IN MEMORIAM
FLORIAN CAJORI
PLANE
TRIGONOMETRY
BY
ARTHUR GRAHAM HALL, Ph.D. (Leipzig)
Pbofessor of Mathematics
University of Michigan
FRED GOODRICH FRINK, M.S. (Chicago)
Professor of Railway Engineering
University of Oregon
NEW YORK
HENRY HOLT AND COMPANY
Copyright, 1W9,
BY
HENRY HOLT AND COMPANY.
CAJORI
NarfajDoli \$xt9S
J. 8. Gushing Co. — Berwick & Smith Co.
Norwood, Mass., U.S.A.
PREFACE
This book has had its origin in the desire of the authors to
meet the mutual demands of mathematicians and engineers for
a treatment that shall more completely supply the needs of the
technological student. It is believed that this has been done by
enriching the subject with applications to physics and engineering,
in such a way as to increase its value at the same time to the
general student. The present volume is, moreover, based upon
a preliminary edition actually used for several terms in the class-
room.
In view of the peculiar situation of trigonometry in the cur-
riculum, the course has been kept of the usual length. The
topics have been arranged, however, in the order of increasing
difficulty, by postponing the more abstract but no less essential
study of the functions of the general angle, until after the arith-
metical solution of triangles. The abundance of exercises and
problems will give the teacher large opportunity for selection.
The discussion of the slide rule is inserted because of the
increasing employment of this useful instrument.
The authors gratefully acknowledge their indebtedness to
Professor E. J. Townsend and Professor H. L. Rietz, of the
University of Illinois, and Professor A. Ziwet and Professor J. L.
Markley, of the University of Michigan, for valuable criticisms
and suggestions.
ARTHUR G. HALL.
FRED G. FRINK.
Ann Arbor, January, 1909.
iii
CONTENTS
PAET I
PLANE TBIQONOMETBY
CHAPTER I
GEOMETRIC NOTIONS
ARTICLE PAOS
1. General statement 1
2. Directed line segments 1
3. Positive and negative angles 2
Exercise I . . . . ^ 3
4. Rectangular coordinates 3
Exercise II 6
CHAPTER II
THE ACUTE ANGLE
6. The purpose of trigonometry 6
6. Definitions of the trigonometric functions . 6
7. Relations between the ratios 7
8. Signs and limitations in value 7
Exercise III 8
9. Fundamental relations 9
Exercise IV 10
10. Functions of complementary angles 11
11. Functions of 45°, 30°, 60° 12
12. Functions of 0° and 90° •. . . .12
Exercise V 13
13. Variation of the trigonometric functions as the angle varies ... 14
14. Inverse trigonometric functions 15
Exercise VI 16
16. Orthogonal projection .• . . 16
Exercise VII 18
CHAPTER III
RIGHT TRIANGLES
16. Laws for solution 19
17. Area of right triangles . 20
VI
CONTENTS
ARTICLE PAGK
18. Method of solution 20
19. Trigonometric tables 22
20. Errors and checks 23
Exercise VIII 25
21.
22.
23.
24.
25.
27.
28.
29.
30.
31.
CHAPTER IV
LOGARITHMS
Definition of a logarithm . . . 29
Laws of combination , . . .30
Common logarithms 31
Characteristic ............ 32
Mantissa 33
Exercise IX . 34
Interpolation 1 ... 35
Numbers from logarithms 35
Cologarithms . .36
Logarithms of trigonometric functions 36
Exercise X . 37
The slide rule 39
Exercise XI .41
Right triangles solved by logarithms .41
Exercise XII 44
CHAPTER V
THE OBTUSE ANGLE
32. Definitions of the trigonometric functions of obtuse angle
33. Signs and limitations in value
34. Fundamental relations
35. Variation
36. Functions of 180°
37. Functions of supplementary angles
38. Functions of (90° -fa)
Exercise XIII
47
47
48
48
48
48
49
50
CHAPTER VI
OBLIQUE TRIANGLES
39. Formulas for solution 52
40. Law of projections 53
41. Law of sines ............ 63
42. Law of cosines 53
43. Law of tangents 64
44. Angles in terms of sides 65
CONTENTS vii
ARTICLE PAGE
45. Area of oblique triangles ....,,.... 55
46. Numerical solution . . ' 56
47. Case I. Two angles and one side 57
48. Case II. Two sides and an opposite angle 58
49. Case III. Two sides and the included angle 61
50. Case IV. Three sides 61
51. Composition and resolution of forces. Equilibrium . . . . . 63
Exercise XIV 66
CHAPTER VII
THE GENERAL ANGLE
52. General definition of an angle 70
53. Axes, quadrants, etc. ' . . . . . . . . . .71
54. Definitions of the trigonometric functions ....... 71
55. Signs and limitations in value 72
Exercise XV 73
56. Variation of the trigonometric functions ....... 74
57. Graphs of the trigonometric functions 75
58. Functions of 270° and 360° 78
Exercise XVI 79
69. Fundamental relations 79
60. Line representations of the trigonometric functions . . . . . 80
