27
.60807 .76594 1.3056 .79388
33
28
.60830 .76640 1.3048 .79371
32
29
.60853 .76686 1.3040 .79353
31
3O
.60876 .76733 1.3032 .79335
30
31
.60899 .76779 1.3024 .79318
29
32
.60922 .76825 1.3017 .79300
28
33
.60945 .76871 1.3009 .79282
27
34
.60968 .76918 1.3001 .79264
26
35
.60991 .76964 1.2993 .79247
25
36
.61015 .77010 1.2985 .79229
24
37
.61038 .77057 1.2977 .79211
23
38
.61 061 .77 103 1.2970 .7? 193
22
39
.61 084 .77 149 1.2962 .79 176
21
4O
.61 107 .77 196 1.2954 .79 158
20
41
.61 130 .77242 1.2946 .79140
19
42
.61 153 .77 289 1.2938 .79 122
18
43
.61 176 .77335 .2931 .79105
17
44
.61199 .77382 .2923 .79087
16
45
.61222 .77428 .2915 .79069
15
46
.61245 .77475 .2907 .79051
14
47
.61 268 .77 521 .2900 .79 033
13
48
.61 291 .77 568 .2892 .79 016
12
49
.61314 .77615 1.28S4 .78998
11
50
.61337 .77661 1.2876 .78980
1O
51
.61 360 .77 708 1.2869 .Z8_%4
9
52
.61383 .77754 1.2861 ./*94^
8.
53
.61406 .77801 1.2853 .78926
7
54
.61 429 .77 848 1.2846 .78 90S
6
55
.61451 .77895 1.2838 .78891
5
56
.61 474 .77 941 1.2830 .78 873
4
57
.61497 .77988 1.2822 .78855
3
58
.61520 .78035 1.2815 .78837
2
59
.61543 .78082 1.2807 .78819
1
00
.61 566 .78 129 1.2799 .78 801
O
/
cos cot tan sin
/
52
NATURAL FUNCTIONS
38
f
sin tan cot cos
/
o
.61566 .78129 1.2799 .78801
60
1
.61 589 .78 175 1.2792 .78 783
59
2
.61 612 .78 222 1.2784 .78 765
58
3
.61 635 .78 269 1.2776 .78 747
57
4
.61658 .78316 1.2769 .78729
56
5
.61681 .78363 1.2761 .78711
55
6
.61 704 .78 410 1.2753 .78 694
54
7
.61 726 .78 457 1.2746 .78 676
53
8
.61749 .78504 1.2738 .78658
52
9
..61772 .78551 1.2731 .78640
51
1O
.61 795 .78 598 1.2723 .78 622
50
11
.61818 .78645 1.2715 .78604
49
12
.61 841 .78 692 1.2708 .78 586
48
13
.61 864 .78 739 1.2700 .78 568
47
14
.61 887 .78 786 1.2693 .78 550
46
15
.61 909 .78 834 1.2685 .78 532
45
16
.61932 .78881 1.2677 .78514
44
17
.61 955 .78 928 1.2670 .78 496
43
18
.61 978 .78 975 1.2662 .78 478
42
19
.62001 .79022 1.2655 .78460
41
2O
.62 024 .79 070 1.2647 .78 442
4O
21
.62046 .79117 1.2640 .78424
39
22
.62069 .79164 1.2632 .78405
38
23
.62092 .79212 1.2624 .78387
37
24
.62115 .79259 1.2617 .78369
36
25
.62138 .79306 1.2609 .78351
35
26
.62 160 .79 354 1.2602 .78 333
34
27
.62183 .79401 1.2594 .78315
33
28
.62 206 .79 449 1.2587 .78 297
32
29
.62229 .79496 1.2579 .78279
31
30
.62251 .79544 1.2572 .78261
30
31
.62 274 .79 591 1.2564 .78 243
29
32
.62297 .79639 1.2557 .78225
28
33
.62 320 .79 686 1.2549 .78 206
27
34
.62 342 .79 734 1.2542 .78 188
26
35
.62 365 .79 781 1.2534 .78 170
25
36
.62388 .79829 1.2527 .78152
24
37
.62411 .79877 1.2519 .78134
23
38
.62433 .79924 1.2512 .78116
22
39
.62456 .79972 1.2504 .78098
21
40
.62479 .80020 1.2497 .78079
2O
41
.62502 .80067 1.2489 .78061
19
42
.62524 .80115 1.2482 .78043
18
43
