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PLANE TRIGONOMETRY
BY
ELMER A. LYMAN
MICHIGAN STATE NORMAL COLLEGE
AND
EDWIN C. GODDARD
UNIVERSITY OF MICHIGAN
3»<C
ALLYN AND BACON
Boston anti Chicago
Copyright, 1899, by
ELMER A. LYMAN and EDWIN C. GODDARD.
Korisooti ^rees
J. S. Gushing & Co. — Berwick & Smith
Norwood Mass. U.S.A.
PREFACE.
The need felt by tlie authors in their class-room for a text-book
furnishing sufficient material in analytical trigonometry, and also
in the solution of the triangle, is responsible for the appearance
of this book. American text-books, for the most part, treat this
latter, practical, part of the subject fully; English text-books
elaborate the former, theoretical, part ; but no book available
seems to meet both needs adequately. To do that is the first aim
of the present work. Nearly everything in the book has been
worked out in the class-room, and tried by that sure test.
Once under way the work grew, and other features demanded
attention. For some unaccountable reason nearly all books, in
the proof of the formulae for functions of a ± p, treat the same
line as both positive and negative in the same discussion, thus
vitiating the proof ; and in many cases proofs are given for acute
angles, and are then supposed to be established without further
discussion for all angles. Some books, indeed, suggest that the
student can draw other figures and show that the formula holds
in all cases. As a matter of fact the student cannot show any-
thing of the kind ; and if he could, the proof would still apply
only to conditions the same as in those figures actually drawn,
and not to all the other indefinite number of possible combina-
tions of conditions. These difficulties have been avoided by so
stating the proofs that the language applies to figures involving
any angles, and to avoid drawing an indefinite number of such
figures, as would be necessary fully to establish the formulae
geometrically, resort has been made to the algebraic proof for the
general case (see page 58).
Inverse functions have been introduced early, and used through-
out the work, so as to familiarize the student with that important
800555
iv PREFACE.
notation. From the beginning, wherever computations are intro-
duced they are made by means of logarithms. The average stu-
dent, using logarithms for a short time and only at the end of
the subject, goes away and straightway forgets what manner of
things they are. It is hoped, by dint of much practice, extended
over as long a time as possible, to give the student a command
of logarithms that will stay. The fundamental formulae of trigo-
nometry must be memorized. There is no substitute for this.
To assist in thus fixing formulae in mind, considerable oral work
has been introduced, and frequent lists of review problems in-
volving all principles and formulae previously developed. These
lists serve the further purpose of throwing the student on his
own resources, and compelling him to find in the problem itself,
and not in any model solution, the key to its solution, thus devel-
oping power, instead of mere ability to imitate. Enough prob-
lems are provided so that different selections may be assigned to
different members of a class, or to classes in different years. It
is not expected that each student will be able to solve all the
problems in the time usually given to the subject. Articles
marked * {see Art. *26) may be omitted unless the teacher finds
time for them without 7ieglecting the rest of the work. Do not assign
too much work at first. Make sure the student has complete mastery
of the fundamental formulce.
Special attention is called to the fact that in the solution of
triangles, divisions and subdivisions into cases have been aban-
doned, and the student is thrown on his own resources to select
from the three possible sets of formulae those leading to the solu-
tions from the given data. Long experience has shown that this
tends to clearness and simplicity. The use of checks is insisted
upon in all computations.
No complete acknowledgment of help received could here be
made. The authors are under obligation to many who have con-
tributed general hints, and to several who, after going over the
manuscript and proof with care, have given valuable suggestions.
The standard works of Levett and Davison, Hobson, Henrici and
Treutlein, and others, have been freely consulted, and while many
of the problems have been prepared by the authors in their class-
PREFACE. V (
room work, they have not hesitated to take, from such standard
collections as writers generally have drawn upon, any problems \
that seemed better adapted than others to the work. Quality ]
has not been knowingly sacrificed to originality in making this j
book. Corrections and suggestions will be gladly received at any i
time. 1
E. A. L. i
E. C. G. 5
OCTOBEB, 1899. 1
CONTENTS.
Chapter I. Angles — Measurement of Angles.
^.^^
PAGE
Angles ; magnitude of angles 1
Rectangular axes ; direction . . . * 2
Measurement; sexagesimal and circular systems of measurement;
the radian 3
Examples 6
Chapter II. The Trigonometric Functions.
Function defined 8
The trigonometric functions 9
Fundamental relations 11
Examples 14
Functions of 0°, 30°, 45°, 60°, 90° 15
Examples 18
Variations in the trigonometric functions 19
Graphic representation of functions , .22
Examples 27
Chapter in. Functions of any Angle — Inverse
Functions.
Relations of functions of - 0, 90° ± B, 180° ± Q, 270° ± ^ to the
functions of ^ 29
Inverse functions , , , , .35
Examples 36
Review 38
Chapter IV. Computation Tables.
Natural functions 40
Logarithms 40
Laws of logarithms 42
Use of tables 45
Cologarithms 49
Examples 50
vii
viii CONTENTS.
Chapter V. Applications.
PA6B
Measurements of heights and distances 51
Common problems in measurement 52
Examples 54
Chapter VI. General Formula. — Trigonometric Equa-
tions AND Identities.
Sine, cosine, tangent of a ± )8 . . 56
Examples . . . ', . . .... . .59
Sin ± sin cfy, cos 6 ± cos ^ 61
Examples 62
Functions of the double angle . . . . ... .63
Functions of the half angle . .64
Examples , . .64
Trigonometric equations and identities 66
Method of attack 66
Examples ... 67
Simultaneous trigonometric equations 69
Examples . . . . .70
Chapter VII. Triangles.
Laws of sines, tangents, and cosines . 72
Area of the triangle 76
Solution of triangles 76
Ambiguous case 78
Model solutions 80
Examples 83
Applications 84
Review 86
PLANE TRIGONOMETRY.
CHAPTER I.
ANGLES — MEASUREMENT OF ANGLES.
1. Angles. It is difficult, if not impossible, to define an
angle. This difficulty may be avoided by telling how it
is formed. If a line revolve about one of its points^ an angle
is generated^ the magnitude of the angle depending on the
amount of the rotation.
Thus, if one side of the angle ^, as OR^ be originally in
the position OX^ and be revolved about the point to the
position in the figure, the
angle XOR is generated.
OX is called the initial line,
and any position of OR the
terminal line of the angle
formed. The angle 6 is
considered positive if gener-
ated hy a counter-clockwise
rotation of OR, and hence negative if generated hy a clockwise
rotation. The magnitude of depends on the amount of
rotation of OR, and since the amount of such rotation may
be unlimited, there is no limit to the possible magnitude of
angles, for, evidently, the revolving line may reach the posi-
tion OR by rotation through an acute angle 6, and, likewise,
by rotation through once, twice, •••, w times 360°, plus the
acute angle 6, So that XOR may mean the acute angle
(9, ^ -f- 360°, e + 720°, .-, O-^n- 360°.
1
i?>'
Fig. 1.
£ PLANE TRIGONOMETRY.
In reading an angle, read first the initial line, then the
terminal line. Thus in the figure the acute angle XOE, or
xr, is a positive angle, and ROX^ or rx^ an equal negative
angle.
Ex. 1. Show that if the initial lines for \, f, ^/, — ^, right angles are
the same, the terminal lines may coincide.
2. Name four other angles having the same initial and terminal lines
as ^ of a right angle ; as f of a right angle ; as f of a right angle.
2. Rectangular axes. Any plane surface may be divided
by two perpendicular straight lines XX^ and YY' into four
portions, or quadrants.
XX' is known as the x-axis,
YY' as the y-axis^ and the two
together are called axes of refer-
X ence. Their intersection is the
origin^ and the four portions of
the plane surface, XOY, YOX',
y" X'OY', Y'OX, are called respec-
Fjq, 2. tively the first, second, third, and
fourth quadrants. The position of
any point in the plane is determined when we know its dis-
tances and directions from the axes.
3. Any direction may be considered positive. Then the
opposite direction must be negative. Thus, if AB represents
any positive line, BA is an equal nega-
tive line. Mathematicians usually
consider lines measured in the same direction as OX or OY
(Fig. 2) as positive. Then lines measured in the same direc-
tion as OX' or OY' must he negative.
The distance of any point from the ?/-axis is called the
abscissa, its distance from the a;-axis the ordinate, of that
point ; the two together are the coordinates of the point,
usually denoted by the letters x and y respectively, and
written (x, y).
Y
p'
P
,/
N
p"
Y'
P'"
ANGLES — MEASUREMENT. 3
When taken with their proper signs, the coordinates define completely
the position of the point. Thus, if the point P is + a units from YY',
and + h units from XX', any convenient
unit of length being chosen, the position of
P is known. For we have only to measure
a distance ON equal to a units along OX,
and then from N measure a distance h
units parallel to OY, and we arrive at the
position of the point P, (a, &). In like
manner we may locate P', (— «, ^), in the
second quadrant, P", (—a, — Z>), in the
third quadrant, and P'", (a, — &), in „ ^
the fourth quadrant.
Ex. Locate (2, -2); (0,0); (-8, -7); (0, 5); (-2, 0); (2, 2);
(m, n).
4. If OX is the initial line, is said to be an angle of the
first, second, third, or fourth quadrant, according as its ter-
minal line is in the first, second, third, or fourth quadrant.
It is clear that as OR rotates its quality is in no way affected,
and hence it is in all positions considered positive, and its ex-
tension through 0, OB', negative.
The student should notice that the initial line may take any position
and revolve in either direction. While it is customary to consider the
counter-clockwise rotation as forming a positive angle, yet the condi-
V. ^'^' tions of a figure may be such
• / that a positive angle may be
\/ generated by a clockwise rota-
yu ^ ^ ""^B' /V /4^ *io^- Thus the angle X072 in
-2^' j^'r \x each figure may be traced as
p . a positive angle by revolving
the initial line OX to the posi-
tion OR. No confusion can result if the fact is clear that when an
angle is read XOP, OX is considered a positive line revolving to the
position OR. OX' and OR' then are negative lines in whatever direc-
tions drawn. These conceptions are mere matters of agreement, and the
agreement may be determined in a particular case by the conditions of
the problem quite as well as by such general agreements of mathema-
ticians as those referred to in Arts. 3 and 4 above.
5. Measurement. All measurements are made in terms
of some fixed standard adopted as a unit. This unit must
4 PLANE TRIGONOMETRY.
be of the same kind as the quantity measured. Thus, length
is measured in terms of a unit length, surface in terms of a
unit surface, weight in terms of a unit weight, value in terms
of a unit value, an angle in terms of a unit angle.
The measure of a given quantity is the number of times it
contains the unit selected.
Thus the area of a given surface in square feet is the
number of times it contains the unit surface 1 sq. ft. ; the
length of a road in miles, the number of times it contains
the unit length 1 mi. ; the weight of a cargo of iron ore in
tons, the number of times it contains the unit weight 1 ton ;
the value of an estate, the number of times it contains the
unit value f 1.
The same quantity may have different measures, according
to the unit chosen. So the measure of 80 acres, when the
unit surface is 1 acre, is 80, when the unit surface is 1 sq. rd.,
is 12,800, when the unit surface is 1 sq. yd., is 387,200.
What is its measure in square feet ?
6. The essentials of a good unit of measure are :
1. That it be invariable, i.e. under all conditions bearing
the same ratio to equal magnitudes.
2. That it be convenient for practical or theoretical pur-
poses.
3. That it be of the same kind as the quantity measured.
7. Two systems of measuring angles are in use, the sexa-
gesimal and the circular.
The sexagesimal system is used in most practical applica-
tions. The right angle, the unit of measure in geometry,
though it is invariable, as a measure is too large for con-
venience. Accordingly it is divided into 90 equal parts,
called degrees. The degree is divided into 60 minutes, and
the minute into 60 seconds. Degrees, minutes, seconds, are
indicated by the marks ° ' ", as 36° 20' 15''.
The division of a right angle into hundredths, with subdivisions into
hundredths, would be more convenient. The French have proposed such
MEASUREMENT OF ANGLES.
a centesimal system, dividing the right angle into 100 grades, the grade
into 100 minutes, and the minute into 100 seconds, marked ^^ ''\ as 508
70^ 28^\ The great labor involved in changing mathematical tables,
instruments, and records of observation to the new system has prevented
its adoption.
8. The circular system is important in theoretical con-
siderations. It is based on the fact that for a given angle
the ratio of the length of its arc to the length of the radius
of that arc is constant, i.e. for a fixed
angle the ratio arc : radius is the same
no matter what the length of the
radius. In the figure, for the angle ^,
OA OB OQ
AA' BB' CQ'
That this ratio of arc to radius for a fixed angle is constant
follows from the established geometrical principles :
1. The circumference of any circle is 2 tt times its radius.
2. Angles at the centre are in the same ratio as their arcs.
The Radian. It follows that an angle whose arc is equal
in length to the radius is a constant angle for all circles,
since in four right angles, or the perigon, there are always
2 7r such angles. This constant angle.,
ivhose arc is equal in length to the radius.,
is taken as the unit angle of circular
measure., and is called the radian. From
the definition we have
4 right angles = 360°
2 right angles = 180°
2 TT radians,
TT radians.
Fig. 6.
TT
1 right angle = 90° = — radians.
TT is a numerical quantity, 3.14159+, and not an angle. When we
speak of 180° as tt, 90° as ^, etc., we always mean tt radians, ^ radians, etc.
6 PLANE TRIGONOMETRY.
9. To change from one system of measurement to the
other we use the relation,
2 TT radians = 360°.
. •. 1 radian = i^ = 57^.2958- ;
TT
i.e. the radian is 57°.3, approximately.
Ex. 1. Express in radians 75° 30'.
75° 30' = 75°.5 ; 1 radian = 57°.3.
.-. 75° 30' = — = 1.317 radians.
57.3
2. Express in degree measure 3.6 radians.
1 radian = 57°.3.
.-. 3.6 radians = 3.6 x o7°.3 = 206° 16' 48".
EXAMPLES.
1. Construct, approximately, the following angles : 50°, — 20°, 90°,
179°, -135°, 400°, -380^ 1140°, | radians, | radians, --radians,
q -If)
3 IT radians, — ^ radians, — — ^ radians. Of which quadrant is each
angle? ^ ^
2. What is the measure of :
(a) f of a right angle, when 30° is the unit of measure ?
(b) an acre, when a square whose side is 10 rds. is the unit ?
(c) m miles, when y yards is the unit ?
3. What is the unit of measure, when the measure of 2^ miles is 50?
4. The Michigan Central R.R. is 535 miles long, and the Ann Arbor
R.R. is 292 miles long. Express the length of the first in terms of the
second as a unit.
5. What will be the measure of the radian when the right angle is
taken for the unit ? Of the right angle when the radian is the unit ?
6. In which quadrant is 45°? 10°? -60°? 145°? 1145°? -725°?
Express each in right angles ; in radians.
7. Express in sexagesimal measure
J, ^. 1, 6.28, I, 1^, -i^, radians.
O 12 TT o 3
EXAMPLES. 7
8. Express in each system an interior angle of a regular hexagon ;
an exterior angle.
9. Find the distance in miles between two places on the earth's
equator which are 11° 15' apart. (The earth's radius is about 3963 miles.)
10. Find the length of an arc which subtends an angle of 4 radians
at the centre of a circle of radius 12 ft. 3 in.
11. An arc 15 yds. long contains 3 radians. Find the radius of the
circle.
12. Show that the hour and minute hands of a watch turn through
angles of 30^ and 6° respectively per minute ; also find in degrees and in
radians the angle turned through by the minute hand in 3 hrs. 20 mins.
13. Find the number of seconds in an arc of 1 mile on the equator ;
also the length in miles of an arc of 1' (1 knot).
14. Find to three decimal places the radius of a circle in which the
arc of 71° 36' 3''.6 is 15 in. long.
15. Find the ratio of - to 5°.
6
16. What is the shortest distance measured on the earth's surface
from the equator to Ann Arbor, latitude + 42° 16' 48"?
17. The difference of two angles is 10°, and the circular measure of
their sum is 2. Find the circular measure of each angle.
18. A water wheel of radius 6 ft. makes 30 revolutions per minute.
Find the number of miles per hour travelled by a point on the rim.
CHAPTER II.
THE TRIGONOMETRIC FUNCTIONS.
10. Trigonometry, as the word indicates, was originally
concerned with the measurement of triangles. It now
includes the analytical treatment of certain functions of
angles, as well as the solution of triangles by means of cer-
tain relations between the functions of the angles of those
triangles.
11. Function. If one quantity depends upon another for
its value, the first is called a function of the second. It
always follows that the second quantity is also a function of
the first ; and, in general, functions are so related that if one
is constant the other is constant, and if either varies in value,
the other varies. This relation may be extended to any
number of mutually dependent quantities.
Illustration. If a train moves at a rate of 30 miles per
hour, the distance travelled is a function of the rate and
time, the time is a function of the rate and distance, and the
rate is a function of the time and distance.
Again, the circumference of a circle is a function of the
radius, and the radius of the circumference, for so long as
either is constant the other is constant, and if either changes
in value, the other changes, since circumference and radius
are connected by the relation (7=2 irR.
Once more, in the right triangle
NOP, the ratio of any two sides is
a function of the angle a, because
p" N' N ^ ^^^ ^^® right triangles of which a is
FiQ. 7. one angle are similar, i,e. the ratio
8
THE TRIGONOMETRIC FUNCTIONS.
9
of two corresponding sides is constant so long as a. is con-
stant, and varies if « varies.
Thus, the ratios
NP ^ N'P' ^ WP''
OP
and
ojsr
NP
OP'
ON'
N'P'
OP"
ON" ,
depend on a for their values, i.e. are functions of a.
12. The trigonometric functions. In trigonometry six
functions of angles are usually employed, called the trigono-
metric functions.
By definition these functions are the six ratios between the
sides of the triangle of reference of the given angle. The
triangle of reference is formed by drawing, from some point in
the initial line., or the initial line produced^ a perpendicular to
that line meeting the terminal line of the angle.
Fig. 8.
Let a be an angle of any quadrant. Each triangle of
reference of a, NOP, is formed by drawing a perpendicular
to OX, or OX produced, meeting the terminal line OB in P.
10 PLANE TRIGONOMETRY.
If « is greater than 360°, its triangle of reference would
not differ from one of the above triangles.
It is perhaps worthy of notice that the triangle of reference might be
defined to be the triangle formed by drawing a perpendicular to either
side of the angle, or that side produced, meet-
ing the other side or the other side produced.
In the figure, NOP is in all cases the triangle
of reference of a. The principles of the fol-
N
\ "^ ,''0 P 2f
j ,,.-'jv- lowing pages are the same no matter which
^''P of the triangles is considered the triangle of
Fig. 9. reference. It will, however, be as well, and
perhaps clearer, to use the triangle defined
under Fig. 8, and we shall always draw the triangle as there described.
13. The trigonometric functions of a (Fig. 8) are called
the sine^ cosine^ tangent^ cotangent^ secant^ and cosecant of a.
These are abbreviated in writing to sin a, cos a, tan «, cot a,
sec a, CSC «, and are defined as follows :
sin a = P^^ = ^, whence y = r sin a ;
hyp. r ^ '
base a? i
cos a = i: — = ~9 whence x = r cos a ;
hyp. r '
tan a = ^^—^ = -> whence y = x tan a ;
base oc ^ '
cot a = = —J whence x = y cot a;
perp. y ^ '
sec a = —^ = —9 whence r — x sec a;
base a? '
CSC a = — ^ = -9 whence r — y esc a.
perp. y ^
1 — cos a and 1 — sin a, called versed-sine a and coversed-sine a, respec-
tively, are sometimes used.
Ex. 1. Write the trigonometric functions of f3, NPO (Fig. 8), and
compare with those of a above.
The meaning of the prefix co in cosine, cotangent, and cosecant
appears from the relations of Ex. 1. For the sine of an angle equals the
cosine, i.e. the complement-sine, of the complement of that angle ; the tangent
THE TRIGONOMETRIC FUNCTIONS.
11
of an angle equals the cotangent of its complementary angle, and the secant
of an angle equals the cosecant of its complement-
ary angle.
2. Express each side of triangle ABC in
terms of another side, and some function of an
angle in all possible ways, as a = 6 tan A, etc. Fig. 10.
14. Constancy of the trigonometric functions. It is iiiipor-
taiit to notice why these ratios are functions of the angle, i.e.
are the same for equal angles and different for unequal
angles. This is shown by the principles of similar triangles.
\
Fig. 11.
In each figure show that in all possible triangles of refer-
ence for a the ratios are the same, but in the triangles of
reference for a and a', respectively, the ratios are different.
The student must notice that sin a is a single symbol. It is the name
of a number, or fraction, belonging to the angle a ; and if it be at any
time convenient, we may denote sin « by a single letter, such as o, or x.
Also, sin^a is an abbreviation for (sin «)'-^, i.e. for (sin a) x (sin «).
Such abbreviations are used because they are convenient. Lock, Ele-
mentary Trigonometry.
15. Fundamental relations. From the definitions of Art. 13
the following reciprocal relations are apparent :
sin a =
a =
tana
CSC a
1
sec a'
1
cot a
Also from the definitions.
tana =
sm g
cos a
C8C a
sm a
1
sec a = 1
cos a
1
cot a
cot a —
tan oL
cos a
sin a
12 ^ PLANE TRIGONOMETRY.
From the right triangle NOP, page 9,
y'^ -\- x^ = T^ \
/2 ^2
whence (1) U-j^'L^l^
From (1) sin^a+cos^ a=l; sma= Vl — cos^ a; cos cc=?
(2) tan2a + l = sec2a; ^^/^ «= -y/sec^ cc— 1 ; sec a = ?
(3) l+cot2a = csc2a; cota=Vcsc^ a—1 ; esc a = ?
The foregoing definitions and fundamental relations are of
the highest importance, and must he mastered at once. The
student of trigonometry is helpless without perfect familiarity
with them.
These relations are true for all values of a, positive or negative, but
the signs of the functions are not in all cases positive, as appears from
the fact that in the triangles of reference in Fig. 8 x and y are sometimes
negative. The equations sin a = ± Vl — cos^ a, tan a=± Vsec^ a—1,
cot a = ± Vcsc^ ct — 1, have the double sign ± . Which sign is to be used
in a given case depends on the quadrant in which a lies.
16. The relations of Art. 15 enable us to express any
function in terms of any other, or when one function is
given, to find all the others.
Ex. 1. To express the other functions in terms of tangent :
.inct- ^ - ^ - ^^"^^ •
CSC a VI + cot2 a VH- tan^ a
1 1
tana
sec a = VI + tan2 a ;
sec a Vl + tan2a
tan a = tan a ;
C8C«^^l+**»^«.
tan a
THE TRIGONOMETRIC FUNCTIONS.
13
In like manner determine the relations to complete the following
table ;
sm a
cos a
tan«
cot a
tan a
sm a
cos a
tana
cot a
sec a
CSC a
VI + tan2 a
1
Vl + tan"'^ a
tan a
1
tan a
Vl + tan2 a
Vl + tan2 a
tan a
2. Given sin a = f ; find the other functions.
a=Vl -^5 = ^V7; tan
= fV7;
\V7 V7
^ r- 14/- 14
cot a = —^ = ^ V7 ; sec a = = — = f V7 ; esc « = - = -•
fV7
iV7 V7
3; Given tan (f> + cot ^ = 2 ; find sin <^.
tan <f) +
tan <f>
2, tan2 <^ - 2 tan <^ + 1 = 0, tan <^ = 1.
.♦. sin <f> =
tan <^
Vl + tan2 <^
= iV2.
Or, expressing in terms of sine directly, ?11L2_(_ ^ = 2,
cos <^ sin <^
sin^ <^ + cos^ (fi = 2 sin <^ cos <fi, sin^ <^ — 2 sin ^ cos <f> + cos^ ^ = ;
whence sin <^ — cos <^ = 0, sin <f> = cos <f>. .*. sin ^ = ^ V2.
4. Prove sec^ x — sec^ x = tan^ x + tan^ x.
sec^x — sec^x = sec^ a: (sec^ a: — 1) = (1 + tan^ x) tan^ a: = tan^a: + tan* a:.
5. Prove sin^ y + cos® ?/ = 1 — 3 sin^ y cos^ y.
sin® y + cos® y = (sin^ y + cos^ y) (sin* y — sin^ y cos^ ?/ + cos* y)
= (sin^ 2/ + cos^ ?/)2 — 3 sin^ y cos^ y = 1 —3 sin^ ^ cos^ y.
14 PLANE TRIGONOMETRY.
6. Prove -i^2^ + _22t^ = sec. CSC. + 1.
1 — cot z 1 — tan z
sing cos 2
tan z cot z _ cos z sin 2
cot . 1 — tan z I _ cos 2
COS.
cos . (sin . — COS .) sin z (cos z — sin z)
_ sin^ . — cos^ . _ sin^ . + sin . cos z + cos^ .
sin . cos . (sin . — cos z) sin z cos .
1 + sin . cos . 1.1 ,1
= —^. = h 1 = sec . esc 2 + 1.
sm . cos . sin z cos z
In solving problems like 3, 4, 5, and 6 above, it is usually safe, if no
other step suggests itself, to express all other functions of one member
in terms of sine and cosine. The resulting expression may then be re-
duced by the principles of algebra to the expression in the other member
of the equation. For further suggestions as to the solution of trigono-
metric equations and identities see page 66.
EXAMPLES.
1. Find the values of all the functions of a, if sin a = | ; if tan a = f ;
if sec r}t = 2 ; if cos a = ^V3 ; if cot a = | ; if esc ot = V2.
2. Compute the functions of each acute angle in the right triangles
whose sides are : (1) 3, 4, 5; (2) 8, 15, 17; (3) 480, 31, 481 ; (4) a,b,c;
yr-^ 2 xy x^ + y^
(5) ^, ^ ^ , x+y.
X — y X — y
3. If cos a = j\, find the value of si^<^ + ^^^^^ .
cos a — cot a
4. If 2 cos a = 2 — sin ct, find tan a.
5. If sec^ a csc^ a — 4 = 0, find cot a.
6. Solve for sin ^ in 13 sin /? + 5 cos^ ^ = 11.
Prove
7. sin* <;^ — cos* <^ = 1 — 2 cos^ <^.
8. (sin a + cos a) (sin a — cos a) = 2 sin^ a — 1.
9. (sec a + tan a) (sec a — tan a) = 1.
10. cos2 y8 (sec2 13-2 sin2 ^) = cos* jS + sin* (3.
cos V
11. tan V + sect'
12.
1 — sin V
sin w 1 + cos w
1 — cos w sin w
13. (sec^ + l)(l-cos^) = tan2^cosA
FUNCTIONS OF CERTAIN ANGLES. 15
14. sin* t — siii2 1 = cos* t — cos^ t.
15. -ilH^ + 1+^ = sec^^ (CSC fi + 1).
1 — Sin y8 smfi
16. (tan A + cot Ay = sec2 ^ csc^ ^.
17. sec^ ar — sin^ a; = tan^ a: + cos^ x.
In the triangle ABC, right angled at C,
18. Given cos A = ^y BC = 45, find tan B, and AB.
19. If cos A = ^l ~ ""l and AB = m^ + n% find ^ C and ^C.
20. If ^ C = m + n, £C = m — n, find sin A, cos 5.
21. In examples 18, 19, 20, above, prove sin^ ^4 4- cos^ .4 = 1 ;
1 + tan2 A = sec2 A .
17. Functions of certain angles. The trigonometric func-
tions are numerical quantities which may be determined for
any angle. In general these values are taken from tables
prepared for the purpose, but the principles already studied
enable us to calculate the functions of the following angles.
18. Functions of O''. If a be a very small angle, the
value of y is very small, and
decreases as a diminishes.
Clearly, when a approaches
0° as a limit, ^ likewise ap-
proaches 0, and X approaches r, so that when a = 0°,
^ = 0, and X = r.
.-. «mO° = ^ = 0, co^ 0° = — i— = QO,
r
r
tanO"" = ^ = 0, C8C 0° = -r^ = 00.
X sin 0°
In the figure of Art. 18, by diminishing a it is clear that we can make
y as small as we please, and by making a small enough, we can make the
value of y less than any assignable quantity, hoivever small, so that sin a ap-
proaches as a limit 0. This is what we mean when we say sin 0° = 0.
In like manner, it is evident that, by sufficiently diminishing a we can
make cot a greater than any assignable quantity. This we express by
saying cotO° = co.
€OtO°
±
tanO°
8ecO°
1
cos 0°
nRn 0°
1
16
PLANE TRIGONOMETRY.
19. Functions of 30°. Let NOP be the triangle of refer-
,22 ence for an angle of 30°. Make
triangle NOP' = NOP. Then
POP' is an equilateral triangle
(why?), and ON bisects PP'.
Hence
Also X = Vr^ — y^ = V3^— y V3.
c%G 30° = 2,
Fig. 13
nn 30° = ^ = ^ = ^'
r 2^ 2
r 2y ^
^an 30°
y
= -4^=iV3,
«/V3 V3
se<? 30° = I V3,
co^30°=V3.
20. Functions of 45°. Let NOP be the triangle of refer-
ence. If angle NOP = 45°, OPN^ 45°.
Then
y = x^ and r = Va^^ + 3/^ = V2 a;^ = a; V2.
tn 45° = ^-
sm
V2
iV2,
cos 45'
a: a;
-^=i-V2,
^ a:V2
Find cot 45°, sec 45°, esc 45°
ia^i 45° = ^ = - = 1.
.T X
FUNCTIONS OF CERTAIN ANGLES.
17
21. Functions of 60°. The functions of 60° may be com-
puted by means of the figure, or
they may be written from the func-
tions of the complement, or 30°.
Let tlie student in both ways show
that
sm60°=iV3, cos 60° =1
tan 60° = Vs.
Compute also the other functions of 60°.
Fig. 15.
22. Functions of 90°. If « be an angle very near 90°,
the value of x is very small, and de-
creases as a increases toward 90°.
Clearly when a approaches 90° as a
limit, X approaches 0, and ?/ ap-
proaches r, so that when
"^¥ ^ a = 90°, x=0, y = r.
, •. sin 90° = 1, cos 90° = 0, tan 90° = oo .
Fig. 16.
Compute the other functions. Also find the functions of
90° from those of its complement, 0°.
23. It is of great convenience to the student to remember
the functions of these angles. They are easily found by
recalling the relative values of the sides of the triangles of
reference for the respective angles^ or the values of the other
functions may readily be computed by means of the funda-
mental relations, if the values of the sine and cosine are
remembered, as follows :
a
0°
30°
45°
60°
90°
sine
cosine
iVo
K/4
1 rz
2 Vo
iV2
W2
iVO
18 PLANE TRIGONOMETRY.
ORAL WORK.
1. Which is greater, sin 45° or I sin 90=? sin 60° or 2 sin 30°?
2. ^ From the functions of 60°, find those of 30° ; from the functions of
90°, those of 0°. Why are the functions of 45° equal to the co-functions
of 45°?
3. Given sin A = |, find cos A ; tan A.
4. Show that sin B esc ^ = 1 ; cos C sec C = 1 ; cot x tan x = 1.
5. Show that sec2 - tan^ = csc2 - cot^ 6 = sin2 $ + cos^ 6.
6. Show that tan 30° tan 60° = cot 60° cot 30° = tan 45°.
7. Showthattan60°sin2 45° = cos30°sin90°.
8. Show that cos a tan a = sin a ; sin ^ cot /3 = cos 13.
9. Show that 1 -tan2 30° ^ ^^^ g^^ ^ ^ ^os 0°.
l-|-tan2 30° ^
10. Show that (tan y + coty) sin y cos y = 1.
EXAMPLES.
1. Show that sin 30° cos 60° + cos 30° sin 60° = sin 90°.
2. Show that cos 60° cos 30° + sin 60° sin 30° = cos 30°.
3. Show that sin 45° cos 0° - cos 45° sin 0° = cos 45°.
4. Show that cos2 45° - sinHS" = cos 90°.
5. Show that tan 45° + tan 0° ^ ^^^^^o.
1 - tan 45° tan 0°
IiA= 60°, verify
i^-i-
6. sin^^ ,^-cos^
7. tan"^
_ /l — cos A
2 ~ >'l + cos^'
8. cos^ = 2cos2^-l = l-2sin2:^.
2 2
licc = 0°,l3 = 30°, y = 45°, 8 = 60°, e = 90°, find the values of
9. sin 13 + cos 8.
10. cos y8 + tan 8.
11. sin ^ cos S + cos ;8 sin 8 — sin e.
12. (sin 13 + sin e) (cos a + cos 8) — 4 sin a (cos y + sin e) .
VARIATIONS IN THE FUNCTIONS.
19
24. Variations in the trigonometric functions.
Signs. Thus far no account has been taken of the signs of
the functions. By the definitions it appears that these de-
pend on the signs of a;, ?/, and r. Now r is always positive,
and from the figures it is seen that x is positive in the first
8&H. +
Csc. +
X-
(X-)
(r+)
Cot. +
Sin.
Cos.
Tan.
Cot.
Sec.
Csc.
\y-)
Cos. +
Fia. 17.
and fourth quadrants, and ^ is positive in the first and
second. Hence
For an angle in the first quadrant all functions are positive^
since a:, ^, r are positive.
In the second quadrant x alone is negative., so that those
functions whose ratios involve a:, viz. cosine., tangent^ co-
tangent^ secant., are negative; the others, sine and cosecant.,
are positive.
In the third quadrant x and y are both negative., so that
those functions involving r, viz. sine., cosine., secant., cosecant.,
are negative ; the others, tangent and cotangent., ?iVQ positive.
In the fourth quadrant y is negative^ so that sine^ tangent.,
cotangent, cosecant are negative., and cosine and secant., positive.
Values. In the triangle of reference of any angle, the
hypotenuse r is never less than x or y. Then if r be taken of
any fixed length, as the angle varies, the base and perpen-
dicular of the triangle of reference may each vary in length
X 11
from to r. Hence the ratios - and - can never be greater
r r °
r r
than 1, nor if x and y are negative, less than —1; and — > -
X y
20
PLANE TRIGONOMETRY.
cannot have values between + 1 and — 1. But the ratios
^ and - may vary without limit, i.e. from + oo to — oo.
X y
Therefore the possible values of the functions of an angle
are :
sine and cosine between + 1 and — 1,
i.e. sine and cosine cannot he numerically greater than 1;
tangent and cotangent between + oo and — oo,
i.e. tangent and cotangent may have any real value ;
secant and cosecant between + oo and + 1, and — 1 and — oo,
i.e. secant and cosecant may have any real values., except
values between + 1 and — 1.
These limits are indicated in the following figures. The
student should carefully verify.
Sin. + 1
Cos. -
Tan. —00
90°
Y
\Sin 0=±0 o /.
-X 180 X-
1,4
Sin.
Cos.
Tan.
-1
-0
+ 00
+ 1
+
+00
X
0, +1,-0 360
-1
+
F'
370°
Fig. 18.
25. In tracing the changes in the values of the functions as
a changes from 0° to 360°, consider the revolving line r as
of fixed length. Then x and y may have any length between
and r.
y
Sine. At 0°, sin «="=- = 0. As a increases through
r r
y ^
the first quadrant, y increases from to r, whence - increases
from to 1. In passing to 180° sin a decreases from 1 to 0,
VARIATIONS IN THE FUNCTIONS. 21
since y decreases from r to 0. As « passes through 180°, y
changes sign, and in the third quadrant decreases to nega-
tive r, so that sin a. decreases from to — 1. In the fourth
quadrant y increases from negative r to 0, and hence sin a
increases from — 1 to 0.
Cosine depends on changing values of x. Show that,
as a increases from 0° to 360°, cos « varies in the four
quadrants as follows: 1 to 0, to — 1, — 1 to 0, to 1.
Tangent depends on changing values of both y and x.
At 0°, ^ = 0, a: = r, at 180°, y = 0,x = -r,
at 90°, x = 0,y = r, at 270°, x=0,y = -r,
V
Hence tan 0° = -^ = - = 0. As a passes to 90°, y increases
X r
to r, and x decreases to 0, so that tan a increases from to oo.
As a passes through 90°, x changes sign, so that tan a
changes from positive to negative by passing through oo.
In the second quadrant x decreases to negative r, y to 0, and
tan a passes from — oo to 0. As a passes through 180°,
tana changes from minus to plus by passing through 0,
because at 180° y changes to minus. In the third quadrant
tana passes from to oo, changing sign at 270° by passing
through 00, because at 270° x changes to plus. In the fourth
quadrant tan a passes from — oo to 0.
Cotangent. In like manner show that cot a passes through
the values oo to 0, to — oo, oo to 0, to — oo, as a passes
from 0° to 360°.
Secant depends on x for its value. Noting the change
in X as under cosine, we see that secant passes from 1 to oo,
— oo to — 1, — 1 to — 00, 00 to 1.
Cosecant passes through the values oo to 1, 1 to oo,
— 00 to — 1, — 1 to — 00.
The student should trace the changes in each function
fully, as has been done for sine and tangent, giving the
reasons at each step.
22
PLANE TRIGONOMETRY.
a
0° to 90°
90° to 180°
180° to 270°
270° to 360°
sin
to 1
1 to
- to - 1
- 1 to -
cos
1 to
- to - 1
-1 to -0
to 1
tan
to 00
- 00 to -
to 00
- 00 to -
cot
00 to
- to - 00
00 to
- to - 00
sec
1 to GO
— 00 to — 1
— 1 to — 00
00 to 1
CSC
00 to 1
1 to 00
— 00 to — 1
— 1 to —00
* 26. Graphic representation of functions. These variations
are clearly brought out by graphic representations of the
functions. Two cases will be considered : I, when a is a
constant angle ; II, when a is a variable angle.
I. When a is a constant angle.
The trigonometric functions are ratios, pure numbers.
By so choosing the triangle of reference that the denomi-
nator of the ratio is a side of unit length, the side forming
the numerator of that ratio will be a geometrical representa-
tion of the value of that function, e.g. if in Fig. 19 r = 1,
then sin a = ? = ^=^. This may be done by making a a
central angle in a circle of radius 1, and drawing triangles
of reference as follows :
Fio. 19.
GRAPHIC REPRESENTATION OF FUNCTIONS. 23
In all the figures A OF = a, and
BP BP j.r>
OB OB ^j,
BP AD AD .J,
''""'^OB^OA^ 1 =^^'
OA BO EC T,n
OP OB OB r.j.
'''''- OB=OA= 1 =^^'
OP 00 00 rtn
It appears then that, by taking a radius 1,
sine is represented by the perpendicular to the initial line,
drawn from that line to the terminus of the arc sub-
tending the given angle;
cosine is represented by the line from the vertex of the
angle to the foot of the sine ;
tangent is represented by the geometrical tangent drawn
from the origin of the arc to the terminal line, produced
if necessary;
cotangent is represented by the geometrical tangent drawn
from a point 90° from the origin of the arc to the
terminal line, produced if necessary;
secant is represented by the terminal line, or the terminal
line produced, from the origin to its intersection with
the tangent line ;
cosecant is represented by the terminal line, or the terminal
line produced, from the origin to its intersection with
the cotangent line.
24
PLANE TRIGONOMETRY.
These lines are not the functions^ but in triangles drawn
as explained their lengths are equal to the numerical values
of the functions, and in this sense the lines may be said to
represent the functions. It will be noticed also that their
directions indicate the signs of the functions. Let the
student by means of these representations verify the results
of Arts. 24 and 25.
II. When a is a variable angle.
Take XX' and YY' as axes of reference, and let angle
units be measured along the ic-axis, and values of the func-
tions parallel to the ?/-axis, as in Art. 3. We may write
corresponding values of the angle and the functions thus :
a=0°, 30°, 45% 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°,
sin«=0, i, iV2, iV3, 1, iV3, iV2, i, 0, _ i, _ i V2,
a= 240°, 270°, 300°, 315°, 330°, 360°, -30°, -45°, -60°, -90°, etc.,
sina=-|V3, -1, -iV3, -^^2, -\, 0, -^ -iV2, -|V3, -1, etc.
These values will be sufficient to determine the form of the
curve representing the function. By taking angles between
those above, and computing
the values of the function, as
given in mathematical tables,
the form of the curve can be
/^ determined to any required
degree of accuracy. Reduc-
ing the above fractions to
decimals, it will be convenient
to make the ?/-units large in
comparison with the a:-units.
In the figure one a^-unit repre-
sents 15°-, and one y-unit 0.25.
Measuring the angle values along the rr-axis, and from these
points of division measuring the corresponding values of sin a
parallel to the ^-axis, as in Art. 3, we have, approximately.
Curves of Sine and Cosecant.
Cosecant
Fig. 20.
GRAPHIC REPKESENTATION OF FUNCTIONS. 25
OX^ = 30° = 2 units, OX^ = 45° =3 units,
Xi Fi = 1 =2 units, Xg Fg = 0. 71 = 2. 84 units,
OXg =60° =4 units, etc.,
X^Y^=0.S6 = 3.44 units, etc.
We have now only to draw through the points F^, Fg, Fg,
etc., thus determined, a continuous curve, and we have the
sine-curve or sinusoid.
The dotted curve in the figure is the cosecant curve. Let
the student compute values, as above, and draw the curve.
In like manner draw the cosine and secant curves, as
follows :
i
Curves of Cosine and Secant,
Cosine
Secant ~
FiG. 21.
Tangent curve. Compute values for the angle a and for
tan a, as before ;
a = 0°, 30°, 45°, 60°, 90°, 120°, 136°, 150°, 180°, 210°, 226°, 240°, 270°,
tan a = 0, \VS, 1, V3, ±oo, - V3, -1, -^VS, 0, |\/3, 1, V3, ±oo,
a = - 30°, - 45°, - 60°, - 90°, etc.,
tan a = - ^ V3, - 1, - V3, ± oo, etc.
Then lay off the values of a and of tan a along the x^ and
parallel to the ?/-axis, respectively. It will be noted that,
26
PLANE TRIGONOMETRY.
as a approaches 90°, tan a increases to oo, and when a passes
90°, tan a is negative. Hence the value is measured parallel
Curves of Tangent and Cotangent.
Tangent
Cotangent
Fig. 22. '
to the ^-axis downward, thus giving a discontinuous curve,
as in the figure.
27. The following principles are illustrated by the curves :
1. The sine and cosine are continuous for varying values
of the angle, and lie within the limits + 1 and — 1. Sine
changes sign as the angle passes through 180°, 360°, ••♦,
n 180°, while cosine changes sign as the angle passes through
90°, 270°, •••, (2 7^ + 1) 90°. Tangent and cotangent are
discontinuous, the one as the angle approaches 90°, 270°, •••,
(2?i + l) 90°, the other as the angle approaches 180°, 360°, •-,
7il80°, and each changes sign as the angle passes through
these values. The limiting values of tangent and cotangent
are + oo and — oo.
2. A line parallel to the ?/-axis cuts any of the curves in
but one point, showing that for any value of a there is but
one value of any function of a. But a line parallel to the
a?-axis cuts any of the curves in an indefinite number of
points, if at all, showing that for any value of the function
there are an indefinite number of values, if any, of a.
GRAPHIC REPRESENTATION OF FUNCTIONS. 27
3. The carves afford an excellent illustration of the varia-
tions in sign and value of the functions, as a varies from
to 360°, as discussed in Art. 25. Let the student trace these
changes.
4. From the curves it is evident that the functions are
periodic^ i.e. each increase of the angle through 360° in the
case of the sine and cosine, or through 180° in the case of
the tangent and cotangent, produces a portion of the curve
like that produced by the first variation of the angle within
those limits.
5. The difference in rapidity of change of the functions
at different values of a is important, and reference will be
made to this in computations of triangles. (^See Art. 64,
Case III.) A glance at the curves shows that sine is chang-
ing in value rapidly at 0°, 180°, etc., while near 90°, 270°,
etc., the rate of change is slow. But cosine has a slow rate
of change at 0°, 180°, etc., and a rapid rate at 90°, 270°, etc.
Tangent and cotangent change rapidly throughout.
Ex. Let the student discuss secant and cosecant curves.
ORAL WORK.
1. Express in radians 180°, 120°, 45°; in degrees, J^ radians, 2 it,
Itt, f tt.
2. If ^ of a right angle be the unit, what is the measure of | of a
right angle? of 90°? of 135°?
3. Which is greater, cos 30° or I cos 60°? tan - or cot-? sin ^ or cos- ?
^ ^ 6 3 4 4
4. Express sin a in terms of sec a ; of tan a ; tan a in terms of cos a ;
of sec a.
5. Given sin a = |, find tan a. If tan cc = 1, find sin a, esc a, cot a ;
also tan 2 a, sin 2 a, cos 2 a.
6. If cos a = i, find sin -, tan -•
2 2
7. In what quadrant is angle t, if both sin t and cos t are minus ? if
sin t is plus and cos t minus ? if tan t and cot t are both minus ? if sin <
and CSC t are of the same sign ? Why ?
8. Of the numbers 3, ^, — 5, — 4, a, — 6, oo, 0, which may be a value
of sin JO ? of sec j9 ? of tan p ? Why ?
28 PLANE TKIGONOMETRY.
EXAMPLES.
1. If sin 26° 40' = 0.44880, find, correct to 0.00001, the cosine and
tangent.
2. If tan a = VS, and cot fi = | Vs, find sin a cos ^ — cos a sin /i.
3 Evaluate si" ^0° cot 30° - cos eO'^ tan 60°
sin 90° cos 0°
Prove the identities :
4. tan^(l -cot2^) + cot^(l -tan2^) = 0.
5. (sin^ + sec^)2 +(cos^ + csc^)2 =(1 + sec^ csc^)^.
6. sin2 X cos a: esc a; — cos^ x esc x sin^ x + cos* x sec a: sin a? = sin^ x cos x
4- cos^ X sin x.
7 . tan^ w + cot^ w = sec^ w csc^ w — 2.
8. sec^ V + cos2 V = 2 -{- tan^ v sin^ v.
9. cos2« + 1 = 2cos3«sec^ + sin2^
10. csc2 1 — sec2 1 = cos2 « csc^ t - sin2 < sec^ t.