Exercise XVII 82
61. Periodicity of the trigonometric functions 83
62. Functions of (k • ^± a) , . . .83
Exercise XVIII 87
CHAPTER VIII
FUNCTIONS OF TWO ANGLES
63. Formulas for sin (a + /3) and cos (a + /S) . . = . . . .89
64. Extension of the addition formulas 90
Exercise XIX 91
65. Subtraction formulas 91
66. Formulas for tan (a ±/3), cot (a ± (S) . . . . . . .92
Exercise XX .... 92
67. Functions of twice an angle 93
68. Functions of half an angle . . . , . . . . .93
Exercise XXI .... . . . . . . . .94
69. Conversion formulas for products 95
Exercise XXII 96
70. Conversion formulas for sums and differences 97
71. Multiple angles 97
Exercise XXIII 97
viii CONTENTS
CHAPTER IX
ANALYTIC TRIGONOMETRY
AETIOLE PAGE
6
72. Limits of - — t, and - — -, as 6 approaches zero 99
sin d tan 0'
Examples 101
73. De Moivre's theorem 101
Examples 103
74. Graphical representation of complex numbers 103
Examples 108
75. Exponential values of the trigonometric functions 109
Examples 110
76. Hyperbolic functions 110
77. Exponential and trigonometric series Ill
Examples 115
78. Computation of trigonometric tables ........ 116
79. Proportional parts 116
80. General inverse functions . . . . , . , . . .117
81. Logarithmic values of inverse functions 118
Examples 119
Review Exercises ' . . 120
Formulas 130
Answers 137
Index , . 147
TRIGONOMETRIC AND LOGARITHMIC TABLES
I. Common logarithms of numbers 3
II. Logarithms of the trigonometric functions . . . . . .25
III. Natural trigonometric functions ........ 71
IV. Squares and square roots ..... o .... 91
TRIGONOMETRY
GREEK ALPHABET
Letters
Names
Letters
Names
Letters
Names
A o
Alpha '
I I
Iota
P P
Rho
B^
Beta
K/c
Kappa
S(r s
Sigma
Ty
Gamma
AX
Lambda
Tt
Tau
A5
Delta
Mm
Mu
Tu
Upsilon
Ee
Epsilon
N p
Nu
$0
Phi
Zf
Zeta
S^
Xi
Xx
Chi
H^
Eta
Oo
Omicron
^,/.
Psi
ee
Theta
Htt
Pi
w
Omega
TRIGONOMETRY
PART I
PLANE TRIGONOMETRY
CHAPTER I
GEOMETRIC NOTIONS
1. General statement. It is assumed that the student is well
versed in those theorems of elementary geometry concerning
angles, arcs, and triangles. It is particularly desirable that he
be familiar with the measurement of angles and with the proper-
ties of similar triangles.
While the review thus suggested is left to the student, certain
more advanced geometric ideas are treated in the remaining arti-
cles of this chapter.
Throughout the course the student should make continual,
careful, and intelligent use of such drawing instruments as are
included in the equipment at technical schools. In case such sets
are not available, as in more general classes, there should be pro-
vided at least a straightedge, with graduated scale, a protractor,
and a pair of compasses or dividers.
2. Directed line segments. A point which moves from one
position to a second, without changing its direction of motion,
traces a directed line segment. Directed line segments are always
read with regard to their direction, from the initial extremity to
the terminal extremity.
Two line segments are equal if they have the same length and
direction, whether their lines are coincident or parallel. Either
of two line segments having the same length but opposite direc-
tions is said to be the negative of the other. If one direction is
taken as positive, the opposite direction is negative.
1
GEOMETRIC NOTIONS
iXiius, An FigCli,.
E F K H
HK^ BA = - AB,
Fio.i. CD='2BA = -2AB.
If ^ is the initial point and B the terminal point, the line
segment is read AB, and in this notation the positive or negative
direction of the segment is expressed without the aid of a prefixed
+ or — . In case the segment is represented by a single symbol,
as the letter «, the direction must be indicated in some further
manner, as by a prefixed + or — , or by an arrowhead in the
figure.
Two line segments lying in the same line are added by
placing the initial point of the second upon the terminal point
of the first, each retaining its proper direction. The sum is the
segment extending from the initial point of the first to the termi-
nal point of the second. Subtraction is performed by reversing
the direction of the subtrahend and adding. Line segments
having the same or opposite directions may all be transferred
to a common line. Their addition and subtraction thus cor-
respond exactly to the algebraic addition and subtraction of posi-
tive and negative numbers.
If A, B^ denote three points arranged in any order along
a straight line, then
AB^BQ=AQ,
and AB^BC^CA = ^.