.62547 .80163 1.2475 .78025
17
44
.62570 .80211 1.2467 .78007
16
45
.62592 .80258 1.2460 .77988
15
46
.62615 .80306 1.2452 .77970
14
47
.62638 .80354 1.2445 .77952
13
48
.62660 .80402 1.2437 .77934
12
49
.62683 .80450 1.2430 .77916
11
50
.62 706 .80 498 1.2423 .77 897
1O
51
.62 728 .80 546 1.2415 .77 879
9
52
.62 751 .80 594 1.2408 .77 861
8
53
.62 774 .80 642 1.2401 .77 843
7
54
. .62 796 .80 690 1.2393 .77 824
6
55
.62819 .80738 1.23S6 .77806
5
56
.62 842 .80 786 1.2378 .77 788
4
57
.62864 .80834 1.2371 .77769
3
58
.62887 .80882 1.2364 .77751
2
59
.62909 .80930 1.2356 .77733
1
6O
.62932 .80978 1.2349 .77715
O
/
cos cot tan sin
/
51
39
/
sin tan cot cos
/
O
.62932 .80978 1.2349 .77715
60
1
.62955 .81027 1.2342 .77696
59
2
.62977 .81075 1.2334 .77678
58
3
.63000 .81123 1.2327 .77660
57
4
.63 022 .81 171 1.2320 .77 641
56
5
.63045 .81220 1.2312 .77623
55
6
.63 068 .81 268 1.2305 .77 605
54
7
.63 090 .81 316 1.2298 .77 586
53
8
.63 113 .81 364 1.2290 .77 568
52
9
.63 135 .81 413 1.2283 .77 550
51
10
.63 158 .81 461 1.2276 .77 531
50
11
.63180 .81510 1.2268 .77513
49
12
.63 203 .81 558 1.2261 .77 494
48
13
.63 225 .81 606 1.2254 .77 476
47
14
.63248 .81655 1.2247 .77458
46
15
.63 271 .81 703 1.2239 .77 439
45
16
.63 293 .81 752 1.2232 .77 421
44
17
.63 316 .81 800 1.2225 .77 402
43
18
.63 338 .81 849 1.2218 .77 384
42
19
.63 361 .81 898 1.2210 .77 366
41
20
.63 383 .81 946 1.2203 .77 347
40
21
.63 406 .81 995 1.2196 .77 329
39
22
.63 428 .82 044 1.2189 .77 310
38
23
.63451 .82092 1.2181 .77292
37
24
.63473 .82141 1.2174 .77273
36
25
.63 496 .82 190 1.2167 .77 255
35
26
.63 518 .82 238 1.2160 .77 236
34
27
.63 540 .82 287 1.2153 .77 218
33
28
.63 563 .82 336 1.2145 .77 199
32
29
.63 585 .82 385 1.2138 .77 181
31
30
.63608 .82434 1.2131 .77162
3O
31
.63 630 .82 483 1.2124 .77 144
29
32
.63653 .82531 1.2117 .77125
28
33
.63 675 .82 580 1.2109 .77 107
27
34
.63698 .82629 1.2102 .77088
26
35
.63 720 .82 678 1.2095' .77 070
25
36
.63742 .82727 1.2088 .77051
24
37
.63765 .82776 1.2081 .77033
23
38
.63 787 .82 825 1.2074 .77 014
22
39
.63 810 .82 874 1.2066 .76 996
21
4O
.63832 .82923 1.2059 .76977
2O
41
.63854 .82972 1.2052 .76959
19
42
.63877 .83022 1.2045 .76940
18
43
.63 899 .83 071 1.2038 .76 921
17
44
.63 922 .83 120 1.2031 .76 903
16
45
.63944 .83169 1.2024 .76884
15
46
.63966 .83218 1.2017 .76866
14
47
.63989 .83268 1.2009 .76847
13
48
.64011 .83317 1.2002 .76828
12
49
.64 033 .83 366 1.1995 .76 810
11
50
.64056 .83415 1.1988 .76791
10
51
.64078 .83465 1.1981 .76772
9
52
.64 100 .83 514 1.1974 .76 754
8
53
.64 123 .83 564 1.1967 .76 735
7
54
.64145 .83613 1.1960 .76717
6
55
.64 167 .83 662 1.1953 .76 698
5
56
.64 190 .83 712 1.1946 .76 679
4
57
.64212 .83761 1.1939 .76661
3
58
.64234 .83811 1.1932 .76642
2
59