11. The sine of an angle is ^„ ~ ^ i find the other functions.
12. If tan^ + sin J. = m, tan ^ — sin ^ = n, prove m'^ — n^ = 4:Vmn.
Solve for one function of the angle involved the equations :
13. sin^ + 2cos^ = 1. 16. 2sin2ar + cosa; - 1 = 0.
cosa_3 17. sec^a; — 7tana: — 9 = 0.
tana 2 18. 3 cscy + lOcoty - 35 = 0.
15. \/3csc2^ = 4cot^. 19. sin^i; -|cosu- 1 = 0.
20. a sec^ zo + b tan w + c — a = 0.
21. K ^HLd = V2, *HLd = V3, find A and 5.
sin ^ tan B
22. Find to five decimal places the arc which subtends the angle of
1° at the centre of a circle whose radius is 4000 miles.
23. If CSC A = f V3, find the other functions, when A lies between
— and TT.
24. In each of two triangles the angles are in G. P. The least angle
of one of them is three times the least angle of the other, and the sum of
the greatest angles is 240°. Find the circular measure of each of the
angles.
CHAPTER III.
FUNCTIONS or ANY ANGLE — INVERSE FUNCTIONS.
28. By an examination of the figure of Art. 24 it is seen
that all the fundamental relations between the functions hold
true for any value of a. The table of Art. 16 expresses the
functions of a, whatever be its magnitude, in terms of each
of the other functions of that angle if the ± sign be prefixed
to* the radicals.
The definitions of the trigonometric functions (Art. 12)
apply to angles of any size and sign, but it is always possible
to express the functions of any angle in terms of the func-
tions of a positive acute angle.
The functions of any angle ^, greater than 360°, are the
same as those of ^ ± w • 360°, since 6 and 6 ±n • 360° have
the same triangle of reference. Thus the functions of 390°,
or of 750°, are the same as the functions of 390° — 360°, or
of 750°— 2-360°, i.e. of 30°, as is at once seen by drawing a
figure. So also the functions of —315°, or of —675° are
the same as those of - 315° + 360°, or of - 675° + 2-360°,
i.e. of 45°.
For functions of angles less than 360° the relations of this
chapter are important.
29. To find the relations of the functions of — ^, 90° ± ^,
180° ± 6, and 270° ±6 to the functions of 6, 6 being any angle.
Four sets of figures are drawn, I for d an acute angle, II
for Q obtuse. III for an angle of the third quadrant, and
IV for d an angle of the fourth quadrant.
In every case generate the angles forming the compound
angles separately, i.e. turn the revolving line first through
30 PLANE TRIGONOMETRY,
(a) (6)
(c)
x'
^r'
III
in
III
IV
IV
Fig. 23.
t
\r'
y
Y\
\
oW ]
rCA
y
M~
1%'i''
IV
FUNCTIONS OF ANY ANGLE.
31
/
\
Ix'
1^
\ /
\ -f
t
V
\#/
y
V"
N
y
K^.
//
, \?.io
— fl /
y
\
y
x\
c^y
*>0
+8
II
II
III
t
[V
ry^
y
/.X
>
<^'y'
vs=
<
y
»0
=9
IV
^
+^
a;'A/1
N\ X'
I v
f
y
/
y
/f/
'''\
/
\
IV
Fig. 23.
32 PLANE TRIGONOMETRY.
0°, 90°, 180°, or 270°, and then from this position through
^, or — ^, as the case may be. Form the triangles of refer-
ence for (a) the angle (9, (6) - ^, (c) 180° ± (9, (^d) 90° ± (9,
(e) 270° ±^.
The triangles of reference (a), (6), (<?), (c?), and (e), in
each of the four sets of figures, I, II, III, IV, are similar,
being mutually equiangular, since all have a right angle and
one acute angle equal each to each. Hence the sides x^ y^ r
of the triangles (a) are homologous to x\ y\ r' of the cor-
responding triangles (5) and (c), but to ?/', x\ 7•^ of the
corresponding triangles (c?) and {e). For the sides x of
triangle {a) and x^ of the triangles (6) and (c) are opposite
equal angles, and hence are homologous, but the sides y^ are
opposite this same angle in triangles (c?) and (e), and there-
fore sides y^ of (c?) and (e) are homologous to x of (a).
Attending to the signs of x and a;', y and ?/' in the similar
triangles {a) and (6),
sin(-^)=|^ = -|=-sin^,
cos(-6>)=^ = ^ =cos(9,
tan (- 6>) = ^= - ^ = - tan^.
^ a?' X
Also in the similar triangles {a) and (c),
sin (180° - (9) = ^ = ^ = sin (9,
r
= = — cos r,
cos (180°-^) = =^
tan (180° - ^)= ^ = - ^ = - tan(9,
In like manner show that
sin (180° + (9) = -sin (9,
cos (180° + (9) = - cos ^,
tan (180° + ^)= tan ^.
FUNCTIONS OF ANY ANGLE. 33
Again, in the similar triangles (a) and ((^),
sin (90° + (9) = ^ = - = cos^,
cos (90° 4- ^) = ^ = - - = - sin ^,
r
.f
tan (90° + (9) = ^ = - -= - cot ^.
Show that
sin (90°-^)= cos ^, C
cos(90°-6»)=sin(9, | ^
tan (90° -(9) = cot (9.
Finally, from the similar triangles (a) and (e), show that
sin (270° ± (9)=- cos 6^,
cos(270°±^)=±sin^,
tan (270° ±^)=Tcot^.
From the reciprocal relations the student can at once
write the corresponding relations for secant, cosecant, and
cotangent.
30. Since in each of the four cases x\ y' of triangles
(6) and (<?) are homologous to x^ y of triangle (a)^ while
x\ y' of the triangles (cT) and (e) are homologous to y^ x
of triangle (a), we may express the relations of the last
article thus :
The functions of -j ^^o , n correspond to the same functions
of 6^ while those of \ QrT/^o , n correspond to the co functions
of 6, due attention being paid to the signs.
The student can readily determine the sign in any given
case, whether 6 be acute or obtuse, by considering in what
quadrant the compound angle, 90° ± 0, 180° ± 6, etc., would
84 PLANE TRIGONOMETRY.
lie if 6 were an acute angle, and prefixing to the correspond-
ing functions of the signs of the respective functions for
an angle in that quadrant. Thus 90° + ^, if ^ be acute, is
an angle of the second quadrant, so that sine and cosecant
are plus, the other functions minus. It will be seen that
sin (90° + (9)= + cos (9, cos (90° + (9)= - sin^, etc., and this
will be true whatever be the magnitude of 6. It will assist
in fixing in the memory these important relations to notice
that when in the compound angle 6 is measured from the
?/-axis, as in 90° ± ^, 270° ± ^, the functions of one angle
correspond to the co-functions of the other, but when in the
compound angle 6 is measured from the a;-axis, as in ± ^,
180° ± 6^ then the functions of one angle correspond to the
same functions of the other.
These relations, as has been noted in Art. 28, can be
extended to angles greater than 360°, and it may be stated
generally that
function (9 = ± function (2 ^ • 90° ± (9),
function (9 = ± co-function [(2 n + 1) 90° ± (9].
Computation tables contain angles less than 90° only. The chief
utility of the above relations will be the reduction of functions of angles
greater than 90° to functions of acute angles. Thus, to find tan 130° 20',
look in the tables for cot 40° 20', or for tan 49° 40'. Why ?
Ex. 1. What angles less than 360° have the same numerical cosine
as 20°?
cos 20° = - cos (180° ± 20°) = cos (360° - 20°).
.-. 200°, 160°, 340° have the same cosine numerically as 20°.
2. Find the functions of 135° ; of 210°.
sin 135° = sin (90° + 45°) = cos 45° = ^v^,
cos 135° = cos (180° - 45°) = - cos 45° = - i V2, etc.
sin 210° = sin (180° + 30°) = - sin 30° = - \.
Let the student give the other functions for each angle.
INVERSE FUNCTIONS. 35
ORAL WORK.
1. Determine the sine and tangent of each of the following angles :
30°, 120°, - 30°, - 60°, f TT, 2f tt, - 135°, - ir.
2. Which is the greater, sin 30° or sin(- 30°)? tan 135° or tan 45°?
cos 60° or cos( - 60°) ? sin 22° 30' or cos 67° 30' ?
3. What positive angle has the same tangent as — ? the same sine
as 50°? ^
4. If tan^ = -l, findsin^.
5. Find sin 510°, cos(- 60°), tan 150°.
6. Reduce in two ways to functions of a positive acute angle, cos 122°
tan 140° 30', sin (-60°). '
7. Find all positive values of x, less than 360°, satisfying the fol-
lowing equations : cos x — cos 45°, sin 2 a: = sin 10°, tan 3 a; = tan 60°,
sin X = sin 30°, tan x = tan 135°.
8. What angles are determined when (a) sine and cosine are + ?
(b) cotangent and sine are — ? (c) sine + and cosine — ? (</) cosine —
and cotangent + ?
INVERSE FUNCTIONS.
31. That a is the sine of an angle 6 may be expressed in
two ways, viz., sin 6 = a^ or, inversely, 9 = sin""i a, the latter
being read, 6 equals an angle whose sine is a, or, more briefly,
is the anti-sine of a.
The notation sin~ia, cos"^ a, tan-^a, etc., is not a fortunate one, but
is so generally accepted that a change is not probable. The symbol may
have been suggested from the fact that if ax = b, then x = a~^ b, whence,
by analogy, if sin ^ = a, ^ = sin"i a. But the likeness is an analogy only,
for there is no similarity in meaning. Sin"^ a is an angle 0, where sin = a,
and is entirely different from (sin a)-^ = . In Europe the symbols
sin a
arc sin a, arc cos a, etc., are employed.
32. Principal value. We have found that in sin 6 = a^
for any value of ^, a can have but one value ; but in
6 = sin~i a, for any value of a there are an indefinite number
of values of 6 (Art. 27, 2).
Thus, when sin (9 = a, if a = J, (9 may be 30°, 150°, 390°,
510°, - 330°, etc., or, in general, wtt +(- 1)"30°.
In the solution of problems involving inverse functions.
36
PLANE TRIGONOMETRY.
the numerically least of these angles, called the principal
value^ is always used ; i.e. we understand that sin~i a, tan~i a,
are angles between + 90° and — 90°, while the limits of
cos-la are 0° and 180°.
Thus, sin-i i = 30°, sin-i( - J) = - 30°, cos"! J = 60°,
cos-i(-i)=120°.
ORAL WORK.
How many degrees in each of the following angles? How many
radians ?
1. cos-i:^?
2
2. tan-il?
3. cot-i(-V3)?
4. sin-i(-iV2)?
5. cos-i(-iV2)?
6. sln-.(_^),
7. tan-iVS?
8. cos-iQ?
9. sin-n?
10. tan-iQ?
11. tan-i(-l)?
12. sin-i(-l)?
Find the values of the functions :
13. sin(tan-ii\/3).
14. tan(cos"^ 1).
15. tan(cot~i[— go]).
16. cos(tan~iGo).
17. sin(sin-i|V2).
18. tan(tan-ia;).
19. cos(sin-iO).
20. sin(cos-i[- 1]).
21. cos(cot-iV3).
22. tan(sin-i[-l]).
23. sin(tan-i[-l]).
Ex. 1. Construct cot-^ f .
Construct the right triangle xyr, so that a: = 4,
2/ = 3, whence angle xr = cot"^ f .
2. Find cos(tan-i ^^).
Let 6 = tan-i j^, whence
tan = xV> and cos = |f
.♦. cos 6 = cos(tan-^ -j^) = if.
and
3. If ^ = csc-i a, prove 6 = cos~i — ^
CSC ^ = a
cos e =Vl--„ = ^"'^ ~ \ or ^ = cos-
. sin ^ = ->
a
.1 Va^
EXAMPLES. 37
EXAMPLES.
1. Construct sin-^f, tan-ij^, cos-i(— ^).
2. Find tan(sin~ix^j), sin(tan-iy\).
3. If ^ = sin-i a, prove = tan-^ —
VI -a2
4. Show that sin"^ a = 90° — cos"^ a.
5. Prove tan-i\/3 + cot-iV3=^.
6. Prove tan-ifsin '^\ = cos-^^^.
7. What angles, less than 360°, have the same tangent numerically
as 10°?
8. Given tan 143° 22' = - 0.74357 ; find, correct to 0.00001, sine and
cosine.
9. If cot2(90° + /?) + csc(90° - /8) - 1 = 0, find tan fi.
10. Find all positive values of x, less than 360°, when sin x = sin 22° 30' ;
when tan 2 a; = tan 60°.
11. When is sin x = possible, and when impossible ?
12. Verify sin-i | + cos'i— + tan-i V3 = sin-i ^.
13. What values of x will satisfy sin-i(.r2 - x)= 30° ?
14. If tan2 e - sec2 a = 1, prove sec $ + tan^ ^ esc ^ = (3 + tan2 a)^.
15. Prove sin ^ (1 + tan A)+ cos ^ (1 + cot A) = sec ^ + esc A.
16. Solve the simultaneous equations :
sin-i(2 X + Sy)=30° and 3 a: + 2 y = 2.
17. Verify (a) tan60° = V ^ -cosli
' 1 + cos 1'.:
(6) cos 60° =
i2!.
120°'
1 -tan2 30°
l+tan2 30°*
(c) 2 sin2 60° = 1 - cos 120°.
18. Show that the cosine of the complement of - equals the sine of
6
the supplement of -•
38 PLANE TRIGONOMETRY.
REVIEW.
Before leaving a problem the student should review and master all
principles involved.
1. Construct cos'^xV 5 sin-i(— |); tan-i2.
2. Find cos (sin-i f ) ; tan (cos-i [ — i] ) .
3. Prove cot"^ a = cos~^ ^
VI +a2
4. Given a = cot-i|, find tan a + sin (90° + a).
5. Find tan ( sin-i| + cos-^: — )•
6. State the fundamental relations between the trigonometric func-
tions in terms of the inverse functions. Thus,
1
sin~i« = csc~^-, sin~ia = cos~^Vl — aK etc.
a
7. Find all the angles, less than 360°, whose cosine equals sin 120°.
8. Given cot~i 2.8449, find the sine and cosine of the angle, correct
to 0.0001.
9. If tan2 (180° -0)- sec (180° + (9) = 5, find cos 0.
irx T£ • n 9 £ J tan^^ + cos^^
10. If sm 6 = ^, find -— ' --
^' tan2^-cos2^
11. Is sin X — 2 cos x + Ssina; — 6 = 0a possible equation ?
12. Verify (a) sin 60°= ^ tan 30° ,
^ ^ ^ l + tan230°
(b) 2 cos2 60° = 1 + cos 120°.
(c) cos 60° - cos 90° = 2 cos2 30° - 2 cos2 45°.
13. If sin X = — ^K^_± 1 — find sec x and tan x.
a^ + 2ab + 2 b^
14. Prove 1 + sin ^ - cos ^ _^ 1 + sin^ + cos^^ ^^^^^^
1 + sin ^ + cos $ 1 + sin ^ — cos
15. Prove
cos 45° + cos 135° + cos 30° + cos 150° - cos 210° + cos 270° = sin 60°.
16. If tan = prove that
Va2 _ Ij2
sin ^(1 + tan 6) + cos ^(1 + cot ^ - sec = |-
17. Solve sin2 x + sin^ (x + 90°) + sin2 (^ ^ i80°) = 1.
EXAMPLES. 39
18. Given cos^ a = msina — n, find sin a.
19. If sin2y3=-A^, find^.
2 sec p
20. Given tan 238° =1.6, find sin 148°.
21. Prove tan-i m + cot-i m = 90°.
22. Find sin (sin~ij9 + cos~ijo).
23. Solve cot2 ^ (2 esc ^ - 3) + 3 (esc ^ - 1) = 0.
24. Prove sin^ a sec^ ^ + tan^ /? cos^ a = sin^ a + tan2 p.
25. Prove cos^ F + sin^ F = 1 - 3 sin^ F + 3 sin^ F.
26. What values of A satisfy sin 2 A = cos 3 ^ ?
27. If tan C = ^^ ~ '^'^ , and tan D =\ ^ - cos C ^ ^^^^ ^^^ ^ .^ ^^^^^^
ofm. ''^ M+cosC
28. If sin a: — cos x + 4 cos^ a: = 2, find tan x ; sec a:.
29. Does the value of sec x, derived from sec^ x = — — - — - — , give a
possible value of a:? 1 - cos x
30. Prove
[cot (90° - ^ ) - tan (90° + A)] [sin (180° -A) sin (90° + /I )] = 1 .
31. Prove (1 +sin^)2[cot^ +2sec^(l -csc^)] + csc^ cos^^ = 0.
32. Given sin a: = m sin y, and tan x = n tan y, find cos x and cos y.
33. Given cot 201° = 2.6, find cos 111°.
34. Find the value of
cos-H + sin-HV^ + csc-i(- 1)+ tan-U - 2cot-iV3.
35. Solve 2 cos^d + 11 sin ^ - 7 = 0.
36. Prove
cos2 B + cos2 {B + 90°) + cos2 (5 + 180°) + cos2(5 + 270°) = 2.
CHAPTER IV.
COMPUTATION TABLES.
33. Natural functions. It has been noted that the trigo-
nometric functions of angles are numbers^ but the values
were found for only a few angles, viz. 0°, 30°, 45°, 60°,
90°, etc. In computations, however, it is necessary to know
the values of the functions of any angle, and tables have
been prepared giving the numerical values of the functions
of all angles between 0° and 90° to every minute. In
these tables the functions of any given angle, and co^i-
versely the angle corresponding to any given function, can
be found to any required degree of accuracy ; e.g. by look-
ing in the tables we find sin 24° 26'= 0.41363, and also
1 .6415 = tan 58° 39'. These numbers are called the natural
functions., as distinguished from their logarithms, which are
called the logarithmic functions of the angles.
Ex. 1. Find from the tables of natural functions :
sin35n4'; cos 54° 46'; tan 78° 29'; cos 112° 58'; sin 135°.
2. Find the angles less than 180° corresponding to :
sin-i 0.37865; cos-i 0.37865; tan-i 0.58670 ; cos"! 0.00291 ; sin-^O
34. Logarithms. The arithmetical processes of multi-
plication, division, involution, and evolution, are greatly
abridged by the use of tables of logarithms of numbers
and of the trigonometric ratios, which are numbers. The
principles involved are illustrated in the following table :
Write in parallel columns a geometrical progression having
the ratio 2, and an arithmetical progression having the dif-
ference 1, as follows :
40
LOGARITHMS.
41
G. P.
A. P.
1
2
1
4
2
8
3
16
4
32
5
64
6
128
7
256
8
512
9
1024
10
2048
11
4096
12
8192
13
16384
14
32768
15
655S6
16
131072
17
262144
18
524288
19
1048576
20
It will be perceived that the numbers in
the second column are the indices of the
powers of 2 producing the corresponding
numbers in the first column, thus : 2^ = 64,
211 = 2048, 218 = 262144, etc. The use of
such a table will be illustrated by examples.
Ex. 1. Multiply 8192 by 128.
From the table, 8192 = 2^% 128 = 2'. Then by
actual multiplication, 8192 x 128 = 1048576, or by the
law of indices, 21^ x 2^ = 220 = 1048576 (from table).
Notice that the simple operation of addition is sub-
stituted for multiplication by adding the numbers in
the second column opposite the given factors in the
first column. This sum corresponds to the number
in the first column which is the required product.
2. Divide 16384 by 512.
16384 -4- 512 = 32, which corresponds to the result
obtained by use of the table, or 2^^ - 2^ = 2^ = 32.
The operation of subtraction takes the place of
division.
3. Find V262144.
>>^62144
2^^ = 03 —
In the table, 262144 is opposite 18. 18 -- 6 = 3,
which is opposite 8, the required root ; i.e. simple division takes the
place of the tedious process of evolution.
4. Cube 64. 6. Find ^^^2768.
5. Multiply 256 by 4096.
7. Divide 1048576 by 32768.
35. The above table can be made as complete as desired
by continually inserting between successive numbers in the
first column the geometrical mean, and between the opposite
numbers in the second, the arithmetical mean, but in prac-
tice logarithms are computed by other methods. The num-
bers in the second column are called the logarithms of the
numbers opposite in the first column. 2 is called the base of
this system, so that the logarithm of a number is the exponent
by which the base is affected to produce the number.
42 PLANE TRIGONOMETRY.
Thus, the logarithm of 512 to the base 2 is 9, since
29 = 512.
Logarithms were invented by a Scotchman, John Napier, early in the
seventeenth century, but his method of constructing tables was different
from the above. See Encyc. Brit, art. ^'•Logarithms,'''' for an exceedingly
interesting account. De Morgan says that by the aid of logarithms the
labor of computing has been reduced for the mathematician to about
one-tenth part of the previous expense of time and labor, while Laplace
has said that John Napier, by the invention of logarithms, lengthened
the life of the astronomer by one-half.
Columns similar to those above might be formed with any
other number as base. For practical purposes, however, 10
is always taken as the base of the system, called the common
system^ in distinction from the natural system^ of which the
base is 2.71828 •••, the value of the exponential series (^Higher
Algebra) . The natural system is used in theoretical discus-
sions. It follows that common logarithms are indices^ positive
or negative^ of the powers of 10.
Thus, 103 = 1000 ; i.e. log 1000 = 3 ;
10-2 = i- = 0.01; i.e. log0.01 = -2.
36. Characteristic and mantissa. Clearly most numbers
are not integral powers of 10. Thus 300 is more than the
second and less than the third power of 10, so that
log 300 = 2 plus a decimal.
Evidently the logarithms of numbers generally consist of
an integral and a decimal part, called respectively the charac-
teristic and the mantissa of the logarithms.
37. Characteristic law. The characteristic of the loga-
rithm of a number \^ independent of the digits composing
the number, but depends on the position of the decimal
point, and is found by counting the number of places the first
significant figure in the number is removed from the units''
place, being positive or negative according as the first significant
LOGARITHMS. 43
figure is at the left or the right of units' place. This follows
from the fact that common logarithms are indices of powers
of 10, and that 10", n being a positive integer, contains n -f- 1
places, while 10~" contains n—1 zeros at the right of units'
place. Thus in 146.043 the first significant figure is two
places at the left of units' place ; the characteristic of log
146.043 is therefore 2. In 0.00379 the first significant digit
is three places at the right of units' place, and the charac-
teristic of log 0.00379 is - 3.
To avoid the use of negative characteristics, such charac-
teristics are increased by 10, and —10 is written after the
logarithm. Thus, instead of log 0.00811 = 3.90902, write
7.90902 — 10. In practice the — 10 is generally not written,
but it must ahvays be remembered and accounted for in the
result.
Ex. Determine the characteristic of the logarithm of :
1; 46; 0.009; 14796.4; 230.001; lO^ x 76; 0.525; 1.03; 0.000426.
38. Mantissa law. The mantissa of the logarithm of a
number is hidependent of the position of the decimal point,
but depends on the digits composing the number, is always
positive^ and is found in the tables.
For, moving the decimal point multiplies or divides a
number by an integral power of 10, i.e. adds to or subtracts
from the logarithm an integer, and hence does not affect the
mantissa. Thus,
log 225.67 = log 225.67,
log 2256.7 = log 225.67 X 101 = log 225.67 -f 1,
log 22567.0 = log 225.67 x 102 ^ i^g 225.67 + 2,
log 22.567 = log 225.67 x lO-i = log 225.67 +(- 1),
log 0. 22567 = log 225. 67 x 10-3 = log 225. 67 + ( - 3),
so that the mantissae of the logarithms of all numbers com-
posed of the digits 22567 in that order are the same, .35347.
MoviTig the decimal point affects the characteristic only.
The student must remember that the mantissa is always positive.
44 PLANE TRIGONOMETRY.
Log 0.0022567 is never written - 3 +.35347, but 3.35347, the minus
sign being written above to indicate that the characteristic alone is nega-
tive. In computations negative characteristics are avoided by adding
and subtracting 10, as has been explained.
39. We may now define the logarithm of a number as the
index of the power to which a fixed number, called the base,
must be raised to produce the given number.
Thus, a^ = 5, and x = logab (where log J) is read logarithm
of b to the base a') are equivalent expressions. The relation
between base, logarithm, and number is always
(base)^°^ = number.
To illustrate: log28 = 3 is the same as 2^ = 8; log381 = 4 and
3*= 81 are equivalent expressions ; and so are log^QlOOO = 3
and 103 = 1000, and logio0.001= -3 and 10-3= 0.001.
Find the value of :
log464; log5l25; log3243; log«(«)^; log27 3 ; log^l.
40. From the definition it follows that the laws of indices
apply to logarithms, and we have :
I. The logarithm of a product equals the sum of the loga-
rithms of the factors.
II. The logarithm of a quotient equals the logarithm of the
dividend minus the logarithm of the divisor.
III. The logarithm of a power equals the index of the
power times the logarithm of the number.
IV. The logarithm of a root equals the logarithm of the
number divided by the index of the root. ,
For if a^ =^n and a^ = m,
■ then n xm = a^"*"^ .*. log nm = x-\-y = log n + log w;
and n-^m = a^~y, .-.log— = a; — ^ = logn — logm;
m
also n""— (a^y= a^^ .-. log n"" =rx = r xlogn;
_ Z_ _ -J
finally, Vn = Va^ = a% .*. log</n = - = - log w.
LOGARITHMS. 45
EXAMPLES.
Given log 2 = 0.30103, log 3 = 0.47712, log 5 = 0.69897, find :
10. logV||.
11,
1. log 4.
4. log 9.
7. log 153.
2. log 6.
5. log 25.
8. logf.
3. log 10.
6. logVS.
9. logl5x9.
•-Vl
X o**
xlO
USE OF TABLES.
41. To find the logarithm of a number.
First. Find the characteristic, as in Art. 37.
Second. Find the mantissa in the tables, thus :
(a) When the number consists of not more than four
figures.
In the column N of the tables find the first three figures,
and in the row N the fourth figure of the number. The
mantissa of the logarithm will be found in the row opposite
the first three figures and in the column of the fourth figure.
Illustration. Find log 42.38.
The characteristic is 1. (Why ?)
In the table in column N find the figures 423, and on the
same page in row N the figure 8. The last three figures of
the mantissa, 716, lie at the intersection of column 8 and
row 423. To make the tables more compact the first two
figures of the mantissa, 62, are printed in column only.
Then log42.38 = 1.62716.
Find log 0.8734 = 1.94121,
log 3.5 = log 3.500 = 0.54407,
log 36350 =4.56050.
(5) When the number consists of more than four figures.
Find the mantissa of the logarithm of the number com-
posed of the first four figures as above. To correct for the
remaining figures we interpolate by means of the principle of
proportional parts, according to which it is assumed that, for
differences small as compared with the numbers, the diff'ereiices
46 PLANE TRIGONOMETRY.
hetiveen several numbers are proportional to the differences be^
tween their logarithms.
The theorem is only approximately correct, but its use
leads to results accurate enough for ordinary computations.
Ex. 1. To find log 89.4562.
As above, mantissa of log 894500 = 0.95158,
mantissa of log 894600 = 0.95163,
.-. log 894600 - log 894500 = O.OOOOo, called the tabular difference.
Let log 894562 - log 894500 = x hundred-thousandths.
Now, by the principle of proportional parts,
log 894562 - log 894500 ^ 894562 - 894500
log 894600 - log 894500 ~ 894600 - 894500'
X 6'^
or - = — ^, whence x = .62 of 5 = 3.1
5 100
.-. log 89.4562 = 1.95158 + 0.00003 = 1.95161,
all figures after the fifth place being rejected in five-place tables. If,
however, the sixth place be 5 or more, it is the practice to add 1 to the
figure in the fifth place. Thus, if a; = 0.0000456, we should call it
0.00005, and add 5 to the mantissa.
2. Find log 537.0643.
To interpolate we have x : 9 = 643 : 1000, i.e. x = 5.787 ;
.-. log 537.0643 = 2.72997 -f 0.00006.
3. Find log 0.0168342 = 2.22619.
4. Find log 39642.7 = 4.59816.
42. To find the number corresponding to a given logarithm.
The characteristic of the logarithm determines the posi-
tion of the decimal point (Art. 37).
(a) If the mantissa is in the tables, the required number
is found at once.
Ex. 1. Find log~^ 1.94621 (read, the number whose logarithm is
1.94621).
The mantissa is found in the tables at the intersection of row 883 and
column 5.
.-. log-i 1.94621 = 88.35,
the characteristic 1 showing that there are two integral places.
LOGARITHMS. 47
(5) If the exact mantissa of the given logarithm is not in
the tables, the first four figures of the corresponding num-
ber are found, and to these are annexed figures found by
interpolating by means of the principle of proportional
parts, as follows :
Find the two successive mantissas between which the given
mantissa lies. Then, by the principle of proportional parts,
the amount to be added to the four figures already found is
such a part of 1 as the difference between the successive
mantissse is of the difference between the smaller of them
and the given mantissa.
2. Find log-i 1.43764.
Mantissa of log 2740 = 0.43775
of log 2739 = 0.43759
Differences 1 16
Mantissa of log required number = 0.43764
of log2739 = 0.43759
Differences x 5
By p. p. a; : 1 = 5 : 16 and x = ^^- 0.3125.
Annexing these figures, log-i 1.43764 = 27.3931+.
3. Find log-i T.48762.
The differences in logarithms are 14, 6.
... a: = A = .4284-,
14:
and log-i 1.48762 = 0.307343+.
4. Find log 891.59; log 0.023; log^; log 0.1867; log V2.
5. Find log-i 2.21042 ; log-i 0.55115; log-i 1.89003.
43. Logarithms of trigonometric functions. These might
be found by first taking from the tables the natural func-
tions of the given angle, and then the logarithms of these
numbers. It is more expeditious, however, to use tables
showing directly the logarithms of the functions of angles
less than 90° to every minute. Functions of angles greater
than 90° are reduced 'to functions of angles less than 90° by
48 PLANE TRIGONOMETRY.
the formulae of Art. 29. To make the work correct for
seconds, or any fractional part of a minute, interpolation
is necessary by the principle of proportional parts, thus :
Ex. 1. Find log sin 28° 32' 21".
In the table of logarithms of trigonometric functions, find 28° at the
top of the page, and in the minute column at the left find 32'. Then
under log sin column find log sin 28° 32' = 9.67913 - 10
log sin 28° 33 = 9.67936 - 10
Differences 1' 23
By p. p. a; : 23 = 21" : 60", i.e. a; = — x 23 = 8.4.
60
.-. log sin 28° 32' 21" = 9.67913 + 0.00008 - 10
= 9.67921 - 10.
Whenever functions of angles are less than unity, i.e. are decimals
(as sine and cosine always are, except when equal to unity, and as tan-
gent is for angles less than 45°), the characteristic of the logarithm will
be negative, and, accordingly, 10 is always added in the tables, and it
must be remembered that 10 is to be subtracted. Thus, in the example
above, the characteristic of the logarithm is not 9, but 1, and the log-
arithm is not 9.67913, as written in the tables, but 9.67913 - 10.
2. Find log cos 67° 27' 50".
In the table of logarithms at the foot of the page, find 67°, and in the
minute column at the right, 27'. Then computing the difference as
above, x = 25.
But it must be noted that cosine decreases as the angle increases
toward 90°. Hence, log cos 67° 27' 50" is less than log cos 67° 27', i.e.
the difference 25 must be subtracted, so that
log cos 67° 27' 50" = 9.58375 - 0.00025 - 10
= 9.58350 - 10.
44. To find the angle when the logarithm is given, find the
successive logarithms between which the given logarithm
lies, compute by the principle of proportional parts the
seconds, and add them to the less of the two angles corre-
sponding to the successive logarithms. This will not neces-
sarily be the angle corresponding to the less of the two
logarithms ; for, as has been seen, the number, and, therefore,
the logarithm, may decrease as the angle increases.
LOGARITHMS. 49
Ex. 1. Find the angle whose log tan is 9.88091.
log tan 37° 14' = 9.88079 - 10
log tan 37° 15' = 9.88105 - 10
Differences 60" 26
log tan 37° 14' = 9.88079 - 10
log tan angle required = 9.88091 — 10
Differences x" 12
.-. a: : 60 = 12 : 26, or x" = if x 60" = 28", approximately, and the
angle is 37° 14' 28".
2. Find the angle whose log cos = 9.82348.
We find x = ^x 60" = 26", and the angle is 48° 14' 26".
3. Show that log cos 25° 31' 20" = 9.95541 ;
log sin 110° 25' 20" = 9.97181 ;
log tan 49° 52' 10" = 0.07418.
4. Show that the angle whose log tan is 9.92501 is 40° 4' 40" ; whose
log sin is 9.88365 is 49° 54' 20" ; whose log cos is 9.50828 is 71° 11' 50".
45. Cologarithms. In examples involving multiplications
and divisions it is more convenient, if n is any divisor, to
add log - than to subtract log n. The logarithm of - is
called the cologarithm of n. Since
log - = log 1 — log n=0 — log n,
it follows that colog n = — log n^ i.e. logn subtracted from
zero. To avoid negative results, add and subtract 10.
Ex.1. Find colog 2963.
log 1 = 10.00000 -
log 2963= 3.47173
-10
.-. colog 2963= 6.52827-
-10
2. Find colog tan 16° 17'.
log 1 = 10.00000 -
log tan 16° 17'= 9.46554-
-10
-10
.-. colog tan 16° 17' = 0.53446
50 PLANE TRIGONOMETRY.
By means of the definitions of the trigonometric functions, the parts
of a right triangle may be computed if any two parts, one of them being
a side, are given. Thus,
■ B given a and A in the rt. triangle ABC.
Then c = a -^ sin A, b = a ^ tan A ,
B = 90° -A.
Again, if a and b are given, then
tan ^ =-,c = a^ sin A^ and B = 90°-A-
b
3. Given c = 25.643, B = 37° 25' 20", compute the other parts.
^ = 90° - 37° 25' 20" = 52° 34' 40".
a = c cos B. b = a tan B.
log c = 1.40897 log a = 1.30889
log cos B = 9.89992 log tan B = 9.88376
log a = 1.30889 log b = 1.19265
.-. a = 20.365. .-. b = 15.583.
Check: c^ = a^ -\- b^ = 20.365^ + 15.583^ = 657.57 = 25.6432.
4. Given b = 0.356, B = 63° 28' 40", compute the other parts.
A = 26° 31' 20".
h h
a =
sin B tan B
log b = 9.55145 log b = 9.55145
colog sin B = 0.04829 colog tan B = 9.6981B
log c = 9.59974 log a = 9.24961
c = 0.3979 a = 0.1777
Check: c^ - a^ = 0.1583 - 0.03157 = 0.12673 = b^
EXAMPLES.
Compute the other parts :
1. Given a = 9.325, A = 43° 22' 35".
2. Given c = 240.32, a = 174.6.
3. Given 5 = 76° 14' 23", a = 147.53.
4. Given a = 2789.42, b = 4632.19.
5. Given c = 0.0213, A = 23° 14".
6. Given & = 2, c = 3.
CHAPTER V.
APPLICATIONS.
46. Many problems in measurements of heights and dis-
tances may be solved by applying the preceding principles.
By means of instruments certain distances and angles may
be measured, and from the data thus determined other
distances and angles computed. The most common instru-
ments are the chain^ the transit^ and the compass.
The chain is used to measure distances. Two kinds are in
use, the engineer'' s chain and the Gunter^s chain. They each
contain 100 links, each link in the engineer's chain being
12 inches long, and in the Gunter's 7.92 inches.
Fig. 26.
The transit is the instrument most used to measure hori-
zontal angles, and with certain attachments to measure verti-
cal angles. The figure shows the form of the instrument.
51
52
PLANE TRIGONOMETRY.
The mariner^ 8 compass is used to determine the directions,
or hearings, of objects at sea. Each quadrant is divided
into 8 parts, making the 32 points of the compass, so that
each point contains 11° 15^
Z^A
Fig. 27.
tS^
Fig. 28.
47. The angle between the horizontal plane and the line
of vision from the eye to the object is called the angle of
elevation, or of depression, according
as the object is above or below the
Eievaiiony^^^ observer.
It is evident that the elevation
angle of B, as seen from A, is equal
to the depression angle of J., as seen from B, so that in the
solution of examples the two angles are interchangeable.
PROBLEMS.
48. Some of the more common problems met with in
practice are illustrated by the following :
To find the height of an object
when the foot is accessible.
The distance BC, and the eleva-
tion angle B are measured, and x
is determined from the relation ^
X = BC tan B, Fig. 29.
APPLICATIONS.
63
Ex. 1. The elevation angle of a cliff measured from a point 300 ft.
from its base is found to be 30°. How high is the cliff?
Then
BC = 300, B = 30°.
a; = 300 • tan 30° = 300 • ^ V3 = 100 V3.
2. From a point 175 ft. from the foot of a tree the elevation of the
top is found to be 27° 19'. Find the height of the tree.
The problem may be solved by the use of natural functions, or of
logarithms. The work should be arranged for the solution before the
tables are opened. Let the student complete.
Then
BC =
175.
B--
= 27° 19'.
x = BC tan B.
Or by
natural functions,
logBC =
BC = 175
log tan B =
tan £ = 0.5165
logx =
.-. X = 90.3875.
.-. X = 90.39.
To find the height of an object
when the foot is inaccessible.
Measure BB\ 6 and 0'.
Then x =
BC BB'-hB'C
cot 6 cot 9
But B' C = x cot 6', whence substituting,
BB'
cot 6- cot 6'"
which is best solved by the use of the natural functions of
e and 6'.
3. Measured from a certain point at its base the elevation of the
peak of a mountain is 60°. At a distance of one mile directly from this
point the elevation is 30°. Find the height of the mountain.
BB' = 5280 ft., e = 30°, 0' = 60°.
^ ^ y + 5280
cot 30°
5280
But y = xcot 60°.
X =
cot 30° - cot 60'
= 4572.48 ft.
54
PLANE TRIGONOMETRY.
In surveying it is often necessary to make measurements
across a stream or other obstacle too wide to be spanned by
a single chain.
To find the distance from O to a
point B on the opposite side of a
stream.
At O measure a right angle, and
take CA a convenient distance.
Measure angle A^ then
Fi«-3i. " BC=CA.t^nA.
4. Find CB when angle A = 47° 16', and CA = 250 ft.
5. From a point due south of a kite its elevation is found to be
42° 30'; from a point 20 yds. due west £>
from this point the elevation is 36° 24'.
How high is the kite above the ground ?
^^ = a:, cot 42° 30',
^C = re. cot 36° 24',
AC^-AB^ = BC^ = 400.
.'. a;2 (cot2 36° 24' - cot^ 42° 30') = 400,
whence
^2 _ J00_ and a: = .f^ = 24.84 yds.
.6489
.805
Fig. 32.
EXAMPLES.
1.. What is the altitude of the sun when a tree 71.5 ft. high casts
a shadow 37.75 ft. long ?
2. What is the height of a balloon directly over Ann Arbor w^hen
its elevation at Ypsilanti, 8 miles away, is 10° 15'?
3. The Washington monument is 555 ft. high. How far apart are
two observers who, from points due east, see the top of the monument
at elevations of 23° 20' and 47° 30', respectively?
4. A mountain peak is observed from the base and top of a tower
200 ft. high. The elevation angles being 25° 30' and 23° 15', respec-
tively, compute the height of the mountain above the base of the tower.
5. From a point in the street between two buildings the elevation
angles of the tops of the buildings are 30° and 60°. On moving across
APPLICATIONS. 55
the street 20 ft. toward the first building the elevation angles are found
to be each 45°. Find the width of the street and the height of each
building.
6. From the peak of a mountain two towns are observed due south.
The first is seen at a depression of 48° 40', and the second, 8 miles farther
away and in the same horizontal plane, at a depression of 20° 50'. What
is the height of the mountain above the plane ?
7. A building 145 ft. long is observed from a point directly in front
of one corner. The length of the building subtends tan-i 3, and the
height tan-i 2. Find the height.
8. An inaccessible object is observed to lie due N.E. After the ob-
sen^er has moved S.E. 2 miles, the object lies N.N.E. Find the distance
of the object from each point of observation.
9. Assuming the earth to be a sphere with a radius of 3963 miles,
find the height of a lighthouse just visible from a point 15 miles distp,nt
at sea.
10. The angle of elevation of a tower 120 ft. high due north of an
observer was 35° ; what will be its angle of elevation from a point due
west from the first point of observation 250 ft. ? Also the distance of
the observer from the base of the tower in each position ?
11. A railway 5 miles long has a uniform grade of 2° 30' ; find the rise
per mile. What is the grade when the road rises 70 ft. in one mile?
(The grade depends on the sine of the angle.)
12. The foot of a ladder is in the street at a point 30 ft. from the
line of a building, and just reaches a window 22^ ft. above the ground.
By turning the ladder over it just reaches a window 36 ft. above the
ground on the other side of the street. Find the breadth of the street.
13. From a point 200 ft. from the base of the Forefathers' monument
at Plymouth, the base and summit of the statue of Faith are at an eleva-
tion of 12° 40' 48" and 22° 2' 53", respectively ; find the height of the
statue and of the pedestal on which it stands.
14. At a distance of 100 ft. measured in a horizontal plane from the
foot of a tower, a flagstaff standing on the top of the tower subtends an
angle of 8°, while the tower subtends an angle of 42° 20'. Find the
length of the flagstaff.
15. The length of a string attached to a kite. is 300 ft. The kite's
elevation is 56° 6'. Find the height of the kite.
16. From two rocks at sea level, 50 ft. apart, the top of a cliff is ob-
served in the same vertical plane with the rocks. The angles of eleva-
tion of the cliff from the two rocks are 24° 40' and 32° 30'. What is the
height of the cliff above the sea ?
CHAPTER VI.
GENERAL FORMULA — TRIGONOMETRIC EQUATIONS
AND IDENTITIES.
49. Thus far functions of single angles only have been
considered. Relations will now be developed to express
functions of angles which are sums, differences, multiples,
or sub-multiples of single angles in -terms of the functions
of the single angles from which they are formed.
First it will be shown that,
sin (a ± p) = sin a cos p ± cos a sin p,
cos (a ± p) = cos a cos p T sin a sinpe
tan g ± tan p
1 T tan a tan p
The following cases must be considered :
1. a, y8, a + yS acute angles.
2. a, y8, acute, but a + yS an obtuse angle.
3. Either a, or y8, or both, of any magnitude, positive or
negative.
The figures apply to cases 1 and 2.
tan (a ± p)
Let the terminal line revolve through the angle «, and
then through the angle ^, to the position OB, so that angle
56
GENERAL FORMULA. 57
XOB = a-{- 13. Through any point F in OB draw perpen-
diculars to the sides of a, DP and (7P, and through C draw
a perpendicular and a parallel to OX, MO and JVC,
Then the angle QCA = a (why?), and CNP is the triangle
of reference for angle QCF = 90° + a.
CNP is sometimes treated as the triangle of reference for angle CPN.
The fallacy of this appears when we develop cos (« + ^), in which PC
would be treated as both plus and minus.
Now sm(« + ^)=sinXO£ = |^ = ^+^,
or expressiftg in trigonometric ratios,
^MC 00^. NP CP_
OC' OP CP' OP
= sin a cos ^ + sin (90° + a) sin ff.
Hence, since sin (90° 4- a) = cos a, we have
sin (a 4- yQ) = sin a cos ^ + cos a sin ff.
In like manner
cos(« + ^) = cosXO^ = — = — + -^
or expressing in trigonometric ratios,
OM 00 ON OP
00 OP OP OP
= cos a cos P + cos (90° + a) sin yS.
And since cos (90° + a) = — sin a, we have
cos (a + y8) = cos a cos yS — sin a sin y5.
It will be noted that the wording of the demonstration ap-
plies to both figures, the only difference being that when a + /3
is obtuse OD is negative. ON is negative in each figure.
50. In the case, when a, or /3, or both, are of any magni-
tude, positive or negative, figures may be constructed as
before described by drawing through any point in the terminal
line of P a perpendicular to each side of a, and through the foot
of the perpendicular on the terminal line of a a perpendicular
and a parallel to the initial line of a. Noting negative lines,
68 PLANE TRIGONOMETRY.
the demonstrations already given will be found to apply for
all values of a and y8.
To make the proof complete by this method would require an unlim-
ited number of figures, e.g. we might take a obtuse, both a and (i obtuse,
either or both greater than 180°, or than 360°, or negative angles, etc.
Instead of this, however, the generality of the proposition
is more readily shown algebraically, as follows :
Let a^ = 90° + a be any obtuse angle, and a, yS, acute
angles.
Then ^
sin Qa! + iS) = sin (90° + a + yS) = cos (a + ^S)
= cos a cos /3 — sin a sin y8
= sin (90° + a) cosy8 + cos (90° + a) sinyS(why?)
= sin a' cos /3 + cos ct' sin y(3.
In like manner, considering any obtuse angle ^' = 90° + yS,
it can be shown that
sin (a' + yS') = sin ex! cos y8' + cos aJ sin/3^
Show that cos (a' + ^8') = cos a' cos fi' — sin a' sin /3^
By further substitutions, e.g. a" = 90° ± a', 0" = 90° ± yS^
etc., it is clear that the above relations hold for all values,
positive or negative, of the angles a and yS.
Since a and may have any values, we may put — yS for y8,
and sin(a+ [— yS])
= sin (a — yS) = sin a cos ( — y5) + cos a sin ( — yS)
= sin a cos yS — cos a sin yS (why ?) .
Also cos (a — I3}= cos a cos( — yQ) — sin a sin ( — /3)
= cos a cos yS + sin a sin /3,
Finally,
tan r + iS^ — ^"^ C^ ^ )^) _ sin ct cos y9 ± cos ct sin /S
cos(a±/3) cosctcosyS^ sinasinyS
sinctcosy8 cos«siny8
cos a cos /S cos a cos yS tan a ± tan y8
cosctcosy^ sin a sin y8 1 qp tan a tan y8
cos a cos /3 cos a cos yS
EXAMPLES. 59
ORAL WORK.
By the above formulae develop :
1. sin (2A +SB). 7. sin 90° = sin (45° + 45°).
2. cos (90° -5). 8. cos 90°.
3. tan (45° + <^). 9. tan 90°.
4. sin 2 yl = sin (A + A), 10. sin (90° + /? + y).
5. cos 2^. 11. cos (270° - m - n).
6. tan (180° + C). 12. tan (90° + m + n).