3. Positive and negative angles. If a line rotates (in a plane)
about one of its points, an angle is generated, of which the origi-
nal position of the line is the initial side and the final position the
terminal side. A distinction may be made according as the rota-
tion is clockwise or counter-clockwise about the vertex. The
counter-clockwise direction is chosen as positive. Angles are
always read with regard to their direction of rotation ; thus if
OA is the initial side and OB the terminal side, tlie angle is read
AOB. This notation includes the direction or sign of the angle,
and the -f- or — should not be prefixed. In case the angle is
represented by a single symbol, as by the Greek letter a, the
direction must be indicated in some further manner, as by a pre-
fixed -f or — , or by a curved arrow in the figure.
POSITIVE AND NEGATIVE ANGLES 3
Just as with line segments, reversing the direction multiplies
the angle by - 1 ; thus BOA = -AOB.
Two angles are added by placing them in the same plane with
a common vertex, the initial side of the second coincident with
the terminal side of the first, each retaining its own direction.
The sum is the angle from the initial side of the first to
the terminal side of the second. Subtraction
is performed by reversing the subtraliend and
adding.
In Fig. 2,
AOB + BOO=AOC,
AOO-BOC=AOB. ^:;^^
EXERCISE I
Solve the following problems graphically :
1. On a train running 40 miles an hour, a man walks 4 miles an hour.
Find the speed of the man with reference to the ground, (a) if he walks
toward the front ; (6) if he walks toward the rear of the train.
2. The man's speed with reference to the ground is 10 miles an hour.
What is the speed of the train (a) if he is walking 5 miles an hour toward
the front; (b) if lie is running 8 miles an hour toward the rear?
3. On June 1 the price of corn was 50 cents, and during the succeeding
ten days it fluctuated as follows : rose 2 cts., rose 3, fell 1, fell 2, fell 5, fell 3,
rose 2, rose 2, rose 3, rose 1. Find the price on June 11.
4. During a football game the progress of the ball from the middle of the
field was north 40 yards, south 25, south 5, south 10, south 30, north 50, north
10, north 20. Find the resulting position of the ball.
Combine graphically, using a protractor :
5. 45° + 30° ; 90° + 45° ; 40° + 35° + 50°.
6. 60° - 45° ; 90° - 50° ; 180° - 120°.
7. 30° + 80° + 55° ; 40° + 60° - 30° ; 60° - 20° + 70° - 90°.
8. 40° - 70° + 15°; 65° -f 15° - 90°; 75° - 180°.
4. Rectangular coordinates. If two mutually perpendicular
straight lines are chosen, and a positive direction on each, the
position of any point in their plane is determined by giving its
perpendicular distances from these fixed lines. The two lines are
called the axes of coordinates, and are usually taken so that one
GEOMETRIC NOTIONS
r^^
O^^-x.
-^x
^A
is horizontal and the other vertical. The point of intersection of
the axes is called the origin. The two determining data for any
point are called its coordinates. The horizontal distance from the,
axis Oy to the point is the abscissa of the point, and the vertical
.y distance from the axis OX
to the point is the ordinate
of the point. The point
^^-jP^ C^^ whose abscissa is x and
J^ ordinate y is denoted by
the notation (a:, ^). Be-
cause it is convenient to lay
off the abscissa of a point
upon the axis OX and the
ordinate upon the axis OY^
these axes are referred to
^^^- ^- as the axes of abscissas and
ordinates respectively. When x denotes the abscissa and y the.
ordinate of the point, the axes may be referred to as the X-axis
and the iF-axis respectively.
The distance from the origin to the point is called the radius
vector of the point. It is known whenever the abscissa and the
ordinate are given, since the three form, respectively, the hypote-
nuse, base, and altitude of a right triangle.
The abscissa of a point should always be read from the F-axis
to the point. The direction from left to right is chosen as posi-
tive. Therefore all points at the right of the y"-axis have positive
abscissas, and all points at the left, negative abscissas. The ordi-
nate of a point is always read from the JT-axis to the point. The
upward direction is chosen as positive. Hence all points above
the X-axis have positive ordinates, and all points below, negative
ordinates. The radius vector is always read from the origin to
the point, and is always considered positive.
It will be noticed that the abscissa and the ordinate are equal
to the projections of the radius vector on the X-axis and P"-axis,
respectively; these projections will henceforth be used inter-
changeably for the coordinates themselves.*
* The foot of the perpendicular dropped from a point upon a given line is said to
be the orthogonal or orthographic projection of the point on the line. The projec-
tion of a line segment on a given line is the segment from the projection of the ini-
tial point of the given segment to that of the terminal point. This kind of projec-
tion will be used exclusively throughout this book, unless otherwise expressly
stated.
RECTANGULAR COORDINATES
The two axes divide the whole plane into four portions, known
as the first, second, third, and fourth quadrants, beginning with
the upper right-hand quadrant and
numbering counter-clockwise about
the origin.
If two points, P and Q, lie in a
line through the origin, their coordi-
nates, with the radii vectores, form
two similar triangles. If the abscissa,
ordinate, and radius vector of P are
X, y, V, respectively, those of Q are
Jcx, ky^ kv. Fig. 4.