.64256 .83860 1.1925 .76623
1
6O
.64279 .83910 1.1918 .76604
O
/
cos cot tan sin
/
50
78
NATURAL FUNCTIONS
40
/
sin tan cot cos
/
.64279 .83910 1.1918 .76604
60
1
.64301 .83960 1.1910 .76586
59
2
.64323 .84009 1.1903 .76567
58
3
.64346 .84059 1.1896 .76548
57
4
.64368 .84108 1.1889 .76530
56
5
.64390 .84158 1.18S2 .76511
55
6
.64412 .84208 1.1875 .76492
54
7
.64435 .84258 1.1868 .76473
53
8
.64457 .84307 1.1861 .76455
52
9
.64479 .84357 1.1854 .76436
51
1O
.64501 .84407 1.1847 .76417
50
11
.64524 .84457 1.1840 .76398
49
12
.64546 .84507 1.1833 .76380
48
13
.64568 .84556 1.1826 .76361
47
14
.64590 .84606 1.1819 .76342
46
15
.64612 .84656 1.1812 .76323
-45
16
.64 635 .84 706 1.1806 .76 304
44
17
.64 657 .84 756 1.1799 .76 286
43
18
.64679 .84806 1.1792 .76267
42
19
.64701 .84856 1.1785 .76248
41
2O
.64723 .81-906 1.1778 .76229
4O
21
.64746 .84956 .1771 .76210
39
22
.64 768 .85 006 1.1764 .76 192
38
23
.64790 .85057 .1757 .76173
37
24
.64 812 .85 107 .1750 .76 154
36
25
.64834 .85157 .1743 .76135
35
26
.64856 .85207 .1736 .76116
34
27
.64878 .85257 1.1729 .76097
33
28
.64901 .85308 1.1722 .76078
32
29
.64923 .85358 1.1715 .76059
31
3O
.64945 .85408 1.1708 .76041
30
31
.64967 .85458 1.1702 .76022
29
32
.64 989 .85 509 1.1695 .76 003
28
33
.65011 .85559 1.1688 .75984
27
34
.65 033 .85 609 1.1681 .75 965
26
35
.65055 .85660 1.1674 .75946
25
36
.65 077 .85 710 1.1667 .75 927
24
37
.65 100 .85 761 1.1660 .75 908
23
38
.65 122 .85 811 1.1653 .75 889
22
39
.65 144 .85 862 1.1647 .75 870
21
4O
.65 166 .85912 1.1640 .75851
20
41
.65 188 .85 963 1.1633 .75 832
19
42
.65 210 .86 014 1.1626 .75 813
18
43
.65 232 .86064 1.1619 .75 794
17
44
.65254 .86115 1.1612 .75775
16
45
.65 276 .86 166 1.1606 .75 756
15
46
.65 298 .86 216 1.1599 .75 738
14
47
.65 320 -86 267 1.1592 .75 719
13
48
.65342 .86318 1.1585 .75700
12
49
.65364 .86368 1.1578 .75680
11
5O
.65386 .86419 1.1571 .75661
10
51
.65408 .86470 1.1565 .75642
9
52
65430 .86521 1.1558 .75623
8
53
.65452 .86572 1.1551 .75604
7
54
.65474 .86623 1.1544 .75585
, 6
55
.65496 .86674 1.1538 .75566
5
56
.65518 .86725 1.1531 .75547
4
57
.65 540 .86 776 1.1524 .75 528
3
58
.65562 .86827 1.1517 .75509
2
59
.65584 .86878 1.1510 .75490
1
6O
.65606 .86929 1.1504 .75471
o
/
cos cot tan sin
/
49
41
/
sin tan cot cos
/
.65606 .86929 1.1504 .75471
60
1
.65 628 .86 980 1.1497 .75 452
59
2
.65650 .87031 1.1490 .75433
58
3
.65 672 .87 082 1.1483 .75 414
57
4
.65 694 .87 133 1.1477 .75 395
56
5
.65 716 .87 184 1.1470 .75 375
55
6
.65 738 .87 236 1.1463 .75 356
54
7
.65 759 .87 287 1.1456 .75 337
53
8
.65781 .87338 1.1450 .75318
52
9
.65803 .87389 1.1443 .75299
51
1O
.65 825 .87 441 1.1436 .75 280
50
11
.65 847 .87 492 1.1430 .75 261
49
12
.65 869 .87 543 1.1423 .75 241
48
13
.65 891 .87 595 1.1416 .75 222
47.