Ex. 1. Find sin 75°.
sin 75° = sin (45° + 30°) = sin 45° cos 30° + cos 45° sin 30°
= ^.^ + ^.Ul±^ = 0.9659.
y/2. 2 V2 2 2\/2
2. Find tan 15°.
tan 45° - tan 30°
tan 15° = tan (45° - 30°)
i-X
1 + tan 45° tan 30°
^ = ^-^ = 2 - V3 = 0.2679.
1 + J_ V3 + 1
V3
3. Prove !iEM_£2iM=2.
sin A cos A
Combining" ^^^ ^ ^ ^^^ ^ —cos 3 ^ sin ^ _ sin (3^4 — A^
sin A cos A sin A cos A
_ sin 2 A _ si n (A -\- A) _ sin A cos A + cos A sin A _ g
sin A cos A smA cos A sin A cos A
4. Prove tan-^ a + tan-^ b = tan-^ -^-^ —
1 — ah
Let a — tan-^a, /8 = tan-i&, y = tan-^ ^ "^ *
Hence, tan a = a, tan (3 = b, tan y = -^^^ -•
Then a -\- fi = y, and hence tan (« + )8) = tany.
Expanding, tan « + tan /? ^ ^^^
1 — tan a tan yt?
Substituting,
g + 6 __a_±A.
1 - a6 1 - a6*
60 PLANE TRIGONOMETRY.
EXAMPLES.
1. Find cos 15°, tan 75°.
2. Prove cot (a ± (3) = ^ot^^cot^Tl .
, ^ ^^"^ cot)8±cota
3. Prove geometrically sin (« + /?) = sin a cos )8 + cos a sin j8,
and cos (a + )8) = cos a cos ^ — sin a sin j3,
given (a) a acute, y8 obtuse ;
(b) a, P, obtuse ;
(c) a, /3, either, or both, negative angles.
4. Prove geometrically tan (a + B) = ^^^<^J^^^^P
^ J V A-y l-tanatan/3
Verify the formula by assigning values to a and fi, and finding the
values of the functions from the tables of natural tangents.
5. Prove cos (a + jS) cos (a — ft) = cos^ a — sin*^ fi.
6. Show that tan a + tan j8 = sin (a + p\
cos a cos p
7. Given tan a = i, tan )8 = f , find sin (a + )8)
8. Given sin 280° = s, find sin 170°.
9. If a = 67° 22', jS = 128° 40', by use of the tables of natural func-
tions verify the formulae on page 56.
Prove tan-i ^^ "^ ^ = tan-V^+ tan-^Va.
=tan-iV3.
13. If a + /3 = <o, prove cos^ « + cos^ j8 — 2 cos a cos j8 cos cu = sin* w.
14. Solve i sin ^ = 1 — cos 0.
15. Prove sin (A + B) cos A — cos (A + E) sin A = sin J5.
16. Prove cos {A + B) cos {A-B)-\- sin(^ + B) sin(^ -B) = cos 2 B.
17. Prove sin (2 a- ft) cos {a -2 ft)
- cos (2 a- ft) sin (a - 2 j8) = sin (a + /S).
18. Prove sin(n — l)acos(n4-l)a + cos(n-l)asin(n + l)a= sin2n«.
19. Prove sm (135° - 0) + cos (135° + ^) = 0.
1-V^
.. Prove tan-
b^/3
2b-x
xV3
!. Prove sec~
I "" - sin
-lE.
y/a^-x^
a
ADDITION— SUBTRACTIOX FORMULA. 61
20. Prove 1 - tan^ « tan^ R = cos^ jg - si^^ ^.
cos* a cos* /?
21. Prove t^°« + tan^ ^ j^^ „ ^^^ ^
cot a + cot p
22.
tan* f ^ - «U l-2sin«cosci,
\4 / 1 4- 2 sin a cos a
51. The following formulae are very important and should
be carefully memorized. They enable us to change sums
and differences to products, i.e. to displace terms by factors.
sine + siii<|» = 2 sin^^cos^^,
sine - sin<t> = 2cos-^t_?sin— =-5,
COS0 + COS«|> = 2cos-^^cos— ^5
COS - cos<}> = - 2 sin -i-^ sin — ^•
Since sin (« + y8) = sin « cos y8 + cos a sin /8,
and sin (« — )9) = sin a cos y8 — cos a sin ^,
then sin (a + y8) + sin (a — /8) = 2 sin a cos ^9, (1)
and sin (a + /S) — sin (a — y8) = 2 cos a sin ^. (2)
Also since cos (« + yS) = cos a cos y8 — sin a sin yS,
and cos(a — y8)= cosacosy8+ sinasinyS,
then cos (a + /3) + cos (a — /3) = 2 cos a cos y8, (3)
and cos(a + yS)— cos(a — /3)= — 2sinasin/S. (4)
Put a-\-^ = e
and a — 13 = cl>
2a = + <j>, and a = ^±-i,
2/3 = ^-^, andyg = ^-=^.
A
Substituting in (1), (2), (3), (4), we have the above
formulae.
62 PLANE TRIGONOMETRY.
EXAMPLES.
1. Prove ?HL24±iHi| = tan ^.
cos 2 ^ + cos ^ 2
By formulae of last article the first member becomes
2 sm — cos -
2 2 3^
= tan
o 3(9 e 2
2 cos — cos -
2 2
2 p sin ct 4- 2 sin 3 g + sin 5 ct _ sin3 a
sin 3 cc + 2 sin 5 ct + sin 7 a sin 5 a
(sin ct 4- sin 5 g) + 2 sin 3 (z _ 2 sin 3 a cos 2 « + 2 sin 3 ct
(sin 3 ct + sin 7 a) + 2 sin 5 a 2 sin 5 cc cos 2 ct + 2 sin 5 a
_ (cos 2 (^ + 1) sin 3 ot _ sin 3 a
(cos 2 a + 1) sin 5 ct sin 5 a
3. Prove ^^" ^^^ - 2 g)Hh sin (4 ^ - 2 ^) ^ ^^^
cos (4^ -25)+ cos (45 -2^) ^ ^
o- 4^-25+45-2^ 4^-25-45+2^
2 sin cos ■
2 2
o 4yl -25 + 45-2^1 4^-25-45 + 2^
2 cos ;^ COS ■
2 2
= ^-niM+^ = tan(^+5).
COS (.1+5) ^ ^
4. Prove sin 50° - sin 70° + sin 10° = 0.
2 cos ^^° "^ '^^° sill ^^'^ ~ ^^"^ = 2 cos 60° sin ( - 10°) = - sin 10°.
2. 2 ' ^
5. Prove ^^^^""^"^^-^^^^^^^"'^^ + ^^^^^^Q"^Q^=cot6(^cot5tt.
sin4asin3ct— sin2()isiri 5 a + sin4otsin7a
By (3) and (4), p. 61,
cos 5 ct + cos a — cos 9a — cos 5 ot + cos 11 ct + cos 9 ct
cos a — cos 7 a — cos 3 a + cos 7 a + cos 'S a — cos 11 a
cos a + cos 11a 2 cos 6 a cos 5 a ^^4. « ^ «^f k «
= ! = — — = cot 6 a cot a.
cos a — cos 11a 2 sin 6 a sin 5 a
ORAL WORK.
By the formulae of Art. 51 transform :
6. cos 5 a + cos a. 8. 2 sin 3 d cos $.
7. cos a — cos 5 a. 9. sin 2 a — sin 4 a.
FUNCTIONS OF THE DOUBLE ANGLE. 63
10. cos 9^ cos 2^. 16. cos(30°+2<^)sin(30°-</)).
Q
11. sin $ + sin -. 17. sin (2 r + s) + sin (2 r - s).
12. sin 75° sin 15°. 18. cos (2 )8 - a) - cos 3 a.
13. cos7i>-cos2;7. 19. sin 36° + sin 54°.
14. cos(2o + 3o)sin(2p-3o).
^ ^ ^^ ^ -^ ^^ 20. cos 60° + cos 20°.
15 • ?i ' L
' ^^^2 ~^"^2' 21. sin 30° + cos 30°.
• Prove: 22. ?HL^^±^ = tan«4J?cot^^:^.
sin a — sin y8 2 2
23 cos « + cos ^ ^ cotgL±^cot^^-^.
cos /? — cos a 2 2
2^^ sin^ + siny^^^^^+j^^
cos X + COS y 2
25. sin a:- sin y ^ _ ^ot^i^^.
cos a: — cos y 2
26. cos 55° + sin 25° = sin 85°.
Simplify: 27. sin^ + sin 2 B + sin 3 ^^
cos B + COS 2B +COS 3 5
23 sin C - sin 4 C + sin 7 C - sin 10 C
cos C — COS 4 C + cos 7 C — cos 10 C
52. Functions of an angle in terms of those of the half angle.
If in sin (a + /3) = sin a cos yS + cos a sin jS, a = j3,
then sin (a + a) = sin 2 a = 2 sin a cos a.
In like manner
cos (a + a) = COS 2 a = cos^ a - sin^ a
= 2 cos^ a - 1 ^
= l-2sin»a;
and tan 2 a =
1 - tan* a
64 PLANE TRIGONOMETRY.
ORAL WORK.
Ex. Express in terms of functions of half the given angles :
1. sin 4 a. 4. cos a:. 6. sm(2p — q).
2. cos3». . Q 7. cos (30° + 2 6).
5. sm^.
3. tan5^ 2 8. sin (a; + ?/).
9. From the functions of 30° find those of 60° ; from the functions of
45°, those of 90°.
53. Functions of an angle in terms ^ those of twice the angle.
By Art. 52, cos a = 1 - 2 sin2 ^ = 2 cos2^ - 1.
-^ 2 2
2sin^| =
= 1 — cos «,
and
2cos2^
«-r
. a
sm-
a
cos-
-COS a
2 '
-^
... u.|=
^^1-cosa
^ 1 + COS a
1=^4
1 + cos a
Explain the significance of the ± sign before the radicals.
Express in terms of the double angle the functions of
120°; 50°; 90°, with proper signs prefixed.
Ex. 1. Express in terms of functions of twice the given angles each
of the functions in Examples 1-8 above.
2. From the functions of 45° find those of 22° 30' ; from the functions
of 36°, those of 18° (see tables of natural functions).
3. Find the corresponding functions of twice and of half each of the
following angles, and verify results by the tables of natural functions :
Given sin 26° 42' = 0.4493,
tan 62° 24' = 1.9128,
cos 21° 34' = 0.9300.
-4
4. Prove tan-iA/^— -^^^ = ?. 5. 2 tan-i a; = tan-J ^^
+ cos X 2 1 — x^
EXAMPLES. 65
6. Ji Af B, C are angles of a triangle, prove
sin ^ + sin C + sinjB = 4 cos — sin — sin -^
2 2 2
7. K cos2 a + cos2 2a + cos^ 3 a = 1, then
cos a cos 2 a cos 3 a = 0.
8. Prove cot ^ — cot 2 ^ = esc 2 ^.
2
tan
9. Prove
tan
(H)
(M)
1 — tan 2
2
l + tan|j
10 tang _ - 2 sin ^ ^
tan (a + <^) sin (2 a + <^) + sin <^
U. lfv = t2^n-i2^I ±^' + ^^^^^\ prove 2:2 = sin 2y.
12. Prove tan-i Vl + x^- 1 ^an-i -1^ = g tan-i x.
X 1 - a;2 2
13. If y = sin-i - ^ prove x = tan w.
14. Prove cos2 « + cos2 j8 - 1 = cos (a + fS) cos (a - ^).
15. Prove V(cos a - cos ^) 2 -f- (sin a - sin ^8)2 = 2 sin £LZL^.
16. Prove sin-i V-^— = tan-i J- = - cos-i ^^
^a+a; ^a 2 a +
17. Prove cos^ - cos^ <^ = sin (<^ + 6) sin (<l> - 0).
18. Prove tan ^ + tan (A + 120°) + tan (A - 120°) = 3 tan 3 ^.
19. Prove tan a — tan - = tan - sec a.
2 2
20. 3tan-ia = tan-i ^"~^^
1 - 3 a2
21. cos2 3 A (tan2 3 ^ - tan2 ^) = 8 sin2^ cos 2 ^.
22. 1 + cos 2 (^ - B) cos 2 5 = cos2^ + cos2 (A -2 B).
23. cot2fE + ^U2csc26l-
\4 2/ 2csc2^ +
sec
seed
66 PLANE TRIGONOMETRY.
TRIGONOMETRIC EQUATIONS AND IDENTITIES.
54. Identities. It was shown in Chapter I that
sin2 e + cos2 = 1
is true for all values of 0, and in Chapter VI, that
sin (a + /3) = sin a cos /3 + cos a sin jS
is true for all values of a and /3. It may be shown that
sin 2 A "^
= tan-4
1 + cos 2 J.
is true for all values of A^ thus :
sin 2 A _ 2 sin A cos A (by trigonometric transf orma-
1 + cos2J.~l + 2cos2J.-l tion)
= J (by algebraic transformation J
= tsinA (by trigonometric transformation).
Such expressions are called trigonometric identities. They
are true for all values of the angles involved.
55. Equations. The expression
2 cos^ a — 3 cos a + 1 =
is true for but two values of cos a, viz. cos a= ^ and 1, i.e.
the expression is true for a = 0°, 60°, 300°, and for no other
positive angles less than 360°. Such expressions are called
trigonometric equations. They are true only for particular
values of the angles involved.
56. Method of attack. The transformations necessary at
any step in the proof of identities, or the solution of equa-
tions, are either trigonometric^ or algebraic; i.e. in prov-
ing an identity, or solving an equation, the student must
choose at each step to apply either some principles of algebra,
or some trigonometric relations. If at any step no algebraic
operation seems advantageous, then usually the expression
METHOD OF ATTACK. 67
should be simplified by endeavoring to state the different
functions involved in terms of a single function of the angle,
or if there are multiple angles^ to reduce all to functions of a
single angle.
Algebraic
Transformations
Trigonometric, f Single function
to change to a 1 Single angle
No other transformations are needed, and the student will
be greatly assisted by remembering that the ready solution
of a trigonometric problem consists in wisely choosing at
each step between the possible algebraic and trigonometric
transformations. Problems involving trigonometric func-
tions will in general be simplified by expressing them entirely
in terms of sine and cosine.
EXAMPLES.
T -D sin 3 ^ cos 3 ^ n
1. Prove — : — : — = 2.
smA cos A
By algebra.
sin 3^ cos 3 ^ _ sin 3 ^ cos ^ — cos 3 J sin A
sin A cos A sin A cos A
... ., sm (3 ^ — ^ ) sm 2 ^
by trigonometry, = — r^^- 7^ = -: — ;; 7
sin ^ cos A smA cos A
_ 2 sin ^ cos A _ n
sin A cos A
Or, by trigonometry,
sin SA cos 3 ^ _ 3 sin ^ - 4 sin» A 4 cos^ A - 3 cos ^
sin -4 cos -4 sin^ cos J.
by algebra, =3-4 sin2 A — 4: cos^ ^ + 3
= 6 - 4(sinM + cosM)= 2.
sec 8 ^ - 1 tan 8 ^
2. Prove
sec 4 ^ - 1 tan 2 ^
No algebraic operation simplifies. Two trigonometric changes are
needed. 1. To change the functions to a single function, sine or cosine.
2. To change the angles to a single angle, 8 yl, 4 ^, or 2 ^.
68 PLANE TRIGONOMETRY.
By trigonometry and algebra,
1 - cos 8 ^ sin 8 ^
P cos 8 ^ _ cos 8 e _ tavx <f ^ ^
^ l-cos4^-sin2|> - ^^^p. \
cos 4^ cos 2^ iiV^^-v^^
K^ oin.nK,.o cos 4 ^(1 - cos 8 0) sin 8 6 cos^. cr _ <^ ^^ _
by algebra, — ^^ -^ — ^ = ♦ o^ * '^ "= ii-^"*- •? /5«r~
1 — cos 4 sm2^ / . &^-»^ _ j_r
by trigonometry, /^ . ^
COS 4 ^(1 - 1 + 2 sin2 4 ^) Z ^ sin 4 ^ cos 4 ^ cos 2 g . '"' iT&II^T^r^
l-l + 2sin22^ /<" sin2^ ' . , - . «
by algebra, /^^^ = 2 cos 2 ^ ; ~ C«^ «r^ . ^!.ui-^&c.
ysin 2 - /L
/ _, S»Vt, frO" C4H7 2. £
and X sin 40 = 2 sin 2 cos 2ft — Co^t^ ' "sT^ItI
which is a trigonometric identity.
J
3. Solve 2 cos2 + 3 sin = 0. |
By trigonometry, 2(1 - sin^ 0) + 3 sin = 0, 3
a quadratic equation in sin 0. -,
"I
By algebra, 2sin20 - 3sin0 - 2 = 0, ]
and (sin0-2)(2sin0 + l) = O. ^
.*. sin = 2, or — ^. Verify.
The value 2 must be rejected. Why? '
.-. = 210°, and 330° are the only positive values less than 360° that ■
satisfy the equation.
■ '\
4. Solve sec — tan = 2. i
Here tan = — 0.75, .-. from the tables of natural functions, *,
= 143° 7' 48", or 323° V 48". j
Find sec 0, and verify. I
5. Solve 2 sin sin 3 - sin^ 2 = 0. j
By trigonometry, cos 2 — cos 4 — sin^ 2 = 0, I
also cos20-cos2 20 + sin22 0-sin220 = O. j
By algebra, cos 2 0(1 - cos 2 0) = 0. |
.-. cos 2 = or 1, i
and 2 = 90°, 270°, 0°, or 360°, j
* whence = 45°, 135°, 0°, or 180°. Verify. i
TRIGONOMETRIC EQUATIONS. 69
Or, by trigonometry,
2 sin ^(3 sin ^ - 4 sin^ ^) - 4 sin2 cos^ d = ;
by trigonometry and algebra,
6 sin2 ^ - 8 sin* ^ - 4 sin2 ^ + 4 sin4 ^ = 0;
by algebra, 2 sin^ ^ - 4 sin'* ^ = 0,
and 2 sin^ ^(1 - 2 sin2 $) = 0.
.-. sin ^ = 0, or ± V|,
and 6 = 0°, 180°, 45°, 135°, 225°, or 315°.
The last two values do not appear in the first solution, because only
angles less than 360° are considered, and the solution there gave values
of 2 0, which in the last two cases would be 450° and 630°.
Solve : 6. tan ^ = cot ^. 8. 2 cos 2 ^ - 2 sin ^ = 1.
7. sin^ ^ + cos ^ = 1. 9. sin 2 d cos = sin 0,
Prove: 10. 2cot2^ = cot^ — tan^.
11. cos 2 a: + cos 2 y = 2 cos (x + y) cos (x — y).
12. (cos a + sin ay = 1 + sin 2 a.
57. Simultaneous trigonometric equations.
13. Solve cos (x -h y)+ cos (x - y) = 2,
sin - + sin ^ = 0.
2 2
By trigonometry,
cos x cos y — sin x smy + cos x cos y + sin ar sin y = 2,
so that
cos a: cos y = 1 ;
also,
^
-cosar Jl-cosy_Q
2^2
and
.'. cos X = COS y.
Substituting,
cos2 a: = 1,
COS a; = ± 1.
.-. X = 0°, or 180°,
and
y = ar = 0°, or 180°. Verify,
70 PLANE TKIGONOMETRY
14. Solve for R and F.
W — Fsini — R cos i = 0,
W + F cos i — R sin i = 0.
To eliminate F,
Wcos i — Fshi i cos i — R cos^i = 0,
W sin i + jPcos i sin i — R sin^ i = 0.
Adding, TF(sin i + cos i) — R(sm^ i + cos^ i) = 0.
.-. R = W(sm i + cos i).
Substituting, W — Fsini — W((sin i + cos z)cos i =
ET _ W — Ty(sin i + cos i) cos z
' * • sin i
Jl W = S tons, and i = 22° 30', compute F and jR.
i2 = 3(0.3827 + 0.9239)= 3.9198.
F ^ 3 - 3(0.3827 + 0.9239)0.9239 ^_iqoa
0.3827 ■ *
Solve :
15. 472 cot e - 263 cot <^ = 490, 307 cot 6 - 379 cot <^ = 0.
16. sin 2 a: + 1 = cos a; + 2 sin a;.
17. cos2 e + sin ^ = 1.
18. If 2;^(cos2^-sin2^)-2asin^cos^ + 26sin^cos^ = 0, prove
^ = itan-i-?-^.
a — b
Prove :
19. tan y =(1 -{■ sec y) tan ^'
20. 2 cot-i X = csc-i ^ "^ ^^ -
2a:
21. sin(<^ + 45°) + sin (<^ + 135°) = V2 cos <^.
22 cos V + cos 3 y _ 1
cos 3 y + cos 5v 2 cos 2 w — sec 2 1'
23. cos 3 a: — sin 3 a: = (cos x + sin a;) (1 — 2 sin 2 x).
Solve :
24. sin 2 ^ + sin ^ = cos 2 ^ + cos 6.
25. 4 cos(^ + 60°) - V2 = Ve - 4 cos (^ + 30°).
26. tan 2 ^ = tan 0-1.
27. cos ^ + cos 2 ^ + cos 3 ^ = 0.
TRIGONOMETRIC EQUATIONS. 71
28. sin 2 a; + V3 cos 2 a; = 1.
29. 3 tan'-^jo + 8 cos^p = 7.
30. Determine for what relative values of P and W the following
equation is true :
cos2^- — cos^-i=0.
2 W 2 2
31. Compute N from the equation iV+ -— cos a — — sin a — TV cos a = 0,
o o
when W = 2000 pounds and a satisfies the equation 2 sin ct = 1 + cos a.
32. sin 9 — tan <j>(cos 6 + sin 0) = cos 0, sin ^ — tan ^ cos ^ = 1.
Prove :
33. coi{t + 15°) - tan (t - 15°) = ^ ^^^ ^ <
2 sin 2 < + 1
34. sin-i f — sin~i y\ = sin"^ ^.
35. tan(^ + ^UV ^+^"^^ .
U 2/ >l-sino>
36. 2 sin-i i = cos-i 1.
37. If sin ^ is a geometric mean between sin B and cos B, prove
cos 2^ =2sin(45 - 5) cos (45 + B).
38. Prove sin (a + y8 + y) = sin a cos y8 cos y + cos « sin ^ cos y
+ cos « cos y8 sin y — sin a sin j8 sin y.
Also find cos(a + /S + y).
39. Prove tan((. + /3 + y) :^ ^^" ^ + ^^" ^ + ^^^ V " *^^ ^ *^" ^ ^^^ ^.
1 —tan a tan ^— tan )8 tan y— tan y tan «
If a, jS, and y are angles of a triangle, prove
40. tan a + tan ^ + tan y = tan « tan y8 tan y.
41. cot- + cot^4-cot^ = cot-cot^cotX
2 2 2 2 2 2
If a + /8 + y = 90°, prove
42. tan a tan ^ + tan ^ tan y + tan y tan a = 1.
Prove :
43. sin na = 2 sin (n — 1) a cos a — sin (n — 2)a.
44. cos na = 2 cos (n — 1) a cos a — cos (n — 2)a.
45. tann«= tan (n - 1)« + tan « ,
1 — tan (n — 1) a tan a
CHAPTER VII.
TRIANGLES.
58. In geometry it has been shown that a triangle is
determined, except in the ambiguous case, if there are given
any three independent pai^s, as follows :
I. Two angles and a side.
II. Two sides and an angle,
(<x) the angle being included by the given sides,
(5) the angle being opposite one of the given sides (am-
biguous case).
III. Three sides.
The angles of a triangle are not three independent parts, since they
are connected by the relation A ■\- B + C = 180°.
The three angles of a triangle will be designated A^ B, O,
the sides opposite, a, b, c.
But the principles of geometry do not enable us to compute
the unknown parts. This is accomplished by the following
laws of trigonometry :
I. Law of Sines, 5«L4 = !ilL? = !iH-2.
a c
II. Law of Tangents, tan :^ (^ - ^) ^ a-|
•^ ^ tani-(^ + ^) a-\-b
III. Law of Cosines, cos A = — \^ , ~" ^ , etc.
2oc
59. Law of Sines. In any triangle the sides are propor-
tional to the sines of the angles opposite.
Let ABQ be any triangle, p the perpendicular from B
on h. In I (Fig. 34), C is an acute, in II, an obtuse, in III,
72
LAW OF SINES — OF TANGENTS.
73
a right angle. The demonstration applies to each triangle,
but in II, ^mACB=^\nDOB (why?); in III, sinC=l
(why?).
P
Now sin A — —'> ,'. v = c sin A,
c ^
P
sin C= —1 .'. p = a sin (7.
Equating values of ^, cs>vn.A = a sin (7,
sin A sin C
or, =
a c
By dropping a perpendicular from A^ or (7, on a, or (?, show
sin B sin C sin ^ sin B
, or
whence
he a
sin A sin B sin (7
6 '
60. Law of Tangents. The tangent of half the difference
of two angles of a triangle is to the tangent of half their sum,
as the difference of the sides opposite is to their sum,
a __ sin A
h~
By Art. 59,
or
sin j5
By composition and division,
g - 5 ^ sin J. — sin ^ ^ 2 cos IQA + B) sin i ( J. - ^)
a 4- ^ ~ sin ^ + sill ^ ~ 2 sin i^ ( Jl + ^) cos ^(^A — B^
^ tan^(J.-jg) ,
tan|-(^ + J5)'
tan ^{A — B) _ a—b
tani(^H-5)~a + 6'
74
PLANE TRIGONOMETRY.
61. Law of Cosines. The cosine of any angle of a triangle
is equal to the quotient of the sum of the squares of the adjacent
sides less the square of the opposite side, divided hy twice the
product of the adjacent sides.
In each figure a^=p'^-\-DC^
= c^-AD^ + (h-AI>y
(in Fig. 34, II, DC is negative ; in III, zero)
= c2 - AB^ ^P-2b'Al) + Aiy^
= h'^-\-c^-2h-AD.
But
AD = ccosA, .'. a^ = P-\-(^-2boGOsA;
\osA^'I±^-Z^.
2 be
Prove that cos B = ^^ + g^-^^
2aG '
and
2ab
62. Though these formulae may be used for the solution
of the triangle, they are not adapted to the use of loga-
rithms (why?). Hence we derive the following:
Since cos J. = 2 cos^ -4-1 = 1-2 sin2^,
2 2
we have
nA A
2 cos^^ = 1 + cos A, and 2 sin^- = 1 - cos -4.
z 2
LAW OF COSINES.
75
From the latter
0.0-^1 524-c2_^2 25^-62-^2 _|.^2
^^^"2=^- 2hc = Ihc
2 be 2 be
Let a + 5 + c = 2s, then a-\-b — e=a + b-}-c—2c=2s—2e;
i.e. a + b — e=2(s — c^.
In like manner, a — b -\- c = 2 (^s — b^,
— a + 5 + (?=2(s-a).
Substituting,
Show that
also
From
show that
also
o„in2^_2(«-
6).2Cs-c)
-„m 2-
2 be
.•.sin| = V^-
sinf=?
8in|= ?
2cos2^ = l + cos^,
COS
^ 9
cos- = ?
and
Also derive the formulae
~f-'
^i'^^W^-
tan:? = ?
tanf=?
76 PLANE TRIGONOMETRY.
63. Area of the triangle. In the figures of Art. 59 the
area of the triangle ABC= A = ^pb.
But p=csinA. .*. A = ^hc sin A, (i)
c sin B
Again, by law of sines, h =
sin O
g^sin^ sin^
2 sin
c^sin^sinj^
Substituting, A =
Zt sill w
(why?). (ii)
2 sin (^4-^)
<x A A
Finally, since sin^ = 2 sin — cos — , we have from (i)
ju a
A 11 o • -4. A z ^ls(s — a)Cs—b^(s — c)
A = lbc ' 2 sin— cos — = bc\-^ ^ — -^^ ^
2 2 2 ^ be ' be
or A = Vs(s — a)(s — 6)(s — <?). (iii)
Find A ; (1) Given a = 10, 6 = 12, C = 45°.
(2) Given a = 4, 6 = 5, c = 6.
(3) Given a = 2, B = 45°, C = 60°.
SOLUTION OF TRIANGLES.
64. For the solution of triangles we have the following
f ormulee, which should be carefully memorized :
T sin^ _ sin B _ sin C
a h c
II. tan|(^-^) = ^^tan|(^ + B).
III. sm — = \^ ^ ^, or cos — = ^-^ — ^»
2 ^ be 2 ^ be
ortan:|=V^ ^^H^-o) .
2 ^ s(s- a)
IV. li=lbcs\nA = ^^^r^^/^^ = V8i8-aK8-bKs-c).
^ 2 sm (A 4- B)
SOLUTION OF TRIANGLES. 77
Which of the above formulae shall be used in the solution
of a given triangle must be determined by examining the
parts known, as will appear in Art. 69. It is always pos-
sible to express each of the unknown parts in terms of three
known parts.
In solving triangles such as Case I, Art. 58, the law of
sines applies; for, if the given side is not opposite either
given angle, the third angle of the triangle is found from
the relation A -h B -\- = 180°, and then three of the four
quantities in — — = ^^ — being known, the solution gives
a
the fourth.
In Case II (5) the law of sines applies, but in II (a) two
only of the four quantities in ^^2 — = EE — are known.
a
Therefore, we resort to the formula
tan i(^ -B) = ^nitanK^ + B),
in which all the factors of the second member are known.
In Case III, tan ^ = >|^ ~ ^^ ^^ ~ ""^ is clearly applicable,
A S yS — d)
and is preferred to the formulae for sin— and cos — ; for,
first, it is more accurate since tangent varies in magnitude
from to 00, while sine and cosine lie between and 1.
(See Art. 2T, 5.)
Let the student satisfy himself on this point by finding, correct to
seconds, the angle whose logarithmic sine is 9.99992, and whose loga-
rithmic tangent is 1.71668. Does the first determine the angle ? Does
the second?
And, second, it is more convenient, since in the complete
solution of the triangle by sin -- nx logarithms must be taken
A A
from the table, by cos — seven^ and by tan — but four.
The right triangle may be solved as a special case by the
law of sines, since sin (7=1.
T8
PLANE TRIGONOMETRY.
65. Ambiguous case. In geometry it was proved that a
triangle having two sides and an angle opposite one of them
of given magnitude is not always determined. The marks
of the undetermined or ambiguous triangle are :
1. The parts given are two sides and an angle opposite one,
2. The given angle is acute.
3. The side opposite this angle is less than the other given
side.
When these marks are aJl present, the number of solutions
must be tested in one of two ways :
((^) P>om the figure it is apparent that there will be no
solution when the side opposite is less than the perpendicular
p ; one solution when side a equals p ; and two solutions when
a is greater than p.
M. b A b O A b C C
No Solution, One Solution, Two Solutions,
Fig. 35.
And since sin^ = — , it follows that there will be no solu-
c
tion, one solution, two solutions, according as sin A = —
< c
(5) A good test is found in solving by means of loga-
rithms ; and there will be no solutions, one solution, two solu-
tions, according as log sin O proves to be impossible, zero,
possible, i.e. as log sin Q is positive, zero, or negative. This
results from the fact that sine cannot be greater than unity,
whence log sine must have a negative characteristic, or be
zero.
66. In computations time and accuracy assume more than
usual importance. Time will be saved by an orderly arrange-
ment of the formulae for the complete solution, before open-
ing the book of logarithms, thus :
SOLUTION OF TRIANGLES. 79
Given J., B^ a.
Solve completely.
= 180°-(^-h^),
, a sin -B « sin (7 A 1 1 • />
sin A sin J. ^
180°
log a= log a =
A-\-B =
log sin jB = log sin C =
.-. C =
colog sin A = colog sin A =
log 6 = log c =
.-.6= .-.0 =
Check:
loga =
l0g(5-&) =
log ft =
log(5-c) =
log sin C =
colog 5 =
colog 2 =
^ colog (5 — a) =
logA =
2)
.•.A =
logtan:| =
.-. A =
67. Accuracy must be secured by checks on the work at
every step ; e.g. in adding columns of logarithms, first add
up, and then check by adding down. Too much care can-
not be given to verification in the simple operations of
addition, subtraction, multiplication, and division. A final
check should be made by using other formulse involving the
parts in a different way, as in the check above. As far as
possible the parts originally given should be used through-
out in the solution, so that an error in computing one part
may not affect later computations.
68. The formulae should always be solved for the unknown
part before using^ and it should be noted whether the solu-
tion gives one value, or more than one, for each part; e.g.
the same value of sin^ belongs to two supplementary angles,
one or both of which may be possible, as in the ambiguous
case.
" 69. Write formulae for the complete solution of the fol-
lowing triangles, showing whether you find no solution, one
solution, two or more solutions, in each case, with reasons for
your conclusion :
80
PLANE TRIGONOMETRY.
a
b
c
A
B
C
1.
81° 26' 28''
44° 11' 20"
54° 22' 12"
2.
78.54
63° 18' 20"
41° 30' 18"
3.
135.82
26.89
53° 28' 30"
4.
0.75
0.85
0.95
5.
243
562
36° 15' 40"
6.
38.75
25.92
63° 50' 10"
7.
0.058
78° 15'
33° 46'
8.
2986
1493
30°
9.
48
50
26° 15'
MODEL SofctJTIONS.
1. Given a = 0.785, b = 0.85, c = 0.633. Solve completely.
tan
^a/5
6)0
c) ^ B
— , tan 2
4
(s — a){s — c)
tan
4
(s-a)(s-b)
2 ~ '^ s(s-a) ' 2~^ s(s-b) ' 2 ^ s{s-c)
Check: A + B ■}- C = 180°. A = Vs(s - a){s - b)(s - c).
a = 0.735
b = 0.85
c = 0.633
2 )2.268
s = 1.134
s-a = 0.349
s-b = 0.284
s - c = 0.501
log(s-&)= 9.45332
log (s - c) = 9.69984
cologs= 9.94539
colog (s -a)= 0.45717
2 )19.55572
log tan ^^= 9.77786
^A =30° 56' 49"
A = 61° 53' 38"
Check: log(s-a)= 9.54283
A = 61° 53' 38" log (s - &) = 9.45332
5 = 72° 46' 4" colog s = 9.94539
C = 45° 20' 20" colog (s-c)= 0.30016
180° 0' 2"
2 )19.24170
logtan^C= 9.62085
22° 40' 10"
C = 45° 20' 20"
hC
log(s-a)
log (s - c)
colog s
colog (s — b)
log tan ^ J5
IB
B
logs
log (s - a)
log (s - b)
log (s - c)
log A
A
= 9.54283
= 9.69984
= 9.94539
= 0.54668
2)19.73474
= 9.86737
= 36° 23' 2"
= 72° 46 4"
= 0.05461
= 9.54283
= 9.45332
= 9.69984
2 )18.75060
= 9.37530
= 0.2373
Solve :(1) Given a = 30, & = 40, c = 50.
(2) Given a = 2159, b = 1431.6, c = 914.8.
(3) Given a = 78.54, b = 32.56, c = 48.9.
SOLUTION OF TRIANGLES.
81
2. Given A = 57° 23' 12", C = 68° 15' 30", c = 832.56. Solve completely.
csin J.
B
180
54° 21' 18".
sin C
(A + C)
b =
csinB
sm C
Check : tan ^ A
^ be sin A .
^^(s-bXs-c)
logc = 2.92042
log sin A = 9.92548
colog sin C = 0.03204
log a
Check .
2.87794
a= 754.98
a= 754.98
b= 728.38
c= 832.56
log c = 2.92042
log sin B = 9.90990
colog sin C = 0.03204
log6 = 2.86236
b= 728.38 A
s{s — a)
logb= 2.86236
logc= 2.92042
log sin ^ = 9.92548
log2A= 5.70826
= 510811^255405.5
2 )2315.92
s = 1157.96
s-a= 402.98 log(s-6)= 2.63304
s- b= 429.58 log (s - c) = 2.51242
5-c= 325.40 colog s= 6.93634
colog (s- a) = 7.39471
2 )19.47651
log tan i^ = 9.73826
^^=28° 41' 38"
A = 57° 23' 16"
Solve :
(1) Given a = 215.73, B = 92° 15', C = 28° 14'.
(2) Given b = 0.827, A = 78° 14' 20", B = 63° 42' 30".
(3) Given b = 7.54, c = 6.93, B = 54° 28' 40".
3. Given a = 25.384, c = 52.925, 5 = 28° 32' 20". Solve completely.
("Why not use the same formulae as in Example 1, or 2?)
tan
C-A
£-:^tan.^+^
b =
csin^
2 c + a 2
180° -B = C-\-A= 151° 27' 40".
sinC'
Check: b =
I ac sin B.
asiiiB
sin J
.'. ^(C + A)= 75° 43' 50".
c= 52.925 log (c- a) = 1.43998 .-. l(C-A)= 54° 7'38"
a= 25.384 colog (c + a) =8.10619 ^(C+A)= 75°43'50"
c+a= 7SM9 logtanKC+^) = 0.59460 ^^^^^^^ C=129°51'28"
c-a= 27.541 log tan i(C-^) = 0.14077 subtracting, ^= 21°36'12"
log a = 1.40456
log c = 1.72366
log sin 5 = 9.67921
colog sin C= 0.1 1484
log & = 1.51771
b= 32.939
Check: log a = 1.40456
log sin 5 = 9.67921
colog sin .4 =0.43395
log 6 = 1.51772
logc = 1.72366
log sin 5 = 9.67921
log 2 A =2.70743
A=511!^=254.965
82 PLANE TRIGONOMETRY.
Solve : (1) Given a = 0.325, c = 0.426, B = 48° 50' 10".
(2) Given b = 4291, c = 3194, A = 73° 24' 50".
(3) Given b = 5.38, c = 12.45, A = 62° 14' 40".
4. Ambiguous eases. Since the required angle is found
in terms of its sine, and since sin a = sin (180° — a), it fol-
lows that there may be two values of a, one in the first, and
the other in the second quadrant, ^eir sum being 180°. In
the following examples the student should note that all the
marks of the ambiguous case are present. The solutions will
show the treatment of the ambiguous triangle having no
solution, one solution, two solutions.
(a) Given 5 = 70, c = 40, C= 47° 32' 10''. Solve. Why
ambiguous ?
. p ^ ^ sin (7 log 5 =1.84510
^^^ c * logsin (7= 9.86788
cologc = 8.89794
log sin^ = 0.11092
.'. B is impossible, and there is no solution. Why?
Show the same by sin > -•
(5) Given a = 1.5, c = 1.7, A = 61° 55' 38". Solve.
. ^^ csinA log c= 0.23045
^^^ a ' logsin^= 9.94564
colog«= 9.82391
logsin (7= 0.00000
a=90°
and there is one solution. Why ? Show the same by
sin A -
work.
sin J. = -. Solve for the remaining parts and check the
SOLUTION OF TRIANGLES. 83
(c) Given a = 0.235, b = 0.189, B = 36° 28' 20^'. Solve.
. . a sin ^ b sin O
sin A = — = ? e = —-.
sin^
log a =9. 37107 log 6 = 9. 27646 9. 27646
log sin ^ = 9. 77411 log sin 0=9. 99772 or 9. 28774
colog b = 0.72354 colog sin B = 0.22589 0. 22589
log sin ^ = 9.86872 log c = 9.50007 or 8.79009
A = 47° 39' 25'' c = 0.31628 or 0.06167
or 132° 20' 35".
... (7 =95° 52' 15" or 11° 11' 5".
Solve for A, and check. Show the same by sin B < —
Solve :
(1) Given 6 = 216.4, c= 593.2, B= 98° 15'.
(2) Given a = 22, 6 = 75, 5 = 32° 20'.
(3) Given a = 0.353, c= 0.295, A = 46° 15' 20".
(4) Given a = 293.445, b = 450, A = 40° 42'.
(5) Given b = 531.03, c= 629.20, ^=34° 28' 16".
Solve completely, given :
a
h
c
A
B C
L 50
60
78° 27' 47"
2.
10
11
93° 35'
3. 4
5
6
4.
10
109° 28' 16"
38° 56' 54"
5. 40
51
49° 28' 32"
6. 352.25
513.27
482.68
7. 0.573
0.394
112° 4'
8. 107.087
56° 15'
48° 35'
9.
V2
117°
45°
10. 197.63
246.35
34° 27'
11. 4090
3850
3811
12. 3795
73° 15' 15" 42° 18' 30"
13.
234.7
185.4
84° 36'
14.
26.234
22.6925
49° 8' 24"
15. 273
136
72° 25' 13"
84 PLANE TRIGONOMETRY.
APPLICATIONS.
70. Measurements of heights and distances often lead to
the solution of oblique triangles. With this exception, the
methods of Chapter V apply, as will be illustrated in the
following problems.
The hearing of a line is the angle it makes with a north
and south line, as determined by the^magnetic needle of the
mariner's compass. If the bearing does not correspond to
any of the points of the compass, it is usual to express it
thus: N. 40° W., meaning that the line bears from N. 40°
toward W.
EXAMPLES.
1. When the altitude of the sun is 48°, a pole standing on a slope
inclined to the horizon at an angle of 15° casts a shadow directly down
the slope 44.3 ft. How high is the pole?
2. A tree standing on a mountain side rising at an angle of 18° 30'
breaks 32 ft. from the foot. The top strikes down the slope of the moun-
tain 28 ft. from the foot of the tree. Find the height of the tree.
3. From one corner of a triangular lot the other corners are found to
be 120 ft. E. by N., and 150 ft. S. by W. Find the area of the lot, and
the length of the fence required to enclose it.
4. A surveyor observed two inaccessible headlands, A and B. A was
W. by N. and B, N.E. He went 20 miles N., when they were S.W. and
S. by E. How far was A from B ?
5. The bearings of two objects from a ship were N. by W. and N.E.
by N. After sailing E. 11 miles, they were in the same line W.N.W.
Find the distance between them.
6. From the top and bottom of a vertical column the elevation angles
of the summit of a tower 225 ft. high and standing on the same hori-
zontal plane are 45° and 55°. Find the height of the column.
7. An observer in a balloon 1 mile high observes the depression angle
of an object on the ground to be 35° 20'. After ascending vertically and
uniformly for 10 mins., he observes the depression angle of the same object
to be 55° 40'. Find the rate of ascent of the balloon in miles per hour.
8. A statue 10 ft. high standing on a column subtends, at a point
100 ft. from the base of the column and in the same horizontal plane, the
same angle as that subtended by a man 6 ft. high, standing at the foot
of the column. Find the height of the column.
9. From a balloon at an elevation of 4 miles the dip of the horizon
is 2° 33' 40". Required the earth's radius.
TRIANGLES — APPLICATIONS. 85
10. Two ships sail from Boston, one S.E. 50 miles, the other N.E. by
E. 60 miles. Find the bearing and distance of the second ship from the
first.
11. The sides of a valley are two parallel ridges sloping at an angle of
30°. A man walks 200 yds. up one slope and observes the angle of eleva-
tion of the other ridge to be 15°. Show that the height of the observed
ridge is 273.2 yds.
12. To determine the height of a mountain, a north and south base
line 1000 yds. long is measured ; from one end of the base line the sum-
mit bears E. 10° N., and is at an altitude of 13° 14'. From the other end
it bears E. 46° 30' N. Find the height of the mountain.
13. The shadow of a cloud at noon is cast on a spot 1600 ft. due west
of an observer. At the same instant he finds that the cloud is at an ele-
vation of 23° in a direction W. 14° S. Find the height of the cloud and
the altitude of the sun.
14. From the base of a mountain the elevation of its summit is 54° 20'.
From a point 3000 ft. toward the summit up a plane rising at an angle
of 25° 30' the elevation angle is 68° 42'. Find the height of the mountain.
15. From two observations on the same
meridian, and 92° 14' apart, the zenith
angles of the moon are observed to be
44° 54' 21" and 48° 42' 57". Calling the
earth's radius 3956.2 miles, find the dis- , ,, , ^
, , ,, { iC.\/\Z= Zenith angle
tance to the moon. ' ^ \^ ^ v
16. The distances from a point to three
objects are 1130, 1850, 1456, and the angles
subtended by the distances between the three objects are respectively
102° 10', 142°, and 115° 50'. Find the distances between the three objects.
17. From a ship A running N.E. 6 mi. an hour direct to a port dis-
tant 35 miles, another ship B is seen steering toward the same port, its
bearing from A being E.S.E., and distance 12 miles. After keeping on
their courses 1^ hrs., B is seen to bear from A due E. Find B's course
and rate of sailing.
18. From the mast of a ship 64 ft. high the light of a lighthouse is
just visible when 30 miles distant. Find the height of the lighthouse,
the earth's radius being 3956.2 miles.
19. From a ship two lighthouses are observed due N.E. After sailing
20 miles E. by S., the lighthouses bear N.N.W. and N. by E. Find the
distance between the lighthouses.
20. A lighthouse is seen N. 20° E. from a vessel sailing S. 25° E. A
mile further on it appears due N. Determine its distance at the last
observation.
EXAMPLES FOR ^VIEW.
In connection with each problem the student should review
all principles involved. The following list of problems will then
furnish a thorough review of the book. In solving equations,
find all values of the unknown angle less than 360° that satisfy
the equation.
1. If tan « = }, tan /? = i, show that tan (/3 - 2 a) = -j^.
2. Prove tan a + cot a = 2 esc 2 a.
A A A A
3. From the identities sin^ — |- cos^— = 1, and 2 sin — cos— = sin A.
2 2' 22
prove 2 sin — = ± Vl + siii A ±Vl — sin A,
and 2 cos — = ± Vl + sin A T Vl - sin^.
4. Remove the ambiguous signs in Ex. 3 when A is in turn an angle
of each quadrant.
5. A wall 20 feet high bears S. 59° 5' E. ; find the width of its shadow
on a horizontal plane when the sun is due S. and at an altitude of 60°.
6. Solve sin a: + sin 2 a: + sin 3 a; = 1 + cos x + cos 2 x.
7. Prove tan-i i + tan-i i = ^.
8. If ^ = 60°, B = 45°, C = 30°, evaluate
tan A + tan B + tan C
tan A tan B -!- tan B tan C + tan C tan A
9 Prove ^Q^ (^^ + ^) <^os C _ 1 — tan A tan B
cos (A + C) cos B 1 — tan A tan C
10. Solve completely the triangle whose known parts are b = 2.35,
c = 1.96, C = 38° 4:5' A.