EXERCISE II
1. Plot the points (2, 3), (-3, 5), (-2, -4), (1, -3), (3, 0), (0, 4),
(-5,0), (0, -2), (0,0).
2. Plot the points (3, 2), (6, 4), (12, 8).
3. Plot the points (0, 5), (4, 3), (-3, 4), (0, -5).
4. Find both graphically and by computation the radius vector of each
point in examples 1, 2, 3. In what quadrant does each point lie ?
5. Describe the location of all points whose abscissas equal 3 ; whose ordi-
nates equal 5 ; whose radii vectores equal 6.
6. Write the coordinates of the vertices of a square of side a, if the axes
of coordinates are taken along two sides ; along its diagonals.
7. The hour hand of a clock is 5 inches long. Find the coordinates of its
tip referred to the horizontal and vertical diameters of its face at three o'clock ;
at six ; at eight ; at half-past ten.
8. Compare the location of the points (2, 3), (3, 2), ( — 2, 3), ( — 2, —3),
(3, —2). Describe the directions of their radii vectores.
9. Find the distance from (2, 5) to (6, 9) ; from (-3, 2) to (4, 5).
10. Find the direction of the line joining (6, 1) to (8, 3) ; (4, 1) to (1, 4) ;
(6, 3) to (1,3); (-2,4) to (1,1).
11. A man starts from the southwest corner of a government township and
goes in turn to the northwest corner of section 36 ; northwest corner section
22 ; southeast corner section 3 ; northeast corner section 5 ; thence to the point
of beginning. Show by sketch the shortest route along section lines, and com-
pute the cross-country distances.
12. A city is laid out in checker-board fashion. The streets are eight to
the mile and look to the cardinal points of the compass. It is proposed to in-
troduce two diagonal (45"") streets extending through the busiest corner to the
outskirts. What distances will be saved thereby in driving from this corner to
each of the points specified below? Nine blocks east and six blocks north;
5 W. and 7 S. : 10 W. and 10 N. : 1 E. and 14 S.
CHAPTER II
THE ACUTE ANGLE
5. Purpose of trigonometry. One of the principal objects of
trigonometry is the computation of triangles. We have learned
from elementary geometry that a triangle is determined when we
know any three of its parts (sides and angles), at least one of them
being a side. These data enable us to construct the triangle ; but
elementary geometry does not teach us how to calculate the re-
maining parts. The reason is that sides and angles are expressed
in different units. It is, therefore, desirable to measure angles not
only in degrees, but also by means of lines, or rather by means of
ratios of lines. This can be done as shown in the following
articles.
6. Definitions of the trigonometric functions.. Suppose the
acute angle A OB ( = «) to be placed on a system of axes of coor-
dinates with its vertex at the origin and its
initial side OA extending along the X-axis
toward the right. Its terminal side OB will
fall in the first quadrant. (See Fig. 5.)
Any point P in its terminal side possesses
one and only one set of related distances
(two coordinates and radius vector).
1 ^ > X Its abscissa x(^=OM)^ its ordinate
^ (=zMP), and its radius vector
V (= OF) are all positive and con-
tY
O
M
Fig. 5.
nected by the relation o . o o
•^ TT -\- y^ — v^.
The six ratios between these three distances are of frequent
occurrence and prime importance. They serve, indeed, the pur-
pose mentioned in Art. 5, and are given the following names and
accompanying abbreviations :
^ = sine of a = sin a,
= cosine of a = cos a,
= tangent of a = tan a,
= cosecant of a = esc a,
= secant of a = sec a,
= cotangent of a = cot a.
FUNCTIONAL RELATION
7. Relations between the ratios and the angle. The three re-
lated distances of any other point P^ in the terminal side of the
angle a are connected with the determining distances of P by the
relation , , ,
X y V
It follows that the values of the six ratios defined in Art. 6 do
not depend at all upon the particular choice of the point in the
terminal side of the angle, but are determined solely and definitely
by the size of the angle. A number that is determined in value
by the value of some other number is said to be a function of that
other number. The six ratios are therefore called trigonometric
functions of the angle.
This relation between the ratios and the angle is, moreover, a
mutual one, such that a knowledge of one of the ratios leads to a
knowledge of the angle.* Thus if we have given tan a = J, we
can construct the angle a as follows : On the system of axes OX
and OF plot the point P whose coordinates are (3, 2), using any
convenient scale. Draw the line OA from the origin through the
point P ; then is XOA the required angle a, in consequence of the
definitions of Art. 6.
It appears still further that from the knowledge of any one of
the six trigonometric functions the remaining five can be found.
Thus in the foregoing example we know by the * Pythagorean
proposition that on the scale employed the hypotenuse or radius
vector is V9 H- 4 = Vl3. We have then at once
sin a
cos a
Vl3
3
2
tan„ = -.
Vl3
seca = -— ,
cot '^= -,
VT3
CSC « = —-—.