14
.65913 .87646 1.1410 .75203
46
15
.65 935 .87 698 1.1403 .75 184
45
16
.65956 .87 749 1.1396 .75 165
44
17
.65 978 .87 801 1.1389 .75 146
43
18
.66000 .87852 1.1383 .75 126
42
19
.66022 .87904 1.1376 .75107
41
2O
.66044 .87955 1.1369 .75088
4O
21
.66066 .88007 1.1363 .75069
39
22
.66088 .88059 1.1356 .75050
38
23
.66109 .88110 1.1349 .75030
37
24
.66131 .88162 1.1343 .75 QU^
36
25
.66153 .88214 1.1336 .74992
35
26
.66175 .88265 1.1329 .74973
34
27
.66197 .88317 1.1323 .74953
33
28
.66218 .88369 1.1316 .74934
32
29
.66240 .88421 1.1310 .74915
31
30
.66262 .88473 1.1303 .74896
30
31
.66 284 .88 524 1.1296 .74 876
29
32
.66306 .88576 1.1290 .74857
28
33
.66327 .88628 1.1283 .74838
27
34
.66349 .88680 1.1276 .74818
26
35
.66371 .88732 1.1270 .74799
25
36
.66393 .88784 1.1263 .74780
24
37
.66414 .88836 1.1257 .74760
23
38
.66436 .88888 1.1250 .74741
22
39
.66458 .88940 1.1243 .74722
21
4O
.66480 .88992 1.1237 .74703
2O
41
.66501 .89045 1.1230 .74683
19
42
.66523 .89097 1.1224 .74664
18
43
.66545 .89149 1.1217 .74644
17
44
.66566 .89201 1.1211 .74625
16
45
.66588 .89253 1.1204 .74606'
15
46
.66610 .89306 1.1197 .74586
14
47
.66632 .89358 1.1191 .74567
13
48
.66653 .89410 1.1184 .74548
12
49
.66675 .89463 1.1178 .74528
11
50
.66697 .89515 1.1171 .74509
1O
51
.66718 .89567 1.1165 .74489
9
52
.66 740 .89 620 1.1158 .74 470
8
53
.66762 .89672 1.1152 .74451
7
54
.66783 .89725 1.1145 .74431
6
55
.66805 .89777 1.1139 .74412
5
56
.66827 .89830 1.1132 .74392
4
57
.66848 .89883 1.1126 .74373
3
58
.66870 .89935 1.1119 .74353
2
59
.66891 .89988 1.1113 .74334
1
6O
.66913 .90040 1.1106 .74314
O
/
cos cot tan sin
/
48
NATURAL FUNCTIONS
79
42
/
sin tan cot cos
/
o
.66913 .90040 1.1106 .74314
60
1
.66935 .90093 1.1100 .74295
59
2
.66956 .90146 1.1093 .74276
58
3
.66 978 .90 199 1.1087 .74 256
57
4
.66999 .90251 1.1080 .74237
56
5
.67021 .90304 1.1074 .74217
55
6
.67043 .90357 1.1067 .74198
54
7
.67064 .90410 1.1061 .74178
53
8
.67086 .90463 1.1054 .74159
52
9
.67107 .90516 1.1048 .74139
51
1O
.67 129 .90 569 1.1041 .74 120
50
11
.67151 .90621 1.1035 .74100
49
12
.67172 .90674 1.1028 .74080
48
13
.67194 .90727 1.1022 .74061
47
14
.67 215 .90 781 1.1016 .74 041
46
15
.67 237 .90 834 1.1009 .74 022
45
16
.67258 .90887 1.1003 .74002
44
17
.67 280 .90 940 1.0996 .73 983
43
18
.67301 .90993 1.0990 .73963
42
19
.67323 .91046 1.0983 .73944
' 41
20
.67344 .91099 1.0977 .73924
4O
21
.67366 .91153 1.0971 .73904
39
22
.67 387 .91 206 1.0964 .73 885
38
23
.67409 .91259 1.0958 .73865
37
24
.67430 .91313 1.0951 .73846
36
25
.67 452 .91 366 1.0945 .73 826
35
26
.67473 .91419 1.0939 .73806
34
27
.67495 .91473 1.0932 .73787
33
28
.67516 .91526 1.0926 .73767
32
29
.67 538 .91 580 1.0919 .73 747
31
3O
.67 559 .91 633 1.0913 .73 728
30
31
.67 580 .91 687 1.0907 .73 708
29
32
.67 602 .91 740 1.0900 .73 688
28
33
.67 623 .91 794 1.0894 .73 669
27
34