11. Find the functions of 18°, 36°, 54°, 72°.
Let x = 18°. Then 2a;=36°, 3x = 54°, and 2x + 3a = 90°.
P
12. If cot a = -, find the value of
sin a + cos a + tan a + cot a + sec a + esc a.
86
EXAMPLES FOR REVIEW. 87
T « -r, sin 3 « sin 2 ^ — sin 3 i8 sin 2 a -, , ^ o
13. Prove — ^— ■, — ^ : — -^-^ = 1 + 4 cos a cos B.
sin 2 a sin p — sin 2 /j sin a '
14. From a ship sailing due N., two lighthouses bear N.E. and
N.N.E., respectively; after sailing 20 miles they are observed to bear
due E. Find the distance between the lighthouses.
15. Solve 1 — 2 sin a: = sin 3 x,
16. Prove sin-i\'— ^ = tan-i-\p.
^a + b ^b
17. If cos ^ — sin ^ = \/2 sin 6, then cos ^ + sin ^ = V2 cos 9,
18. Solve completely the triangle ABC, given a = 0.256, b = 0.387,
C = 102° 20'.5.
2 cos 2 cc - 1
19. Prove tan (30° + a) tan (30°- a) =
2 cos 2 a + 1
20. Solve tan (45° - 0) + tan (45° + ^) = 4.
21. Prove sin^ a cos^ /8 - cos^ a sin^ ft = sin^ a - sin^ ^.
22. Prove cos^ a cos^ /3 - sin^ a sin^ ^ = cos^ « - sin^ p..
23. A man standing due S. of a water tower 150 feet high finds its
elevation to be 72° 30' ; he walks due W. to A street, where the elevation
is 44° 50' ; proceeding in the same direction one block to B street, he finds
the elevation to be 22° 30'. What is the length of the block between A
and B streets.
24. Prove tan-i - 4- tan-i - + tan-i i + tan-i i = -•
3 5 7 8 4
25. If P = 60°, Q = 45°, R = 30°, evaluate
sin P cos Q + tan P cos Q
sin P cos P + cot P cot R
26. If cos (90° + «) = -!, evaluate 3 cos 2 a + 4 sin 2 a.
27. If sin B + sin C = m, cos J5 + cos C = n, show that tan — ^ — = — .
2 n
28. Show that sin 2 )3 can never be greater than 2 sin )8.
29. Prove sin-^ | + sin-^ ^ = tan-^ f f .
30. Solve cot-ix + sin-i- V5 = ^•
o 4
31. Solve sin-^x + sin-i(l — x)= cos~^ar.
32. A man standing between two towers, 200 feet from the base of
the higher, which is 90 feet high, observes their altitudes to be the same ;
70 feet nearer the shorter tower he finds the altitude of one is twice that
of the other. Find the height of the shorter tower, and his original
distance from it.
88 PLANE TRIGONOMETRY.
33. Solve cos 3 /3 + 8 cos^ p = 0.
34. Solve cot m — tan (180 + m) = sin m + sin (90" — m),
35. Solve ljzi!:Bi = 2 cos 2 1.
1 + tan «
36. Prove cot^ + cot 5 =^^-(A±M.
sm A sm B
37. Prove cot P - cot Q = - ^^"^^7 ^^ -
sin P sm Q
38. In the triangle ABC prove
a = 6 sin C + c sin 5,
6 = c sin J. + a sin C,
c = a sin .B + 5 sin ^4.
39. Solve completely the triangle, given
a = 927.56, b = 648.25, c = 738.42.
40. Prove cos^ a - sin (30° + a) sin (30° - «) = f .
-, -D X o J. cos 2 a; — cos 4 a:
41. Prove tan 3 x tan a; = —
cos 2 a; + cos 4 x
42. Simplify cos (270° + «) + sin (180° + a)+ cos (90° + a).
43. Simplify tan (270° -$)- tan (90° + 6)+ tan (270° + 0).
44. Solve cos 3 <^ — cos 2 <^ + cos ^ = 0.
45. Solve cos ^ + cos 3 ^ + cos 5 ^ + cos 7 ^ = 0.
46. The topmast of a yacht from a point on the deck subtends the
same angle a, that the part below it does. Show that if the topmast be
a feet high, the length of the part below it is a cos 2 a.
47. A horizontal line AB is measured 400 yards long. From a point
in A B Si balloon ascends vertically till its elevation angles at A and B
are 64° 15' and 48° 20', respectively. Find the height of the balloon.
sin a
48. If cos d) = n sin a, and cot<^= '* prove cos B=
tan^ Vl + n2cos2a
49. Find cos 3 a, when tan 2 a = — f .
50. Solve completely the triangle, given a = 0.296, B = 28° 47'.3,
C = 84° 25'.
51. Evaluate sin 300° + cos 240° + tan 225^
52. Evaluate sec IS - esc ^ + tan 1^.
O O O
EXAMPLES FOR REVIEW. 89
53. If tan^ = ^^^"^^^V-^"^^^^^Y
cos « COS y — cos /8 sin y
and tan <t> = sin a sin y - sin ^ cos y^
cos « sin y — cos ^ cos y
show that tan(^ + <^) = tan(a + (3).
54. If tan 466° 15' 38" = - ^^, find the sine and cosine of 233" 7' 49".
55 Prove ^^^ ^ ~ ^^^ ^ sec a — tan a
sec a + tan a esc a + cot a
56.
Prove cos((.-3^)-cos(3«-^) ^ ^ sin(a - fl).
sin 2 « + sin 2 ^ ^ '^^
57. Prove sin 80° = sin 40° + sin 20°.
58. Prove cos 20° = cos 40° + cos 80°.
59. Prove 4 tan-i - - tan"! — = S
5 239 4
60. From the deck of a ship a rock bears N.N.W. After the ship
has sailed 10 miles E.N.E., the rock bears due W. Find its distance
from the ship at each observation.
61. Find the length of an arc of 80° in a circle of 4 feet radius.
62. Given tan 6 = ^, tan <f} = -^^, evaluate sin(^ + <^) + cos(^ — <^).
63. If tan ^ = 2 tan <^, show that sin(^ + <^) = 3 sin(^ - <^).
64. Prove cos(a + j8)cos(a-^) + sin(a + /8)sin(a-fi)=i^I*^^.
^ '^ '^ l+tan2^
65. Solve 4 cos 2 ^ + 3 cos ^ = 1.
66. Solve 3 sin a = 2 sin (60° - a).
67. Prove (sin a - esc «)2 — (tan a - cot a) ^ + (cos a — sec «) 2= 1.
68. Prove 2(sin^ a + cos^ a) + 1 = 3(sin^ a + cos* a).
69. Prove esc 2 y8 + cot 4 /? = cot )8 - esc 4 p.
70. If tan » = — , cos 2 g = — , then esc ^^-^ = SVlS.
^ 12 ^ 625 2
71. Solve completely the triangle, given
a = 0.0654, 6 = 0.092, 5 = 38°40'.4.
72. Solve completely the triangle, given
& = 10, c = 26, J5 = 22°37'.
73. A railway train is travelling along a curve of | mile radius at the
rate of 25 miles per hour. Through what angle (in circular measure)
will it turn in half a minute ?
90 PLANE TRIGONOMETRY.
74. Express the following angles in circular measure :
63°, 4° 30', 6° 12' 36".
75. Express the following angles in sexagesimal measure :
6 8 ' 64 '
76. Ji A, B, C are angles of a triangle, prove
ABC
cos A + cos -B + cos C = 1 + 4 sin — sin — sin -•
77. Prove sin 2 x + sin 2 y + sin 2 2; = 4 sin a; sin y sin z, when a?, y, z
are the angles of a triangle.
78. Prove sec a = 1 + tan a tan -•
79. Prove sin^ (« + j8) - sin^ (a _ ^) = sin 2 a sin 2 j8.
80. Prove cos^ (a + /3) - sin^ (^a - f3)= cos 2 cc cos 2 ^.
81. Prove sinl9 ;> + sin 17 p ^ 2 cos 9;,.
sm 10 p + sm 8j9
82. Consider with reference to their ambiguity the triangles whose
known parts are :
(a) a = 2743, b = 6452, B = 43° 15' ;
(b) a = 0.3854, c = 0.2942, C=:38°20^
(c) &= 5, c = 53, 5 = 15°22';
(d) a = 20, b = 90, A= 63° 28'.5.
83. From a ship at sea a lighthouse is observed to bear S.E. After
the ship sailed N.E. 6 miles the bearing of the lighthouse is S. 27° 30' E.
Find the distance of the lighthouse at each time of observation.
84. Prove sin (^ + 3 <^) + sin (3 ^ + <^) ^ 3 cos (0 + <^).
sin 2 d + sin 2 <^ v ^ v'y
85. Prove cos 15° - sin 15° = — •
V2
86. Show that cos (a + )8) cos (a — /3)= cos^ a - sin^ p
= cos2)8 — sin^a.
87. Show that tan (a + 45°) tan (a - 45°) = ^sin^ot-l
^ ^ ^ ^ 2cos2a-l
88. Solve sin (x + y) sin (x — y)= ^, cos (x + y) cos (x — y) = 0.
89. Prove l + sin«-cos« ^ ^^^ a
1 + sin a + cos a 2
EXAMPLES FOR REVIEW. 91
90. Prove tan 2 ^ + sec 2 ^ = cos + sin 6
cos ^ — sin ^
91. If tan <}>=-, then a cos 2^ + &sin2d> = a,
a
92. Prove sin-i-i + cot-i3 = ^'
93. Solve cos A + cos 7 A = cos 4 A.
94. Two sides of a triangle, including an acute angle, are 5 and 7,
the area is 14 ; find the other side.
95. Show that 3cos3g-2cose-cos5g ^ ^^^ ^ ^
sm 5 ^ — 3 sm 3 ^ + 4 sin 6
96. A regular pyramid stands on a square base one side of which is
173.6 feet. This side makes an angle of 67° with one edge. What is
the height of the pyramid ?
97. From points directly opposite on the banks of a river 500 yards
wide the mast of a ship lying between them is observed to be at an eleva-
tion of 10° 28'.4 and 12° 14'.5, respectively. Find the height of the mast.
98. Show that (sin 60° - sin 45°) (cos 30° + cos 45°) = sin2 30°.
99. Find x if sin-i x + sin-i ^ = ^.
2 4
100. Trace the changes in sign and value of sin a + cos a as a
changes from 0° to 360°.
FIVE-PLACE
LOGARITHMIC AND TRIGONOMETRIC
TABLES
ADAPTED FROM GAUSS'S TABLES i
i
BY j
ELMER A. LYMAN J
MICHIGAN STATE NORMAL COLLEGE •
AND ^
EDWIN C. GODDARD |
UNIVERSITY OF MICHIGAN i
>J«io
ALLYN AND BACON
Boston antJ Chicago
V/^
COPYRIGHT, 189 9, BY
ELMER A. LYMAN and
EDWIN C. GODDARD.
Nortoooti iPwaa
J. S. Cashing & Co. — Berwick & Smith
Norwood Mass. U.S.A.
TABLE I.
THE COMMON LOGARITHMS OF NUMBERS
FROM 1 TO 10009.
N.
L.
1
2
3
4
5
6
7
8 9
P.P.
100
00000
043
087
130
173
217
260
303
346 389
lOI
432
475
518
561
604
647
689
732
775 817
44 43 42
I02
860
903
945
988
*03o
*072
*ii5 *i57 *i99 *242
I
2
103
01 284
326
368
410
452
494
536
tl
620 662
4/4 4/3 4/2
8,8 8,6 8,4
13,2 12,9 12,6
17,6 17,2 16,8
22,0 21,5 21,0
26,4 25,8 25,2
30,8 30,1 29,4
35/2 34,4 33,6
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248
254
260
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309
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333
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376
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467
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479
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509
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534
540
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600
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159
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171
177
183
189
195
201
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7
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213
219
225
231
237
243
249
255
261
267
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273
279
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308
314
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332
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362
368
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380
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198
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4
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181
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212
217
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258
263
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288
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308
313
318
323 328
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374 379
384
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399
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409
414
420
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206
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240
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231
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376
381
386
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2
3
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5 6
7
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3
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6
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856
861
866
871
875
880
885
890
895
899
910
904
909
914
918
923
928
933
938
942
947
911
952
957
961
966
971
976
980
985
990
995
5
912
999 *oo4 ^009 *oi4 *oi9
*023
*028
*o33
*038
*042
I
o.S
1,0
913
96047
052
057
061
066
071
076
080
085
090
2
914
095
099
104
109
114
118
123
128
133
137
3
4
1.5
2,0
2,5
915
142
147
152
156
161
166
171
175
180
185
916
190
194
199
204
209
213
218
223
227
232
6
3/0
917
237
242
246
251
256
261
265
270
275
280
7
3/5
918
284
289
294
298
303
308
313
317
322
327
8
4/0
919
332
336
341
346
3So
355
360
365
369
374
9
4.5
920
379
384
388
393
398
402
407
412
417
421
921
426
431
435
440
445
450
454
459
464
468
922
473
478
483
487
492
497
501
506
5"
515
923
520
525
530
534
539
544
548
553
558
562
924
567
572
577
581
586
591
595
600
605
609
925
614
619
624
628
633
638
642
647
652
656
926
661
666
670
675
680
685
689
694
699
703
927
708
713
717
722
727
731
736
741
745
750
928
755
759
764
769
774
778
783
788
792
797
929
802
806
811
816
820
825
830
834
839
844
930
848
^53
858
862
867
872
876
881
886
890
931
895
900
904
909
914
918
923
928
932
937
4
932
^il
946
951
956
960
965
970
97^
979
984
I
1/2
1/6
2
933
988
993
997 *002 *007
*OII
*oi6
^021 ^025 ^030
934
97035
039
044
049
053
058
063
067
072
077
2
3
4
935
081
086
090
095
100
104
109
114
118
123
936
128
132
137
146
151
155
160
165
169
2/4
2 8
937
174
179
183
188
192
197
202
206
211
216
7
938
220
225
230
234
239
243
248
253
257
262
te
939
267
271
276
280
285
290
294
299
304
308
9
940
313
317
322
327
331
336
340
345
350
354
941
359
364
368
373
377
382
387
391
396
400
942
405
410
414
419
424
428
433
437
442
447
943
451
456
460
465
470
474
479
483
488
493
944
497
502
506
511
516
520
525
529
534
539
945
543
548
552
557
562
566
571
575
580
585
946
589
594
598
603
607
612
617
621
626
630
947
635
640
644
649
653
658
663
667
672
676
948
681
685
690
695
699
704
708
713
717
722
949
727
731
736
740
745
749
754
759
763
768
950
772
777
782
786
791
795
800
804
809
813
N.
L.
1
2
3
4
5
6
7
8
9
P.P.
N.
L.
1
2
3
4
5
6
7
8
9
P.P.
950
97772
in
782
786
791
795
800
804
809
813
951
818
823
827
832
836
841
845
850
855
859
952
864
868
873
877
882
886
891
896
900
905
953
909
914
918
923
928
932
937
941
946
950
954
955
959
964
968
973
978
982
987
991
996
955
98 000
005
009
014
019
023
028
032
037
041
956
046
050
055
059
064
068
073
078
082
087
957
091
096
100
105
109
114
118
123
127
132
958
137
141
146
ISO
155
159
164
168
173
177
959
182
186
191
195
200
204
209
214
218
223
960
227
232
236
241
245
250
254
259
263
268
961
272
277
281
286
290
295
299
304
308
313
5
962
318
322
327
331
336
340
345
349
354
358
I
0,5
I
963
363
367
372
376
381
385
390
394
399
403
2
964
408
412
417
421
426
430
435
439
444
448
3
4
5
lis
2,0
2,5
9^1
453
457
462
466
471
475
480
484
489
493
966
498
502
507
5^^
516
520
525
529
534
538
6
3/0
967
543
547
552
556
561
565
570
574
579
583
7
3/5
968
588
592
597
601
605
610
614
619
623
628
8
4/0
969
632
637
641
646
650
655
659
664
668
673
9
4,5
970
677
682
686
691
695
700
704
709
713
717
971
722
726
731
735
740
744
749
758
762
972
767
771
776
780
784
789
793
798
802
807
973
811
816
820
825
829
834
838
843
847
851
974
856
860
865
869
874
878
883
887
892
896
975
900
905
909
914
918
923
927
932
936
941
976
945
949
954
958
963
967
972
976
981
985
977
989
994
998 *oo3
*oo7
*OI2
*oi6
*02I
*025 *029
978
99 034
038
043
047
052
056
061
065
069
074
979
078
083
087
092
096
100
105
109
114
118
980
123
127
131
136
140
145
149
154
158
162
981
167
171
176
180
185
189
193
198
202
207
4
982
211
216
220
224
229
233
238
242
247
251
983
•255
260
264
269
273
277
282
286
291
295
I
o;t
1/2
2
984
300
304
308
313
317
322
326
330
335
339
2
3
4
9^1
344
348
352
357
361
366
370
374
379
383
986
388
392
396
401
405
410
414
419
423
427
2^4
2/8
987
432
436
441
445
449
454
458
463
467
471
7
8
988
476
480
484
489
493
498
502
506
511
515
3'?
3/6
989
520
524
528
533
537
542
546
550
555
559
9
990
564
568
572
577
581
585
590
594
599
603
991
607
612
616
621
625
629
634
638
642
647
992
651
656
660
664
669
673
677
682
686
691
993
695
699
704
708
712
717
721
726
730
734
994
739
743
747
752
756
760
765
769
774
778
995
782
787
791
795
800
804
808
813
817
822
996
826
830
835
839
843
848
852
856
861
865
997
870
874
878
883
887
891
896
900
904
909
998
913
917
922
926
930
935
939
944
948
952
999
957
961
965
970
974
978
983
987
991
996
1000
00 000
004
009
013
017
022
026
030
035
039
N.
L.
1
2
3
4
5
6
7
8
9
P.P.
NOTES ON TABLES I AND II.
The logarithms of numbers are in general incommensurable.
In these tables they are given correct to five places of decimals.
If the sixth place is 5 or more, the next larger number is used
in the fifth place. Thus log 8102 = 3.908549+; in five-place
tables this is written 3.90855, the dash above the 5 showing
that the logarithm is less than given.
So log 8133 = 3.910251-; in five-place tables this is written
3.91025, the dot above the 5 showing that the logarithm is more
than given.
In the natural functions of the angles (Table II) all numbers
are decimals for sine and cosine (why ?), and for tangent and
cotangent, except where the decimal point is used to indicate
that part of the number is integral. When no decimal point
is printed in the tables it is to be understood. When the
natural function is a pure decimal the characteristic of the
logarithm is negative. Accordingly, in the tables 10 is added,
and in the result this must be allowed for. Thus
nat. sin 44° 20' = 0.69883, log sin 44° 20' = 1.84437,
or, as printed in the tables, 9.84437, which means 9.84437 — ?0.
TABLE n.
THE LOGARITHMIC AND NATURAL SINES, COSESTES,
TANGENTS, AND COTANGENTS OF ANGLES
FROM 0° TO 90°.
»5
Nat. Sin Log. d.
Nat.CoSLog. Nat.TanLog
Log.CotNat.
ooooo
029
058
087
116
646373
6.76476
6.94085
7-06579
0014s
175
204
233
262
7.10270
7.24188
7.30882
7.36682
7.41797
00291
320
349
378
407
746373
7-50512
7.54291
7-57767
7-60985
00436
465
495
524
553
7.63982
7.66784
7.69417
7.71900
7.74248
00582
611
640
669
698
7-76475
7.78594
7.8061$
7-82545
7-8439^
00727
756
785
814
844
00873
902
931
960
7.861O0
7.87870
7.89509
7.91088
7.92612
7.94084
7-95508
7.96887
7.98223
7.99520
01018
047
076
105
134
8.00779
8.02002
8.03192
8.04350
8.ot;4'78
01 164
193
222
251
280
8.0J57i
8.07650
8.0S696
8.09718
8.10717
01309
338
367
396
425
8.1 1693
8.12647
8.13581
8.14495
8-15391
01454
483
513
542
571
8.16268
8.17128
8.17971
8.18798
8.19610
01600
629
658
687
716
745
8.20407
8.21 189
8.21958
8.22713
8.23456
8.24186
30103
17609
12494
9691
7918
6694
5800
51 15
4576
4139
3779
3476
3218
2997
2802
2633
2483
2348
2227
2 1 19
2021
1930
1848
1773
1704
1639
1579
1524
1472
1424
1379
1336
1297
1259
1223
1 190
1158
1 128
IIOO
1072
1046
1022
999
976
954
934
914
896
877
860
843
827
812
797
782
769
755
743
730
loooo 0.00000
000 0.00000
000 0.00000
000 0.00000
000 0.00000
loooo 0.00000
000 0.00000
000 0.00000
000 0.00000
000 0.00000
loooo 0.00000
99999 0.00000
999 0.00000
999 0.00000
999 0.00000
99999 0.00000
999 0.00000
999 9-99999
999 9-99999
998 9-99999
99998 9.99999
998 9-99999
998 9-99999
998 9-99999
998 9-99999
99997
997
997
997
996
9.99999
9.99999
9.99999
9.99999
9-99998
99996
996
996
995
995
9.99998
9.99998
9.99998
9.99998
9-99998
99995
995
994
994
994
9.99998
9.99998
9.99997
9-99997
9.99997
99993
993
993
992
992
9-99997
9-99997
9.99997
9.99997
9-99996
99991
991
991
990
990
9-99996
9.99996
9-99996
9-99996
9-99996
99989
989
989
9-99995
9-9999$
9-99995
9-99995
9-99995
99987 9-99994
987 9-99994
986 9-99994
986 9.99994
985 9.99994
985 9.99993
029
058
087
116
6.46373
6.76476
6.94085
7-06579
0014s
175
204
233
262
7.16270
7.24188
7.30882
7.36682
7.41797
00291
320
349
378
407
7.46373
7.50512
7.54291
7-57767
7.60986
00436
46s
495
524
553
7-63982
7.66785
7.69418
7.71900
7.74248
00582
611
640
669
7.76476
7-7859$
7.80615
7-82546
7-84394
00727
756
785
815
844
7.86167
7.87871
7.89510
7.91089
7-92613
00873
902
931
960
7.94086
7-95510
7.96889
7-98225
7-99522
047
076
105
135
8.00781
8.02004
8.03194
8.04353
8.05481
01 164
193
222
251
280
8.06581
8.07653
8.08700
8.09722
8.10720
01309
338
367
396
425
8.11696
8.12651
8.13585
8.14500
8-15395
01455
484
513
542
571
8.16273
8-17133
8.17976
8.18804
8.19616
01600
629
658
687
716
746
8.20413
8.21 195
8.21964
8.22720
8.23462
8.24192
30103
17609
12494
9691
7918
6694
5800
5115
4576
4139
3779
3476
3219
2996
2803
2633
2482
2348
2228
21 19
2020
193 1
1848
1773
1704
1639
1579
1524
1473
1424
1379
1336
1297
1259
1223
1 190
1159
1128
IIOO
1072
1047
1022
998
976
955
934
915
895
878
860
843
828
812
797
782
769
756
742
730
3-53627
3.23524
3-05915
2.93421
3437-7
171B.9
1145-9
859.44
2.83730
2.75812
2.691 18
2.63318
2.58203
687.55
572.96
491.11
429.72
381.97
2.53627
2.49488
2.45709
2.42233
2.39014
2.36018
2.33215
2.30582
2.28100
2.25752
343-77
312.52
286.48
264.44
^5:55
229.18
214.86
202.22
190.98
180.93
2.23524
2.21405
2.19385
2.17454
2.15606
171.89
163.70
156.26
149.47
143.24
2.13833
2.12129
2.10490
2.0891 1
2.07387
137-51
132.22
127.32
122.77
118.54
2.05914
2.04490
2.03111
2.01775
2.00478
114-59
110,89
107.43
104.17
lOI.II
1.99219
1.97996
1.96806
1.95647
1.94519
1. 93419
1.92347
1. 91300
1.90278
1.89280
98.218
95.489
92.908
90.463
88.144
85.940
83.844
81.847
79-943
78.126
1.88304
1.87349
1.86415
1.85500
1.84605
76.390
74.729
73-139
71.615
70.153
1.83727
1.82867
1.82024
1.81196
1.80384
68.750
67.402
66.105
64.858
63-657
1.79587
1.78805
1.78036
1.77280
1-76538
1.75808
62.499
61.383
60.306
59.266
58.261
57-290
Nat. Cos Log. d.
Nat. Sin Log
Nat.CotLog
89'
d. Log.TanNat.
>i^
Nat. Sin Log. d.
Nat. Cos Log
Nat .Tan Log.
c.d.
Log. Cot Nat,
01745
774
803
832
862
8.24186
8.24903
8.25609
8.26304
8.26988
01891
920
949
978
02007
8.27661
8.28324
8.28977
8.29621
8.30255
02036
065
094
123
152
8.30879
8.31495
8.32103
8.32702
8.33292
02181
211
240
269
298
8-33875
8.34450
8.35018
8.35578
8.36131
02327
356
385
414
443
8.36678
8.37217
8.37750
8.38276
8.38796
25
26
27
28
_?9
30
31
32
33
_31
35
36
37
38
39
40
41
42
43
44_
45
46
47
48
49_
50
SI
52
53
54
02472
501
530
560
589
8.39310
8.39818
840320
840816
841307
02618
647
676
705
734
8.41792
8.42272
8.42746
843216
8.43680
02763
792
821
850
879
8.44139
8.44594
845044
845489
8.45930
02908
938
967
996
03025
846366
8.46799
8.47226
8.47650
8.48069
03054
083
112
141
170
8.48485
848896
849304
849708
8.50108
03199
228
257
286
316
8.50504
8.50897
8.51287
8.51673
8.52055
03345
374
403
432
461
490
8.52434
8.52810
8.53183
8.53552
8.53919
8.54282
717
706
695
684
673
663
653
644
634
624
616
608
599
590
583
575
568
560
553
547
539
533
526
520
514
508
502
496
491
485
480
474
470
464
459
455
450
445
441
436
433
427
424
419
416
411
408
404
400
396
393
390
386
382
379
376
373
369
367
363
99985
984
984
983
983
9.99993
9-99993
999993
9-99993
9-99992
01746
775
804
833
862
99982
982
981
980
980
9.99992
9.99992
9-99992
9-99992
9.99991
01891
920
949
978
02007
99979
979
978
977
977
9.99991
9.99991
9.99990
9-99990
9-99990
02036
066
095
124
153
99976
976
975
974
974
9-99990
9-99989
9-99989
9.99989
9-99989
02182
211
240
269
99973
972
972
971
970
9.99988
9.99988
9.99988
9.99987
9-99987
02328
357
386
415
444
8.24192
8.24910
8.25616
8.26312
8.26996
8.27669
8.28332
8.28986
8.29629
8.30263
8.30888
8.31505
8.321 12
8.327x1
8.33302
8.33886
8.34461
8.35029
8-35590
8.36143
8.36689
8.37229
8.37762
8.38289
8.38809
99969
969
968
967
966
9-99987
9.99986
9.99986
9.99986
9.99985
02473
502
531
560
589
99966
965
964
963
963
9.99985
9-99985
9-99984
9-99984
9.99984
02619
648
677
706
735
99962
961
960
959
959
9-99983
9-99983
9.99983
9-99982
9.99982
02764
793
822
851
881
8-39323
8.39832
840334
840830
841321
8.41807
8.42287
842762
8.43232
8.43696
844156
99958
957
956
955
954
9.99982
9.99981
9.99981
9.99981
9.99980
02910
939
968
997
03026
99953
952
952
951
950
9.99980
9.99979
9.99979
9.99979
9-99978
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084
114
143
172
8.4461 1
8.45061
8.45507
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9-99977
9-99977
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9-99976
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230
259
288
317
99944
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942
941
940
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9.99976
9-99975
9-99975
9-99974
9-99974
9-99974
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376
405
434
463
492
8.48917
8.49325
8.49729
8.50130
8.50527
8.50920
8.51310
8.51696
8.52079
8.52459
8.52835
8.53208
8.53578
8.53945
8.54308
718
706
696
684
673
663
654
643
634
625
617
607
599
591
584
575
568
561
553
546
540
533
527
520
514
509
502
496
491
486
480
475
470
464
460
455
450
446
441
437
432
428
424
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404
401
397
393
390
386
383
380
376
373
370
367
363
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57.290
56.351
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52.882
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50.549
49.816
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49.104
48.412
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45.829
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42.964
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41.916
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40.917
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40.436
39.965
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38.618
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57713
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38.188
37.769
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36.956
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55844
55389
54939
54493
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36.178
35.801
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34.715
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34.368
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33.694
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32.730
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31.821
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31.242
30.960
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412
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47541
47165
46792
46422
46055
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29.882
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28.877
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88°
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Nat. S
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Nat.Tan Log.
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Log. Cot Nat.
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8.54282
360
357
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9.99972
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8.55382
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27.937
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8.60698
314
311
310
307
305
303
301
299
297
295
292
291
289
287
285
284
281
280
1.39302
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8.60973
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9-99964
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8.61282
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9-99963
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8.61319
1.38681
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8.61589
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9.99963
133
8.61626
1-38374
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8.61894
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9.99962
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8.61931
1.38069
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8.62196
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301
298
296
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9.99962
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8.62234
1.37766
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9.99961
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8.62535
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9.99960
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8.63426
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8.63968
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8.67039
270
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267
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8.67841
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9-99951
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8.67890
264
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1.32110
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9.99950
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8.68154
1.31846
20.819
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8.68367
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9.99949
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249
248
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Nat. Cos Log. d.
Nat. S
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c.d.
Log. Tan Nat.
f
87°
3°
Nat. Sin Log. d.
Nat. Cos Log,
Nat.TanLog. c.d.
Log. Cot Nat,
05234
263
292
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35°
8.71880
8.72120
8-72359
8.72597
8.72834
05379
408
437
466
495
8.73069
8-73303
8-73535
8.73767
8.73997
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8.74226
8-74454
8.74680
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8-75353
8.75575
8.75795
8.76015
8.76234
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8.76451
8.76667
8.76883
8.77097
8.77310
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8.77522
8.77733
8.77943
8.78152
8.78360
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192
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8.78568
8.78774
8.78979
8.79183
8.79386
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279
308
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8.79588
8.79789
8.79990
8.80189
8.80388
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8.80585
8.80782
8.80978
8.81 173
8.81367
06540
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8.81560
8.81752
8.81944
8.82134
8.82324
06685
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8.82513
8.82701
8.82888
8.83075
8.83261
06831
860
889
918
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8.83446
8.83630
8.83813
8.83996
8.84177
8.84358
240
239
238
237
235
234
232
232
230
229
228
226
226
224
223
222
220
220
219
217
216
216
214
213
212
211
210
209
208
208
206
205
204
203
202
201
201
99
99
97
97
96
95
94
93
92
92
90
90
89
88
87
87
86
85
84
83
83
81
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861
860
858
857
9.99940
9-99940
9-99939
9.99938
9.99938
05341
270
299
328
357
241
239
239
99855
854
852
851
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9-99937
9.99936
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416
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9-99934
9-99933
9.99932
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9-99931
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562
591
620
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9-99930
9.99929
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9.99926
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9.99926
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9.99925
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9.99924
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9-99923
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9.99923
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9-99922
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9.99921
817
9.99920
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9-99920
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9.99919
812
9.99918
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9.99914
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8.81264
8.81459
8.81653
8.81846
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14.990
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14.606
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Nat. Cos Log. d. Nat. Sin Log. Nat.CotLog. c.d. Log. Tan Nat
86°
Nat. Sin Log. d. Nat. Cos Log.
Nat.Tan Log.
Log. Cot Nat.
06976
07005
034
063
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8.84358
8.84539
8.84718
8.84897
8.85075
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150
179
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8.85252
8.85429
8.85605
8.85780
8.85955
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295
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8.88161
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8.88654
8.88817
8.88980
8.89142
8.89304
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8.89464
8.89625
8.89784
8.89943
8.90102
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8.90260
8.90417
8.90574
8.90730
8.90885
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165
194
223
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8.91040
8.91 195
8.91349
8.91502
8.91655
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310
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368
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8.91807
8.91959
8.921 10
8.92261
8.92411
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8.92561
8.92710
8.92859
8.93007
8-93154
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8.93301
600
8.93448
629
8.93594
658
687
8.93885
716
8.94030
181
179
179
178
177
177
176
175
175
173
173
173
171
171
171
169
169
169
167
168
166
166
165
164
164
163
163
162
162
160
161
159
159
159
158
157
157
156
155
155
155
154
153
153
152
152
151
151
150
150
149
149
148
147
147
147
146
146
145
145
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752
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9.99894
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8.84464
8.84646
8.84826
8.85006
8.85185
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9.99890
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8.85363
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9.99889
8.85540
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9.99888
197
8.85717
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9.99887
227
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9.99886
256
8.86069
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9.99885
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8.86243
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9.99884
314
8.86417
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9.99883
344
8.86591
729
9.99882
373
8.86763
727
9.99881
402
8.86935
99725
9.99880
07431
8.87106
723
9.99879
461
8.87277
721
9.99879
490
8.87447
719
9.99878
519
8.87616
716
9.99877
548
8.87785
99714
9.99876
07578
8.87953
712
9.99875
607
8.88120
710
9.99874
636
8.88287
70B
9.99873
665
8.88453
705
9.99872
695
8.88618
99703
9.99871
07724
8.88783
701
9.99870
753
8.88948
699
9.99869
782
8.891 1 1
696
9.99868
812
8.89274
694
9.99867
841
8.89437
99692
9.99866
07870
8.89598
689
9.99865
899
8.89760
687
9.99864
929
8.89920
685
9.99863
958
8.90080
683
9.99862
987
8.90240
99680
9.99861
08017
8.90399
678
9.99860
046
8.90557
676
9.99859
075
8.90715
673
9.99858
104
8.90872
671
9.99857
134
8.91029
99668
9.99856
08163
8.91 185
666
999855
192
8.91340
664
9.99854
221
8.91495
661
9.99853
251
8.91650
659
9.99852
280
8.91803
99657
9.99851
08309
8.91957
654
9.99850
339
8.92110
652
9.99848
368
8.92262
649
9.99847
397
8.92414
647
9.99846
427
8.92565
99644
9.99845
08456
8.92716
642
9.99844
485
8.92866
639
9.99843
514
8.93016
637
9.99842
544
8.93165
635
9.99841
573
8.93313
99632
630
627
625
622
619
9.99840
9.99839
9.99838
9.99837
9.99836
9.99834
08602
632
661
690
720
749
8.93462
8.93609
8.93756
8.93903
8.94049
8.94195
82
80
80
79
78
n
77
76
76
74
74
74
72
72
71
71
70
69
69
68
67
67
66
65
65
65
63
63
63
61
62
60
60
60
59
58
58
57
57
56
55
55
55
53
54
53
152
52
51
51
50
50
49
48
49
47
47
47
46
46
1.15536
1.15354
1.15174
1.14994
1.14815
14.301
.241
.182
.124
.065
1.14637
1.14460
1.14283
1.14107
1.13931
14.008
13.951
.894
.838
.782
1.13757
1.13583
1.13409
1.13237
1.13065
13.727
.672
.617
.563
.510
1.12894
1.12723
1.12553
1.12384
1.12215
13.457
.404
.352
.300
.248
1.12047
1.11880
1.11713
1.11547
1.11382
13.197
.146
.096
.046
12.996
1.11217
1.11052
1.10889
1.10726
1.10563
12.947
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.850
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.754
1.10402
1.10240
1.10080
1.09920
1.09760
12.706
.659
.612
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1.09601
1.09443
1.09285
1.09128
1.08971
12.474
.429
.384
.339
.295
1.08815
1.08660
1.08505
1.08350
1.08197
12.251
.207
.163
.120
.077
1.08043
1.07890
1.07738
1.07586
1.07435
12.035
11.992
.950
.909
1.07284
1.07134
1.06984
1.06835
1.06687
11.826
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.745
.705
.664
1.06538
1.06391
1.06244
1.06097
1.05951
1.05805
11.625
.585
.546
•507
.468
.430
Nat. Cos Log. d.
Nat. Sin Log. Nat. Cot Log
86^
c.d. Log. Tan Nat.
' Nat. Sin Log. d. Nat. Cos Log.
Nat.Tan Log.
c.d.
Log. Cot Nat.
08716
745
774
803
831
8.94030
8.94174
8.94317
8.94461
8.94603
08860
889
918
947
976
8.94746
8.94887
8.95029
8.95170
8.95310
09005
034
063
092
121
8.95450
8.95589
8.95728
8.95867
8.96005
09150
179
208
237
8.96143
8.96280
8.96417
8.96553
8.96689
09295
324
353
382
411
8.96825
8.96960
8.97095
8.97229
8.97363
09440
469
498
527
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09585
614
642
671
700
8.97496
8.97629
8.97762
8.97894
8.98026
8.98157
8.98288
8.98419
8.98549
8.98679
09729
758
707
816
845
09874
903
932
961
990
10019
048
077
106
135
9.00082
9.00207
9.00332
9.00456
9.00581
10164
192
221
250
279
9.00704
9.00828
9.00951
9.01074
9.01196
10308
337
366
395
424
453
9.01318
9.01440
9.01561
9.01682
9.01803
9.01923
8.98808
8.98937
8.99066
8.99194
8.99322 I
8.99450
8.99577
8.99704
8.99830
8.99956
99619
617
614
612
609
999834
999833
9.99832
9.99831
9.99830
08749
778
807
837
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8.94195
8.94340
8.94485
8.94630
8.94773
99607
604
602
599
596
9.99829
9.99828
9.99827
9.99825
9.99824
08895
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983
09013
8.94917
8.95060
8.95202
8.95344
8.95486
99594
591
588
586
583
9.99823
9.99822
9.99821
9.99820
9.99819
09042
071
lOI
130
159
8.95627
8.95767
8.95908
8.96047
8.96187
99580
578
575
572
570
9.99817
9.99816
9.99815
9.99814
9.99813
09189
218
247
277
306
8.96325
8.96464
8.96602
8.96739
8.96877
99567
564
562
559
556
9.99812
9.99810
9.99809
9.99808
9.99807
09335
365
394
423
453
8.97013
8.9715?
8.97285
8.97421
8.97556
99553
551
548
545
542
9.99806
9.99804
9.99803
9.99802
9.99801
09482
5"
541
570
600
8.97691
8.97825
8.97959
8.98092
8.98225
99540
537
534
531
528
9.99800
9.99798
9.99797
9.99796
9.99795
09629
658
688
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746
8.98358
8.98490
8.98622
8.98753
8.98884
99526
523
520
517
514
9.99793
9.99792
9.99791
9.99790
9.99788
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834
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8.9901$
8.9914$
8.99275
8.99405
8.99534
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506
503
500
9.99787
9.99786
9.99785
9.99783
9.99782
09923
952
981
lOOII
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8.99662
8.99791
8.99919
9.00046
9.00174
99497
494
491
488
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9.99781
9.99780
9.99778
9-99777
9.99776
10069
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128
158
187
9.00301
9.00427
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9.00679
9.00805
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479
476
473
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9.99775
9.99773
9.99772
9.99771
9.99769
102 16
246
275
305
334
9.00930
9.01055
9.01179
9.01303
9.01427
99467
464
461
458
455
452
9.99768
9.99767
9.99765
9.99764
999763
9.99761
10363
393
422
452
481
510
9.01550
9.01673
9.01796
9.01918
9.02040
9.02162
05805
05660
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05370
05227
C.430
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05083
04940
04798
04656
04514
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04373
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03953
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11.059
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10.988
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10.883
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10.712
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10.546
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10.385
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10.229
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0.99954
0.99826
10.078
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9.9893
601
0.99699
0.99573
0.99447
0.99321
0.99195
9.9310
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9.8734
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164
0.99070
0.98945
0.98821
0.98697
0.98573
9,7882
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322
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9.6768
0.98450
0.98327
0.98204
0.98082
0.97960
0.97838
9.6493
220
9-5949
679
411
144
Nat. Cos Log. d. Nat. Sin Log. Nat. Cot Log. c.d. Log. Tan Nat. '
84°
6'
f
Nat. Sin Log. d.
|Nat.CoSLog
|Nat.Tan Log.
[:i
Log. Cot Nat.
^^
10453
9.01923
120
99452 9-99761
I05IO 9.02162
0.97838
9.5144
60
I
482
9.02043
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449 9-99760
540 9.02283
0.97717
9.4878
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2
511
9.02163
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540
9.02283
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0.97475
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56
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9.02520
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0.97234
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626
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424 9-99749
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0.96758
9.2806
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10742
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10805 9-03361
0.96639
9.2553
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771
9.03226
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834 9-03479
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0.96521
302
49
12
800
9-03342
415 9-99745
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9.03690
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lb
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0.95935
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9.03920
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0.95819
9.0821
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0.95703
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42
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9.04149
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0.95587
338
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9.04262
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0.95357
8.9860
39
22
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383 999731
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0.95242
623
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23
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9.04603
380 9.99730
187 9-04873
115
0.95127
387
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147
9.04715
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217 9.04987
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114
0.95013
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36
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9.04828
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8.8919
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205
9.04940
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370 9-99726
276 9.05214
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0.94786
686
34
27
234
9.05052
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367 9-99724
305 9-05328
0.94672
455
33
28
2b3
9.05164
III
364 9.99723
335 9-05441
113
0-94559
225
32
29
30
291
9-05275
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360 9-99721
364 9-05553
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0-94447
8.7996
31
30
1 1320
9-05386
99357 9-99720
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0-94334
8.7769
31
349
9.05497
354 999718
423 9-05778
0.94222
542
29
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109
109
109
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344 9-99714
511 9.061 13
III
0.93887
8.6870
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25
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11465
9-05937
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0.93776
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8.5989
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39
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580
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327 9-99707
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1 1609
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109
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9.00004
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107
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314 9.99701
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783
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0.92572
280
14
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812
9.07231
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300 9.99695
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0.92464
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13
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840
9-07337
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106
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297 999693
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0.92357
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50
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9.07548
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0.92142
8.3450
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51
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12013 9.07964
0.92036
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9
52
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8.2434
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9.08176
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8.1837
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258 9.99677
249 9.08810
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0.91190
640
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187
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Nat. Cos Log. d.
Nat. Sin Log.
Nat. Cot Log.
C.d.
Log. Tan Nat.
f
83^
f
Nat. S
in Log.
d.
Nat. Cos Log.
Nat.Tan Log.
c.d.
Log. Cot Nat.
"
12187
9.08589
103
103
102
99255
999675
12278
9.08914
105
0.91086
8.1443
60
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216
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504
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215
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9.10049
0.89951
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0.89647
789
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12620
9.10106
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9.99651
12722
9.10454
0.89546
7.8606
45
lb
649
9.10205
197
9.99650
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0.89445
424
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9.99648
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9.10656
0.89344
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9.10402
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9.10756
0.89244
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9.10501
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9.10856
100
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0.89144
7.7882
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12764
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9.99643
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9.99638
958
9.11254
0.88746
171
37
24
880
9.10990
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999637
988
9.11353
0.88647
7.6996
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25
12908
9. 1 1087
99163
9-99635
13017
9.11452
0.88548
7.6821
35
2b
937
9.1 1 184
160
9-99633
047
9.11551
0.88449
647
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27
966
9.11281
97
96
97
96
96
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95
95
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9.99632
076
9.11649
0.88351
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9-99630
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9.11747
9.11845
0.88253
301
32
29
13024
9.11474
148
9.99629
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0.88155
129
31
30
13053
9.1 1570
99144
9-99627
13165
9.11943
0.88057
7.5958
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31
081
9.11666
141
9.99625
195
9.12040
0.87960
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29
32
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9.11761
137
9.99624
224
9.12138
0.87862
618
28
33
139
9.11857
133
9.99622
254
9.12235
0.87765
449
27
34
168
9.11952
129
9.99620
284
9.12332
96
96
96
95
95
95
95
95
94
95
94
94
93
94
93
93
93
93
92
92
93
91
92
0.87668
281
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9.12047
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9.99618
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9.12428
0.87572
7.5113
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226
9.12142
122
9.99617
343
0.87475
7.4947
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37
254
9.12236
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94
94
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118
9-99615
372
9.12621
0.87379
781
23
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283
9.12331
114
9.99613
402
9.12717
0.87283
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22
39
312
9.12425
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9.99612
432
9.12813
0.87187
451
21
20
40
13341
9.12519
99106
9.99610
13461
9.12909
0.87091
7.4287
41
370
9.12612
102
9.99608
491
9.13004
0.86996
124
19
42
399
9.12706
098
9.99607
521
9.13099
0.86901
7.3962
18
43
427
9.12799
93
93
93
93
094
9.99605
550
9.13194
0.86806
800
17
44
456
9.12892
091
9.99603
580
9.13289
0.86711
639
lb
45
13485
912985
99087
9.99601
13609
9.13384
0.86616
7.3479
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46
514
9.13078
083
9.99600
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9.13478
0.86522
319
14
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9.13171
93
92
92
92
92
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9.99598
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9.13573
0.86427
160
13
48
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9.13263
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9.99596
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9.13667
0.86333
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9-13355
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9-99595
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9.13761
0.86239
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50
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9.13447
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9.99593
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9.14041
0.85959
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9.99588
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9.14134
0.85866
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9.99586
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9.14227
0.85773
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9-99584
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9.14320
0.85680
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9-13994
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9.99582
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9.14412
0.85588
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831
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9.99581
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9.14504
0.85496
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9-99579
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0.85403
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9.14266
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999577
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9.14688
0.85312
304
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9-14356
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9-99575
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0.85220
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Nat. Cos Log.
d.
Nat. Sin Log.
Nat. Cot Log.
c.d. 'Log. Tan Nat.
r
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r
Nat. Sin Log. d.
Nat. Cos Log
Nat.Tan Log.
c.d.
Log. Cot Nat.