V13
The properties and relations of these functions and their more
immediate applications in pure and applied mathematics constitute
the subject-matter of trigonometry. The science takes its name
from its origin in the attempts of the ancient peoples to measure
triangles.
8. Signs and limitations in value. When any acute angle is
placed on the axes of coordinates in the manner prescribed in
*Tbis statement, as well as some others in the present chapter, will require
some modification when we extend the consideration to angles greater than 90°.
8 THE ACUTE ANGLE
Art. 6, its terminal side will always fall in the first quadrant.
The abscissa and ordinate, as well as the radius vector, of every
point in the terminal side will all be positive. It follows that
their ratios, comprising the six trigonometric functions named in
Art. 6, are all positive.
In no case can the abscissa or the ordinate of a point be greater
than the radius vector. Indeed, save in the exceptional cases con-
sidered in Art. 12, they are less than the radius vector. Conse-
quently, sin a and cos a cannot be greater than unity and sec a and
CSC a cannot be less than unity.
EXERCISE III
By careful construction and measurement find the approximate
values of the following functions :
1. cos 60°. 2. tan 30°. 3. esc 45°.
4. cot 35°. 5. sin 20°. 6. sec 50°.
Construct each of the following angles and find by measure-
ment the values of all its functions, given
7. sin a = f . 8. cos a = y\. 9. tan a = ^\.
10. cot a = y«3. 11. sec a = ^^. 12. esc a = f i.
13. cos a = ^f . 14. sin a - |^.
15. For what angle is the tangent equal to the cotangent ? For what angle
is the sine equal to the cosine ?
16. Show that the direct functions (sin a, tan a, sec a) are greater or less
than the corresponding complementary functions (cos a, cot a, esc a) respec-
tively, according as the angle a is greater or less than 45°.
17. Can sin a and tana be equal? When do they approach equality?
18. Show that tan a • cot a does not depend on a. Show that the same is
true of sin a • esc a.
19. Show that cos a • see a does not depend on a. Show the same for
csc2 « - eot^ a.
20. Does sin2 a + eos^ a depend on a ? Does see^ a — tan^ a ?
21. Before reading Art. 11, find the values of the sine, cosine, and tangent
of 30°, 45°, and 60°.
9. Fundamental relations. The Pythagorean theorem
a;2 -J- ?/2 _ ^2^
FUNDAMENTAL RELATIONS 9
and the definitions of Art. 6 yield certain fundamental relations
between the six trigonometric functions of a single angle. Thus,
we have directly from the definitions
sma . ^ ^
sec a = -, (2^
cos a
cota = - — -• (3)
tana
Again, by division,
tena = ^"^, (4)
cos a
and cota=^^^. (5)
sin a
Dividing by v^ both members of the equation
2/2 -f ^2 _ ^2^
we have {t\ j^{^ ^x
whence sin^ a + cos^ a = 1 . (6)
Dividing, in like manner, by a^ and by y^ respectively, we
obtain
tan^ a + 1 = sec^ a, (T)
cot^ a + 1 = csc^ a. (8) .
These eight equations constitute the fundamental relations of
trigonometry. Of these the fifth, seventh, and eighth may be
derived algebraically from the other five. By means of these
equations it is possible to express all six functions in terms of any
one of them. If the value of any one is given, the values of the
others can be found. Simple numerical examples of the latter
kind are more quickly solved by the geometrical method of Art. 7.
Example i. To find the remaining functions of the acute
angle whose tangent is -f^.
(1) G-eometric Method. The right triangle OMP (Art. 6,
^^^' 5) has sides of relative length ;«/ = 5, a; = 12, whence on the
same scale v = 13. Thus the sine is -j^, etc.
10 THE ACUTE ANGLE
(2) Analytic Method. Given tana = ^5_^ Then by the for-
mulas of Art. 9,
1 12 ,. ^- 13.
13
cot a = = — ; sec a = Vl + tiin^ « - zj^ ,
tan « 5 12
CSC a = VI + cot^ CL— r
1 12
cos a = = —7 ; sm a
sec a 13 esc a 18
Example 2. To express all the functions of a in terms of
sec «. By the use of the appropriate formulas of Art. 9, we
obtain
1 . /^ 9 - Vsec^ « — ] ;
cosa = ; sina = VI — cos^a =
sec a sec a
1 sec a
csca = -r— - = — =^=i' tan«= Vsec2«-l;
sin « Vsec^a— 1
cot 06 =
tana Vsec^a-l
Example 3. Verify the following relation by reducing the
first member to the second :
tanSyg 1 o
^ — 1 = sec /3.
sec /3 — 1
By means of the formulas of Art. 9, we have
tan'-^ ^ -< sec2 yS — 1^ o,ii o
W^ -1 = S — -- 1 = sec/3+1 -1 = sec ^.
sec p — 1 sec p — 1
EXERCISE IV
By means of the formulas of Art. 9, find the values of the
remaining functions of each of the following angles, given
1. sina = -iV5. 2. cot/3 = ff- 3. cosy = ||. 4. tany^^^.