.67 645 '.91 847 1.0888 .73 649
26
35
.67 666 .91 901 1.0881 .73 629
25
36
.67 688 .91 955 1.0875 .73 610
24
37
.67709 .92008 1.0869 .73590
23
38
.67 730 .92 062 1.0862 .73 570
22
39
.67 752 .92 116 1.0856 -.73 551
21
40
.67 773 .92 170 1.0850 .73 531
2O
41
.67795 .92224 1.0843 .73511
19
42
.67 816 .92 277 1.0837 .73 491
18
43
.67837 .92331 1.0831 .73472
17
44
.67859 .92385 1.0824 .73452
,16
45
.67880 .92439 1.0818 .73432
15
46
.67901 .92493 1.0812 .73413
14
47
.67923 .92547 1.0805 .73393
13
48
.67 944 .92 601 1.0799 .73 373
12
49
.67965 .92655 1.0793 .73353
11
50
.67 987 .92 709 1.0786 .73 333
1O
51
.68008 .92763 1.0780 .73314
9
52
.68029 .92817 1.0774 .73294
8
53
.68051 .92872 1.0768 .73274
7
54
.68 072 .92 926 1.0761 .73 254
6
55
.68093 .92980 1.0755 .73234
5
56
.68 115 .93 034 1.0749 .73 215
4
57
.68 136 .93 088 1.0742 .73 195
3
58
.68 157 .93 143 1.0736 .73 175
2
59
.68 179 .93 197 1.0730 .73 155
1
60^
.68 200 .93 252 1.0724 .73 135
O
/
cos cot tan sin
/
47
43
/
sin tan cot cos
/
.68200 .93252 1.0724 .73135
6O
1
.68221 .93306 1.0717 .73116
59
2
.68242 .93360 1.0711 .73096
58
3
.68 264 .93 415 1.0705 .73 076
57
4
.68285 .93469 1.0699 .73056
56
5
.68 306 .93 524 1.0692 .73 036
55
6
.68327 .93578 1.0686 .73016
54
7
.68 349 .93 633 1 .0680 .72 996
53
8
.68 370 .93 688 1.0674 .72 976
52
9
.68 391 .93 742 1.0668 .72 957
51
1O
.68412 .93797 1.0661 .72937
50
11
.68434 .93852 1.0655 .72917
49
12
.68455 .93906 1.0649 .72897
48
13
.68476 .93961 1.0643 .72877
47
14
.68497 .94016 1.0637 .72857
46
15
.68518 .94071 1.0630 .72837
45
16
.68 539 .94 125 1.0624 .72 817
44
17
.68 561 .94 180 1.0618 .72 797
43
18
.68582 .94235 1.0612 .72777
42
19
.68603 .94290 1.0606 .72757
41
2.0
.68624 .94345 1.0599 .72737
40
21
.68645 .94400 1.0593 .72717
39
22
.68666 .94455 1.0587 .72697
38
23
.68688 .94510 1.0581 .72677
37
24
.68 709 .94 565 1.0575 .72 657
36
25 .68 730 .94 620 1.0569 .72 637
35
26 .68751 .94676 1.0562 .72617
34
27 .68 772 .94 731 1.0556 .72 597
33
28
.68 793 .94 786 1.0550 .72 577
32
29
.68814 .94841 1.0544 .72557
31
30
.68835 .94896 1.0538 .72537
30
31
.68 857 .94 952. 1.0532 .72 517
29
32
.68878 .95007 1.0526 .72497
28
33
.63_S.9_9 .95062 1.0519 .72477
27
34
.68920 .95118 1.0513 .72457
26
35
.68941 .95173 1.0507 .72437
25
36
.68962 .95229 1.0501 .72417
24
37
.68 983 .95 284 1.0495 .72 397
23
38
.69004 .95340 1.0489 .72377
22
39
.69025 .95395 1.0483 .72357
21
40
.69046 .95451 1.0477 .72337
2O
41
.69067 .95506 1.0470 .72317
19
42
.69 088 .95 562 1.0464 .72 297
18
43
.69 109 .95 618 1.0458 .72 277
17
44
.69 130 .95 673 1.0452 .72 257
16
45
.69 151 .95 729 1.0446 .72 236
15
46
.69172 .95785 1.0440 .72216
14
47
.69 193 .95 841 1.0434 .72 196
13
48
.69 214 .95 897 1.0428 .72 176
12
49
.69235 .95952 1.0422 .72156
11
50
.69256 .96008 1.0416 .72136
1O
51
.69 277 .96 064 1.0410 .72 116
9
52
.69 298 .96 120 1.0404 .72 095
8