13917 9-I4356
89
90
89
90
89
88
99027 9.99575
14054
9.14780
0.85220 7.1154
60
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946 9-14445
023 9-99574
084
91
91
91
91
91
0.85128 004
59
2
975 9.14535
019 9-99572
113
9.14963
0.85037 7.0855
58
3
14004 9.14624
015 9-99570
143
9.15054
57
4
5
033 9.14714
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173
9.15145
0.84855 558
56
55
1406 I 9.14803
99006 9.99566
14202
9.15236
0.84764 7.0410
b
090 9.14891
89
89
88
002 9.99565
232
9.15327
0.84673 264
54
7
119 9.14980
98998 9.99563
262
9.15417
90
0.84583 117
53
8
148 9.15069
994 9-99561
291
9.15508
91
90
90
89
90
89
89
88
0.84492 6.9972
52
9
177 9-15157
88
88
990 9.99559
321
9.15598
0.84402 827
51
10
14205 9-15245
98986 9.99557
14351
9.15688
0.84312 6.9682
50
II
234 9-15333
88
982 9.99556
381
9.15777
0.84223 538
49
12
263 9.15421
87
88
978 9-99554
410
9.15867
0.84133 395
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292 9-15508
973 9-99552
440
9.15956
0.84044 252
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14
320 9.15596
87
87
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87
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969 9.99550
470
9.16046
0.83954 110
46
15
14349 9-15683
98965 9.99548
14499
9.16135
0.83865 6.8969
45
lb
378 9-15770
961 9.99546
529
9.16224
0.83776 828
44
17
407 9-15857
957 9-99545
559
9.16312
89
00
0.83688 687
43
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436 9-15944
953 9-99543
588
9.16401
0.83599 548
42
19
464 9.16030
86
87
86
948 9-99541
618
9.16489
88
88
0.83511 408
41
20
14493 9.16116
98944 9.99539
14648
9.16577
0.83423 6.8269
40
21
522 9.16203
940 9-99537
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0.83335 131
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22
551 9-16289
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936 9-99535
707
9.16753
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0.83247 6.7994
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580 9.16374
931 9.99533
737
9.16841
87
88
87
87
87
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0.83159 856
37
24
608 9.16460
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927 9-99532
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9.16928
0.83072 720
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14637 9.16545
98923 9-99530
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9.17016
0.82984 6,7584
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666 9.16631
85
85
84
85
84
84
84
84
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84
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83
83
83
83
82
919 9-99528
826
9.17103
0.82897 448
34
27
695 9.16716
914 9-99526
856
9.17190
0.82810 313
33
28
723 9.16801
910 9.99524
886
9.17277
0.82723 179
32
29
752 9.16886
906 9.99522
915
9.17363
87
86
0.82637 045
31
1478 I 9.16970
98902 9.99520
1494s
9.17450
0.82550 6.6912
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31
810 9.17055
897 9.99518
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9-17536
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0.82464 779
29
32
838 9-17139
893 9.99517
15005
9.17622
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0.82378 646
28
33
867 9.17223
889 9.99515
034
9.17708
86
0.82292 514
27
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896 9.17307
884 9.99513
064
9.17794
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0.82206 383
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25
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14925 9.17391
98880 9.99511
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876 9.99509
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85
85
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9.18136
0.81864 863
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863 9.99503
213
9.18221
0-81779 734
21
20
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15069 9.17807
98858 9.99501
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9.18306
0.81694 6.5606
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097 9.17890
854 9.99499
272
9.18391
0.81609 478
19
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126 9.17973
849 9.99497
302
0.81525 350
18
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155 9-18055
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845 9.99495
332
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0.81440 223
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184 9.18137
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82
841 9.99494
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9.18644
0.81356 097
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15212 9.18220
98836 9.99492
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9.18728
0.81272 6.4971
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46
241 9.18302
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832 9.99490
421
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0.81188 846
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270 9.18383
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827 9.99488
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299 9.18465
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0.80771 225
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805 9.99478
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0.80688 103
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800 9.99476
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0.80605 6.3980
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80
796 9.99474
660
9.19478
0.80522 859
6
15500 9.19033
98791 9-99472
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9.19561
0.80439 6.3737
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529 9.19113
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787 9-99470
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9.19643
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0.80357 617
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9-19725
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586 9-19273
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778 9.99466
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9.19807
0.80193 376
2
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615 9-19353
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773 9.99464
809
9.19889
0.80111 257
1
643 9-19433
769 9.99463
838
9.19971
0.80029 138
Nat. Cos Log. d.
Nat. Sin Log.
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C.d. Log. Tan Nat.
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81°
r
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Nat. Cos Log.
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15643 9.19433
80
98769
9.99462
15838
9.19971
82
0.80029 6.3138
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9.99460
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9-20053
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0-79947 019
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701 9-19592
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9-99458
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9-20134
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0.79866 6.2901
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730 9.19672
79
79
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9.20216
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0.79784 783
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15787 9.19830
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9-99454
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9.20297
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0.79703 666
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98746
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816 9.19909
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9-99450
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9.20459
0.79541 432
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845 9.19988
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9.20540
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0.79460 316
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873 9.20067
78
78
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78
78
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9.99446
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9.20621
0-79379 200
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9.20701
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0.79299 085
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0.79218 6.1970
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9.99440
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0.79138 856
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16017 9.20458
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9-99436
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9-99434
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9.21102
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0.78898 515
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16074 9-20613
98700
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0.78818 6.1402
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16
103 9.20691
695
9.99429
316
9.21261
79
80
0.78739 290
44
17
132 9.20768
690
9.99427
346
9-21341
0.78659 178
0.78580 066
43
lb
160 9.20845
77
77
77
686
999425
376
9.21420
79
79
79
42
19
20
189 9.20922
681
999423
405
9.21499
0.78501 6.0955
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40
16218 9.20999
98676
9.99421
16435
9.21578
0.78422 6.0844
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246 9.21076
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9.99419
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9.21657
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0.78343 734
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0.78264 624
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9.21893
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9.21971
0.78029 6.0296
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390 9.21458
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9.99409
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9.22049
0.77951 188
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419 921534
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9-99407
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9.22127
0.77873 080
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447 9.21610
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9.99404
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0.77795 5.9972
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476 9.21685
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9.99402
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30
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16505 9.21761
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0.77639 5.9758
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533 9.21836
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624
9.99398
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9.22438.
77
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0.77562 651
29
32
562 9.21912
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9.99396
794
9.22516
0.77484 545
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591 9.21987
614
9-99394
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9-22593
0.77407 439
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620 9.22062
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9.22670
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0.77330 333
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16648 9.22137
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677 9.22211
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9.22824
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16792 9.22509
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0.76717 502
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878 9.22731
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565
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9-23359
0.76641 400
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16935 9-22878
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73
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0.76565 298
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0.76490 5.8197
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17078 9.23244
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250 9.23679
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Nat. Cos Log. d.
Nat. Sin Log.
Nat. Cot Log.
c.d.! Log. Tan Nat.
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Nat. Sin Log. d.
Nat. Cos Log. d.
Nat.TanLog.
c.d
Log. Cot Nat.
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17365 9-23967
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71
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72
71
71
71
70
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70
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663 9.24706
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0.75294 617
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422 9.241 10
471 9-99331
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0.75221 521
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0.75147 425
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479 9-24253
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0.75074 329
0.75000 5.6234
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98455 9.99324
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537 9.24395
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0.74927 140
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565 9.24466
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594 9-24536
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0.74781 5.5951
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623 9.24607
435 9-99315
98430 9.99313
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903 9.25292
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0.74708 857
0.74635 5.5764
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17651 9.24677
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680 9.24748
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0.74563 671
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420 9-99308
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0.74490 578
48'
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737 9-24888
414 999306
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18023 9-25582
0.74418 485
47
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766 9.24958
409 9.99304
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2
053 9-25655
73
72
72
0.74345 393
46
15
17794 9-25028
98404 9.99301
18083 9-25727
0.74273 5.5301
45
lb
823 9.25098
399 9.99299
"3 9-25799
0.74201 209
44
17
852 9.25168
394 9.99297
3
143 9-25871
72
0.74129 118
43
l8
880 9.25237
389 9.99294
173 9.25943
72
0.74057 026
42
19
909 9-25307
383 9.99292
2
203 9.26015
72
71
0.73985 5-4936
41
20
17937 9-25376
98378 9.99290
18233 9.26086
0.73914 5-4845
40
21
966 9.25445
373 9.99288
3
263 9.26158
7^
0.73842 755
39
22
995 9.25514
368 9.99285
293 9.26229
0.73771 665
38
23
18023 9.25583
362 9.99283
"^
323 9.26301
0.73699 575
37
24
25
052 9.25652
357 9.99281
3
353 9.26372
71
0.73628 486
36
1808 I 9.25721
98352 9.99278
18384 9.26443
0.73557 5-4397
35
2b
109 9.25790
347 9-99276
2
414 9.26514
71
0.73486 308
34
27
138 9-25858
69
68
341 9.99274
3
2
444 9.26585
0.73415 219
33
28
166 9.25927
336 9.99271
474 9.26655
70
71
71
0.73345 131
32
29
195 9-25995
68
68
331 9.99269
2
3
504 9.26726
0.73274 043
31
30
18224 9.26063
98325 9.99267
18534 9-26797
0.73203 5-3955
30
31
252 9.26131
68
320 9.99264
564 9.26867
0.73133 868
29
32
281 9.26199
68
315 9.99262
"
594 9.26937
71
0.73063 781
28
33
309 9.26267
68
310 9.99260
3
2
3
2
624 9.27008
0.72992 694
27
34
338 9.26335
68
67
68
304 9-99257
654 9.27078
70
0.72922 607
26
35
18367 9.26403
98299 9.99255
18684 9.27148
0.72852 5.3521
25
3^
395 9-26470
294 9.99252
714 9.27218
70
69
0.72782 435
24
37
424 9.26538
67
67
67
67
67
67
67
66
67
66
288 9.99250
745 9.27288
0.72712 349
23
3»
452 9.26605
283 9.99248
3
2
775 9-27357
0.72643 263
22
39
40
481 9.26672
277 9.99245
98272 9.99243
805 9.27427
70
69
69
69
69
69
69
68
0-72573 178
21
20
18509 9-26739
18835 9.27496
0.72504 5.3093
41
538 9.26806
267 9.99241
865 9.27566
0.72434 008
19
42
567 9.26873
261 9.99238
3
895 9.27635
0-72365 5-2924
18
43
595 9-26940
256 9.99236
3
2
925 9-27704
0.7220 839
17
44
45"
624 9.27007
250 9-99233
955 9.27773
0.72227 755
lb
18652 9.27073
98245 9.99231
18986 9.27842
0.72158 5.2672
15
4b
681 9.27140
240 9.99229
3
19016 9.2791 1
0.72089 588
14
47
710 9.27206
67
66
234 9.99226
046 9.27980
0.72020 505
13
48
738 9.27273
229 9.99224
3
2
076 9.28049
0.71951 422
12
49
7(y7 9-27339
66
66
223 999221
106 9.28117
69
68
0.71883 339
n
50
18795 9.2740g
98218 9.99219
19 136 9.28186
0.71814 5.2257
10
51
824 9.27471
66
212 9.99217
3
166 9.28254
0.71746 174
9
52
852 9.27537
%
207 9.99214
197 9.28323
0.71677 092
8
53
881 9.27602
20I 9.99212
3
2
3
227 9.28391
68
0.71609 on
7
54
910 9.27668
66
65
65
66
196 9.99209
257 9.28459
68
6H
0.71541 5.1929
6
55
18938 9-27734
98190 9.99207
19287 9.28527
9.71473 5.1848
5
5^^
967 9.27799
185 9.99204
317 9-28595
67
68
0.71405 767
4
57
995 9-27864
179 9.99202
347 9.28662
0.71338 686
3
5«
19024 9.27930
65
65
174 9.99200
3
378 9.28730
68
0.71270 606
2
il
052 9.27995
168 9-99197
408 9.28798
67
0.71202 526
I
081 9.28060
163 9-99195
438 9.28865
0.7113S 446
Nat. Cos Log. d. 1
Nat. Sin Log. d. 1
Nat. Cot Log.
c.d.
Log.TanNat.
/
79'
ir
r
Nat. Sin Log. d.
Nat. Cos Log
d.
Nat.TanLog.
c.d.
Log. Cot Nat.
1908 1 9.28060
1
64
65
65
64
64
65
64
64
64
64
63
64
64
63
63
64
63
63
63
98163 9.99195
3
2
19438 9.28865
68
0.71 135 5.1446
60
I
109 9.28125
157 9-99192
468 9.28933
67
67
67
67
67
6(S
0.71067 366
5Q
2
138 9.28190
152 9.99190
3
498 9.29000
0.71000 286
58
3
167 9.28254
146 9-99187
529 9.29067
0.70933 207
57
4
195 9.28319
140 9-99185
3
559 9-29134
0.70866 128
56
5
19224 9.28384
98135 9-99182
19589 9.29201
0.70799 5.1049
55
6
252 9.28448
129 9.99180
3
619 9.29268
0.70732 5.0970
54
7
281 9.28512
124 9.99177
649 9.29335
0.70665 892
53
8
309 9.28577
118 9-99175
3
2
3
680 9.29402
0.70598 814
S2
9
338 9.28641
112 9-99172
710 9.29468
67
66
0.70532 736
51
50
10
19366 9.28705
98107 9-99170
19740 9.29535
0.70465 5.0658
II
395 9.28769
loi 9.99167
770 9.29601
67
66
0.70399 581
4Q
12
423 9.28833
096 9.99165
3
801 9.29668
0.70332 504
48
13
452 9.28896
090 9.99162
831 9-29734
66
0.70266 427
47
14
15
481 9.28960
084 9.99160
98079 9.99157
3
861 9.29800
66
66
66
0.70200 350
46
45
19509 9.29024
19891 9.29866
0.70134 5.0273
I6
538 9.29087
^3 9-99155
3
921 9.29932
0.70068 197
44
17
566 9.29150
067 9.99152
952 9.29998
66
0.70002 121
43
l8
595 9-29214
061 9.99150
3
2
3
982 9.30064
66
0.69936 045
42
19
623 9.29277
056 9-99147
20012 9.30130
65
66
0.69870 4.9969
41
20
19652 9.29340
98050 9.99145
20042 9.30195
0.69805 4.9894
40
21
680 9.29403
044 9.99142
073 9.30261
^5
0.69739 819
39
22
709 9.29466
039 9.99140
3
103 930326
0.69674 744
38
23
737 9-29529
033 9-99137
133 9-30391
0.69609 669
37
24
25
766 9-29591
63
60
027 9-99135
3
164 9-30457
65
65
64
%
%
64
i^
64
64
64
64
i^
64
64
63
64
63
63
63
63
^3
63
63
62
0.69543 594
3b
19794 9-29654
98021 9.99132
20194 9-30522
0.69478 4.9520
35
2b
823 9.29716
53
016 9.99130
3
3
224 9.30587
0.69413 446
34
27
851 9.29779
010 9.99127
254 9-30652
0.69348 372
33
28
880 9.29841
6^
004 9.99124
285 9.30717
0.69283 298
32
29
30
908 9-29903
63
60
97998 9.99122
3
315 9.30782
0.69218 225
31
19937 9-2996<5
97992 9-99119
20345 9.30846
0.69154 4.9152
30
31
965 9.30028
6'>
987 9.99117
3
2
376 9.30911
0.69089 078
29
32
994 9-30090
61
981 9.99114
406 9.30975
0.69025 006
28
33
20022 9.30151
6'^
975 9-99112
3
3
436 9.31040
0.68960 4.8933
27
34
051 9.30213
62
61
969 9.99109
466 9.31104
0.6880 860
26
25
35
20079 9-30275
97963 9.99106
20497 9.31168
0.68832 4.8788
3&
108 9.30336
958 999104
3
527 9.31233
0.68767 716
24
37
136 9-30398
61
952 9-99IOI
557 9.31297
0.68703 644
23
3«
165 9-30459
Ao
946 9.99099
3
3
588 9.31361
0.68639 573
22
39
193 9-30521
6i
61
61
61
61
61
60
61
60
61
60
61
60
60
940 999096
618 9-31425
0.68575 501
21
40
20222 9.30582
97934 9-99093
20648 9.31489
0.68511 4.8430
20
41
250 9.30643
928 9-99091
679 9-31552
0.68448 359
19
42
279 9-30704
922 9.99088
3
709 9.31616
0.68384 288
18
43
307 9-30765
916 9.99086
3
3
739 9-31679
0.68321 218
17
44
336 9.30826
910 9.99083
770 9-31743
0.68257 147
lb
45
20364 9.30887
97905 9.99080
20800 9.31806
0.68194 4.8077
15
46
393 9-30947
899 9.99078
830 9.31870
0.68130 007
14
47
421 9.31008
893 9-99075
3
861 9.31933
0.68067 4.7937
13
48
450 9.31068
887 9.99072
3
891 931996
0.68004 867
12
49
478 9.31129
881 9.99070
3
921 9.32059
0.67941 798
11
50
20507 9.31 189
97875 9.99067
20952 9.32122
0.67878 4.7729
10
SI
535 9-31250
869 9.99064
3
982 9.32185
0.67815 659
9
52
563 9.31310
863 9.99062
3
3
2
21013 9.32248
0.67752 591
8
53
592 9.31370
60
60
857 9-99059
043 9-32311
0.67689 522
7
54
55
620 9-31430
851 9-99056
073 9-32373
63
62
0.67627 453
b
20649 9-31490
97845 9.99054
21104 9.32436
0.67564 4.7385
5
56
(>77 9-31549
60
839 9-99051
3
134 9.32498
t?
0.67502 317
4
57
706 9.31609
833 9-99048
3
164 9.32561
0.67439 249
3
58
734 9-31669
827 9.99046
195 9.32623
62
0.67377 181
2
hi
763 9.31728
6^
821 9.99043
3
225 9.32685
62
0.67315 114
I
791 9.31788
815 9.99040
3
256 9.32747
0.67253 046
Nat. Cos Log. d.
Nat. Sin Log.
d.
Nat. Cot Log.
c.d.
Log.TanNat.
/
78'
12°
t
Nat. Sin Log. d.
Nat. Cos Log. d.
Nat.TanLog.
c.d.
Log. Cot Nat.
20791 9.31788
59
6n
97815 9.99040
21256 9.32747
%
0.67253 4.7046
60
I
820 9.31847
809 9-99038
3
3
2
286 9.32810
0.67190 4.6979
59
2
848 9.31907
59
59
59
59
59
59
58
59
59
58
58
59
58
58
58
58
57
58
57
58
57
57
58
57
57
%
57
803 9-99035
316 9.32872
61
0.67128 912
S8
3
877 9.31966
797 9-99032
347 9-32933
6^^
0.67067 84s
57
4
90s 9.32025
791 9.99030
3
3
2
377 9-32995
62
50
0.67005 779
56
55
5
20933 9-32084
97784 9.99027
21408 9.33057
0.66943 4.6712
6
962 9.32143
778 9-99024
438 9-33119
61
0.66881 646
S4
7
990 9.32202
772 9.99022
3
3
3
2
469 9-33180
50
0.66820 580
53
8
21019 9.32261
766 9.99019
499 9-33242
61
0.66758 514
52
9
10
047 9-32319
760 9.99016
529 9-33303
21560 9-33365
62
61
0.66697 448
51
21076 9.32378
97754 9-99013
0.66635 4-6382
50
II
104 9.32437
748 9-9901 1
3
3
3
2
3
3
3
2
590 9-33426
61
0.66574 317
49
12
132 9.32495
742 9-99008
621 9.33487
61
0.66513 252
48
13
161 9.32553
735 999005
651 9-33548
61
0.66452 187
47
14
189 9.32612
729 9.99002
682 9.33609
61
61
0.66391 122
46
15
21218 9.32670
97723 9-99000
2 17 1 2 9.33670
0.66330 4.6057
45
lb
246 9.32728
717 9-98997
743 9-33731
61
0.66269 4.5993
44
17
27s 9-32786
711 9-98994
773 933792
0.66208 928
43
i8
303 9.32844
705 9.98991
804 9-33853
60
0.66147 • 864
42
19
331 9.32902
698 9.98989
3
3
3
2
834 9-33913
61
fin
0.66087 800
41
20
21360 9.32960
97692 9.98986
21864 9-33974
0.66026 45736
40
21
388 9.33018
686 9.98983
895 9-34034
61
0.65966 673
39
22
417 9-33075
680 9.98980
925 9-3409$
f)n
,0.6590$ 609
38
23
445 9-33133
673 9-98978
3
3
3
2
956 9-34155
60
0.65845 546
37
24
25-
474 9-33190
667 9-98975
986 9.34215
61
60
0-65785 483
36
21502 9.33248
97661 9.98972
22017 9-34276
0.65724 4.5420
35
26
530 9-33305
655 9.98969
047 9-34336
fin
0.65664 357
34
27
559 9-33362
648 9.98967
3
3
3
3
078 9-34396
°° 0.65604 294
33
28
587 9-33420
642 9.98964
108 9-34456
60 \ 0-65544 232
32
29
6i6 9-33477
636 9.98961
139 9-34516
60
0.65484 169
31
21644 9-33534
97630 9.98958
22169 9-34576
0.65424 4.5107
30
31
672 9-33591
623 9-98955
200 9.34635
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29
32
701 9.33647
3
3
3
3
3
231 9-34695
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28
33
729 9-33704
611 9.98950
261 9-34755 ''. :X 0.65245 922
27
34
758 9-33761
57
57
56
57
56
56
57
^i
^l
^l
56
56
56
55
56
55
56
55
56
55
55
55
55
55
55
55
55
604 9-98947
292 9.34814
2^ 1 0.65186 860
26
25
35
21786 9.33818
97598 9-98944
22322 9.34874
59
59
59
6n
0.65126 4-4799
36
814 9-33874
592 9-98941
353 9-34933
0.65067 737
24
37
843 9-33931
585 9-98938
383 9-34992
0.65008 676
23
3B
871 9-33987
579 9-98936
3
3
3
3
3
2
414 9-35051
0.64949 615
22
39
899 9.34043
573 9-98933
444 9-351"
59
59
59
59
58
59
59
58
59
58
59
58
58
58
58
58
58
57
0.64889 555
21
40
21928 9.34100
97566 9.98930
22475 9-35170
0.64830 4.4494
20
41
956 9.34156
560 9-98927
505 9.35229
0.64771 434
19
42
985 9.34212
553 9-98924
536 9-35288
0.64712 373
18
43
22013 9-34268
547 9-98921
567 9-35347
0.64653 313
17
44
041 9-34324
541 9-98919
3
3
3
3
3
3
3
597 9-35405
0.64595 253
16
15
45
22070 9.34380
97534 9-98916
22628 9.35464
0.64536 4-4194
46
098 9-34436
528 9.98913
658 9-35523
0.64477 134
14
47
126 9.34491
521 9.98910
689 9-35581
0.64419 075
13
48
155 9-34547
515 9-98907
719 9-35640
0.64360 015
12
49
183 9.34602
508 9.98904
750 9-35698
0.64302 4.3956
II
50
22212 9.34658
97502 9.98901
22781 9.35757
0.64243 4-3897
10
SI
240 9-34713
496 9.98898
811 9.35815
0.64x85 838
9
S2
268 9.34769
489 9-98896
3
3
3
842 9-35873
0.64127 779
8
S3
297 9-34824
483 9.98893
0.64069 721
7
54
325 9-34879
476 9.98890
903 9.35989
0.64011 662
b
^5
55
22353 9-34934
97470 9.98887
22934 9-36047
0.63953 4-3604
S6
382 9.34989
463 9.98884
3
3
3
3
3
964 9.36105
0.63895 546
4
S7
410 9.35044
457 9-98881
995 9-36163
0.63837 488
3
,S8
438 9-35099
450 9-98878
23026 9.36221
0.63779 430
2
S9
467 9-35154
444 9-98875
056 9-36279
0.63721 372
I
60
495 9-35209
437 9-98872
087 9-36336
0.63664 315
Nat. Cos Log. d.
Nat. Sin Log. d.
Nat. Cot Log.
c.d.
Log.TanNat.
/
77'
Nat. Sin Log. d.
13^
Nat. Cos Log. d.
Nat.TanLog. c.d
Log. Cot Nat.
22495
523
552
580
608
9-35209
935263
935318
9-35373
9-35427
22637
665
693
722
750
9-35481
9-35536
9-35590
935644
9-35698
22778
807
835
863
892
9-35752
9.35806
9-35860
9-35914
9-35968
22920
948
977
23005
033
9.36022
9-36075
9.36129
9.36182
9-36236
23062
090
118
146
175
9.36289
9-36342
9-36395
9-36449
9.36502
23203
231
260
288
316
9-36555
9.36608
9.36660
9-36713
9-36766
23345
373
401
429
458
9.36819
9-36871
9-36924
9.36976
9.37028
23486
514
542
571
599
9-37081
9-37133
9-37185
9-37237
9-37289
23627
656
684
712
740
9-37341
9-37393
9-37445
9-37497
9-37549
23769
797
825
853
882
9-37600
9-37652
937703
9-37755
9.37806
23910
938
966
995
24023
9-37858
937909
9-37960
9.3801 1
9.38062
24051
079
108
136
164
192
9-381 13
9.38164
9.38215
9.38266
9-38317
9-38368
97437
430
424
417
411
9.98872
9.98869
9.98867
9.98864
9.98861
97404
398
391
384
378
9-98858
9-98855
9.98852
9.98849
9.98846
97371
365
358
351
345
9-98843
9.98840
9.98837
9-98834
9.98831
97338
331
325
318
3"
9.98828
9.98825
9.98822
9.98819
9.98816
97304
298
291
284
278
9-98813
9.98810
9.98807
9.98804
9.98801
97271
264
257
251
244
9.98798
9-98795
9.98792
9.98789
9.98786
97237
230
223
217
210
9-98783
9.98780
9.98777
9-98774
9.98771
97203
196
189
182
176
9.98768
9-98765
9.98762
9-98759
9-98756
97169
162
15s
148
141
9-98753
9.98750
9-98746
9-98743
9-98740
97134
127
120
"3
106
9-98737
9-98734
9.98731
9.98728
9-98725
97100
093
086
079
072
9.98722
9.98719
9.98715
9.98712
9-98709
97065
058
051
044
037
030
9.98706
9.98703
9.98700
9.98697
9-98694
9.98690
23087
117
148
179
209
9-36336
9-36394
9-36452
936509
9.36566
23240
271
301
332
363
9.36624
9.36681
9-36738
9-36795
9.36852
23393
424
455
485
516
9-36909
9-36966
937023
9-37080
9-37137
23547
578
608
639
670
9-37193
9-37250
9-37306
9-37363
9-37419
23700
731
762
793
823
9-37476
937532
937588
9-37644
9-37700
23854
885
916
946
977
9-37756
9-37812
9.37868
9-37924
9.37980
24008
039
069
100
131
9-38035
9.38091
9.38147
9.38202
9-38257
24162
193
223
254
285
9-.38313
9.38368
9-38423
9-38479
9-38534
24316
347
377
408
439
9.38589
9-38644
9-38699
9-38754
9.38808
24470
501
532
562
593
9-38863
9.38918
9.38972
9.39027
9-39082
24624
655
686
717
747
9-39136
9.39190
9-39245
9-39299
9-39353
24778
809
840
871
902
933
9-39407
9.39461
9-39515
9-39569
9.39623
9.39677
0.63664
0.63606
0.63548
0.63491
0-63434
4-331S
257
200
143
086
0.63376
0.63319
0.63262
0.63205
0.63148
4.3029
4.2972
916
859
803
0.63091
0.63034
0.62977
0.62920
0.62863
4.2747
691
635
580
524
0.62807
0.62750
0.62694
0.62637
0.62581
4.2468
413
358
303
248
0.62524
0.62468
0.62412
0.62356
0.62300
4.2193
139
084
030
4.1976
0.62244
0.62188
0,62132
0.62076
0.62020
4.1922
868
814
760
706
0.61965
0.61909
0.61853
0.61798
0.61743
4-1653
600
547
493
441
0.61687
0.61632
0.61577
0.61521
0.61466
4.1388
335
282
230
178
0.61411
0.61356
0.61301
0.61246
0.61192
4.1126
074
022
4.0970
918
0.61 137
0.61082
0.61028
0.60973
0.60918
4.0867
81S
764
713
662
0.60864
0.60810
0.60755
0.60701
0.60647
4.061 1
560
509
459
408
0.60593
0.60539
o.6o48g
0.60431
0.60377
0.60323
4.0358
308
257
207
158
108
Nat. Cos Log. d.
Nat. Sin Log. d.
76^
Nat. Cot Log,
d. Log. Tan Nat
w
r
Nat. Sin Log. d.
Nat. Cos Log. d.
Nat.TanLog.
c.d.
Log. Cot Nat.
24192 9.38368
50
51
50
51
50
50
51
50
50
50
50
50
50
50
50
97030 9.98690
3
3
24933 9-39677
54
0.60323 4.0108
60
I
220 9.38418
023 9.98687
964 9-39731
0.60269 058
59
2
249 9-38469
015 9.98684
995 9-39785
0.60215 009
58
3
277 9-38519
008 9.98681
3
25026 9.39838
53
0.60162 3.9959
57
4
5
305 9-38570
001 9.98678
3
3
056 9-39892
54
53
0.60108 910
56
56
24333 9-38620
96994 9.98675
25087 9-39945
0.60055 3.9861
b
362 9.38670
987 9-98671
118 9.39999
54
0.60001 812
54
7
390 9.38721
980 9.98668
149 9-40052
53
0.59948 763
53
8
418 9.38771
973 9-98665
3
3
180 9.40106
54
0.59894 714
52
9
446 9.38821
966 9.98662
211 9.40159
53
53
0.59841 665
51
60
10
24474 9-38871
96959 9-98659
25242 9.40212
0-59788 3-9617
II
503 9.38921
952 9-98656
3
4
3
273 9.40266
54
53
53
0-59734 568
49
12
531 9.38971
945 998652
304 9.40319
0.59681 520
48
13
559 9-39021
937 9-98649
335 9-40372
0.59628 471
47
14
587 9-39071
930 9.98646
3
3
366 9.40425
53
53
0-59575 423
46
46
16
24615 9.39121
96923 9.98643
25397 9-40478
0.59522 3-9375
I6
644 9.39170
916 9.98640
3
428 9.40531
53
0.59469 327
44
17
672 9.39220
50
909 9.98636
459 9-40584
53
0.59416 279
43
i8
700 9.39270
50
902 9.98633
3
490 9.40636
52
0.59364 232
42
19
728 9.39319
50
894 9.98630
3
3
521 9.40689
53
53
0.5931 1 184
41
40
20
24756 9-39369
96887 9.98627
25552 9.40742
0-59258 3-9136
21
784 9.39418
49
880 9.98623
4
583 9.40795
53
0.59205 089
39
22
813 9-39467
49
873 9.98620
3
614 9.40847
52
0.59153 042
38
23
841 9-39517
50
866 9.98617
3
645 9.40900
53
0.59100 3.8995
37
24
25
869 9-39566
49
49
858 9.98614
3
4
676 9.40952
52
53
0.59048 947
36
24897 9-39615
96851 9.98610
25707 9.41005
0.58995 3.8900
35
26
925 9.39664
844 9.98607
3
738- 9-41057
52
0-58943 854
34
27
954 9-39713
837 9.98604
3
769 9.41109
52
0.58891 807
33
28
982 9.39762
829 9.98601
3
800 9.41161
52
0.58839 760
32
29
25010 9.39811
49
822 9.98597
4
3
831 9.41214
53
52
0.58786 714
31
30
30
25038 9-39860
96815 9-98594
25862 9.41266
0.58734 3.8667
31
066 9.39909
49
807 9.98591
3
893 9.41318
52
0.58682 621
29
32
094 9-39958
49
48
800 9.98588
3
924 9.41370
52
0.58630 575
28
33
122 9.40006
793 9-98584
4
955 9-41422
52
0.58578 528
27
34
151 9.40055
49
48
49
48
786 9.98581
3
3
986 9.41474
52
52
52
51
0.58526 482
0.58474 3-8436
26
26
35
25179 9.40x03
96778 9-98578
26017 9.41526
3^
207 9.40152
77^ 9-98574
4
048 9.41578
0.58422 391
24
37
235 9.40200
764 9.98571
3
079 9.41629
0.58371 345
23
3«
263 9.40249
48
49
48
48
48
48
48
48
47
48
48
756 9-98568
3
no 9.41681
52
0.58319 299
22
39
40
291 9.40297
749 9-98565
3
4
141 9.41733
52
51
52
51
52
51
51
52
51
51
51
51
51
51
51
51
%
50
51
51
50
0.58267 254
21
25320 9-40346
96742 9.98561
26172 941784
0.58216 3.8208
20
41
348 9.40394
734 9-98558
3
203 941836
0.58164 163
19
42
376 9.40442
727 9-98555
3
235 941887
O.58113 118
18
43
404 9.40490
719 9-98551
4
266 9.41939
0.58061 073
17
44
432 9.40538
712 9.98548
3
3
297 9-41990
0.58010 028
lb
45
25460 9.40586
96705 9.98545
26328 9.42041
0-57959 3-7983
16
46
488 9.40634
697 9-98541
4
359 9-42093
0-57907 938
14
47
516 9.40682
690 9.98538
3
390 942144
0.57856 893
13
48
545 9-40730
682 9.98535
3
421 9.42195
0.57805 848
12
49
573 9-40778
675 9-98531
4
3
452 942246
0-57754 804
II
To
50
25601 9.40825
96667 9.98528
26483 9.42297
0-57703 3-7760
SI
629 9.40873
660 9.98525
3
515 9.42348
0.57652 715
9
52
657 9-40921
653 9-98521
546 9-42399
0.57601 671
8
53
685 9.40968
48
47
48
645 9-98518
3
577 9-42450
0-57550 627
7
54
55~
713 9.41016
638 9-98515
3
4'
608 9.42501
0.57499 583
b
25741 9.41063
96630 9.98511
26639 9-42552
0-57448 3-7539
5
5^
769 941111
623 9.98508
3
670 9.42603
0.57397 495
4
57
798 9-4"58
615 9.98505
3
701 9-42653
0.57347 451
3
5«
826 9.41205
47
608 9.98501
4
733 9-42704
0.57296 408
2
18
854 9.41252
48
600 9.98498
3
764 9.42755
0-57245 364
1
882 9.41300
593 998494
4
795 942805
0.5719S 321
U
Nat. Cos Log. d.
Nat. Sin Log. d.
Nat. Cot Log.
c.d.
Log. Tan Nat.
t
w
15:
Nat. Sin Log. d. Nat. Cos Log. d. Nat.TanLog. c.d.lLog.CotNat
25882
910
938
966
994
941300
941347
941394
9.41441
9.41488
26022
050
079
107
135
941535
941582
941628
9.41675
941722
26163
191
219
247
275
9.41768
941815
941861
9.41908
941954
26303
331
359
387
415
9.42001
9.42047
9.42093
942140
942186
26443
471
500
528
556
942232
942278
942324
942370
942416
26584
612
640
668
696
942461
942507
942553
9.42599
9.42644
26724
752
780
808
836
942690
9.42735
9.42781
9.42826
942872
26864
892
920
948
976
942917
9.42962
943008
9.43053
943098
27004
032
060
088
116
943143
9.43188
943233
9.43278
943323
27144
172
200
228
2c;6
943367
943412
943457
943502
943546
27284
312
340
368
396
943591
943635
943680
9.43724
943769
27424
452
480
508
536
564
9-43813
943857
9.43901
943946
943990
944034
96593
585
578
570
562
9.98494
9.98491
9.98488
9.98484
9.98481
96555
547
540
532
524
9.98477
9.98474
9.98471
9.98467
9.98464
96517
509
502
494
486
9.98460
998457
9-984$3
9.98450
9.98447
96479
471
463
456
448
998443
9.98440
9.98436
9.98433
9.98429
96440
433
425
417
410
9^98426
9.98422
9.98419
9.98415
9.98412
96402
394
386
379
371
9.98409
9.98405
9.98402
9.98398
9.98395
96363
355
347
340
332
9.98391
998388
9.98384
9.98381
9.98377
96324
316
308
301
293
9.98373
9.98370
9.98366
998363
9.98359
96285
277
269
261
253
9.98356
998352
9.98349
9.98345
9.98342
96246
238
230
222
214
9.98338
9.98334
9-98331
9.98327
9.98324
96206
198
190
182
174
9.98320
9.98317
9.98313
9.98309
9.98306
96166
158
150
142
134
126
9.98302
9.98299
9.98295
9.98291
9.98288
9.98284
26795
826
857
888
920
9.42805
942856
942906
9.42957
9.43007
26951
943057
982
9.43108
27013
943158
044
9.43208
076 9.43258
27107 943308
138 9.43358
169
943408
201
943458
232
943508
27263 9.43558
294
9.43607
326 943657
357
943707
388 943756
27419
9.43806
451
9.43855
482 9.43905
513
943954
545
9.44004
27576 944053
607
9.44102
638
9.44151
670
944201
701
9.44250
27732 9.44299
764 944348
795
944397
826
9.44446
858 944495
27889
944544
921
9.44592
952
944641
983 9.44690
28015 944738
28046 9.44787
077
944836
109
9.44884
140
9.44933
172
9.44981
28203 9.45029
234
9.45078
266
9.45126
297 945174
329
9.45222
28360 945271
391
9.45319
423
945367
454
9.45415
486 945463
28517
549
580
612
643
675
9455"
9.45559
9.45606
9.45654
945702
945750
0.57195
0.57144
0.57094
0.57043
0.56993
3-7321
277
234
191
148
0.56943
0.56892
0.56842
0.56792
0.56742
3.7105
062
019
3.6976
933
0.56692
0.56642
0.56592
0.56542
0.56492
3.6891
848
806
764
722
0.56442
0.56393
0.56343
0.56293
0.56244
3.6680
638
596
554
512
0.56194
0.5614$
0.56095
0.56046
0.55996
3.6470
429
387
346
30s
0.55947
0.55898
0.55849
0.55799
0-55750
3.6264
222
181
140
100
0.55701
0.55652
0.55603
0.55554
0.55505
3.6059
018
3-5978
937
897
0.55456
0.55408
0-55359
0.55310
0.55262
3.5856
816
776
736
696
0.55213
0.55164
0.551 16
0.55067
0.55019
3.5656
616
576
536
497
0.54971
0.54922
0.54874
0.54826
0.54778
3-5457
418
379
339
300
0.54729
0.54681
0.54633
0.54585
0.54537
3.5261
222
183
144
105
0.54489
0.54441
0.54394
0.54346
0.54298
0.54250
3-5067
028
3.4989
951
912
874
Nat. Cos Log. d. Nat. Sin Log. d. Nat.Cot Log. c.d. Log.TanNat
74'
16 '
' Nat. Sin Log. d. Nat. Cos Log. d.
Nat.TanLog. c.d
Log. Cot Nat,
27564
592
620
648
676
9.44034
9.44078
9.44122
9.44166
9.44210
27704
731
759
787
815
9.44253
9.44297
9-44341
944385
9.44428
27843
871
899
927
955
9.44472
9.44516
9-44559
9.44602
9.44646
27983
2801 1
039
067
095
9.44689
9-44733
9.44776
9.44819
9.44862
28123
150
178
206
234
9.44905
9.44948
9.44992
945035
9-45077
28262
290
318
346
374
9.45120
9-45163
9.45206
9.45249
9.45292
28402
429
457
485
513
9-45334
9-45377
9-45419
945462
9-45504
28541
569
597
625
652
9-45547
9-45589
9-45632
9-45674
9-45716
28680
708
736
764
792
9-45758
9-45801
9-45843
9-45885
9-45927
847
875
903
931
9.45969
9.4601 1
9-46053
9-46095
9.46136
28959
987
29015
042
070
9.46178
9.46220
9.46262
9-46303
9-46345
29098
126
154
182
209
237
9.46386
9.46428
9.46469
9-46511
9-46552
9-46594
96126
118
no
102
094
9.98284
9.98281
9-98277
9.98273
9.98270
96086
078
070
062
054
9.98266
9.98262
9-98259
9-98255
9.98251
96046
037
029
021
013
9.98248
9.98244
9.98240
9.98237
9-98233
96005
95997
989
981
972
9.98229
9.98226
9.98222
9.98218
9.98215
95964
956
948
940
931
9.9821 1
9.98207
9.98204
9.98200
9.98196
95923
915
907
898
890
9.98192
9.98189
9.98185
9.98181
9.98177
95882
874
865
857
9.98174
9.98170
9.98166
9.98162
9.98159
95841
832
824
816
807
9-98155
9.98151
9.98147
9.98144
9.98140
95799
791
782
774
766
9-98136
9.98132
9.98129
9.98125
9.98121
95757
749
740
732
724
9.981 17
9-98113
9.981 10
9.98106
9.98102
95715
707
698
690
681
9.98098
9.98094
9.98090
9.98087
9-98083
95673
664
656
647
639
630
9.98079
9-98075
9.98071
9.98067
9-98063
9.98060
28675
706
738
769
801
9-45750
9-45797
9-45845
9.45892
9-45940
28832
864
895
927
958
9-45987
9-46035
9.46082
9.46130
9.46177
28990
29021
053
084
116
9.46224
9.46271
9.46319
9-46366
9.46413
29147
179
210
242
274
9.46460
9-46507
9-46554
9.46601
9.46648
29305
337
368
400
432
•9.46694
9.46741
946788
9-46835
9.46881
29463
495
526
558
590
9.46928
9-46975
9.47021
9.47068
9.47114
29621
653
685
716
748
9.47160
9.47207
9-47253
9.47299
9-47346
29780
811
843
875
906
9-47392
9-47438
9-47484
9-47530
9-47576
29938
970
30001
033
065
9.47622
9.47668
9.47714
9-47760
9.47806
30097
128
160
192
224
947852
9-47897
9-47943
947989
9-48035
30255
287
319
351
382
9.48080
9.48126
9.48171
9.48217
9.48262
30414
446
478
509
541
573
9.48307
9-48353
9.48398
9-48443
9.48489
948534
0.54250
0.54203
0.54155
0.54108
0.54060
3-4874
836
798
760
722
0.54013
0-53965
0.53918
0.53870
0.53823
3.4684
646
608
570
533
0.53776
0.53729
0.53681
0-53634
0-53587
3-4495
458
420
383
346
0.53540
0.53493
0.53446
0.53399
0.53352
3-4308
271
234
197
160
0.53306
0.53259
0.53212
0.53165
0.53119
3.4124
087
050
014
3.3977
0.53072
0.53025
0.52979
0.52932
0.52886
3-3941
904
868
832
796
0.52840
0-52793
0-52747
0.52701
0.52654
3-3759
723
687
652
616
0.52608
0.52562
0.52516
0.52470
0.52424
3-3580
544
509
473
438
0.52378
0.52332
0.52286
0.52240
0-52194
3-3402
367
332
297
261
0.52148
0.52103
0.52057
0.5201 1
0.51965
3-3226
191
156
122
087
0.51920
0.51874
0.51829
0.51783
0.51738
3-3052
017
3-2983
948
914
0.51693
0.51647
0.51602
0-51557
0.51466
3-2879
845
811
m
743
709
Nat. Cos Log. d.
Nat. Sin Log. d.
73^
Nat. Cot Log
c.d. Log. Tan Nat
17°
Nat. Sin Log. d. Nat. Cos Log. d. Nat.TailLog. c.d.lLog. Cot Nat.
29237
265
293
321
348
9.46594
946635
9.46676
9.46717
9.46758
29376
404
432
460
487
9.46800
946841
946882
9.46923
9.46964
10 ' 29515
543
571
599
626
9.4700$
947045
9.47086
947127
9.47168
29654
682
710
737
765
9.47209
9.47249
9.47290
947330
947371
29793
821
849
876
904
9.4741 1
947452
9.47492
947533
947573
29932
960
987
30015
043
947613
947654
9.47694
947734
9.47774
30071
098
126
154
182
9.47814
947854
9.47894
947934
947974
30209
237
265
292
320
948014
9.48054
9.48094
948133
9.48173
30348
376
403
431
459
30486
514
542
570
597
9.48213
948252
9.48292
948332
948371
94841 1
9.48450
9.48490
9.48529
9.48568
30625
653
680
708
736
9.48607
9.48647
9.48686
9.48725
9.48764
30763
791
819
846
874
902
948803
9.48842
948881
9.48920
9.48959
9.48998
95630
622
613
605
596
9.98060
9.98056
9.98052
9.98048
9.98044
95588
579
571
562
554
9.98040
9.98036
9.98032
9.98029
9.98025
95545
536
528
519
5"
9.98021
9.98017
9.98013
9.98009
9-98005
95502
493
485
476
467
9.98001
9-97997
9-97993
9.97989
9-97986
95459
450
441
433
424
9.97982
9.97978
9-97974
9.97970
9.97966
95415
407
398
389
380
9.97962
9-97958
9-97954
9-97950
9-97946
95372
363
354
345
337
9-97942
997938
9-97934
9-97930
9-97926
95328
319
310
301
293
9.97922
9.97918
9.97914
9.97910
9-97906
95284
275
266
257
248
9.97902
9.97898
9.97894
9.97890
9.97886
95240
231
222
213
204
9.97882
9-97878
9.97874
9-97870
997866
95195
186
177
168
159
9.97861
997857
9-97853
9-97849
9-97845
95150
142
133
124
"5
106
9.97841
9-97837
9-97833
9-97829
9.97825
9.97821
30573
605
637
669
700
9-48534
948579
9.48624
9.48669
9.48714
30732
9-48759
764 9-48804
796 948849
828
9.48894
860
9.48939
30891
948984
923
9.49029
955
9-49073
987 9491 18
31019
9.49163
31051
9.49207
083 949252
"5
9.49296
147
9-49341
178 9-49385
31210
9-49430
242
9-49474
274
9-49519
306 949563
338 949607
31.370
9-49652
402
949696
434
9-49740
466 9.49784
498 9.49828
31530
9.49872
562
9.49916
594
949960
626
9.50004
658 9.50048
31690 9.50092
722
9-50136
754
9.50180
786 9.50223
818
9.50267
31850 9.5031 1
882
9-50355
914
9-50398
946 9.50442
978 9-50485
32010 9.50529
042
9-50572
074
9.50616
106
9-50659
139
950703
32171
9-50746
203
9.50789
235
950833
267 9.50876
299
9.50919
32331
363
396
428
460
492
9-50962
9.51005
9.51048
9.51092
9.5II35
9.51 178
0.51466
0.5I42I
0-51376
0-5I33I
0.51286
3-2709
675
641
607
573
0.51241
0.51 196
0.51151
0.51106
0.51061
3-2539
506
472
438
405
0.51016
0.50971
0.50927
0.50882
0.50837
3-2371
338
305
272
238
0-50793
0-50748
0.50704
0.50659
0.50615
3-2205
172
139
106
073
0.50570
0.50526
0.50481
0-50437
0.50393
3-2041
008
3-1975
943
910
0.50348
0.50304
0.50260
0.50216
0.50172
3-1878
845
813
780
748
0.50128
0.50084
0.50040
0.49996
049952
3.1716
684
652
620
0.49908
0.49864
0.49820
049777
049733
3-1556
524
492
460
429
049689
0.49645
0.49602
0.49558
0.49515
3-1397
366
334
303
271
0.49471
0.49428
0.49384
0.49341
049297
3.1240
209
178
146
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0-49254
0.4921 1
0.49167
0.49124
0.49081
3.1084
053
022
3.0991
961
0.49038
04899S
0.48952
0.48908
0.48865
048822
3.0930
899
868
838
807
m
Nat. Cos Log. d.