Express all the functions of a in terms of
5. tan a. 6. cos a. 7 cot a. 8. sin a.
Find the values of the following expressions:
ft tan a + cot a i 9
9. , when cos a = —
tana — cot a 41
, r, sec a — cos a i • 12
10. , when sm a = —
tan a — sin a 13
FUNCTIONS OF COMPLEMENTARY ANGLES
11
11. ^ ^^"^^ + cotB, when tanB=—-
1 + cos/3 ^ ^ 21
12. cos ^ • tan /? + sin y8 . cot y8, when sec /? = ||.
Express the following in terms of a single function
13.
CSC a
cot a + tan a
in terms of cos a-
, - sin B , 1 + COS B • . ^ ^
14. ^=-— H =!^^ — —^ m terms of esc B.
1 + cos /? sni y8 ^
15. sec y — tan y in terms of sin y.
1 1
16.
+
1 — sin y 1 + sin y
Verify the following identities :
17. cos4y8-sin4yS = 2cos2^-l.
18. cos a ' tan a = sin a.
cot2a
in terms of tan y.
19.
20.
cos'^ a.
1 + cot2 a
^ + L_
tan^yg+l cot2y8 + l
I.
10. Functions of complementary
angles. If, in Fig. 6, Z XOjS is
constructed equal to Z.AOY^
XOB (=fi} and XOA (= a) are
complementary. The triangles
OM'F' and Oi)[fP are similar, the
pairs of corresponding sides being
v' and v^ x' and «/, ?/' and x.
We have then
sin (90°- a) = sin ^ =
cos «,
cos (90° — a) — cos ^ = , = ^ = sin a,
v' V
tan (90° - «) = tan ^ = -^^ = - = cot «,
cot (90°- a) = cot /?= -. = ^= tan a,
y ^
12
THE ACUTE ANGLE
sec (90° - «) = sec /? = -^
CSC a,
CSC (90'
a) = CSC 8= —. = -= sec a.
The prefix " co " is thus seen to be the abbreviation of the
word "complement's." The general theorem may be stated as
follows :
Ant/ trigonometric function of an acute angle is equal to the
corresponding co function of its complementary angle.
Thus, sin 72° = cos 18°, cot 54° = tan 36°, etc.
11. Functions of 45°, 30°, 60°. The exact values of the func-
tions of certain angles are readily found.
(1) If, in Fig. 7, Z XOA = 45°, the triangle OMF is isosceles,
so that x — y =
We have at once
>X
V2
sin 45° = cos 45° = \-\/%.
tan 45° = cot 45° = 1,
sec 45° = CSC 45° = V2.
>X
Fig. 7.
(2) If, 'in Fig. 8, AXOA = ^0\ and "{Y
/. XOQ is constructed equal to —30°,
the triangle QOP is equilateral, so that
y = ^v, x = \^lv.
We have, at once,
sin 30° = |, cos 30° = I V3,
tan 30° = 1 V3, cot 30° = V3,.
sec 30°=|V3, CSC 30° = 2.
(3) In like manner to (2), or by Art. 10, we obtain
sin 60°= J V3, COS 60° =1,
tan 60° = V3, cot 60° = l VB,
sec 60° = 2, CSC 60° = | V3.
12. Functions of 0° and 90°.
(1) As the Z XOA (see Fig. 9) decreases so as to approach
the limit zero, the abscissa x will approach equality to the radius
FUNCTIOA^S OF 0° AND QO'^
13
vector V. If, at the same time, the radius vector remains finite,
the ordinate ?/ will approach the limit zero.
It should be noticed that the cosecant varies in such a manner
that its denominator approaches the limit zero while its numera-
tor remains constantly equal to the
finite number v, so that the value
of the cosecant increases without
limit as the angle approaches zero.
It is then said to become infinite
and is represented by the symbol oo.
The cotangent also becomes infinite
as the angle approaches zero, since
its numerator, although changing,
never exceeds v.
We have, then, sin 0° = 0,
tan 0° = 0,
sec 0° = 1,
(2) In like manner, we obtain
sin 90° = 1, cos 90° =0,
tan 90°= 00, cot 90° = 0,
sec 90° = 00, CSC 90° = 1.
Example. Solve the equation
sec^ 7 — V3 tan 7 = 1.
Expressing wholly in terms of tan 7,
tan^ ry -f 1 — V3 tan 7—1 = 0,
tan^ 7 — V3 tan 7 = 0,
tan 7=0 and V3.
■^1
1^
p
(2)
'1
1
y
•
I
J
\i
X
Fig. 10.
we have
or
whence
Then, by Arts. 11 and 12,
7 = 0° and 60°.
EXERCISE V
1. Express in terms of an angle less than 45° the functions of 75".
2. Express in terms of an angle less than 45° the functions of 65°.
14 THE ACUTE ANGLE
Verify the following for « = 30° ; also for a = 45° ;
3. sin 2 a = 2 sin a cos a.