53
.69319 .96176 1.0398 .72075
7
54
.69340 .96232 1.0392 .72.055
6
55
.69361 .96288 1.0385 .72035
5
56
.69382 .96344 1.0379 .72015
4
57
.69403 .96400 1.0373 .71995
3
58
.69424 .96*457 1.0367 .71974
2
59
.69445 .96513 1.0361 .71954
1
6O
.69466 .96569 1.0355 .71934
O
/
cos cot tan sin
/
46
80
NATURAL FUNCTIONS
44
/
sin tan cot cos
/
o
.69 466 .96 569 1.0355 .71 934
6O
1
.69487 .96625 1.0349 .71914
59
2
.69508 .96681 1.0343 .71894
58
3
.69 529 .96 738 .1.0337 .71 873
57
4
.69549 .96794 1.0331 .71853
56
5
.69 570 .96 850 1.0325 .71 833"
55
6
.69591 .96907 1.0319 .71813
54
7
.69612 .96963 1.0313 .71792
53
8
.69633 .97020 1.0307 .71772
52
9
.69654 .97076 1.0301 .71752
51
10
.69 675 .97 133 1.0295 .71 732
50
11
.69696 .97189 1.0289 .71711
49
12
.69717 .97246 1.0283 .71691
48
13
.69737 .97302 1.0277 .71671
47
14
.69758 .97359 1.0271 .71650
46
15
.69 779 .97 416 1.0265 .71 630
45
16
.69800 .97472 1.0259 .71610
44
17
.69821 .97529 1.0253 .71590
43
18
.69842 .97586 1.0247 .71569
42
19
.69862 .97643 1.0241 .71549
41
2O
.69883 .97700 1.0235 .71529
4O
21
.69904 .97756 1.0230 .71508
39
22
.69925 .97813 1.0224 .71488
38
23
.69946 .97870 1.0218 .71468
37
24
.69966 .97927 1.0212 .71447
36
25
.69987 .97984 1.0206 .71427
35
26
.70008 .98041 1.0200 .71407
34
27
.70029 .98098 1.0194 .71386
33
28
.70049 .98155 1.01SS .71366
32
29
.70070 .98213 1.0182 .71345
31
30
.70091 .98270 1.0176 .71325
30
31
.70112 .98327 1.0170 .71305
29
32
.70 132 .98 384 1.0164 .71 284
28
33
.70153 .98441 1.0158 .71264
27
34
.70174 .98499 1.0152 .71243
26
35
.70195 .98556 1.0147 .71223
25
36
.70215 .98613 1.0141 .71203
24
37
.70 236 .98 671 1.0135 .71 182
23
38
.70257 .98728 1.0129 .71162
22
39
.70277 .98786 1.0123 .71141
21
40
.70298 .98843 1.0117 .71 121
2O
41
.70319 .98901 1.0111 .71100
19
42
.70339 .98958 1.0105 .71080
18
43
.70360 .99016 1.0099 .71059
17
44
.70381 .99073 1.0094 .71039
16
45
.70401 .99131 1.0088 .71019
15
46
.70422 .99189 1.0082 .70998
14
47
.70443 .99247 1.0076 .70978
13
48
.70463 .99304 1.0070 .70957
12
49
.70484 .99362 1.0064 .70937
11
5O
.70505 .99420 1.0058 .70916
1O
51
.70525 .99478 1.0052 .70896
9
52
.70546 .99536 1.0047 .70875
8
53
.70567 .99594 1.0041 .70855
7
54
.70587 .99652 1.0035 .70834
6
55
.70608 .99710 1.0029 .70813
5
56
.70628 .99768 1.0023 .70793
4
57
.70649 .99826 1.0017 .70772
3
58
.70670 .99884 1.0012 .70752
2
59
.70690 .99942 1.0006 .70731
1
6O
.70711 1.0000 1.0000 .70711
O
/
cos cot tan sin
. /
45
n I
7
-
5"'
14 DAY USE
RETURN TO DESK FROM WHICH BORROWED
LOAN DEPT.
This book is due on the last date stamped below, or
on the date to which renewed.
Renewed books are subject to immediate recall.
*EP $3 i960
rrcr
65 -4PM
^25-
LD 21-100m-7,'40 (0936s)
7~1 ^
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^v-^e
9
, 7-
918256
THE UNIVERSITY OF CALIFORNIA LIBRARY
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