Nat. Sin L og, d. |Nat. Cot Log, c.d. Log.TanN at.
72"
18
D
r
Nat. Sin Log. d.
Nat. Cos Log. d.
Nat.TanLog.
c.d.
Log. Cot Nat.
30902 9.48998
39
39
39-
38
39
39
38
39
95106 9.97821
4
5
32492 9.5II78
43
0.48822 3.0777
60
I
929 9.49037
097 9.97817
524 9,51221
0.48779 746
59
2
957 949076
088 9.97812
556 9.51264
43
0.48736 716
58
3
985 9.49115
079 9.97808
4
4
4
588 9.51306
42
43
43
0.48694 686
57
4
5
31012 9.49153
070 9.97804
621 9.51349
0.48651 655
56
55
31040 9.49192
95061 9,97800
32653 9.51392
0.48608 3.0625
6
068 9.49231
052 9-97796
4
685 9.51435
43
0.48565 595
.54
7
095 9.49269
043 9-97792
717 9.51478
0.48522 565
53
8
033 9-97788
. 749 9.51520
0.48480 535
52
9
151 949347
38
38
024 9.97784
5
4
4
4
782 9.51563
43
42
43
43
0.48437 505
51
50
10
3 1 178 949385
95015 9-97779
32814 9.51606
0.48394 3.0475
II
206 9.49424
006 9.97775
846 9.51648
0.48352 445
49
12
233 949462
94997 9-97771
878 9.51691
0.483O9 415
48
13
261 9.49500
988 9-97767
911 9.51734
0.48266 385
47
14
289 9.49539
39
38
38
39
38
38
38
38
38
38
38
38
38
3?
38
38
38
979 9-97763
4
4
5
4
4
4
4
4
5
943 9.51776
43
42
42
43
42
43
42
42
0.48224 356
4b
45
16
31316 9.49577
94970 9.97759
32975 9.51819
0.48181 3.0326
lb
344 949615
961 9-977$4
33007 9.51861
0.48139 296
44
17
372 9.49654
952 997750
040 9.51903
0.48097 267
43
lb
399 949692
943 9-97746
072 9.51946
0.48054 237
42
19
20
427 949730
31454 949768
933 9-97742
104 9.51988
0.48012 208
41
40
94924 9.97738
33136 9.52031
0.47969 3.0178
21
482 9.49806
915 9-97734
169 9.52073
0.47927 149
39
22
510 9.49844
906 9.97729
201 9.52115
0,47885 120
38
23
537 949882
897 9.97725
233 9.52157
43
42
42
42
42
42
42
0.47843 090
37
24
565 9.49920
888 9.97721
4
266 9.52200
0.47800 061
3^
35
25
31593 949958
94878 9-97717
33298 9..52242
0.47758 3.0032
2b
620 9.49996
869 9.97713
4
5
4
4
4
5
0.47716 003
34
27
648 9.50034
860 9.97708
363 9.52326
0.47674 2.9974
33
28
675 9.50072
851 9.97704
395 9.52368
047632 945
32
29
703 9.501 10
842 9.97700
427 9.52410
0.47590 916
31
30
30
31730 9.50148
94832 9.97696
33460 9.52452
0.47548 2.9887
31
758 9-50185
38
823 9-97691
492 9.52494
42
42
0.47506 858
29
32
786 9.50223
814 9-97687
524 9.52536
0.47464 829
28
33
813 9.50261
805 9.97683
4
557 9.52578
0.47422 800
27
34
35
841 9.50298
37
38
38
37
38
795 997679
5
589 9,52620
41
42
42
42
42
41
42
41
42
0.47380 772
2b
25
31868 9.50336
94786 9.97674
33621 9.52661
0.47339 2.9743
36
896 9-50374
777 997670
4
654 9.52703
0.47297 714
24
37
923 9-50411
768 9.97666
686 9.52745
0.47255 686
23
3H
951 9-50449
758 9-97662
5
4
718 9.52787
0.47213 657
22
39
40
979 9-50486
37
37
38
749 9-97657
751 9.52829
0.47171 629
21
20
32006 9.50523
94740 9-97653
33783 9.52870
0.47130 2,9600
41
034 9-50561
730 9.97649
816 9,52912
0.47088 572
19
42
061 9.50598
37
721 9.97645
4
5
848 9.52953
0.47047 544
i8
43
089 9.50635
^8
37
37
712 9.97640
881 9.52995
o.470og 515
17
44
116 9-50673
702 9.97636
4
4
4
5
4
913 9.53037
4-
41
42
41
41
0.46963 487
lb
15
45
32144 9.50710
94693 9.97632
33945 9.53078
0.46922 2,9459
4b
171 9-50747
684 9.97628
978 9.53120
0.46880 431
14
47
199 9-50784
37
37
674 9.97623
34010 9.53161
0.46839 403
13
48
227 9.50821
665 9.97619
043 9.53202
0.46798 375
12
49
50
254 9-50858
37
38
656 9.97615
5
4
4
5
4
4
5
4
075 9.53244
41
42
41
41
41
42
41
0.46756 347
II
32282 9.50896
94646 9.97610
34108 9.53285
0.46715 2.9319
10
51
309 9-50933
637 9.97606
140 9.53327
0.46673 291
9
52
337 9-50970
37
%
37
627 9,97602
173 9-53368
0.46632 263
8
53
364 9.51007
618 9.97597
205 9.53409
0.46591 23s
7
54
55
392 9.51043
609 9.97593
238 9.53450
0.46550 208
6
5
32419 9.51080
94599 9.97589
34270 9.53492
0.46508 2.9180
5b
447 9.51117
37
590 9.97584
303 9.53533
0.46467 152
4
57
474 9-5"54
37
580 9,97580
41
41
0.46426 125
3
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502 9.51191
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571 9.97576
4
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368 9.53615
0.46385 097
2
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529 9.51227
561 9.97571
400 9.53656
0.46344 070
I
557 9.51264
37
552 9.97567
4
433 9.53697
0.46303 042
Nat.CoSLog. d.
Nat. Sin Log. d.
Nat. Cot Log.
c.d.
Log.Tan Nat.
r
n
19'
Nat. Sin Log. d. Nat. Cos Log. d
Nat.TanLog.
c.d.
Log. Cot Nat.
32557
584
612
639
667
9.51264
9-51301
9-51338
9-51374
32694
722
749
777
804
9-51447
9.51484
9.51520
9-51557
9-51593
32832
859
887
914
942
9.51629
9.51666
9.51702
9.51738
9-51774
32969
997
33024
051
079
9.51811
9.51847
9-51883
9.51919
9-51955
33106
134
161
189
216
9.51991
9.52027
9.52063
9.52099
9-52135
25
26
27
28
30
31
32
33
34
33244
271
298
326
__J53_
33381
408
436
463
490
9.52171
9.52207
9.52242
9.52278
9-52314
9-52350
9-52385
9-52421
952456
9-52492
33518
545
573
600
627
9-52527
9-52563
9-52598
9-52634
9-52669
33655
682
710
737
764
9-52705
9.52740
9-52775
9.5281 1
9.52846
33792
819
846
874
901
9.52881
9.52916
9-52951
9.52986
9.53021
33929
956
983
3401 1
038
953056
9-53092
9.53126
9-53161
9-53196
34065
093
120
147
175
202
9-53231
9.53266
9-53301
9-53336
9-5337?
9-53405
94552
542
533
523
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9-97567
9-97563
9.97558
9.97554
9.97550
94504
495
485
476
466
9-97545
9-97541
9.97536
9.97532
9.97528
94457
447
438
428
418
9.97523
9.97519
9-97515
9.97510
9.97506
94409
399
390
380
370
9.97501
9.97497
9.97492
9.97488
9.97484
94361
351
342
332
322
9-97479
9-97475
9.97470
9.97466
9.97461
94313
303
293
284
274
9.97457
9.97453
9.97448
9.97444
9-97439
94264
254
245
235
225
9-97435
9-97430
9-97426
9.97421
9.97417
94215
206
196
186
176
9.97412
9.97408
9-97403
9-97399
9-97394
94167
157
147
137
127
9-97390
997385
9.97381
9.97376
9-97372
941 18
108
098
088
078
9-97367
9.97363
9.97358
9.973.53
9.97349
94068
058
049
039
029
9.97344
9.97340
9.97335
9.97331
9.97326
94019
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93999
989
979
969
9.97322
9.97317
9.97312
9.97308
9-97303
9-97299
34433
465
498
530
563
9.53697
9.53738
9.53779
9.53820
9.53861
34596
628
661
693
726
9.53902
9-53943
9.53984
954025
9.54065
34758
791
824
8^6
9.54106
9-54147
9.54187
9.54228
9.54269
34922
954
987
35020
052
9.54309
9.54350
9-54390
9-54431
9.54471
35085
118
150
183
216
9.54512
9.54552
9.54593
9.54633
9.54673
35248
281
314
346
379
9.54714
9.54754
9.54794
9.54835
9-54875
35412
445
477
510
543
9-5491$
9-54955
9.5499$
9.5503$
9.55075
35576
608
641
674
707
955"$
9-5515$
9-5519$
9.5523$
9.55275
35740
772
805
838
871
9-55315
9-55355
9-55395
9-55434
9.55474
35904
937
969
36002
035
9.55514
9.55554
9.55593
955633
9.55673
36068
lOI
134
167
199
9-55712
9.55752
9.55791
9.55831
9.55870
36232
265
298
331
364
397
9.55910
9.55949
9.55989
9.56028
9.56067
9.56107
0.46303
046262
0.46221
0.46180
046139
2.9042
015
2.8987
960
933
046098
046057
0.46016
045975
045935
2.8905
878
851
824
797
045894
045853
0.45813
0.45772
0-45731
2.8770
743
716
689
662
0.45691
0.45650
0.45610
045569
0.45529
2.8636
609
582
556
529
0.45488
0.45448
0.45407
o.4$367
045327
2.8502
476
449
423
397
0.45286
0.45246
045206
045165
0.45125
2.8370
344
318
291
265
045085
045045
045005
0.44965
044925
2.8239
213
187
161
135
044885
0.44845
0.44805
0.44765
0.44725
2.8109
083
057
032
006
0.44685
044645
0.44605
0.44566
044526
2.7980
955
929
903
878
0.44486
044446
0.44407
0.44367
044327
2.7852
827
801
776
751
0.44288
0.44248
0.44209
044169
0.44130
2.7725
700
675
650
625
0.44090
044051
04401 1
043972
043933
0.43893
2.7600
575
550
525
500
475
Nat. Cos Log. d.
Nat. Sin Log. d.
70^
Nat. Cot Log. c.d. Log. Tan Nat.
Nat. Sin Log. d.
20^
Nat. Cos Log. d.
Nat.
;.TanLog. c.d. Log. Cot Nat
34202
229
257
284
3"
9-53405
9-53440
9-53475
953509
9-53544
34339
366
393
421
448
9-53578
9-53613
9-53647
9.53682
9-53716
34475
503
530
557
584
9-53751
9-53785
9.53819
953854
9-53888
34612-
639
666
694
721
9-53922
9-53957
9-53991
9-54025
9-54059
34748
775
803
830
857
9-54093
9.54127
9.54161
9-54195
9-54229
34884
912
939
966
993
9-54263
9-54297
9-54335
9-54365
9-54399
35021
048
075
102
130
9-54433
9-54466
9-54500
9-54534
9-54567
35157
184
211
239
266
9-54601
9-54635
9-54668
9-54702
9-54735
35293
320
347
375
402
9-54769
9.54802
9-54836
9-54869
9-54903
35429
456
484
511
538
9-54936
9-54969
955003
9-55036
9-55069
35565
592
619
647
674
9-55102
9-55136
9-55169
9-55202
9-55235
36701
728
755
782
810
837
9.55268
9-55301
9-55334
9-55367
9-55400
9-55433
93969
959
949
939
929
9-97299
9-97294
9-97289
9-97285
9.97280
93919
909
899
889
879
9.97276
9.97271
9.97266
9.97262
9-97257
93869
859
849
839
829
9.97252
9-97248
9-97243
9.97238
9-97234
93819
809
799
789
779
9-97229
9-97224
9.97220
9-97215
9.97210
93769
759
748
738
728
9.97206
9.97201
9.97196
9.97192
9.97187
93718
708
698
688
677
9.97182
9.97178
9-97173
9.97168
9-97163
93667
657
647
637
626
9-97159
9-97154
9.97149
9-97145
9.97140
93616
606
596
585
575
9-97135
9.97130
9.97126
9.97121
9.971 16
93565
555
544
534
524
9.97111
9.97107
9.97102
9-97097
9-97092
93514
503
493
483
472
9-97087
9-97083
9.97078
9-97073
9.97068
93462
452
441
431
420
9-97063
9-97059
9-97054
9.97049
997044
93410
400
389
379
368
358
9-97039
9-9703S
9.97030
9.9702g
9.97020
9.97015
36397
430
463
496
529
9.56107
9.56146
9.56185
9-56224
9-56264
36562
595
628
661
694
9-56303
9-56342
9-56381
9.56420
9-56459
36727
760
793
826
859
9-56498
9-56537
9-56576
9-56615
9-56654
36892
925
958
991
37024
9-56693
9-56732
9.56771
9.56810
9-56849
37057
090
123
157
190
9-56887
9.56926
9-56965
9-57004
9-57042
37223
256
289
322
355
9.57081
9.57120
9.57158
9.57197
9-57235
37388
422
455
488
521
9-57274
9-57312
9-57351
9-57389
9.57428
37554
588
621
654
687
9-57466
9-57504
9-57543
9-57581
9-57619
37720
754
787
820
853
9-57658
9-57696
9-57734
9-57772
9.57810
37887
920
953
986
38020
9-57849
9.57887
9-57925
957963
9.58001
38053
086
120
153
186
9-58039
9-58077
9-58115
9.58153
9-58191
38220
253
286
320
353
386
9-58229
9.58267
9.58304
9-58342
9.58380
9.58418
Nat. Cos Log. d. Nat. Sin Log, d. |Nat. Cot Log
0.43893
0.43854
0.43815
0.43776
0.43736
2.7475
450
425
400
376
0.43697
0.43658
0.43619
0.43580
0.43541
2.7351
326
302
277
253
0.43502
0.43463
0.43424
0.43385
043346
2.7228
204
179
155
130
0.43307
0.43268
0.43229
0.43190
0.43151
2.7106
082
058
034
009
0.43113
0.4307-^
0.43035
0.42996
0.42958
2.6985
961
937
913
0.42919
042880
0.42842
0.42803
0.42765
2.6865
841
818
794
770
0.42726
0.42688
0.42649
0.4261 1
0.42572
2.6746
723
699
675
652
0.42534
0.42496
042457
042419
0.42381
2.6628
605
581
558
534
0.42342
0.42304
0.42266
0.42228
0.42190
2.651 1
488
464
441
418
0.42151
0.42113
0.42075
0.42037
041999
2.6395
371
348
325
302
0.41961
0.41923
0.41885
0.41847
0.41809
2.6279
256
233
210
187
041771
041733
0.41696
041658
0.41620
041582
2.6165
142
119
096
074
051
c.d.|Log.TanNat. '
69
Nat. Sin Log. d.
2r
Nat. Cos Log. d.
Nat.TanLog.
c.d,
Log. Cot Nat,
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
35837
864
891
918
945
9-55433
9-55466
9-55499
9-55532
955564
35973
36000
027
054
081
36108
135
162
190
217
33
33
33
32
9-55597 00
9-55630 i ti
9-55663 ' ti
9-55695 00
9-55728 f
9.55761 f
9-55793 L
9-55826 1 33
9.55858 ! 3^
9-55891 i ^g
36244
271
298
325
352
36379
406
434
461
36515
542
569
596
623
9-55923 00
9-55956 a
9-55988 32
9.56021 ^^
9-56053 3
9-56085 3
9-56118 33
9-56150 3
9.56182 32
9.56215 2
36650
677
704
731
758
956247 I2
9-56279 ^2
9.563" ^
956343
9-56375
36785
812
839
867
894
36921
948
975
37002
029
32
32
32
33
32
32
32
32
9-56568 II
9-56599 3
9.56631 i 3
9.56663 3
956695 i II
9.56408
9.56440
9.56472
9.56504
9.56536
37056
083
no
137
164
9.56727 L
9.56759 i 3^
9.56790 1 3
9.56822 1 II
9.56854 ! II
37191
218
245
272
299
9.56886 3
9.56917 3J
9.56949 3
9.56980 II
9.57012
37326
353
380
407
434
461
9.57044
9.57075
9.57107
9-57138
9-57169
9-57201
9-57232
9-57264 _j
9-57295 3;
9-57326 3^
9.57358 3^
93358
348
337
327
316
9.97015
9.97010
9.97005
9.97001
9.96996
93306 9-96991 1
295
9.96986
285
9.96981
274
9.96976
264 9.96971 1
93253
9.96966
243
9.96962
232
996957
222
9.96952
211
9-96947
93201
9.96942
190
996937
180
9.96932
169 9.96927 !
159
9.96922
93148 9.96917
137
9.96912
127
9.96907
116
9-96903
106
9.96898
93095
l'&
084
074
9.96883
063 9.96878
052
9.96873
93042
9.96868
031
9-96863
020
9-96858
010
9-96853
92999
9.96848
92988 9.96843
978
9.96838
967 9.96833
956
9.96828
945
9.96823
92935
9.96818
924
9-96813
913
9.96808
902
9-96803
892 9.96798
92881 9.96793
870 9.96788
859 9.96783
849 9.96778
838 9-96772
92827 9-96767
816
9.96762
805 9-96757
794
9.96752
784 9-96747
92773
762
751
740
729
718
9.96742
9.96737
9.96732
9.96727
9.96722
9.96717
38386
420
453
487
520
9.58418
9.58455
9.58493
9.58531
9.58569
38553 9.58606
587 9.58644
620
9.58681
654
9-58719
687 9.58757
38721 9.58794
754
9.58832
787 9.58869
821
9-58907
854
9-58944
38888 9.58981
921
9.59019
955
9.59056
988
9-59094
39022 9.59131
39055
9-59168
089 9.59205
122
9-59243
156 9.59280
igo
9-59317
39223
9-59354
257
9-59391
290 9.59429
324
9-59466
357
9-59503
39391
9-59540
425
9-59577
458
9.59614
492
9-59651
526 9.5088
39559
9-59725
593
9.59762
626
9-59799
660
9.59835
694 9-59872
39727
9.59909
761 9.59946
795
9.59983
829 9.60019
862
9.60056
39896
9.60093
930
9.60130
963
9.60166
997
9.60203
40031
9.60240
40065
9.60276
098
9.60313
132
9-60349
166
9.60386
200
9.60422
40234
267
301
335
369
403
9-60459
9.60495
9-60532
9.60568
9.60605
9.60641
0.41582
0.41545
0.41507
0.41469
0.41431
2.6051
028
006
2.5983
961
0.41394
0.41356
0.41319
0.41281
0-41243
2.5938
916
893
871
0.41206
0.41 168
0.41131
0.41093
0.41056
2.5826
804
782
759
737
0.41019
0.40981
0.40944
0.40906
0.40869
2.5715
693
671
649
627
0.40832
0-40795
0.40757
0.40720
0.40683
2.5605
583
561
539
517
0.40646
0.40609
040571
040534
040497
2.5495
473
452
430
408
040460
0.40423
0.40386
0.40349
040312
2.5386
365
343
322
300
0.40275
040238
040201
0.40165
0.40128
2.5279
257
236
214
193
0.40091
0.40054
0.40017
0.39981
0.39944
2.5172
150
129
108
086
0.39907
0.39870
0.39834
0.39797
0.39760
2.5065
044
023
002
2.4981
0.39724
0.39687
0.39651
0.39614
0.39578
2.4960
939
918
897
876
0.39541
O.3950S
0.39468
0.39432
0.39395
0.39359
24855
834
813
792
772
751
Nat. Cot Log. c.d. Log.Tan Nat
Nat.CoSLog. d.
Nat. Sin Log. d.
68^
22°
' Nat. Sin Log. d. Nat. Cos Log. d. Nat.TanLog.
c.d. Log. Cot Nat
37461
488
515
542
569
9-57358
9-57389
9.57420
9-57451
9-57482
37595
622
649
676
703
9-57514
9.57545
9-57576
9-57607
9-57638
37730
757
784
811
838
9-57669
9.57700
9-57731
9.57762
9-57793
37865
892
919
946
973
9.57824
9-57855
9-57885
9.57916
9-57947
37999
38026
053
080
107
9-57978
9.58008
9.58039
9-58070
9.58101
38134
161
188
215
241
9-58131
9.58162
9.58192
9.58223
9-58253
38268
295
322
349
376
9-58284
9-58314
9-58345
958375
9-58406
38403
430
456
483
510
9.58436
9.58467
9.58497
9.58527
9.58557
38537
564
591
617
644
9.58588
9.58618
9.58648
9.58678
9.58709
38671
698
725
752
778
9.58739
9.58769
9-58799
9.58829
9-58859
38805
832
859
886
912
9.58889
9.58919
9.58949
9.58979
9.59009
38939
966
993
39020
046
073
9.59039
9.59069
9.59098
9.59128
9-59158
9.59188
92718
707
697
686
675
9.96717
9.96711
9.96706
9.96701
9.96696
92664
653
642
631
620
9.96691
9.96686
9.96681
9.96676
9.96670
92609
598
587
576
565
9.96665
9.96660
9.96655
9.96650
9.96645
92554
543
532
521
510
9.96640
9.96634
9.96629
9.96624
9.96619
92499
488
477
466
455
9.96614
9.96608
9.96603
9.96598
996593
92444
432
421
410
399
9.96588
9.96582
9.96577
9.96572
9.96567
92388
377
366
355
343
9.96562
9.96556
9.96551
9.96546
9.96541
92332
321
310
299
287
9-96535
9.96530
9-96525
9.96520
9.96514
92276
265
254
243
231
9-96509
9-96504
9.96498
9-96493
9.96488
92220
209
198
186
175
9.96483
9.96477
9.96472
9.96467
9.96461
92164
152
141
130
119
9.96456
9.96451
9.96445
9.96440
9.96435
92107
096
085
073
062
050
9.96429
9.96424
9.96419
9.96413
9.96408
9.96403
40403
436
470
504
538
9.60641
9.60677
9.60714
9.60750
9.60786
40572
9.60823
606
9.60859
640 9.60895
674 9.60931
707
9.60967
40741
9.61004
775
9.61040
809 9.61076
843
9.61112
877 9.61148
409H
9.61184
945
9.61220
979
9.61256
41013
9.61292
047
9.61328
4108 1
9.61364
"5
9.61400
149
9.61436
183 9.61472
217
9.61508
41251
9.61544
285 9.61579
319
9.61615
9.61651
9.61687
41421
9.61722
455
9.61758
490
9.61794
524
9.61830
558 9.61865
41592
9.61901
9.61936
660
9.61972
694
9.62008
728 9.62043
41763 9.62079
797
9.62114
831
9.62150
865 9.62185
899
9.62221
41933
9.62256
968 9.62292
42002
9-62327
036 9.62362
070
9-62398
42105
9-62433
139
9.62468
173
9.62504
207
9-62539
242
9-62574
42276
310
345
379
413
447
9.62609
9-62645
9.62680
9.62715
9.62750
9.62785
0.39359
0.39323
0.39286
0.39250
0.39214
2.4751
730
709
689
668
0.39177
0.39141
0.39105
0.39069
0.39033
2.4648
627
606
586
566
0.38996
0.38960
0.38924
0.38888
0.38852
2.4545
525
504
484
464
0.38816
0.38780
0.38744
0.38708
0.38672
2.4443
423
403
383
362
0.38636
0.38600
0.38564
0.38528
0.38492
2.4342
322
302
282
262
0.38456
0.38421
0.3838.5
0.38349
0.38313
2.4242
222
202
182
162
0.38278
0.38242
0.38206
0.38170
0.38135
2.4142
122
102
083
063
0.38099
0.38064
0.38028
0.37992
0.37957
2.4043
023
004
2.3984
964
0.37921
0.37886
0.37850
0.37815
0.37779
2.3945
925
906
886
867
0.37744
0.37708
0.37673
0.37638
0.37602
2.3847
828
808
789
770
0.37567
0.37532
0.37496
0.37461
0.37426
2.3750
731
712
693
673
0.37391
0.37355
0.37320
0.37285
0.37250
0.37215
2.3654
635
616
597
578
559
Nat. Cot Log. c.d. Log. Tan Nat,
Nat. Cos Log. d. Nat. Sin Log. d
67°
23°
Nat. Sin Log. d.
Nat. Cos Log. d.
Nat Tan Log. c.d.Log. Cot Nat,
39073
100
127
153
180
9.59188
9.59218
959247
959277
9-59307
39207
234
260
287
314
9-59336
9-5936<3
959396
959425
9-59455
39341
367
394
421
448
9-59484
9-59514
9-59543
9-59573
9-59602
39474
501
528
555
581
9-59632
9-5061
9.59690
9.59720
9-59749
39608
635
661
688
715
9-59778
9.59808
9-59837
9-59866
9-59895
39741
768
795
822
848
9-59924
9-59954
9-59983
9,60012
9.60041
39875
902
928
955
082
9.60070
9.60099
9.60128
9.60157
9.60186
40008
035
062
088
115
9.60215
9.60244
9.60273
9.60302
9.60331
40141
168
195
221
248
9-60359
9.60388
9.60417
9.60446
9-60474
40275
301
328
355
381
9-60503
9-60532
9.60561
9-60589
9.60618
40408
434
461
488
514
9.60646
9-60675
9.60704
9.60732
9.60761
40541
567
594
621
647
674
9.60789
9.60818
9.60846
9.60875
9-60903
9-60931
92050
039
028
016
005
9.96403
9-96397
9-96392
9-96387
9.96381
91994
9.96376
982
9-96370
971
9-96365
959
9-96360
948 9.96354
91936 9-96349
925
9.96343
914
9-96338
902
996333
891
9-96327
91879 9.96322
868
9.96316
856 9-9631 1
845
9.96305
833
9-96300
91822
9-96294
810
9.96289
799
9.96284
787 9.96278
775
9-96273
91764 9.96267
752
9.96262
741
9.96256
729
9.96251
718 9.96245
91706 9.96240
694 9.96234
683 9.96229
671
9.96223
660
9.96218
91648
9.96212
636 9.96207
625 9.96201
613 9.96196
601
9.96190
91590
9-96185
578 9-96179
566 9-96174
555
9.96168
543
9.96162
91531
9.96157
519
9.96151
508 9.96146
496 9.96140
484
996135
91472
9.96129
461
9.96123
449
9.96118
437
9.961 12
425
9.96107
9I4I4
402
390
378
366
355
9.96101
9-96095
9.96090
9.96084
9.96079
9-96073
Nat. Cos Log. d.
Nat. Sin
42447
482
516
551
585
9-62785
9.62820
9.62855
9.62890
9.62926
42619
654
688
722
757
9.62961
9.62996
9.63031
9.63066
9.63101
42791
826
860
894
929
9-63135
9.63170
9.63205
9.63240
9-63275
42963
998
43032
067
loi
9.63310
9-63345
9-63379
9.63414
9.63449
43136
170
205
239
274
9.63484
9-63519
9.63553
9.63588
9.63623
43308
343
378
412
447
9.63657
9.63692
9.63726
9.63761
9.63796
43481
516
550
585
620
9.63830
9-63865
9-63899
9-63934
9.63968
43654
689
724
758
793
9-64003
9.64037
9.64072
9.64106
9.64140
43828
862
897
932
966
9.64175
9.64209
9-64243
9.64278
9.64312
44001
036
071
105
140
9-64346
9.64381
9.64415
9-64449
9.64483
44175
210
244
279
314
9.64517
9-64552
9-64586
9.64620
9.64654
44349
384
418
453
488
523
9.64688
9.64722
9.64756
9.64790
9.64824
9.64858
)073 I 523 9.04050 ^' 0.35142 400
Log. d. Nat.CotLog. c.d.|Log.TanNat
0-37215
0.37180
0.37145
0.371 10
0.37074
2.3559
539
520
501
483
0.37039
0.37004
0.36969
0.36934
0.36899
2.3464
445
426
407
0.36865
0.36830
0.36795
0.36760
0.36725
2.3369
351
332
313
294
0.36690
0.36655
0.36621
0.36586
0.36551
2.3276
257
238
220
201
0.36516
0.36481
0.36447
0.36412
0.36377
2.3183
164
146
127
109
0.36343
0.36308
0.36274
0.36239
0.36204
2.3090
072
053
035
017
0.36170
0.36135
0.36101
0.36066
0.36032
2.2998
980
962
944
925
0.35997 2.2907
0.35963
0.35928
0.35894
0.35860
871
853
835
0.35825
0.35791
0-35757
0.35722
0.35688
2.2817
799
781
763
745
0-35654
0.35619
0.35585
0.35551
0.35517
2.2727
709
691
673
655
0.35483
0.35448
0.35414
0.35380
0.35346
2.2637
620
602
584
566
0.35312
0.35278
0.35244
0.35210
0.35176
0.35142
2.2549
531
513
496
478
460
24
f
Nat. Sin Log. d.
Nat. Cos Log. d.
Nat.TanLog.
c.d. Log. Cot Nat.
40674 9.60931
29
91355 9-96073
f.
44523 9.64858
34
34
34
34
34
34
0.35142 2.2460
60
I
700 9.60960
343 9-96067
5
558 9.64892
0.35108 443
59
2
727 9.60988
o9
331 9-96062
593 9-64926
0.35074 425
58
3
753 9.61016
29
28
319 9-96056
A
627 9.64960
0.35040 408
57
4
6
780 9.61045
307 9.96050
5
6
662 9.64994
0.35006 390
56
55
40806 9.61073
91295 9.96045
44697 9.65028
0.34972 2.2373
6
833 9.61 lOI
08
283 9.96039
732 9.65062
0-34938 355
54
7
860 9.61 129
29
08
272 9.96034
767 9.65096
34
0.34904 338
53
8
886 9.61158
260 9.96028
f.
802 9.65130
34
0.34870 320
52
9
913 9.61186
28
248 9.96022
5
6
837 9.65164
34
33
34
34
34
0.34836 303
51
10
40939 9.61214
91236 9.96017
44872 9.65197
0.34803 2.2286
50
II
966 9.61242
oO
224 9.9601 1
5
907 9.65231
0.34769 268
49
12
992 9.61270
03
212 9.96005
942 9.65265
0.34735 251
48
13
41019 9.61298
og
200 9.96000
977 9.65299
0.34701 234
47
14
04s 9.61326
28
08
188 9.95994
I
45012 9-65333
34
33
0.34667 216
46
'45
15
41072 9.61354
91176 9.95988
45047 9-65366
0.34634 2.2199
lb
098 9.61382
164 9.95982
082 9.65400
0.34600 182
44
17
125 9.61411
27
08
152 9-95977
117 9-65434
34
33
11
33
34
34
33
34
33
34
33
0.34566 165
43
l8
151 9.61438
140 9-95971
6
152 9.65467
0.34533 148
42
19
178 9.61466
28
28
128 9.95965
5
6
187 9.65501
0.34499 130
41
40
20
41204 9.61494
91 116 9.95960
45222 9.65535
0.34465 2.2113
21
231 9.61522
28
104 9.95954
6
257 9-65568
0.34432 096
39
22
257 9.61550
28
092 9.95948
5
292 9.65602
0.34398 079
38
23
284 9.61578
28
080 9.95942
5
6
6
,327 9.65636
0.34364 062
37
24
310 9.61606
28
og
068 9-95937
362 9.65669
0-34331 045
36
35
25
41337 9.61634
91056 9.95931
45397 9.65703
0.34297 2.2028
26
363 9.61662
27
og
044 9-95925
5
6
432 9.65736
0.34264 on
34
27
390 9.61689
032 9.95920
467 9.65770
0.34230 2.1994
33
28
416 9.61717
28
28
020 9.95914
f.
502 9.65803
0.34197 977
32
29
443 9-61745
008 9.95908
6
5
5
538 9.65837
34
33
0.34163 960
.31
30
41469 9.61773
90996 9.95902
45573 9.65870
0.34130 2.1943
30
31
496 9.61800
oQ
984 9.95897
608 9.65904
33
34
0.34096 926
29
32
522 9.61828
08
972 9-95891
(=,
643 9.65937
0.34063 909
28
3S
549 9-61856
960 9.95885
5
678 9.65971
0.34029 892
27
34
575 9-61883
27
28
og
948 9.95879
6
5
713 9.66004
33
34
33
0-33996 876
2b
35
41602 9.61911
90936 9.95873
45748 9.66038
0.33962 2.1859
25
36
628 . 9.61939
924 9.95868
784 9.66071
0-33929 842
24
37
655 9.61966
^8
911 9.95862
6
819 9.66104
33
0.33896 825
23
3B
681 9.61994
899 9.95856
6
854 9.66138
34
33
33
34
0.33862 808
22
39
707 9.62021
27
28
887 9.95850
6
5
A
889 9.66171
0.33829 792
21
20
40
41734 9.62049
90875 9-95844
45924 9.66204
0.33796 2.1775
41
760 9.62076
08
863 9.95839
960 9.66238
0.33762 758
19
42
787 9.62104
851 9-95833
A
995 9.66271
0.33729 742
18
43
813 9.62131
08
839 9.95827
5
46030 9.66304
33
34
33
0.33696 725
17
44
45
840 9.62159
27
28
826 9.95821
6
065 9.66337
0.33663 708
lb
41866 9.62186
90814 9.95815
46101 9.66371
0.33629 2.1692
15
46
892 9.62214
802 9.95810
136 9.66404
0.33596 675
14
47
919 9.62241
27
790 9.95804
6
171 9.66437
33
33
33
34
33
0.33563 659
13
48
945 9.62268
27
28
27
778 9-95798
6
206 9.66470
0.33530 642
12
49
972 9.62296
766 9.95792
6
242 9.66503
0.33497 625
II
lo
50
41998 9.62323
90753 9.95786
46277 9.66537
0.33463 2.1609
=^1
42024 9.62350
27
741 9.95780
5
312 9.66570
0.33430 59?
9
52
051 9-62377
28
729 9.95775
348 9.66603
33
33
33
33
0.33397 576
8
,S3
077 9.62405
717 9.95769
A
383 9.66636
0.33364 560
7
54
55
104 9.62432
27
704 9.95763
6
6
418 9.66669
0.33331 543
b
~5
42130 9-62459
90692 9.95757
46454 9.66702
0.33298 2.1527
,0
156 9.62486
27
680 9.95751
5
489 9.66735
0.3.3265 510
4
57
183 9-62513
27
28
668 9.95745
6
525 966768
33
33
33
33
0.33232 494
3
5«
209 9.62541
655 9.95739
6
560 9.66801
0.33199 478
2
^0
235 9-62568
27
643 9-95733
631 9-95728
5
595 9-66834
0.33166 461
I
262 9.62595
27
631 9.66867
0.33133 445
Nat.CoSLog. d.
Nat. Sin Log. d.
Nat. Cot Log.
c.d.
Log.Tan Nat.
r
66'
26°
f Nat. Sin Log. d. Nat. Cos Log. d. Nat.TanLog. c.d. Log.CotNat.
42262
288
315
341
367
9-62595
9.62622
9.62649
9.62676
9-62703
42394
420
446
473
499
9.62730
9.62757
9.62784
9.6281 1
9.62838
42525
552
578
604
631
42657
683
709
736
762
42788
815
841
867
894
42920
946
972
999
43025
27
27
27
27
27
27
27
27
27
27
27
I 26
i 27
; 27
27
j27
26
1 27
I 27
9-63133 26
9-63159 27
9.63186 %
9-63213 : 26
9-63239 ! 27
26
9.62865
9.62892
9.62918
9.62945
9.62972
9.62999
9.63026
9.63052
9.63079
9.63106
43051
077
104
130
156
9.63266
9-63292 i 27
9-63319 26
963345 27
9-63398 ^ 27
9-63425 26
9-63451
9.63478
9-63504
43182
209
235
261
287
9-63531
9-63557
9-63583
9.63610
963636
43313
340
366
392
418
9.63662
9.63689
9-63715
9.63741
963767
43445
471
497
523
549
9-63794
9.63820
9.63846
9-63872
9-63898
43575
602
628
654
680
9.63924
9-63950
9-63976
9.64002
9.64028
43706
733
759
785
811
837
9.64054
9.64080
9.64106
9.64132
9.64158
9.64184
90631
618
606
594
582
9.95728
9-95722
9.95716
9.95710
9-95704
90569 9.95698
557
9-95692
545
9.95686
532
9.95680
520
9-95674
90507
9-95668
495
995663
483
995657
470
9-95651
458
9-95645
90446 9.95639
433
995633
421
9-95627
408
9.95621
396 9.95615
90383 9-95609
371
9-95603
358
9-95597
346
9-95591
334
9-95585
90321
9-95579
309
9-95573
296
9-95567
284 9-95561
271
9-95555
90259 9.95549
246 9-95543
233
9-95537
221
9-95531
208
9-95525
90196
9-95519
183 9-95513
171
9-95507
158
9-95500
14b
9-95494
90133
9.95488
120
9.95482
108
9-95476
095
9-95470
082
9.95464
90070
9-95458
057
9-95452
045
9.95446
032
9.95440
019
9-95434
90007
995427
89994
9-95421
981
■§68
9-95415
9.95409
956 9-95403
89943
9^0
918
905
892
879
9-95397
9-95391
995384
9.95378
995372
995366
46631
666
702
737
772
9.66867
9.66900
9.66933
9.66966
9.66999
46808
843
879
914
950
9.67032
9-67065
9.67098
9.67131
9-67163
46985
47021
056
092
128
9.67196
9.67229
9.67262
9.67295
9-67327
47163
199
234
270
305
9.67360
9-67393
9.67426
9.67458
9.67491
47341
377
412
448
483
9-67524
9-67556
9.67589
9.67622
9-67654
47519
555
590
626
662
9-67687
9.67719
9.67752
9.67785
9.67817
47698
733
769
805
840
9.67850
9.67882
9.67915
9-67947
9.67980
47876
912
948
984
48019
9.68012
9.68044
9.68077
9.68109
9.68142
48055
091
127
163
iq8
9.68174
9.68206
9-68239
9.68271
9-68303
48234
270
306
342
378
9.68336
9.68368
9.68400
9.68432
968465
48414
450
486
521
557
9.68497
9.68529
9.68561
9-68593
9.68626
48593
629
665
701
737
773
9.68658
9.68690
9.68722
9-68754
9.68786
9.68818
0.33133
0.33100
0.33067
0-33034
0.33001
2.1445
429
413
396
380
0.32968
0-32935
0.32902
0.32869
0.32837
2.1364
348
332
315
299
0.32804
0.32771
0.32738
0.32705
0.32673
2.1283
267
251
235
219
0.32640
0.32607
0.32574
0.32542
0-32509
2.1203
187
171
IS5
139
0.32476
0-32444
0.32411
0.32378
0.32346
2.1123
107
092
076
060
0.32313
0.32281
0.32248
0.32215
0.32183
2.1044
028
013
2.0997
981
0.32150
0.32118
0.32085
0.32053
0.32020
2.0965
950
934
918
903
0.31988
0.31956
0.31923
0.31891
0.31858
2.0887
872
856
840
825
0.31826
0.31794
0.31761
0.31729
0.31697
2.0809
794
778
763
748
0.31664
0.31632
0.31600
0.31568
0.31535
2.0732
717
701
686
671
0.31503
0.31471
0.31439
0.31407
0.31374
2.0655
640
625
609
594
0.31342
0.31310
0.31278
0.31246
0.31214
0.31182
2.0579
564
549
533
518
503
Nat. Cot Log. c.d. Log. Tan Nat.
60
59
58
57
5^
55
54
53
52
50
49
48
47
45
44
43
42
40
39
38
37
_36
35
34
33
32
Nat. Cos Log. d.
Nat. Sin Log. d.
64^
<
26
f
Nat. Sin Log. d.
Nat. Cos Log. d.
Nat.TanLog.
c.d. Log. Cot Nat.
43837 9-64184
89879 9-95366
5
48773 9.68818
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
31
32
0.31 182 2.0503
60
I
863 9.64210
05
867 995360
5
809 9.68850
0.31 150 488
59
2
889 9.64236
05
854 9-95354
5
0.31118 473
58
3
916 9.64262
06
841 9.95348
881 9.68914
0.31086 458
57
4
T
942 9.64288
25
05
828 9.95341
7
6
917 9-68946
0.31054 443
5t)
43968 9.64313
89816 9.95335
48953 9-68978
0.31022 2.0428
55
6
06
803 9-95329
f.
989 9.69010
0.30990 413
54
7
44020 9.64365
05
790 9-95323
f.
49026 9.69042
0.30958 398
53
8
046 9.64391
-^6
777 9-95317
062 9.69074
0.30926 383
52
9
072 9.64417
25
06
764 9-95310
6
5
098 9.69106
0.30894 368
51
10
44098 9.64442
89752 9-95304
49134 9.69138
0.30862 2.0353
50
II
124 9.64468
05
739 9-95298
f.
170 9.69170
0.30830 338
49
12
151 9-64494
%
726 9.95292
6
206 9.69202
0.30798 323
48
13
177 9-64519
713 9.95286
242 9.69234
0.30766 308
47
14
203 9.64545
26
%
700 9-95279
7
6
278 9.69266
0.30734 293
4b
45
15
44229 9.64571
89687 9.95273
49315 9-69298
0.30702 2.0278
I6
255 9-64596
674 995267
A
351 9.69329
0.30671 263
44
17
281 9.64622
%
662 9.95261
387 9-69361
0.30639 248
43
i8
307 9-64647
649 9-95254
7
6
423 9-69393
32
32
32
31
0.30607 233
42
19
333 9-64673
25
"6
636 9.95248
6
6
459 9-69425
0.30575 219
4i
20
44359 9-64698
89623 9-95242
49495 9-69457
0.30543 2.0204
40
21
385 9-64724
%
610 9.95236
532 9.69488
0.30512 189
39
22
411 9-64749
597 9-95229
I
6
6
7
5
568 9.69520
32
0.30480 174
38
23
437 9-64775
25
26
25
584 9.95223
604 9-69552
32
0.30448 160
37
24
464 9.64800
571 9-95217
640 9-69584
32
31
32
32
0.30416 145
3t^
35
25
44490 9.64826
89558 9-95211
49677 9.69615
0.30385 2.0130
2b
516 9.64851
545 9-95204
713 9.69647
0.30353 "5
34
27
542 9-64877
25
25
26
25
25
532 9.95198
6
749 9-69679
0.30321 lOI
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28
568 9.64902
519 9-95192
786 9.69710
31
0.30290 086
32
29
30
594 9-64927
506 9-95185
7
6
6
822 9.69742
32
32
31
0.30258 072
31
30
44620 9.64953
89493 9-95179
49858 9.6977^
0.30226 2.0057
31
646 9.64978
480 9-95173
5
894 9.69805
0.30195 042
29
32
672 9.65003
467 9-95167
931 9.69837
0.30163 028
28
33
698 9.65029
25
25
25
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967 9.69868
0.30132 013
27
34
35
724 9.65054
441 9-95154
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50004 9.69900
32
32
0.30100 1.9999
2b
25
44750 9-65079
89428 9.95148
50040 9.69932
0.30068 1.9984
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776 9.65104
415 9-95141
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076 9-69963
31
32
0.30037 970
24
37
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25
25
25
25
25
402 9-95135
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113 9-69995
0.30005 955
23
3a
828 9.65155
389 9-95129
149 9.70026
31
0.29974 941
22
39
854 9-65180
376 9.95122
7
6
185 9.70058
32
31
0.29942 926
21
40
44880 9.65205
89363 9-95116
50222 9.70089
0.29911 1.9912
20
41
906 9.65230
350 9.951 10
258 9.70121
32
0.29879 897
19
42
932 9-65255
337 9-95103
6
295 9-70152
31
32
0.29848 883
18
43
958 9.65281
25
25
25
25
324 995097
331 9.70184
0.29816 868
17
44
984 9.65306
311 9.95090
6
6
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368 9.70215
31
32
0.29785 854
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15
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45010 9.65331
89298 9.95084
50404 9.70247
0.29753 1.9840
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062 9.65381
285 9-95078
441 9.70278
31
31
0.29722 825
14
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272 9-95071
477 9-70309
0.29691 811
13
48
088 9.65406
25
25
25
25
25
25
25
24
259 9.95065
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514 9-70341
32
31
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0.29659 797
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49
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114 9-65431
245 9-95059
7
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550 9-70372
0.29628 782
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45140 9.65456
89232 9.95052
50587 9.70404
0.29596 1.9768
10
51
166 9.6548X
219 9.95046
623 9-70435
31
0.29565 754
9
52
192 9-65506
206 9.95039
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5
660 9.70466
31
0.29534 740
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53
218 9-65531
193 9-95033
696 9-70498
32
31
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0.29502 725
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243 9-65556
180 9.95027
7
6
733 9-70529
0.29471 7"
b
65
45269 9.65580
89167 9.95020
50769 9.70560
0.29440 1.9697
6
56
25
25
153 9-95014
806 9.70592
32
31
0.29408 683
4
57
321 9-65630
140 9-95007
6
6
843 970623
0.29377 669
3
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347 9-65655
373 9-65680
25
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127 9.95001
879 9.70654
31
31
0.29346 654
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114 9-94995
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916 9.70685
0.29315 640
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399 9-65705
25
loi 9-94988
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953 9-70717
32
0.29283 626
Nat. Cos Log. d.