4. cos 2 a = cos^ a — sin^ a.
2 tan a
5. tan2a =
6. cot 2 a =
1 — tan^ a
cot2 a-\
2 cot a
Notice that sin 2 a does not equal 2 sin a, cos 3 a does not equal 3 cos a, etc.
Verify for a = 30° :
7. sin 3 a = 3 sin a — 4 sin* a.
8. cos 3 a = 4 cos^ a — 3 cos a.
Evaluate the following expressions :
9. sin 60° cos 30° - cos 60° sin 30°.
10. cos 60° cos 30° + sin 60° sin 30°.
11. csc2 45° + sin 60° tan 30°.
12. sin 60° tan 45° - sec 30° sin2 45° cot 60°.
Solve each of the following equations for some one function of
a and find the angle in degrees. Verify the results by substitu-
tion in the given equation.
13. tan a - 3 cot a = 0.
14. sec a + 2 cos a = 3.
15. 4 sin2 a + 3 cot^ a = 4.
16. since + 3cosa = 2V2.
17. A ladder 22 feet long leans against a wall, its foot being 11 feet away
from the wall. Find (a) the angle formed by the ladder with the ground;
(6) the height of the top of the ladder above the ground.
18. The diagonal of a rectangle makes an angle of 30° with the long side.
If the length of this side is 14 inches, what is the length of the short side of
the rectangle ? of the diagonal ?
19. The side of a conical pile of sand makes an angle of 45° with the
floor. If the height from the floor is 12 feet, what is the area of the base ?
20. A guy rope (assumed to be straight) has a length of 60 feet and
extends from the top of a mast 30 feet high to the ground. Find the angle
between the rope and the mast.
13. Variation of the trigonometric functions as the angle varies.
Suppose the angle 6 to vary continuously from 0° to 90°. The
VARIATION. INVERSE FUNCTIONS 15
terminal side revolves about the origin from the position OX to
the position OY. li v remains constant, y will increase from
to V, while X will decrease from v to 0. Consequently, the nu-
y
merator of the fraction -(= sin ^) increases from to v. while the
V
denominator remains constant. Hence, while 6 increases from
0° to 90°, sin 6 increases from to 1.
The numerator of the fraction - ( = cos 6) decreases from v to
V
0, while the denominator remains constantly equal to v. Hence,
while increases from 0° to 90°, cos decreases from 1 to 0.
The numerator of the fraction -(= tan 0^ increases from to
X ^
V, while the denominator decreases
from V to 0. Hence, while 6 in-
creases from 0° to 90°, tan ^, be-
ginning with zero, increases with-
out limit as 6 approaches 90°.
We express this by saying that
tan 6 varies from to qo.
The student should trace care-
fully the variation of the other
trigonometric functions and compare the results with the values
found in Arts. 11 and 12. Article 7 should be read again at this
point.
14. Inverse trigonometric functions. The same functional re-
lation is expressed by the two statements, " m is the sine of the
acute angle a" and "a is the acute angle whose sine is m." The
corresponding symbolic notations are
m = sin a, a = arcsin m,*
with the understanding that a is an acute angle and that m is a
positive number not greater than unity. A similar symbolic
relation holds for the other trigonometric functions. It is fre-
quently read " arc-sine m," or "anti-sine m," since two mutually
inverse functions are said each to be the anti-function of the other.
* This notation is universally used in Europe and is fast gaining ground in this
country. A less desirable symbol,
a = sin-i m,
is still found in English and American texts.
The notation a = inv sin m is perhaps better still on account of its general
applicability. (See Art. 80.)
16 THE ACUTE ANGLE
The inverse notation is convenient for the statement of prob-
lems. The purposes of interpretation and manipulation are better
served by transforming to the corresponding direct notation.
Example. Find the value of sin (90° — arccot ^^2)-
In the direct notation the example reads : Given cot a = -^^^
find sin (90° - a). Then, by Arts. 10 and 9,
sin (90° — a) = cos a = ^\.
EXERCISE VI
1. Trace the variation in value of sec 6.
2. Trace the variation in value of esc 0.
Find the values of the following :
3. tan (cos-iyV)- 7. sec (90°- arcsec 2).
4. sin (arccot |). 8. esc (90°— arccsc V2).
5. cos (90°-arctan ^\). 9. sin (2 tan-i 1).
6. cot (90°- sin-i if). 10. cos (2 sin-i I).
Solve the following equations :
11. 2sin2/? + 3cos/8-3 = 0.
12. sec y8 - 2 tan )8 = 0.
13. tan^(2sin^-l)(secj8-V2) = 0.
14. sin y8 (2 cos j8 - V'3) (tan yS - 1) = 0.
Verify the identities :
15. sin^ a + cos* a + sin^ a cos^ a = sin* a — sin^ a + 1.
16. (esc a — cot a) (esc a + cot a) = 1.
17. (tan « + cot a) (sin a -cos a) = 1.
18. 1 - tan* a = 2 sec^ a - sec%.