Nat. Sin Log. d.
Nat. Cot Log.
c.d.
Log.Tan Nat.
f
63
27'
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Nat. Sin Log. d.
Nat. Cos Log. d.|
Nat Tan Log.
c.d.
Log. Cot Nat.
45399 9-65705
24
25
25
25
24
25
25
89101 9.94988
5
50953 9.70717
31
31
31
31
32
31
31
0.29283 1.9626
60
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425 9-65729
087 9.94982
7
5
989 9.70748
0.29252 612
59
2
451 9.65754
074 9.94975
51026 9.70779
0.29221 598
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477 9-65779
061 9.94969
063 9.70810
0.29190 584
57
4
503 9.65804
048 9.94962
7
6
099 9.70841
0.29159 570
56
5
45529 9.65828
89035 9.94956
5II36 9-70873
0.29127 1.9556
55
6
554 9-65853
021 9.94949
6
173 9.70904
0.29096 542
54
7
580 9.65878
008 9.94943
209 9.70935
0.29065 528
53
8
606 9.65902
24
25
25
24
25
88995 9.94936
7
6
7
31
31
31
31
31
31
0.29034 514
52
9
632 9.65927
981 9.94930
283 9.70997
0.29003 500
51
50
10
45658 9-65952
88968 9.94923
5I3I9 9.71028
0.28972 1.9486
II
684 9-65976
955 9-94917
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356 9.71059
0.28941 472
49
12
710 9.66001
942 9.949"
393 9-71090
0.28910 458
48
13
736 9.66025
928 9-94904
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7
6
430 9.71121
0.28879 444
47
14
762 9.66050
25
25
915 9.94898
467 9.71153
32
31
31
31
31
0.28847 430
46
15
45787 9.66075
88902 9.94891
S1503 9.71184
0.28816 1.9416
45
l6
813 9.66099
24
25
888 9.94885
540 9.71215
0.28785 402
44
17
839 9.66124
875 9.94878
7
577 9.71246
0.28754 388
43
l8
865 9.66148
24
862 9.94871
7
6
7
6
614 9.71277
0.28723 375
42
19
891 9.66173
25
24
848 9-94865
651 9.71308
31
31
30
31
31
31
31
31
31
31
31
0.28692 361
41
20
45917 9.66197
88835 9-94858
51688 9.71339
0.28661 1.9347
40
21
942 9.66221
24
25
822 9.94852
724 9.71370
0.28630 333
39
22
968 9.66246
808 9-94845
7
6
761 9.71401
0.28599 319
38
23
994 9.66270
24
795 9-94839
798 9.71431
0.28569 306
37
24
46020 9.66295
25
24
782 9-94832
7
6
835 9.71462
0.28538 292
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25
46046 9.66319
88768 9.94826
51872 9.71493
0.28507 1.9278
35
26
072 9.66343
24
25
755 9.94819
7
6
909 9^71524
0.28476 265
34
27
097 9.66368
741 9-94813
946 9.71555
0.28445 251
33
28
123 9.66392
24
728 9.94806
7
983 9.71586
0.28414 237
32
29
149 9.66416
24
25
715 9.94799
7
6
52020 9.71617
0.28383 223
31
30
46175 9.66441
88701 9.94793
52057 9.71648
0.28352 1.9210
30
31
201 9.66465
24
688 9.94786
I
094 9.71679
0.28321 196
29
32
226 9.66489
24
674 9.94780
131 9.71709
30
31
31
31
31
0.28291 183
28
33
252 9.66513
24
661 9-94773
7
6
7
168 9.71740
0.28260 169
27
34
278 9-66537
24
25
647 9-94767
205 9.71771
0.28229 155
26
35
46304 9.66562
88634 9.94760
52242 9.71802
0.28198 1.9142
25
36
330 9.66586
24
620 9.94753
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279 9.71833
0.28167 128
24
37
355 9.66610
24
607 9.94747
316 9.71863
30
0.28137 115
23
38
381 9.66634
24
593 9-94740
7
6
7
353 9.71894
31
31
30
31
31
31
30
31
31
30
31
30
31
31
30
31
30
31
30
31
30
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30
0.28106 lOI
22
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407 9.66658
24
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580 9-94734
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46433 9.66682
88566 9.94727
52427 9.71955
0.28045 1.9074
20
41
458 9.66706
24
25
553 9-94720
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464 9.71986
0.28014 061
19
42
484 9.66731
539 9-94714
501 9.72017
0.27983 047
18
43
510 9.66755
24
526 9-94707
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538 9.72048
0.27952 034
17
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536 9.66779
24
24
512 9.94700
7
6
575 9.72078
0.27922 020
16
45
46561 9.66803
88499 9.94694
52613 9.72109
0.27891 1.9007
15
46
587 9.66827
24
485 9-94687
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650 9.72140
0.27860 1.8993
14
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613 9.66851
24
472 9.94680
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6
687 9.72170
0.27830 980
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639 9.66875
24
458 9.94674
724 9.72201
0.27799 967
12
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664 9-66899
24
23
445 9.94667
7
7
6
761 9.72231
0.27769 953
II
50
46690 9.66922
88431 9.94660
52798 9.72262
0.27738 1.8940
10
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716 9.66946
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417 9.94654
836 9.72293
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873 9.72323
0.27677 913
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390 9.94640
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0.27646 900
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793 9.67018
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377 9.94634
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0.27616 887
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46819 9.67042
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52985 9-72415
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134 9.72537
0.27463 820
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0.27433 807
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Nat. Sin Log. d.
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Nat. Sin Log. d.
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Nat. Cos Log. d.
Nat.TanLog. c.d. Log. Cot Nat
46947
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9.67185
9.67208
9.67232
9.67256
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9-67374
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255
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9.67468
9.67492
47332
358
383
409
434
9-67515
9-67539
9.67562
9.67586
9.67609
47460
486
511
537
562
9-67633
9-67656
9.67680
9.67703
9.67726
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614
639
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9.67750
9-67773
9.67796
9.67820
9.67843
47716
741
767
793
818
9.67866
9.67890
9.67913
9.67936
9.67959
47844
869
895
920
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9.67982
9.68006
9.68029
9.68052
9.68075
47971
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48022
048
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9.68098
9.68121
9.68144
9.68167
9.68190
48099
124
150
175
201
9.68213
9.68237
9.68260
9.68283
9.68305
252
277
303
328
9.68328
9-68351
9-68374
9.68397
9.68420
48354
379
405
430
456
481
9.68443
9.68466
9.68489
9.68512
9-68534
9.68557
88295
281
267
254
240
9-94593
9-94587
9.94580
9-94573
9-94567
88226
213
199
185
172
9.94560
9-94553
994546
9-94540
9-94533
88158
144
130
117
103
9.94526
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9-94506
9-94499
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9.94492
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9-94465
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9.94438
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9-94404
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840
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9-94390
9.94383
9-94376
9-94369
9.94362
87812
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9-94355
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9-94335
9.94328
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9.94321
9-94314
9-94307
9-94300
9-94293
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659
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631
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9.94286
9.94279
9-94273
9.94266
9-94259
87603
589
575
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9.94252
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9.94238
9.94231
9-94224
87532
518
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490
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9.94217
9.94210
9-94203
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9.94182
53171
208
246
283
320
9-72567
9.72598
9.72628
9-72659
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53358
395
432
470
507
9.72720
9.72750
9.72780
9.7281 1
9.72841
53545
582
620
657
694
9.72872
9.72902
9.72932
9-72963
9.72993
53732
769
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9-73023
9-73054
9.83084
9-73114
9-73144
53920
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54032
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9-73175
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9-7326$
9-73295
54107
145
183
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9-73356
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9.73416
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9-73476
9-73507
9-73537
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54484
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9-73627
9-73657
9-73687
9-73717
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9-73777
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9-73897
54862
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938
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9.73927
9-73957
9-73987
9.74017
9-74047
55051
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165
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9-74077
9.74107
9-74137
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9-74196
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0.26554
1.8482
469
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0.26I63
0.26133
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1-8354
341
329
316
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1.8291
278
265
253
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0.26073
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090
078
065
053
040
Nat. Cos Log. d. Nat. Sin Log. d. Nat. Cot Log. | c.d. Log.Tan Nat.| ^
61°
29°
Nat. Sin Log. d. Nat. Cos Log. d. Nat.Tan Log. c.d
Log. Cot Nat.
48481
506
532
557
5B3
9-68557
a.68580
0.68603
9.68625
9.68648
48608
634
659
684
710
9.68671
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9.68716
9.68739
9.68762
48735
761
786
811
837
9.68784
9.68807
9.68829
9.68852
9.68875
913
938
964
9.68897
9.68920
9.68942
9.68965
9.68987
48989
49014
040
065
090
9.69010
9.69032
969055
9.69077
9.69100
491 16
141
166
192
217
9.69122
9.69144
9.69167
9.69189
9.69212
49242
268
293
318
344
9.69234
9.69256
9.69279
9.69301
9-69323
49369
394
419
445
470
9-69345
9.69368
9.69390
9.69412
969434
49495
521
546
571
596
9.69456
9.69479
9.69501
9.69523
9-69545
49622
647
672
697
723
9.69567
9.69589
9.6961 1
9-69633
9-69655
49748
773
798
824
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9.69677
9.69699
9.69721
9-69743
9.69765
49874
899
924
950
975
50000
9.69787
9.69809
969831
9.69853
9.69875
9.69297
87462
448
434
420
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9.94182
9.94175
9.94168
9.94161
9.94154
87391
377
363
349
335
9.94147
9.94140
9.94133
9.94126
9-94119
87321
306
292
278
264
9.941 12
9.94105
9.94098
9.94090
9.94083
87250
235
221
207
193
9.94076
9.94069
9.94062
9-94055
9.94048
87178
164
150
136
121
9.94041
9-94034
9.94027
9.94020
9.94012
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9.94005
9.93998
9-93991
9-93984
9-93977
87036
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86993
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9-93970
9-93963
9-93955
993948
9.93941
86964
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935
921
906
9.93934
9-93927
9-93920
9.93912
9.93905
86892
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9.93898
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86820
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9.93862
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9-93847
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9-93833
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9.93826
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9.9381 1
9-93804
9-93797
86675
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632
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9.93782
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9-93760
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9-74375
9-74405
9-74435
9.74465
9.74494
55621
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736
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974524
9.74554
9-74583
9.74613
9.74643
55812
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9.74673
9.74702
9.74732
9-74762
9.74791
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117
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9.74821
9.74851
9.74880
9.74910
9.74939
56194
232
270
309
347
9.74969
9.74998
9.75028
9.75058
9.75087
56385
424
462
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539
9.75"7
9.75146
9.75176
9.75205
9.75235
56577
616
654
693
731
9.75264
9-75294
9-75323
9.75353
9-75382
56769
808
846
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9-754"
9-75441
9-75470
9-75500
975529
56962
57000
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9.75558
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9.75647
9.75676
57155
193
232
271
309
9.75705
9-75735
9-75764
9-75793
9-75822
57348
386
425
464
503
9.75852
9.75881
9.75910
9.75939
9.75969
57541
580
619
657
696
9.75998
9.76027
9.76056
9.76086
9.761 15
9.76144
0.25625
0.2559$
0.2556$
0.25535
0.25506
1.8040
028
016
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1.7991
0.25476
0.25446
0.25417
0.25387
0.25357
1.7979
966
954
942
930
0.25327
0.25298
0.25268
0.25238
0.25209
1.7917
905
893
881
0.25179
0.25149
0.25120
0.25090
0.25061
1.7856
844
832
820
808
0.25031
0.25002
0.24972
0.24942
0.24913
1.7796
783
771
759
747
0.24883
0.24854
0.24824
0.2479$
0.24765
1-7735
723
711
699
687
0.24736
0.24706
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0.24647
0.24618
1.7675
663
651
639
627
0.24589
0.24559
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0.24500
0.24471
1.7615
603
591
579
567
0.24442
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532
520
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0.24236
0.24207
0.24178
1.7496
485
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0.24148
0.241 19
0.24090
0.24061
0.24031
1.7437
426
414
402
391
0.24002
0.23973
0.23944
0.23914
0.23885
0.23856
1-7379
367
355
344
332
321
Nat.CoSLog. d. Nat. Sin Log. d. Nat.CotLog. c.d.lLog.TanNat
30
r
Nat. Sin Log. d.
Nat. Cos Log. d.
Nat.TanLog.
c.d.
Log. Cot Nat.
50000 9.69897
86603 9-93753
57735 9-76144
0.23856 1.7321
60
I
02s 9.69919
22
588 9.93746
8
774 9-76173
29
29
0.23827 309
59
2
050 9.69941
573 9-93738
813 9.76202
0.23798 297
58
3
076 9.69963
559 9.93731
7
851 9.76231
29
30
29
29
29
29
29
29
29
29
0.23769 286
57
4
5
loi 9.69984
22
544 9-93724
7
7
Q
890 9.76261
0.23739 274
56
50126 9.70006
86530 9-93717
57929 9.76290
0.23710 1.7262
55
6
151 9.70028
515 9-93709
7
I
968 9-76319
0.23681 251
54
7
176 9.70050
501 9.93702
58007 9.76348
0.23652 239
S3
8
201 9.70072
486 9.93695
046 9-76377
0.23623 228
52
9
227 9.70093
22
471 9-93687
7
7
8
085 9.76406
0.23594 216
SI
50
10
50252 9.701 15
86457 9.93680
58124 9.76435
0.23565 1.7205
11
277 9-70137
442 9-9367$
162 9.76464
0.23536 193
49
12
302 9.70159
427 9.93665
201 9.76493
0.23507 182
48
13
327 9.70180
413 9-93658
8
240 9.76522
29
29
0.23478 170
47
14
352 9.70202
22
398 9-93650
7
I
279 9-76551
0.23449 159
46
15
50377 9.70224
86384 9-93643
58318 9.76580
0.23420 1.7147
45
16
403 9.76245
369 9.93636
357 9-76609
0.23391 13b
44
17
428 9.70267
354 9.93628
7
7
8
7
8
396 9-76639
30
29
29
28
29
29
0.23361 124
43
i8
453 9.70288
340 9.93621
435 9-76668
0.23332 113
42
19
478 9.70310
22
325 9.93614
474 9-76697
0.23303 102
41
20
50503 9.70332
86310 9.93606
58513 9-76725
0.23275 1.7090
40
21
528 9-70353
295 9-93599
552 9-76754
0.23246 079
39
22
553 9-70375
28 T 9.93591
591 9-76783
0.23217 067
38
23
578 9.70396
266 9.93584
7
8
631 9.76812
29
29
29
29
29
29
29
29
29
^8
0.23188 056
.37
24
603 9.70418
21
251 9-93577
670 9.76841
0.23159 045
36
25
50628 9.70439
86237 9-93569
58709 9.76870
0.23130 1.7033
35
26
654 9.70461
222 9.93562
7
8
748 9-76899
0.23101 022
34
27
679 9.70482
207 9-93554
7
8
787 9.76928
0.23072 on
33
28
704 9.70504
192 9-93547
826 9-76957
0.23043 1.6999
32
29
729 9.70525
22
178 9-93539
7
865 9.76986
0.23014 988
31
30
30
50754 9.70547
86163 9.93532
58905 9.77015
0.22985 1.6977
31
779 9.70568
148 9.93525
8
944 9-77044
0.22956 965
29
32
804 9.70590
133 9-93517
983 9-77073
0.22927 954
28
33
829 9.7061 1
119 9-93510
8
59022 9.77101
29
29
0.22899 943
27
34
854 9-70633
21
104 9-93502
7
8
061 9.77130
0.22870 932
26
25
35
50879 9.70654
86089 9.93495
59101 9.77159
0.22841 1.6920
36
904 9.70675
074 9.93487
140 9.77188 1 z
0.22812 90Q
24
37
929 9.70697
21
059 993480
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8
179 9.-772I7
29
0.22783 898
23
3«
954 9-70718
21
045 993472
218 9.77246
0.22754 887
22
39
"40
979 9-70739
22
030 9-93465
8
258 9.77274
29
29
29
29
28
0.22726 875
21
51004 9.70761
86015 9-93457
59297 9-77303
0.22697 1.6864
20
41
029 9.70782
000 9-93450
8
336 9-77332
0.22668 853
19
42
054 9.70803
85985 9-93442
376 9-77361
0.22639 842
18
43
079 9.70824
970 9.93435
8
415 9-77390
0.22610 831
17
44
104 9.70846
21
956 9-93427
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8
454 9-77418
29
29
29
'^8
0.22582 820
16
45
51 129 9.70867
85941 9-93420
59494 9-77447
0.22553 1.6808
15
46
154 9-70888
926 9.93412
7
8
533 9-77476
0.22524 797
14
47
179 9-70909
911 9.93405
573 9-77505
0.22495 786
13
48
204 9.70931
896 9-93397
612 9.77533
29
29
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0.22467 775
12
49
50
229 9.70952
21
21
881 9.93390
7
8
7
8
651 9.77562
0.22438 764
11
51254 9.70973
85866 9.93382
59691 9.77591
0.22409 1.6753
10
51
279 9-70994
851 9-93375
730 9.77619
29
29
29
28
29
08
0.22381 742
9
S2
304 9-71015
836 9-93367
770 9.77648
0.22352 731
8
53
329 9.71036
821 9.93360
8
809 9.77677
0.22323 720
7
54
55
354 9-71058
21
806 9-93352
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7
849 9-77706
0.22294 709
b
5
51379 9-71079
85792 9-93344
59888 9.77734
0.22266 1.6698
56
404 9.71 100
777 9-93337
928 9.77763
0.22237 687
4
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429 9.71 121
762 993329
967 9.77791
29
29
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0.22209 676
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454 9.71 142
747 9-93322
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60007 9.77820
0.22180 665
2
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479 9-71 163
732 9-93314
7
046 9.77849
0.22151 654
1
504 9.71184
1 717 9-93307
086 9.77877
0.22123 643
Nat.CoSLog. d. |Nat. Sin Log. d.
Nat. Cot Log.
C.d.
Log.TanNat.
f
m
31°
f
Nat. Sin Log. d.
Nat. Cos Log. d.
Nat.TanLog.
c.d.
Log. Cot Nat.
51504 9.71184 1 ,^
85717 9-93307
8
60086 9.77877
29
0.22123 1.6643
60
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529 9.71205
702 9.93299
8
126 9.77906
0.22094 632
SQ
2
554 9-71226
21
687 9-93291
I
165 9-77935
29
28
0.22065 621
58
3
579 9-71247
672 9.93284
205 9-77963
0.22037 610
0.22008 599
57
4
604 9.71268
21
657 9-93276
7
8
245 9-77992
29
28
29
56
65
5
51628 9.71289
85642 9-93269
60284 9.78020
0.21980 1.6588
b
653 9-71310
627 9.93261
8
324 9-78049
0.21951 577
54
7
678 9-71331
612 9.93253
7
8
364 9-78077
29
29
0.21923 566
53
8
703 9-71352
597 9-93246
403 9.78106
0.21894 555
52
9
728 9-71373
20
21
582 9.93238
8
7
8
443 9-78135
0-21865 545
51
10
51753 9.71393
85567 9-93230
60483 9.78163
29
28
0.21837 1.6534
0.21808 523
50
II
778 9-71414
551 9-93223
522 9.78192
49
12
803 9-71435 21
536 9-93215
562 9.78220
29
28
0.21780 512
48
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828 9.71456 f.
521 9.93207
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8
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602 9.78249
0.21751 501
47
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852 9.71477
21
21
506 9.93200
642 9.78277
-
29
08
0.21723 490
46
45
15
51877 9.71498
85491 9-93192
60681 9.78306
0.21694 1.6479
lb
902 9.71519
20
476 9-93184
721 9.78334
28
0.21666 469
44
17
927 9-71539
21
461 9-93177
761 9.78363
0.21637 458
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952 9.71560
21
446 9.93169
801 9.78391
08
0.21609 447
42
19
977 9-71581
21
20
431 9.93161
7
Q
841 9-78419
29
0.21581 436
41
20
52002 9.71602
85416 9-93154
60881 9.78448
0.21552 1.6426
40
21
026 9.71622
21
401 9.93146
Q
921 9.78476
29
0.21524 415
39
22
051 9-71643
21
385 9-93138
960 9-78505
0.21495 404
38
23
076 9.71664
21
370 9.93131
61000 9.78533
29
28
0.21467 393
37
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loi 9.71685
355 9-93123
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040 9.78562
0.21438 383
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52126 9-71705
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85340 9-93115
61080 9.78590
O.21410 1.6372
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151 9.71726
325 9.93108
120 9.78618
29
0.21382 361
34
27
175 9-71747
310 9.93100
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160 9.78647
0.21353 351
33
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200 9.71767
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294 9-93092
200 9.78675
29
28
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0.21325 340
32
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225 9.71788
21
20
279 9.93084
7
240 9.78704
0.21296 329
31
30
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52250 9.71809
85264 9-93077
61280 9.78732
0.21268 1.6319
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275 9.71829
21
249 993069
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320 9.78760
29
.0.21240 308
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32
299 9-71850
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234 9-93061
360 9.78789
O.21211 297
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324 9.71870
218 9.93053
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8
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400 9.788i'7
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O.21183 287
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349 9-71891
20
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203 9-93046
440 9.78845
29
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0.21 155 276
26
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52374 9-71911
85188 9.93038
61480 9.78874
O.21126 1.6265
25
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399 9-71932
173 9-93030
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520 9.78902
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157 9-93022
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561 9-78930
29
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0.21070 244
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448 9.71973
142 9.93014
601 9.78959
641 9-78987
O.21041 234
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473 9-71994
20
127 9.93007
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O.21013 223
21
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52498 9.72014
85112 9-92999
61681 9.79015
0.20985 1.6212
20
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522 9.72034
21
096 9-92991
721 9.79043
29
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0.20957 202
19
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547 9-7205$
081 9-92983
761 9-79072
0.20928 191
18
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572 9-72075
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066 9.92976
801 9.79100
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0.20900 181
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597 9-72096
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051 9.92968
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842 9.79128
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29
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0.20872 170
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52621 9.721 16
85035 9.92960
61882 9.79156
0.20844 1.6160
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646 9.72137
020 9.92952
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922 9-79185
0.20815 149
14
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671 9.72157
005 9.92944
962 9.79213
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0.20787 139
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696 9.72177
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84989 9.92936
62003 979241
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0.20759 128
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720 9.72198
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974 9-92929
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043 9-79269
28
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0.20731 118
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52745 9.72218
84959 9-92921
62083 9.79297
0.20703 1.6107
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770 9-72238
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943 9-92913
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124 9.79326
0.20674 097
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794 9-72259
928 9-92905
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164 9-79354
28
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819 9.72279
913 9.92897
204 9.79382
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245 9.79410
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0.20590 066
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52869 9.72320
84882 9.92881
62285 9.79438
0.20562 1.6055
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866 9.92874
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325 9.79466
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918 9.72360
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943 9-72381
836 9.92858
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406 9-79523
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967 9.72401
820 9.92850
8
446 979551
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0.20449 014 I 1
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992 9.72421
80s 9.92842
487 9-79579
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Nat. Cos Log. d.
Nat. Sin. Log. d.
|Nat.CotLog.
C.d.
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u
68^
f
Nat. Sin Log. d.
Nat. Cos Log. d.
Nat.TanLog.
c.d.
Log. Cot Nat.
52992 9.72421
20
84805 9.92842
g
62487 9.79579
28
0.20421 1.6003
60
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53017 9-72441
789 9.92834
8
527 9.79607
28
0.20393 1.5993
59
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041 9.72461
774 9-92826
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568 9-79635
28
0.20365 983
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066 9.72482
759 9.92818
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608 9.79663
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0.20337 972
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091 9.72502
20
743 9.92810
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649 9.79691
28
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0.20309 962
56
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53IIS 9.72522
84728 9.92803
62689 9-79719
0.20281 1.5952
b
140 9.72542
712 9.92795
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730 9-79747
29
28
0.20253 941
54
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164 9.72562
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697 9-92787
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770 9.79776
0.20224 931
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681 9-92779
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811 9.79804
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0.20196 921
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214 9.72602
20
666 9.92771
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852 9-79832
28
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0.20168 911
51
50
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53238 9.72622
84650 9.92763
62892 9.79860
0.20140 1.5900
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263 9.72643
635 992755
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933 9-79888
28
0.201 12 890
49
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288 9.72663
619 9.92747
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973 9-79916
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0.20084 880
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312 9-72683
604 992739
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63014 9.79944
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0.20056 869
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337 9-72703
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588 9.92731
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055 9-79972
28
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0.20028 859
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45
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53361 9.72723
84573 9.92723
63095 9.80000
0.20000 1.5849
lb
386 9-72743
557 9-92715
136 9.80028
28
0.19972 839
44
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411 9.72763
542 9.92707
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177 9.80056
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0.19944 829
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435 9-72783
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526 9.92699
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217 9.80084
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0.19916 818
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460 9.72803
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511 9.92691
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258 9.80112
28
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0.19888 808
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40
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53484 9.72823
84495 9.92683
63299 9.80140
0.19860 1.5798
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509 9.72843
480 9.92675
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340 9.80168
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28
0.19832 788
39
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534 9-72863
464 9.92667
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380 9.80195
0.19805 778
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558 9.72883
19
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448 9.92659
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421 9.80223
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0.19777- 768
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583 9.72902
433 9-92651
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53607 9.72922
84417 9.92643
63503 9.80279
0.19721 1.5747
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632 9.72942
402 9.92635
544 9-80307
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0.19693 737
34
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656 9.72962
386 9.92627
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584 9.80335
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0.19665 727
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681 9.72982
370 9.92619
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625 9.80363
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0.19637 717
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29
705 9-73002
20
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355 9.92611
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666 9.80391
28
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0.19609 707
31
30
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53730 9-73022
84339 9.92603
63707 9.80419
0.19581 1.8697
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754 973041
324 9.92595
748 9.80447
27
28
0.19553 687
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779 9-73061
308 9.92587
789 9-80474
0.19526 677
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804 9.73081
292 9.92579
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830 9.80502
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0.19498 667
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34
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828 9.73101
20
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277 9-92571
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871 9.80530
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0.19470 657
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53853 9-73121
84261 9.92563
63912 9.80558
0.19442 1-5647
25
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877 9-73140
245 9-92555
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8
953 9-80586
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0.19414 637
24
37
902 9.73160
"
230 9-92546
994 9.80614
28
0.19386 627
23
3»
92b 9.73180
20
214 9.92538
8
64035 9.80642
27
28
28
0.19358 617
22
39
40
951 9.73200
19
198 9-92530
8
076 9,80669
0.19331 607
21
20
53975 9-73219
84182 9.92522
641 17 9.80697
0.19303 1-5597
41
54000 9-73239
167 9.92514
8
158 9.80725
08
0.19275 587
19
42
024 9-73259
19
151 9.92506
8
199 9-80753
28
0.19247 577
18
43
049 9-73278
135 9.92498
8
240 9.80781
27
28
08
0.19219 567
17
44
073 9-73298
20
19
120 9.92490
8
9
8
281 9.80808
0.19192 557
lb
45
54097 9-73318
84104 9.92482
64322 9.80836
0.19164 1.5547
15
46
122 9-73337
088 9.92473
363 9.80864
08
0.19136 537
14
47
146 9-73357
072 9.92465
8
404 9.80892
27
'^8
0.19108 527
13
48
171 9-73377
19
20
19
057 9-92457
8
446 9.80919
0.19081 517
12
49
195 9-73396
041 9.92449
8
8
487 9.80947
28
28
0.19053 507
II
50
54220 9-73416
84025 9.92441
64528 9.80975
0.19025 1.5497
10
51
244 9-73435
009 9.92433
8
569 9.81003
27
28
0.18997 487
9
S2
269 9-73455
"
83994 9.92425
9
8
610 9.81030
0.18970 477
8
=^3
293 9.73474
19
978 9.92416
652 9.81058
28
0.18942 468
7
54
317 9-73494
19
962 9.92408
8
8
693 9.81086
27
"8
O.18914 458
b
55
54342 9-73513
83946 9.92400
64734 9.81 1 13
0.18887 1.5448
5
,0
366 9-73533
930 9.92392
775 9.81 141
28
0.18859 438
4
57
391 9-73552
915 9.92384
8
817 9.81 169
27
28
O.18831 428
3
58
415 9-73572
899 9-92376
9
858 9.81 196
0.18804 418
2
ro
440 9-73591
19
883 9.92367
899 9.81224
o«
0.18776 408
I
404 9-73611
2° 1 8b7 9-92359
941 9.81252 1 - \ 0.18748 399
Nat. Cos Log. d.
Nat. Sin Log. d.
Nat. Cot Log.
c.d.
Log.TanNat.
f
57'
33°
1
Nat. Sin Log. d.
Nat. Cos Log
d.
Nat.TanLog.
c.d.
Log. Cot Nat.
"^
54464 9.7361 1
19
83867 9.92359
8
64941 9.81252
0.18748 1.5399
60
I
488 9.73630
851 9.92351
8
982 9.81279
28
0.18721 389
59
2
513 9-73650
19
835 9.92343
8
65024 9.81307
0.18693 379
58
3
537 9.73669
819 9.92335
9
8
Q
065 9.81335
0.18665 369
S7
4
561 9.73689
19
19
20
804 9.92326
106 9.81362
27
28
28
0.18638 359
56
55
5
54586 9.73708
83788 9.92318
65148 9.81390
0.18610 1.5350
6
610 9.73727
772 9.92310
189 9.8I4I8
0.18582 340
54
7
635 9.73747
19
19
20
19
19
20
756 9.92302
9
8
231 9.81445
28
27
28
28
0.18555 330
53
8
659 9.73766
740 9.92293
272 9.81473
0.18527 320
52
9
683 9.73785
724 9.92285
8
8
314 9.81500
0.18500 311
51
50
10
54708 9.73805
83708 9.92277
65355 9.81528
0.18472 1.5301
11
732 9.73824
692 9.92269
397 9.81556
0.18444 291
4Q
12
756 9.73843
676 9.92260
9
8
438 9.81583
27
28
0.18417 282
48
13
781 973863
19
19
660 9.92252
8
9
8
8
8
9
8
8
480 9.81611
0.18389 272
47
14
805 9.73882
645 9.92244
521 9.81638
27
28
0.18362 262
46
45
15
54829 9.73901
83629 9.92235
65563 9.81666
0.18334 1.5253
lb
854 9.73921
19
19
19
19
613 9.92227
604 9.81693
27
28
0.18307 243
44
17
878 9.73940
597 9.92219
646 9.81721
0.18279 233
43
18
902 9.73959
581 9.9221 1
688 9.81748
27
08
0.18252 224
42
19
927 9.73978
565 9.92202
729 9.81776
27
28
0.18224 214
41
20
54951 9-73997
83549 9.92194
65771 9.81803
0.18197 1.5204
40
21
975 9.74017
19
19
19
19
533 9.92186
813 9.81831
0.18169 195
39
22
999 9-74036
517 9.92177
9
8
8
9
8
8
854 9.81858
0!
0.18142 185
38
23
55024 9.74055
501 9.92169
896 9.81886
0.18114 175
37
24
048 9.74074
485 9.92161
938 9.81913
27
28
0.18087 166
36
25
55072 9.74093
83469 9.92152
65980 9.81941
0.18059 1.5 156
35
2b
097 9.74113
19
19
19
19
19
19
19
453 9.92144
66021 9.81968
28
0.18032 147
34
27
121 9.74132
437 9.92136
063 9.81996
0.18004 137
33
2b
145 9.74151
421 9.92127
9
105 9.82023
27
0.17977 127
32
29
169 9.74170
405 9.92119
8
147 9.82051
27
28
0.17949 118
31
30
55194 9.74189
83389 9.921 1 1
66189 9.82078
0.17922 1.5108
30
31
218 9.74208
373 9.92102
9
8
8
230 9.82106
0.17894 099
29
32
242 9.74227
356 9.92094
272 9.82133
s
0.17867 089
28
33
266 9.74246
340 9.92086
314 9.82161
0.17839 080
27
34
291 9.74265
■••y
19
19
19
19
324 9.92077
9
8
9
8
356 9.82188
27
27
28
0.17812 070
26
25
35
55315 9.74284
83308 9.92069
66398 9.82215
0.17785 1.5061
3('
339 9.74303
•292 9.92060
440 9.82243
0.17757 051
24
37
363 9.74322
276 9.92052
482 9.82270
^8
0.17730 042
23
3«
388 9.74341
260 9.92044
524 9.82298
0.17702 032
22
39
40
412 9.74360
••^9
19
19
19
19
244 9.92035
9
8
566 9.82325
27
27
28
0.17675 023
21
55436 9.74379
83228 9.92027
66608 9.82352
0.17648 1.5013
20
41
460 9.74398
212 9.92018
8
8
650 9.82380
0.17620 004
19
42
484 9.74417
195 9.92010
692 9.82407
28
0.17593 1.4994
18
43
509 9.74436
179 9.92002
734 9.82435
0.17565 985
17
44
45
533 9-74455
19
19
163 9.91993
9
8
27
27
28
0.17538 975
lb
15
55557 9.74474
83147 9.91985
66818 9.82489
0.17511 14966
46
581 9.74493
131 9.91976
9
8
860 9.82517
27
27
28
0.17483 957
14
47
605 9.74512
■^y
115 9.91968
902 9.82544
0.17456 947
13
48
630 9.74531
098 9.91959
9
944 9.82571
0.17429 938
12
49
654 9.74549
19
19
19
082 9.91951
9
8
986 9.82599
27
27
28
0.17401 928
11
10
50
55678 9.74568
83066 9.91942
67028 9.82626
0.17374 1.4919
51
702 9.74587
050 9-91934
071 9.82653
113 9.82681
0.17347 910
9
S2
726 9.74606
034 9-91925
27
27
27
0.17319 900
8
SS
750 9.74625
■••9
017 9.91917
155 9.82708
0.17292 891
7
54
775 9.74644
19
18
001 9.91908
9
8
197 9.82735
0.17265 882
b
55
55799 9.74662
82985 9.91900
67239 9.82762
0.17238 14872
5
823 9.74681
^y
969 9.91891
282 9.82790
27
27
0.17210 863
4
'57
847 9-74700
19
953 9.91883
324 9.82817
0.17183 854
3
58
871 9.74719
i"i
936 9.91874
8
366 9.82844
0.17156 844
2
ro
895 9.74737
920 9.91866
409 9.82871
0.17129 835
1
919 9.74756
■••9
904 9.91857
9
451 9.82899
0.17101 826
Nat. Cos Log. d.
Nat. Sin Log
d.
Nat. Cot Log.
cd.
Log.TanNat.
/
56'
34^
' Nat. Sin Log. d.
NatCoSLosf. d.
Nat.TanLog.
d. Log. Cot Nat,
55919
943
968
992
56016
9-74756
9-74775
9.74794
9.74812
9.74831
56040
064
088
112
136
9.74850
9.74868
9-74887 ,
9-74906 ! g
9-74924
56160
184
208
232
256
9-74943
9.74961
9.74980
9.74999
9.75017
56280
305
329
353
377
9-75036
975054
9.75073
9.75091
9.751 10
56401
425
449
473
497
9.75128
9.75147
9-75165
9.75184
9.75202
56521
545
569
593
617
9.75221
975239
9-75258
9-75276
9-75294
56641
665
689
713
736
9-75313
9-75331
9-75350
9-75368
9-75386
56760
784
808
832
856
9-75405
9-75423
9-75441
9-75459
9.75478
Daao
904
928
952
976
9.75496
9-75514
9-75533
9-75551
9-75569
57000
024
047
071
095
9-75587
975605
9-75624
9.75642
9-75660
57119
143
167
191
215
9-75678
9.75696
9-75714
9-75733
9-75751
57238
262
286
310
334
358
9-75769
9-7.5787
9-75805
9-75823
9.75841
9-75859
82904
887
871
855
839
9.91857
9.91849
9.91840
9.91832
9.91823
806
790
773
757
9.91815
9.91806
9.91798
9.91789
9.91781
8274T
724
708
692
675
9.91772
9.91763
9-91755
9.91746
9.91738
82659
643
626
610
593
9.91729
9.91720
9.91712
9.91703
9.91695
82577
561
544
528
511
9.91686
9.91677
9.91669
9.91660
9-91651
82495
478
462
446
429
9.91643
9.91634
9.91625
9.91617
9.91608
82413
396
380
363
347
9-91599
9.91591
9.91582
9-91573
9-91565
82330
314
297
281
264
9-91556
9-91547
9-91538
9-91530
9.91521
82248
231
214
198
181
9.91512
9.91504
991495
9.91486
9.91477
82165
148
132
115
098
9.91469
9.91460
9.91451
9.91442
9-91433
82082
065
048
032
oiS
9.91425
9.91416
9.91407
9.91398
9-91389
81999
982
965
949
932
915
9.91381
9.91372
9-91363
9-91354
9-91345
9-91336
67451
493
536
578
620
9.82899
9.82926
9-82953
9.82980
9.83008
67663
705
748
790
832
9-83035
9.83062
9-83089
9.83117
9-83144
67875
917
960
68002
045
9.83171
9.83198
9.83225
9-83252
9.83280
130
173
215
258
9-8.3307
9-83334
9-83361
9-83388
9.83415
68301
343
386
429
471
9.83442
9.83470
9.83497
9.83524
9.83551
68514
557
600
642
685
9-83578
9.83605
9.83632
983659
9.83686
68728
771
814
857
900
9-83713
9.83740
9.83768
9-83795
9.83822
68942
985
69028
071
114
9.83849
9.83876
9.83903
9-83930
9-83957
69157
200
243
286
329
9.83984
9.84011
9-84038
9-84065
9.84092
69372
416
459
502
545
9.84119
9.84146
9.84173
9.84200
9.84227
69588
631
675
718
761
9.84254
9.84280
9.84307
9-84334
9.84361
69804
847
891
934
977
70021
9.84388
9.84415
9.84442
9.84469
9.84496
9-84523
7101 1.4826
7074 816
7047 807
7020 798
6992 788
6965 1-4779
6938 770
6911 761
6883 751
6856 742
6829 1.4733
6802
6775
6748
6720
724
715
705
696
6693 1.4687
6666 678
6639 669
6612 659
6585 650
6558 1.4641
6530 632
6503 623
6476 614
6449 605
6422 1.4596
6395 586
6368 577
6341 568
6314 559
6287 1.4550
6260 541
6232 532
6205 523
6178 514
6151 1.4505
6124 496
6097 487
6070 478
6043 469
6016 1.4460
5989 451
5962 442
5935 433
5908 424
5881 1.4415
5854 406
5827 397
5800 388
5773 379
5746 1.4370
5720 361
5693 352
5666 344
5639 335
5612 1.4326
558S 317
5558 308
5531 299
5504 290
5477 281
Nat. Cos Log. d,
Nat. Sin Log. d.
66^
Nat. Cot Log.
c.d. Log.TanNat.
{
35
f
Nat. Sin Log. d.
Nat. Cos Log
d.
Nat.TanLog.
c.d. Log. Cot Nat.
57358 9-75859 i 18
81915 9.91336
8
70021 9.84523
0.15477 1.428 1
60
I
381 9-75877 i 18
899 9.91328
9
9
9
9
9
9
8
064 984550
06
0.15450 273
59
2
405 9-75895 18
882 9.91319
107 9.84576
27
27
27
27
27
0.15424 264
58
3
429 9.75913 1 t8
865 9-91310
151 9.84603
0.15397 255
57
4
453 9-75931
18
848 9.91301
194 9-84630
0.15370 246
56
55
5
57477 9-75949
81832 9.91292
70238 9.84657
0.15343 1.4237
6
501 9.75967
iH
815 9.91283
281 9.84684
0.15316 229
54
7
524 9-75985
18
798 9-91274
325 9.847II
0.15289 220
S3
8
548 9-76003
iR
782 9.91266
9
9
9
9
9
9
9
9
9
368 9-84738
26
0.15262 211
S2
9
572 9.76021
18
765 9.91257
412 9-84764
27
27
0.15236 202
51
10
57596 9-76039
T«
81748 9.91248
70455 9.84791
0.15209 1.4193
II
619 9-76057 ! ;«
731 9.91239
499 9-84818
0.15182 185
49
12
643 9.76075
18
714 9.91230
542 9.84845
0.15155 176
48
1.3
667 9-76093
18
698 9.91221
586 9.84872
0.15128 167
47
14
691 9.761 1 1
18
681 9.91212
629 9.84899
26
27
27
0.15101 158
46
45
15
57715 9.76129
TT
81664 9-91203
70673 9.84925
0.15075 1.4150
I6
738 9.76146 ^
647 9.91194
717 9-84952
0.15048 141
44
17
762 9.76164 : 8
631 9.91185
760 9-84979
0.15021 132
43
i8
786 9.76182 1 ;«
614 9.91176
9
804 9.85006
27
0.14994 124
42
19
810 9.76200
18
t8
597 9-91167
9
9
9
8
848 9.85033
27
26
0.14967 115
41
40
20
57833 9-762i«
81580 9.91158
70891 9.85059
0.14941 1.4106
21
857 9-76230 ,,
563 9.91149
935 9-85086
0.14914 097
39
22
881 9.76253
18
546 9.91141
9
9
9
9
9
9
9
9
9
9
9
9
10
9
9
9
9
9
9
9
9
9
9
9
9
9
979 9-85113
"7
27
26
0.14887 089
38
23
904 9.76271
18
530 9.91 132
71023 9.85140
0.14860 080
37
24
928 9.76289
t8
513 9.91123
066 9.85166
27
27
27
05
0.14834 071
36
35
25
57952 9-76307
18
81496 9.91114
71 no 9.85193
0.14807 1.4063
26
976 9-76324
479 9-91 105
154 9.85220
0.14780 054
34
27
999 9-76342
18
462 9.91096
198 9.85247
0.14753 045
33
28
58023 9.76360
18
445 9-91087
242 9.85273
0.14727- 037
32
29
30
047 9-76378
17
18
428 9.91078
285 9-85300
27
27
27
0.14700 028
31
30^
58070 9-76395
8 14 1 2 9.91069
71329 9.85327
0.14673 1.4019
31
094 9.76413
18
395 9.91060
373 9-85354
0.14646 on
29
32
118 9.76431
;?
378 9.91051
417 9.85380
0.14620 002
28
33
141 9-76448
361 9.91042
461 9.85407
27
26
27
27
06
0.14593 1.3994
27
34
35
165 9.76466 1 11
344 9-91033
505 9-85434
0.14566 985
26
58189 9.76484
17
18
81327 9.91023
71549 9.85460
0.14540 1.3976
25
36
212 9.76501
310 9.91014
593 9-85487
0.14513 968
24
37
236 9-76519
18
293 9.91005
637 9-85514
0.14486 959
23
38
260 9-76537
681 9-85540
0.14460 951
22
39
40
283 9-76554
17
18
18
259 9.90987
725 9.85567
27
27
06
0.14433 942
21
20
58307 9-76572
81242 9.90978
71769 9.85594
0.14406 1.3934
41
330 9-76590
17
18
225 9.90969
813 9.85620
0.14380 925
19
42
354 9-76607
208 9.90960
857 9-85647
27
0.14353 916
18
43
378 9.76625
191 9.90951
901 9.85674
06
0.14326 908
17
44
401 9.76642
18
174 9-90942
946 9.85700
27
27
^6
0.14300 899
16
45
58425 9.76660
81 157 9-90933
71990 9.85727
0.14273 1.3891
15
46
449 9-76677
140 9.90924
72034 9.85754
0.14246 882
14
47
472 9.76695
123 9.90915
078 9.85780
0.14220 874
13
48
496 9.76712
^8
17
18
106 9.90906
122 9.85807
0.14193 865
12
49
50
519 9-76730
089 9.90896
9
9
9
9
9
9
167 9-85834
26
0.14166 857
11
58543 9-76747
81072 9.90887
72211 9.85860
0.14140 1.3848
10
■^i
567 9.76765
055 990878
255 9-85887
0.14113 840
9
S2
590 9.76782
18
038 9.90869
299 9-85913
27
27
26
27
0.14087 831
8
';3
614 9.76800
021 9.90860
344 985940
0.14060 823
7
54
55
637 9.76817
17
18
004 9.90851
388 9-85967
0.14033 814
6
58661 9.76835
80987 9.90842
72432 9.85993
0.14007 1.3806
5
S6
684 9.76852
17
18
970 9.90832
9
477 9.86020
0.13980 798
4
708 9.76870
953 990823
521 9.86046
27
27
'>6
0.13954 789
3
58
731 9.76887
17
936 9.90814
9
9
565 9.86073
0.13927 781
2
i
755 9-76904
18
919 9.90805
610 9.86100
0.13900 772
I
779 9-76922
902 9-90796
9
654 9.86126
0.13874 764
Nat.CoSLog. d.
Nat. Sin Log.
d.
Nat. Cot Log.
c.d.
Log.Tan Nat.
f
64
t°
36
t
Nat. Sin Log. d.
Nat. Cos Log. d.
Nat. Tan Log.
c.d.
Log. Cot Nat.
58779 9-76922
80902 9.90796
72654 9.86126
0.13874 1.3764
60
I
802 9.76939
iS
885 9.90787
699 9.86153
06
0.13847 755
59
2
826 9.76957
17
867 9.90777
743 9-86179
0.13821 747
58
3
849 9-76974
850 9.90768
788 9.86206
26
27
'^6
0.13794 739
S7
4
873 9.76991
17
18
17
17
18
833 9.90759
9
9
9
832 9.86232
0.13768 730
56
5
58896 9.77009
80816 9.90750
72877 9-86259
0.13741 1.3722
55
b
920 9.77026
799 9.90741
921 9.86285
%
0.13715 713
54
7
943 9-77043
782 9.90731
966 9.86312
0.13688 705
53
b
967 9.77061
17
17
17
18
765 9.90722
9
9
73010 9.86338
0.13662 697
52
9
990 9.77078
748 9.90713
055 9-86365
27
0.13635 688
51
10
59014 9-77095
80730 9-90704
73100 9.86392
0.13608 1.3680
50
II
037 9-77"2
713 9.90694
. 9
9
144 9.86418
0.13582 672
49
12
061 9-77130
17
696 9.90685
189 9.86445
o(S
0.13555 663
48
13
084 9.77147
679 9.90676
234 9.86471
0.13529 655
47
14
108 9.77164
17
17
18
662 9.90667
9
10
9
278 9.86498
27
26
0.13502 647
46
15
S9131 9.77181
80644 9-90657
73323 9.86524
0.13476 1.3638
45
16
154 9-77199
627 9.90648
368 9.86551
26
05
0.13449 630
44
17
178 9.77216
17
17
18
17
17
17
610 9.90639
9
9
413 9-86577
0.13423 622
43
l8
201 9.77233
593 9.90630
457 9.86603
0.13397 613
42
19
225 9-77250
576 9.90620
9
502 9.86630
26
0.13370 605
41
20
59248 9.77268
80558 9.90611
73547 9-86656
0.13344 1.3597
40
21
272 9.77285
541 9.90602
"6
0.13317 588
39
22
295 9-77302
524 9.90592
9
637 9.86709
27
0.13291 580
38
23
318 9-77319
507 9.90583
681 9.86736
0.13264 572
37
24
342 9.77336
17
17
17
17
489 9.90574
9
9
726 9.86762
27
05
0.13238 564
36
25
59365 9-77353
80472 9.90565
73771 9.86789
0.13211 1.3555
35
2b
389 9-77370
455 9.90555
816 9.86815
0.13185 547
34
27
412 9.77387
438 9.90546
9
9
861 9.86842
o(=,
0.13158 539
33
28
436 9.77405
17
17
17
17
17
420 9.90537
906 9.86868
'^f,
0.13132 531
32
29
459 9-77422
403 9.90527
9
9
951 9.86894
27
2*^
0.13106 522
31
30
59482 9.77439
80386 9.90518
73996 9.86921
0.13079 1.3514
30
31
506 9.77456
368 9.90509
74041 9.86947
%
0.13053 506
29
32
529 9.77473
351 9.90499
9
086 9.86974
0.13026 498
28
33
552 9-77490
334 9-90490
131 9.87000
27
26
"6
0.13000 490
27
34
35
576 9-77507
17
17
17
17
17
17
17
316 9.90480
9
176 9.87027
0.12973 481
2b
59599 9-77524
80299 9.90471
74221 9-87053
p.12947 1.3473
25
3^^
622 9.77541
282 9.90462
9
267 9.87079
0.12921 465
24
37
646 9-7755?
264 9.90452
312 9.87106
06
0.12894 457
23
3«
669 9.77575
247 990443
9
9
10
357 9-87132
^(^
0.12868 449
22
39
40
693 9.77592
230 9.90434
402 9.87158
27
06
0.12842 440
21
59716 9.77609
80212 9.90424
74447 9-87185
0.12815 1.3432
20
41
739 9.77626
195 9-90415
9
492 9.87211
0.12789 424
19
42
763 9.77643
17
178 9-90405
538 9.87238
27
26
0.12762 416
18
43
786 9.77660
TT
160 9.90396
583 9.87264
06
0.12736 408
17
44
809 9.77677
17
143 9-90386
9
628 9.87290
27
06
0.12710 400
lb
45
59832 9.77694
80125 9.90377
74674 9.87317
0.12683 1.3392
16
4b
856 9.77711
17
108 9.90368
9
719 9.87343
06
0.12657 384
14
47
879 9.77728
16
091 9-90358
764 9-87369
27
06
0.12631 375
13
48
902 9.77744
073 9-90349
9
810 9.87396
0.12604 367
12
49
926 9.77761
17
17
17
17
056 9-90339
9
855 9-87422
26
27
0.12578 359
11
50
59949 9-77778
80038 9.90330
74900 9.87448
0.12552 1.3351
10
51
972 9.77795
021 9.90320
9
946 9.87475
0.12525 343
9
.S2
995 9-77812
003 9.9031 1
991 9.87501
06
0.12499 335
8
S3
60019 9.77829
79986 9.90301
75037 9-87527
27
26
■2(^
0.12473 327
7
54
042 9.77846
16
968 9.90292
10
082 9-87554
0.12446 319
b
55
60065 9.77862
79951 9.90282
75128 9.87580
0.12420 1. 33 II
5
St
089 9.77879
17
934 9-90273
9
173 9.87606
%
0.12394 303
4
57
112 9.77896
17
916 9.90263
219 9-87633
0.12367 295
3
5«
135 9.77913
17
899 9-90254
9
264 9-87659
26
0.12341 287
2
;'J?.
158 9.77930
881 9.90244
310 9.87685
26
0.12315 278
I
60
182 9.77946
864 9.90235
9
355 9-8771 1
0.12289 270
Nat. Cos Log. d.
Nat. Sin Log. d.
Nat. Cot Log.
C.d.
Log.TanNat.
lI
63^
Nat. Sin Log. d.
37
Nat. Cos Log. d.
Nat.TanLog. c.d
Log. Cot Nat,
35
36
37
38
40
41
42
43
44
60182
205
228
251
274
9.77946
977963
9.77980
9.77997
9.78013
60298
321
344
367
390
9.78030
9.78047
9.78063
9.78080
978097
60414
437
460
483
506
9.781 13
9.78130
9.78147
9.78163
9.78180
60529
553
576
599
622
9.78197
9.78213
9.78230
9.78246
9.78263
60645
668
691
714
738
9.78280
9.78296
9.78313
9.78329
9.78346
60761
784
807
830
853
9.78362
978379
978395
9.78412
9.78428
60876
899
922
945
968
9-78445
9.78461
9.78478
9.78494
9.78510
60991
61015
038
061
084
9.78527
9-78543
9.78560
9.78576
9.78592
61107
130
176
199
9.78609
9.78625
9.78642
9.78658
9-78674
61222
245
268
291
314
9.78691
9.78707
9.78723
9-78739
9.78756
61337
360
383
406
429
9.78772
9.78788
9.78805
9.78821
9.78837
6145 1
474
497
520
543
566
9-78853
9.78869
9.78886
9.78902
9.78918
9.78934
79864
846
829
811
793
9.9023.5
9.90225
9.90216
9.90206
9.90197
79776
758
741
723
706
9.90187
9.90178
9.90168
9.90159
9.90149
79688
671
653
635
618
9.90139
9.90130
9.90120
9.90111
9.90101
79600
583
565
547
530
9.90091
9.90082
9.90072
9.90063
9.90053
79512
494
477
459
441
9-90043
9.90034
9.90024
9.90014
9.90005
79424
406
371
353
9-8999$
9.89985
9.89976
9.89966
9.89956
79335
318
300
282
264
9.89947
9-89937
9.89927
9.89918
9.89908
79247
229
211
193
176
9.89898
9.89888
9.89879
9.89869
9.89859
79158
140
122
105
087
9.89849
9.89840
9.89830
9.89820
9.89810
79069
051
033
016
78998
9.89801
9.89791
9.89781
9.89771
9.89761
78980
962
944
926
908
9.89752
9.89742
9.89732
9.89722
9.89712
78891
873
855
837
819
801
9.89702
9.89693
9.89683
9.89673
9.89663
9.89653
75355
401
447
492
538
9.8771 1
9.87738
9.87764
9.87790
9.87817
75584
629
675
721
767
9.87843
9.87869
9.87895
9.87922
9.87948
75812
858
904
950
996
9.87974
9.88000
9.88027
9.88053
9.88079
76042
088
134
180
226
9.88105
9.88131
9.88158
9.88184
9.88210
76272
318
364
410
456
9.88236
9.88262
9.88289
9.88315
9.88341
76502
548
594
640'
686
9.88367
988393
9.88420
9.88446
9.88472
76733
779
825
871
918
9.88498
9.88524
9.88550
9.88577
9.88603
76964
77010
057
103
149
9.88629
9.88655
9.88681
9.88707
9.88733
77196
242
289
335
382
9.88759
9.88786
9.88812
9.88838
9.88864
77428
475
521
568
615
9.88890
9.88916
9.88942
9.88968
9.88994
77661
708
754
801
9.89020
9.89046
9.89073
9.89099
9.89125
77895
941
988
78035
082
129
9.89151
9.89177
9.89203
9.89229
989255
9.89281
Nat. Sin Log. d. Nat. Cot Log. c.d, Log.TanNat
0.12289
0.12262
0.12236
0.12210
0.12183
1.3270
262
254
246
238
0.12157
0.12131
0.12105
0.12078
0.12052
1.3230
222
214
206
198
0.12026
0.12000
0.11973
0.11947
0.11921
1.3190
182
175
167
159
0.11895
0.11869
0.11842
0.11816
0.11790
1-3151
143
135
127
119
0.11764
0.11738
0.11711
0.11685
0.11659
1.3111
103
095
087
079
0.11633
0.11607
0.11580
0.11554
0.11528
1.3072
064
056
048
040
0.11502
0.11476
0.11450
0.11423
0.11397
1.3032
024
017
009
001
0.11371
0.11345
0.11319
0.11293
0.11267
1.2993
985
977
970
962
0.11241
0.11214
0.11188
0.11162
0.1 1 136
1.2954
946
938
931
923
O.IIIIO
0.11084
0.11058
0.11032
0.11006
1.2915
907
900
892
0.10980
0.10954
0.10927
0.10901
0.10875
1.2876
869
861
853
846
0.10849
0.10823
0.10797
0.10771
0.1074^
0.10719
I.2fc
830
822
815
807
799
Nat. Cos Log. d
62^
38
f
Nat. Sin Log. d.
Nat. Cos Log. d.
Nat.TanLog.
c.d.
Log. Cot Nat.
61566 9.78934
16
78801 9.89653
78129 9.89281
05
0.10719 1.2799
60
I
589 9.78950
17
16
783 9-89643
TO
175 9.89307
06
0.10693 792
59
2
612 9.78967
765 9-89633
222 9-89333
06
0.10667 784
58
3
635 9-78983
16
747 9-89624
9
269 9.89359
06
0.10641 776
^7
4
658 9.78999
16
t6
729 9.89614
10
316 9-89385
26
26
0.10615 769
56
5
6I68I 9.79015
787 1 1 9.89604
78363 9.894II
0.10589 1.2761
55
b
704 979031
16
694 9.89594
410 9-89437
og
0.10563 753
54
7
726 9-79047
16
676 9.89584
457 9-89463
26
0.10537 746
53
8
749 9.79063
16
658 9-89574
504 9.89489
06
0.10511 738
52
9
10
772 9.79079
16
16
640 9-89564
10
551 9.89515
26
"6
0.10485 731
51
50
61795 9-79095
78622 9.89554
78598 9.89541
0.10459 1.2723
II
818 9.791 1 1
17
16
604 9-89544
645 9.89567
og
0.10433 715
49
12
841 9.79128
586 9.89534
692 9.89593
06
0.10407 708
48
13
864 9.79144
16
568 9.89524
739 9.89619
og
0.10381 700
47
14
887 9.79160
16
t6
550 9.89514
10
9
786 9.89645
26
0.10355 693
46
15
61909 9.79176
78532 9.89504
78834 9.89671
0.10329 1.2685
45
lb
932 9.79192
16
514 9.89495
881 9.89697
Ofi
0.10303 677
44
17
955 9-79208
16
496 9.89485
928 9.89723
og
0.10277 670
43
l8
978 9.79224
16
478 9.89475
975 9.89749
og
0.10251 662
42
19
62001 9.79240
16
16
460 9.89465
10
79022 9.89775
26
'^6
0.10225 655
41
40
20
62024 9.79256
78442 9.89455
79070 9.89801
0.10199 1.26^
0.10173 - "^
21
046 9.79272
16
424 9.89445
TO
117 9.89827
og
39
22
069 9.79288
16
405 9.89435
TO
164 9.89853
'-'6
0.10147 632
38
23
092 9.79304
15
16
16
387 9.89425
TO
212 9.89879
og
0.I0I2I 624
37
24
115 9.79319
369 9.89415
10
259 9.89905
26
26
0.10095 617
36
25
62138 9.79335
78351 9.89405
79306 9.89931
0.10069 1.2609
35
2b
160 9.79351
16
333 9.89395
354 9.89957
401 9.89983
26
0.10043 602
34
27
183 9-79367
16
315 9.89385
-^6
O.IOOI7 594
33
28
206 9.79383
16
297 9.89375
J J
449 9-90009
^f)
0.09991 587
32
29
229 9.79399
16
t6
279 9.89364
10
496 9-90035
26
0^
0.09965 579
31
30
62251 9.79415
78261 9.89354
79544 9.90061
0.09939 1.2572
30
31
274 9.79431
16
243 9.89344
591 9.90086
0.09914 564
29
32
297 9.79447
16
225 9.89334
639 9-90112
og
0.09888 557
28
33
320 9.79463
15
16
16
206 9.89324
686 9.90138
-""^
0.09862 549
27
34
35^
342 9.79478
188 9.89314
10
734 9.90164
26
-^6
0.09836 542
2b
25
62365 9.79494
78170 9-89304
79781 9.90190
0.09810 1.2534
3b
388 9.79510
16
152 9.89294
TO
829 9.90216
^f)
0.09784 527
24
37
411 9.79526
16
134 9.89284
TO
877 9.90242
^(^
0.09758 519
23
3H
433 9.79542
16
116 9.89274
TO
924 9.90268
og
0.09732 512
22
39
456 9.79558
15
t6
098 9.89264
10
972 9-90294
26
og
0.09706 504
21
40
62479 9.79573
78079 9.89254
80020 9.90320
0.09680 1.2497
20
41
502 9.79589
t6
061 9.89244
067 9.90346
%
0.09654 489
19
42
524 9.79605
16
043 989233
115 9-90371
0.09629 482
18
43
547 9.79621
15
16
16
025 9.89223
163 9.90397
"6
0.09603 475
17
44
570 9.79636
007 9.89213
10
211 9.90423
26
"6
0.09577 467
lb
45
62592 9.79652
77988 9.89203
80258 9-90449
0.09551 1.2460
15
4b
615 9.79668
16
970 9.89193
10
306 9-90475
'^(^
0.09525 452
14
47
638 9.79684
952 9.89183
354 9-90501
'>6
0.09499 445
13
48
660 9.79699
934 9.89173
402 9-90527
'>6
0.09473 437
12
49
683 9.79715
16
916 9.89162
10
450 9-90553
25
0.09447 430
II
50
62706 9.79731
77897 9.89152
80498 9.90578
0.09422 1.2423
10
51
728 9.79746
879 9.89142
546 9.90604
og
0.09396 415
9
52
751 9.79762
16
861 9.89132
594 9.90630
og
0.09370 408
8
53
774 9-79778
15
16
16
IS
16
16
843 9.89122
642 9.90656
og
0.09344 401
7
54
796 9-79793
824 9.891 12
II
690 9.90682
26
26
0.09318 393
6
~5"
55
62819 9.79809
77806 9.89101
80738 9.90708
0.09292 1.2386
56
842 9.79825
788 9.89091
786 9.90734
25
26
0.09266 378
4
57
864 9.79840
769 9.89081
TO
834 9.90759
882 9.90785
0.09241 371
3
5B
887 9-79856
751 9.89071
26
0.0921S 364
2
ro
909 9.79872
15
733 9-89060
TO
930 9.9081 I
og
0.09189 356
I
932 9-79887
715 9-89050
978 9.90837
0.09163 349
Nat. Cos Log. d.
Nat. Sin Log. d.
Nat. Cot Log.
c.d.
Log.TanNat.
/
61"
i
39^
■^
Nat. Sin Log. d. 1
Nat. Cos Log. d.|
Nat.TanLog.
=.d. Log.CotNat.|
62932 9-79887
16
77715 9-89050
80978 9-90837
05
0.09163 1.2349
60
I
955 9-79903
15
696 9.89040
81027 9-90863
^f)
0-09137 342
59
2
977 9-79918
678 9.89030
075 9.90889
S
0.091 1 1 334
58
3
63000 9.79934
16
660 9.89020
123 9.90914
0.09086 327
.57
4
022 9.79950
15
16
641 9.89009
171 9-90940
26
0.09060 320
56
5
63045 9.79965
77623 9.88999
81220 9.90966
0.09034 1.2312
55
6
068 9.79981
15
268 9-90992
05
0.09008 305
54
7
090 979996
586 9.88978
316 9.9IOI8
25
06
0.08982 298
53
8
113 9.80012
15
16
15
568 9-88968
364 9.91043
0.08957 290
52
9
10
135 9.80027
550 9.88958
413 9.91069
26
26
0.08931 283
51
50
63158 9.80043
77531 9.88948
8 146 I 9.91095
0.08905 1.2276
II
180 9.80058
513 9-88937
510 9.91 121
26
0.08879 268
49
12
203 9.80074
15
494 9.88927
558 9.91 147
25
26
0.08853 261
48
i.S
225 9.80089
476 9.88917
606 9.91 172
0.08828 254
47
14
248 9,80105
15
16
458 9.88906
655 9.9II98
26
0.08802 247
46
45
15
63271 9.80120
77439 9.88896
81703 9.91224
0.08776 1.2239
i6
293 9.80136
15
15
16
421 9.88886
752 9.91250
nfS
0.08750 232
44
I?
316 9.80151
402 9.88875
800 9.91276
%
0.08724 225
43
i8
338 9.80166
384 9-88861
849 9.9I30I
0.08699 218
42
19
361 9.80182
15
366 9-88855
898 9-91327
26
0.08673 210
41
20
63383 9.80197
77347 9-88844
81946 9-91353
0.08647 1.2203
40
21
406 9.80213
15
329 9.88834
995 9-91379
%
0.08621 196
39
22
428 9.80228
310 9.88824
82044 9-91404
0.08596 189
38
23
451 9.80244
292 9.88813
092 9.91430
^6
0.08570 181
37
24
473 9-80259
15
15
273 9.88803
141 9.91456
26
0.08544 174
36
25
63496 9.80274
77255 9.88793
82190 9.91482
0.08518 1.2167
35
26
518 9.80290
15
236 9.88782
238 9-91507
0.08493 160
34
27
540 9.80305
218 9.88772
287 9-91533
■^f)
0.08467 153
33
28
563 9.80320
199 9.88761
336 9-91559
'^f)
0.08441 145
32
29
30
585 9-80336
15
IS
181 9.88751
385 9-91585
25
0.08415 138
31
63608 9.80351
77162 9.88741
82434 9.91610
0.08390 1.2131
30
31
630 9.80366
144 9.88730
483 9.91636
->f>
0.08364 124
29
32
653 9-80382
15
15
16
15
15
125 9.88720
531 9.91662
^f)
0.08338 117
28
33
675 9-80397
107 9.88709
580 9.91688
25
26
'>6
0.08312 109
27
34
698 9.80412
088 9.88699
629 9.91713
0.08287 102
2b
25
35
63720 9.80428
77070 9.88688
82678 9.91739
0.08261 1.2095
36
742 9.80443
051 9.88678
727 9.91765
06
0.08235 088
24
37
765 9.80458
033 9.88668
776 9.91791
S
0.08209 081
23
38
787 9-80473
014 9.88657
10
825 9.91816
0.08184 074
22
39
810 9.80489
15
15
15
76996 9.88647
874 9.91842
26
25
-^6
0.08158 066
21
40
63832 9.80504
76977 9.88636
82923 9.91868
0.08132 1.2059
20
41
854 9.80519
959 9.88626
972 9.91893
0.08107 052
19
42
877 9-80534
940 9.88615
83022 9.91919
06
0.08081 "045
18
43
899 9-80550
15
15
15
15
15
16
15
15
921 9.88605
071 9.91945
"6
0.08055 038
17
44
922 9.80565
903 9.88594
120 9.91971
25
'^6
0.08029 031
16
45
63944 9.80580
76884 '9.88584
83169 9.91996
0.08004 1.2024
15
46
966 9.80595
866 9.88573
218 9.92022
-^6
0.07978 017
14
47
989 9.80610
847 9.88563
268 9.92048
25
0.07952 009
13
48
64011 9.80625
828 9.88552
317 9.92073
0.07927 002
12
49
033 9.80641
810 9.88542
366 9.92099
26
0.07901 1.1995
II
50
64056 9.80656
76791 9.88531
83415 9.92125
0.07875 1. 1988
10
SI
078 9.80671
772 9.88521
465 9.92150
0.07850 981
9
q2
100 9.80686
15
754 9.88510
514 9.92176
26
0.07824 974
8
S3
123 9.80701
15
735 9-88499
564 9.92202
25
26
'>6
0.07798 967
7
54
55
145 9.80716
15
15
717 9.88489
613 9.92227
0.07773 960
6
64167 9.80731
76698 9.88478
83662 9.92253
0.07747 I-1953
S6
190 9.80746
16
679 9.88468
712 9.92279
"6
0.07721 946
4
S7
212 9.80762
661 9-88457
761 9.92304
0.07696 939
3
S8
234 9.80777
15
642 9.88447
811 9.92330
06
0.07670 932
2
IS
256 9.80792
15
623 9.88436
860 9.92356
25
0.07644 925
I
279 9.80807
^5
604 9-88425
910 9.92381
0.07619 918
_
|Nat. Cos Log. d.
|Nat. Sin Log. d.
Nat. Cot Log
cd
.Log.TanNat
/
60^
Nat. Sin Log. d.
40°
Nat. Cos Log. d. Nat. Tan Log
c.d.
Log. Cot Nat.
64279
301
323
346
368
9.80807
9.80822
9-80837
9.80852
9.80867
64390
412
435
457
479
9.80882
9.80897
9.80912
9.80927
9.80942
64501
524
546
568
590
9.80957
9.80972
9.80987
9.81002
9.81017
64612
635
657
679
701
9.81032
9.81047
9.81061
9.81076
9.81091
64723
746
768
790
812
9.81 106
9.81121
9.81136
9.81151
9.81166
64834
856
878
901
923
9.81180
9.81 195
9.81210
9.81225
9.81240
64945
967
989
6501 1
033
9.81254
9.81269
9.81284
9.81299
9-81314
65055
077
100
122
144
9.81328
9-81343
9-81358
9.81372
9.81387
65166
188
210
232
254
9.81402
9.81417
9.81431
9.81446
9.81461
65276
298
320
342
364
9.81475
9.81490
9.81505
9.81519
9.81534
65386
408
430
452
474
9.81549
9.81563
9.81578
9.81592
9.81607
65496
518
540
562
584
606
9.81622
9.81636
9.81651
9.81665
9.81680
9.81694
76604
586
567
548
530
9.88425
9-88415
9.88404
9.88394
9.88383
765 1 1
492
473
455
436
9.88372
9.88362
9.88351
9.88340
9.88330
76417
398
380
361
342
9.88319
9.88308
9.88298
9.88287
9.88276
76323
304
286
267
248
9.88266
9.88255
9.88244
9-88234
9.88223
76229
210
192
173
154
9.88212
9.88201
9.88191
9.88180
9.88169
76135
116
097
078
059
9.88158
9.88148
9-88137
9.88126
9.88115
76041
022
003
75984
965
9.88105
9.88094
9.88083
9.88072
9.88061
75946
927
908
889
870
9.88051
9.88040
9.88029
9.88018
9.88007
75851
832
813
794
775
9.87996
9.87985
987975
9.87964
9.87953
75756
738
719
700
680
9.87942
9.87931
9.87920
9.87909
9.87898
75661
642
623
604
585
9.87887
9.87877
9.87866
9.87855
9.87844
75566
547
528
509
490
471
987833
9.87822
9.87811
9.87800
9.87789
9.87778
83910
960
84009
059
108
9.92381
9.92407
9-92433
9.92458
9.92484
84158
208
258
307
357
9.92510
9.92535
9.92561
9.92587
9.92612
84407
457
507
556
606
9.92638
9.92663
9.92689
9.92715
9.92740
84656
706
756
806
856
9.92766
9.92792
9.92817
9.92843
9.92868
84906
956
85006
057
107
9.92894
9.92920
9-92945
9.92971
9.92996
85157
207
257
308
358
9.93022
9.93048
9.93073
9.93099
9.93124
85408
458
509
559
609
9-93150
993175
9-93201
9-93227
9-93252
5660
710
761
811
862
9.93278
9.93303
9.93329
9.93354
9.93380
85912
963
86014
064
"5
9.93406
9.93431
9-93457
9.93482
993508
86166
216
267
318
368
9-93533
9-93559
993584
9.93610
9.93636
86419
470
521
572
623
9.93661
9.93687
9.93712
9.93738
993763
86674
725
776
827
878
929
9.93789
9.93814
9.93840
993865
9.93891
9.93916
26
0.07619
0.07593
0.07567
0.07542
0.07516
1.1915
910
903
896
0.07490
0.07465
0.07439
0.07413
0.07388
1.1882
875
868
861
854
0,07362
0.07337
0.07311
0.07285
0.07260
1.1847
840
833
826
819
0.07234
0.07208
0.07183
0.07157
0.07132
1.1812
806
799
792
785
0.07106
0.07080
0.07055
0.07029
0.07004
1.1778
771
764
757
750
0.06978
0.06952
0.06927
0.06901
0.06876
1.1743
736
729
722
715
0.06850
0.06825
0.06799
0.06773
0.06748
1. 1708
702
695
688
681
0.06722
0.06697
0.06671
0.06646
0.06620
1. 1674
667
660
653
647
0.06594
0.06569
0.06543
0.06518
0.06492
1. 1 640
633
626
619
612
0.06467
0.06441
0.06416
0.06390
0.06364
1.1606
599
592
585
578
0.06339
0.06313
0.06288
0.06262
0.06237
1.1571
565
558
551
544
0.0621 1
0.06186
0.06160
o.o6i3g
0.06109
0.06084
1.1538
531
524
517
510
504
Nat. Cos Log. d. Nat. Sin Log. d. Nat. Cot Log. c.d.
49°
Log.TanNat.
41
/
Nat. Sin Log. d.
Nat. Cos Log. d.|
Nat.TanLog.
c.d.
Log. Cot Nat.
65606 9.81694
15
14
15
75471 9-87778
TT
86929 9.93916
--•6
0.06084 1-1504
60
I
628 9.81709
452 9-87767
980 9.93942
0.06058 497
59
2
650 9.81723
433 9-87756
TT
87031 9.93967
0.06033 490
58
3
672 9.81738
414 9-87745
082 9.93993
25
26
0.06007 483
57
4
694 9.81752
14
15
14
15
14
15
14
15
14
14
15
14
15
14
15
14
14
395 9-87734
II
133 9.94018
0.05982 477
56
55
5
65716 9.81767
75375 9-87723
87184 9.94044
0.05956 1. 1470
6
738 9.8I78I
356 9.87712
236 9.94069
0.05931 463
54
7
759 9-81796
337 9-87701
287 9.94095
25
'^6
0.05905 456
53
8
781 9.81810
318 9.87690
338 9.94120
0.05880 450
52
9
803 9.81825
299 9.87679
II
389 9.94146
25
26
0.05854 443
51
10
65825 9.81839
75280 9.87668
87441 9.94I7I
0.05829 1.1436
50
II
847 9.81854
261 9.87657
492 9.94197
2^
0.05803 430
49
12
869 9.81868
241 9.87646
543 9.94222
0.05778 423
48
13
891 9.81882
222 9.87635
595 994248
25
26
0.05752 416
47
14
15
913 9.81897
203 9.87624
II
646 9.94273
0.05727 410
46
45"
65935 9-81911
75184 9.87613
87698 9.94299
0.05701 1. 1403
16
956 9.81926
165 9.87601
749 994324
0.05676 396
44
17
978 9.81940
146 9.87590
801 9.94350
%
0.05650 389
43
l8
66000 9.81955
852 9.94375
0.05625 383
42
19
022 9.81969
107 9.87568
II
TT
904 9.94401
25
2fS
0.05599 376
41
20
66044 9-81983
75088 9-87557
87955 9.94426
0.05574 1.1369
40
21
066 9.81998
14
14
15
14
14
15
14
14
14
15
14
14
15
14
14
14
069 9-87546
88007 9.94452
%
0.05548 363
39
22
088 9.82012
050 9.87535
059 9-94477
0.05523 356
38
2S
109 9.82026
030 9.87524
no 9.94503
25
26
25
0.05497 349
37
24
25
131 9.82041
on 9.87513
12
162 9.94528
0.05472 343
36
66153 9-82055
74992 9.87501
88214 9.94554
0.05446 1.1336
35
26
175 9.82069
973 9.87490
265 9-94579
0.05421 329
34
27
197 9.82084
953 9.87479
317 994604
0.05396 323
33
28
218 9.82098
934 9.87468
369 994630
25
26
0.05370 316
32
29
240 9.821 12
915 9.87457
II
421 9.94655
0.05345 310
31
30
30
66262 9.82126
74896 9.87446
88473 9.94681
0.05319 1.1303
31
284 9.82141
876 9.87434
524 9.94706
0.05294 296
29
32
306 9.82155
857 9.87423
576 9.94732
25
26
0.05268 290
28
33
327 9.82169
838 9.87412
628 9.94757
0.05243 283
27
34
349 9.82184
818 9.87401
II
680 9.94783
88732 9.94808
25
06
0.05217 276
26
25
35
66371 9.82198
74799 9.87390
0.05192 1,1270
36
393 9-82212
780 9.87378
784 9.94834
25
0.05166 263
24
37
414 9.82226
760 9.87367
836 9.94859
O.05141 257
23
38
436 9.82240
15
14
14
14
14
15
14
14
741 9.87356
J J
888 9.94884
0.051 16 250
22
39
458 9-82255
722 9.87345
II
940 9.94910
25
06
0.05090 243
21
40
66480 9.82269
74703 9.87334
88992 9.94935
0.05065 1. 1 237
20
41
501 9.82283
683 9.87322
89045 9.94961
s
0.05039 230
19
42
523 9.82297
664 9.87311
097 9.94986
0.05014 224
18
43
545 9-8231 1
644 9.87300
149 9.95012
25
25
'^6
0.04988 217
17
44
566 9.82326
625 9.87288
II
201 9.95037
0.0403 211
16
T5
45
66588 9.82340
74606 9.87277
89253 995062
0.04938 1.1204
46
610 9.82354
586 9.87266
306 9.95088
%
0.04912 197
H
47
632 9.82368
14
567 9.87255
358 9.95113
0.04887 191
13
48
653 9-82382
548 9.87243
410 9.95139
25
26
25
25
0.04861 184
12
49
50
675 9.82396
14
14
15
528 9.87232
II
463 9.95164
0.04836 178
II
To
66697 9.82410
74509 9.87221
89515 9.95190
0.04810 1.1171
=;i
718 9.82424
489 9.87209
567 9.95215
0.04785 165
9
=;2
740 9.82439
470 9.87198
620 9.95240
0.04760 158
8
S3
762 9-82453
14
14
451 9.87187
672 9.95266
25
26
%
0.04734 152
y
54
783 9.82467
431 9.87175
II
725 9.95291
0.04709 145
6
55
66805 9.82481
74412 9.87164
89777 9.95317
0.04683 1. 1 139
S6
827 9.82495
14
392 9.87153
830 9.9.5342
0.04658 132
4
S7
848 9.82509
14
373 9.87141
883 9.95368
25
0.04632 126
3
S8
870 9.82523
14
353 9.87130
935 9.95393
0.04607 119
2
SQ
891 9-82537
14
334 9.87119
988 9.95418
0.04582 113
I
60
913 9.82551
14
314 9.87107
90040 9.95444
0.04556 106
Nat. Cos Log. d.
Nat. Sin Log. d.
Nat. Cot Log.
C.d
Log.TanNat.
t
m
Nat. Sin Log. d.
42^
Nat. Cos Log. d.
Nat. Tan Log.
c.d. Log. Cot Nat,
66913
935
956
978
999
9.82551
9.82565
9.82579
9.82593
9.82607
67021
043
064
086
107
9.82621
9.82635
9.82649
9.82663
9.82677
67129
151
172
194
215
9.82691
9.82705
9.82719
9.82733
9.82747
67237
258
280
301
323
9.82761
9.82775
9.82788
9.82802
9.82816
67344
366
387
409
430
9.82830
9.82844
9.82858
9.82872
9.82885
67452
473
495
516
538
9.82899
9.82913
9.82927
9.82941
9.82955
67559
580
602
623
645
9.82968
9.82982
9.82996
9.83010
9.83023
67666
688
709
730
752
9.83037
9.83051
9.8306g
9.83078
9.83092
67773
795
816
837
859
9.83106
9.83120
9.83133
9.83147
9^83161
67880
901
923
944
965
9.83174
9.83188
9.83202
9.83215
9.83229
67987
68008
029
051
072
9.83242
9.83256
9.83270
9.83283
9.83297
68093
115
136
157
179
200
9.83310
9.83324
9.83338
9.83351
9.83365
9.83378
74314
295
276
256
237
9.87107
9.87096
9.87085
9.87073
9.87062
74217
9.87050
198
9.87039
178
9.87028
159
9,87016
139
9.87005
74120
9.86993
100
9.86982
080
9.86970
061
9.86959
041
9.86947
74022
9.86936
002
9.86924
73983 9.86913
963 9.86902
944
9.86890
73924
9.86879
904
9.86867
885 9.86855
865
9.86844
846 9.86832
73826
9.86821
806
9.86809
787 9.86798
767 9.86786
747
9.8677§
73728
9.86763
708
9.86752
688
9.86740
669 9.86728
649 9.86717
73629 9.86705
610
9.86694
590
9.86682
570
9.86670
551
9.86659
73531
9.86647
511
9.86635
491
9.86624
472
9.86612
452
9.86600
73432
9.86589
413
9.86577
393
9.86565
373
9.86554
353
9.86542
73333
9.86530
314
9.86518
294
9.86507
274
9.86495
254
9.86483
73234
215
195
175
155
135
9.86472
9.86460
9.86448
9.86436
9.86425
9.86413
90040
093
146
199
251
9-95444
9.95469
9.95495
9.95520
9.95545
90304
9.95571
357
410
9.95622
463 9.95647
516 9.95672
90569 9.95698
621
9.95723
674 9.95748
727
9-95774
781
9.95799
90834 9.95825
887 9.95850
940
9.95875
993
9.95901
91046
9.95926
91099
153
206
259
313
91366
419
473
526
580
9.95952
9.95977
9.96002
9.96028
9.96053
9.96078
9.96104
9.96129
9-96155
9.96180
91633
9.96205
687
9.96231
740
9-96256
794
9.96281
847 9-96307
91901
9.96332
955
92008
9.96383
062
9.96408
116
9.96433
92170
9.96459
224
9.96484
277
9.96510
331
9.96535
385
9.96560
92439
996586
493
9.9661 1
547
996636
601
9.96662
655 9.96687
92709
9.96712
763 9-96738
817 9.96763
872 9.96788
926 9.96814
92980
93034
088
143
197
252
9.96839
9.96864
9.96890
9.96915
9.96940
9.96966
0.04556
0.04531
0.04505
0.04480
0.04455
[.II06
100
093
087
080
0.04429
0.04404
0.04378
0.04353
0.04328
1. 1074
067
061
054
048
0.04302
0.04277
0.04252
0.04226
0.04201
I.I04I
035
028
022
016
0.04175
0.04150
0.04125
0.04099
0.04074
1. 1009
003
1.0996
990
983
0.04048
0.04023
0.03998
0.03972
0-03947
1.0977
971
964
958
951
0.03922
0.03896
0.03871
0.03845
0.03820
1.0945
939
932
926
919
0.03795
0.03769
0.03744
0.03719
0.03693
1.0913
907
900
894
0.03668
0.03643
0.03617
0.03592
0.03567
3881
875
869
862
0.03541
0.03516
0.03490
0.03465
0.03440
1.0850
843
837
831
824
0.03414
0.03389
0.03364
0.03338
0.03313
1.0818
812
805
799
793
0.03288
0.03262
0.03237
0.03212
0.03186
1.0786
780
774
768
761
0.03161
0.03136
0.031 10
0.0308^
0.03060
0.03034
I.07S5
749
742
736
730
724
Nat. Cos Log. d. Nat. Sin Log. d. Nat. Cot Log. c.d. Log.TanNat. /
470
43
D
f
Nat. Sin Log. d.
Nat. Cos Log. d.
Nat.TanLog.
c.d.
Log. Cot Nat.
68200 9.83378
14
13
73135 9.86413
93252 9.96966
25
0.03034 1.0724
60
I
221 9-83392
116 9.86401
306 9.96991
0.03009 717
S9
2
242 9-83405
096 9.86389
360 9.97016
0.02984 711
58
3
264 9.83419
076 9.86377
415 9-97042
25
25
0.02958 705
57
4
5
285 9-83432
056 9.86366
12
469 9.97067
0.02933 699
56
55
68306 9-83446
73036 9.86354
93524 9-97092
0.02908 1.0692
6
327 9-83459 ! \l
016 9.86342
578 9.971 18
25
25
25
26
0.02882 686
54
7
349 9-83473 , .z
370 9.83486 : ;3
72996 9.86330
633 9.97143
0.02857 680
53
8
976 9.86318
688 9.97168
0.02832 674
52
9
391 983500
-t
957 9-86306
11
742 9-97193
0.02807 668
51
50
10
68412 9.83513
13
72937 9-86295
93797 9-97219
0.02781 1.0661
II
434 9-83527
13
14
13
917 9.86283
852 9.97244
25
%
0.02756 655
49
12
455 9-83540
897 9.86271
~.
906 9.97269
0.02731 649
48
13
476 9-83554
877 9.86259
"
961 9.97295
25
25
26
0.02705 643
47
14
497 9-83567
857 9.86247
12
94016 9.97320
0.02680 637
46
45
15
68518 9.83581
14
72837 9.86235
94071 9.97345
0.02655 1.0630
lb
539 983594
^6
817 9.86223
125 9-97371
25
0.02629 624
44
17
561 9.83608
14
797 9-8621 1
'I
180 9.97396
0.02604 618
4S
i8
582 9.83621
^3
13
14
13
13
14
13
14
13
13
777 9.86200
235 9.97421
25
oA
0.02579 6X2
42
19
603 983634
757 9-86188
12
TO
290 9-97447
25
0.02553 606
41
20
68624 9.83648
72737 9.86176
94345 9-97472
0.02528 1.0599
40
21
64s 9.83661
717 9.86164
12
400 9.97497
0.02503 593
39
22
666 9.83674
697 9.86152
455 9-97523
25
25
25
26
0.02477 587
38
23
688 9.83688
677 9.86140
/"
510 9.97548
0.02452 581
37
24
709 9-83701
657 9.86128
12
565 9-97573
0.02427 575
36
35
25
68730 9-83715
72637 9.861 16
94620 9.97598
0.02402 1.0569
2b
751 9-83728
617 9.86104
~
676 9.97624
25
0.02376 562
34
27
772 9-83741
597 9-86092
~
731 9-97649
0.02351 556
33
28
793 9-83755
^4
13
577 9-86080
786 9.97674
26
0.02326 550
32
29
814 9.83768
557 9-86068
12
841 9.97700
25
25
26
0.02300 544
31
30
30
68835 9-83781
13
14
13
13
13
14
13
13
13
14
13
13
13
14
13
13
13
13
72537 9.86056
94896 9.97725
0.02275 1.0538
31
857 9-83795
517 9.86044
952 9-97750
0.02250 532
29
32
878 9.83808
497 9-86032
95007 9-97776
25
25
25
26
0.02224 526
28
33
899 9.83821
477 9.86020
062 9,97801
0.02199 519
27
34
920 9.83834
457 9-86008
12
12
118 9.97826
0.02174 513
26
25
35
68941 9.83848
72437 9-85996
95173 9-97851
0.02149 1.0507
3^
962 9.83861
417 9-85984
229 9.97877
25
25
26
0.02123 501
24
37
983 9-83874
397 9-85972
"
284 9-97902
0.02098 495
23
3a
69004 9.83887
377 9-85960
340 9.97927
0.02073 489
22
39
025 9.83901
357 9-85948
12
395 9-97953
25
0.02047 483
21
40
69046 9.83914
72337 9-85936
95451 9-97978
0.02022 1.0477
41
067 9.83927
317 985924
10
506 9.98003
0.01997 470
19
42
088 9.83940
297 985912
562 9.98029
25
25
25
26
O.01971 464
18
43
109 9-83954
277 9-85900
618 9-980M
0.01946 458
17
44
130 9-83967
257 9.85888
12
673 9.98079
0.01921 452
16
15
45
6915 1 9.83980
72236 9.85876
95729 9-98104
0.01896 1.0446
4b
172 9.83993
216 9.85864
TT
785 9-98130
25
0.01870 440
14
47
193 9.84006
196 9.85851
^3
841 9.98155
0.01845 434
13
48
214 9.84020
^4
13
13
13
13
13
13
14
176 9-85839
I^
897 9.98180
0.01820 428
12
49
235 9-84033
156 9.85827
12
952 9.98206
25
25
0.01794 422
II
To
50
69256 9.84046
72136 9.85815
96008 9.98231
0.01769 1.0416
51
277 984059
116 9.85803
064 9.98256
120 9.98281
0.01744 410
9
52
298 9.84072
095 9-85791
O.01719 404
8
53
319 9-84085
075 9-85779
176 9.98307
25
25
26
0.01693 398
7
54
340 9.84098
055 9.85766
13
12
232 9-98332
0.01668 392
6
"5"
55
69361 9.841 12
72035 9.85754
96288 9.98357
0.01643 1.0385
5b
382 9.84125
13
015 9-85742
344 9.98383
25
25
25
26
0.01617 379
4
57
403 9.84138
13
13
13
13
71995 9-85730
12
400 9.98408
0.01592 373
3
5a
424 9.84151
974 9.85718
457 9-98433
0.01567 367
2
fo
445 9-84164
954 9-85706
13
513 9-98458
0.01542 361
I
466 9.84177
934 9-85693
0.01516 355
Nat. Cos Log. d.
Nat. Sin Log. d.
Nat. Cot Log.
C.d.
Log.TanNat.
r
46'
' Nat. Sin Log. d.
44°
Nat. Cos Log. d. Nat.TanLog.lc.d. Log. Cot Nat
69466
487
508
529
549
9.84177
9.84190
9.84203
9.84216
9.84229
69570
591
612
633
654
9.84242
984255
9.84269
9.84282
9-84295
69675
696
717
737
758
9.84308
9.84321
9.84334
9.84347
9.84360
69779
800
821
842
862
9-84373
984385
9.84398
9.8441 1
9.84424
69883
'904
925
946
966
9-84437
9.84450
9.84463
9.84476
9-84489
69987
70008
029
049
070
9.84502
9-84515
9.84528
9.84540
9.84553
70091
112
132
153
174
9.84566
9.84579
9.84592
9.84605
9.84618
70195
215
236
257
277
9.84630
9.84643
9.84656
9.84669
9.84682
70298
319
339
360
381
9.84694
9.84707
9.84720
9-84733
9-84745
70401
422
443
463
484
9.84758
9.84771
9.84784
9.84796
9.84809
70505
525
546
567
587
9.84822
984835
9.84847
9.84860
9-84873
70608
628
649
670
690
711
9.84885
9.84898
9.8491 1
9.84923
9.84936
9.84949
71934
914
894
873
853
9-85693
9.85681
9.85669
9.85657
9.85645
71833
813
792
772
752
9-85632
9.85620
9.85608
9-85596
9-85583
71732
711
691
671
650
9-85571
9-85559
9-85547
9-85534
9.85522
71630
610
590
569
549
9.85510
9.85497
9-85485
9-85473
9-85460
71529
508
488
468
447
9.85448
9-85436
985423
9.8541 1
9-85399
71427
407
386
366
345
9-85386
9-85374
9.85361
9-85349
9-85337
71325
305
284
264
243
9.85324
9.85312
9.85299
9.85287
9-85274
71223
203
182
162
141
9.85262
9.85250
9.85237
9-85225
9.85212
71121
100
080
059
039
9.85200
9.85187
9-85175
9.85162
9.85150
71019
70998
978
957
937
9-85137
9.85125
9.851 12
9.85100
9.85087
70916
896
875
855
834
9.85074
9.85062
9.85049
9.85037
9.85024
70813
793
772
752
731
711
9.85012
9.84999
9.84986
9.84974
9.84961
9.84949
96569
625
681
738
794
9.98484
9.98509
9-98534
9-98560
9-98585
96850
907
963
97020
076
9.98610
9.98635
9.98661
9.98686
9-9871 1
97133
189
246
302
359
998737
9.98762
9-98787
9.98812
9-98832
97416
472
529
586
643
9.98863
9.98888
9.98913
9-98939
9.98964
97700
756
813
870
927
9.98989
9.99015
9.99040
9-99065
999090
97984
98041
098
155
213
9.991 16
9-99141
9.99166
9.99191
9.99217
98270
327
384
441
499
9.99242
9.99267
9.99293
9.99318
9-99343
98556
613
671
728
786
9.99368
9-99394
9.99419
9-99444
9.99469
98843
901
958
99016
073
9-99495
9.99520
9-99545
9.99570
9.99596
99131
189
247
304
362
9.99621
9.99646
9.99672
9.99697
9.99722
99420
478
536
594
652
9.99747
9.99773
9.99798
9.99823
9.99848
99710
768
826
884
942
lOOOO
9.99874
9.99899
9.99924
9-99949
9-99975
0.00000
0.01516
0.01491
0.01466
0.01440
0.01415
1-0355
349
343
337
331
0.01390
0.01365
0.01339
0.01314
0.01289
1.0325
319
313
307
301
0.01263
0.01238
0.01213
0.01188
0.01162
1.0295
289
283
277
271
0.01137
0.01112
0.01087
0.01061
0.01036
1.0265
259
253
247
241
O.OIOII
0.00985
0.00960
o.oo93g
0.00910
1.0235
230
224
218
212
0.00884
0.00859
0.00834
0.00809
0.00783
1.0206
200
194
188
182
0.00758
0.00733
0.00707
0.00682
0.00657
1.0176
170
164
158
152
0.00632
0.00606
0.00581
0.00556
0.00531
1.0147
141
135
129
123
0.00505
0.00480
0.00455
0.00430
0.00404
1.0117
III
105
099
094
0.00379
0.00354
0.00328
0.00303
0.00278
.0088
082
076
070
064
0.00253
0.00227
0.00202
0.00177
0.00152
1.0058
052
047
041
035
0.00126
O.OOIOI
0.00076
0.00051
0.00025
0.00000
1.0029
023
017
012
006
000
Nat. Cos Log. d.
Nat. Sin Log. d.
4^
Nat. Cot Log.
c.d. Log.TanNat.
■d ■
/
/
UNIVERSITY OF CAUFORNIA LIBRARY