19. sin« a + cos^ a = 1 - 3 sin^ a cos^ a.
20. cos* a - sin* a = l -2 sin^ a.
15. Orthogonal projection. In accordance with the definitions
of Art. 4 (see note, page 4) it follows that the projection of a
line segment on any line is equal to the length of the line segment
multiplied by the cosine of the angle formed by the line segment
with the line of projection. Thus, in Fig. 12, the projection of
AB on RS is
M]V= AB cos a.
ORTHOGONAL PROJECTION
IT
In like manner, the projection of AB on a line perpendicular
to MS (i.e. making + 90° with MS) has the value AB sin a.
These projections are called the components of the
line segment AB along and at right angles
to the direction MS.
In physics, line segments are
often used to represent quanti-
ties that have direction as well as
magnitude ; for example, forces,
velocities, accelerations. The components of the line segment
used to represent a force represent components of the force ; like-
wise for a line segment representing a velocity, acceleration, or
moment. Suppose, for example, that the line segment AB, Fig.
13, represents a force applied to the block m resting on a liorizon-
tal plane. This segment has the component UB parallel to the
plane and the compo-
nent FB perpendicular
to the plane. Segment
FB represents a force
component F^ parallel
to the plane, which
tends to move the block
along the plane ; seg-
ment FB represents a
force component F.^ per-
pendicular to the plane and tending to produce pressure between
the block and plane. Denoting by F the force represented by
AB, we have
F^ = F cos a,
Fy = F sin a.
Example. At a given instant a point is moving in a direc-
tion at an angle of 30° with a given horizontal line with a velocity
of 20 feet per second. Find the component of the velocity along
the line.
Taking the given line as the JT-axis, we have for the compo-
nent v^
v^ = v COS 30°= 20 X I V3 = 17.321 feet per second.
The component along a line perpendicular to the given hori-
zontal line in the plane of motion is
Vy = v sin 30°= 20 x | = 10 feet per second.
18 THE ACUTE ANGLE
EXERCISE VII
The student should draw appropriate figures for each of the following exercises.
1. Find the projections of a line segment 8.5 inches in length on the X-
and Z-axes, (a) when the segment makes an angle of 45° with the Z-axis ;
(b) when it makes an angle of 60° with the F-axis.
2. A crank 16 inches long rotates in a vertical plane. When the crank
makes an angle of 30° with the horizontal diameter of the circle described by
the moving end, what is the distance of the moving end, (a) from the hori-
zontal diameter? (h) from the vertical diameter ?
3. If in Fig. 13 the force jP denoted by AB is 40 pounds, find the compo-
nents F^ and Fy, (a) when a = 30° ; (b) when a = 45°. Discuss the cases
a =r 0° and a = 90°.
4. A steamer is moving at a speed of 18 miles per hour in a direction
north of east, making an angle of 30° with an east and west line. At what
rate is the steamer sailing eastward ? at what rate northward ?
5. A guy wire exerts a pull of 3000 pounds on its anchorage and makes an
angle of 30° with the ground. Find the component of this force, (a) along
the ground; (b) vertical.
6. The eastward and northward components of the velocity of a moving
body are found to be 1^^= 12 miles per hour and Vj,= 12 V3 miles per hour, respec-
tively. Find (a) the magnitude and (b) the direction of the body's velocity.
CHAPTER ITT
RIGHT TRIANGLES
16. Laws for solution. If, in a right triangle, two independent
parts are known, in addition to the right angle, the three remain-
ing parts can be found. Thus two given parts, at least one of
which is a side, determine a right triangle. The formulas needed
in all cases to effect this solution are
five in number. Two are the state-
ments of well-known geometric theo-
rems, while the other three are the
immediate consequences of the defini-
tions of the trigonometric ratios con-
tained in Art. 6.
In Fig. 14 let ACB be a right
triangle, right-angled at 0. We shall
denote the interior angles at the ver-
tices by a, yS, 7, and the lengths of
the sides opposite them by a, 5, c,
respectively. Note that <y = 90°, and
The five formulas are the following :
B
/f^
X
a
X
y
A
b
c
-'
C IS
Fig. 14.
the hypotenuse.
(1)
(2)
(8)
(-1)
(5)
Equation (1) follows from the Pythagorean theorem, and
(2) from the fact that the sum of the angles of a triangle is equal
to two right angles. In order to establish the last three, place
the triangle on the axes of coordinates described in Art. 4, the
side JL(7 extending from the origin to the right along tlie X-axis,
and the hypotenuse lying in the first quadrant, as in Fig. 14.
19
a2+62_
c\
a+p =
90*^
»
a
c
= sin a =
cos
p.
h
c
= cos a =
sin
p.
a
h
= tan a =
cot
p-
20 RIGHT TRIANGLES
Then 5, a, c, are respectively the abscissa, ordinate, and radius
vector of j5, a point on the terminal side of the angle a, which is
conventionally placed.
It follows at once from Art. 6 that
a
- — sm a,
c
h
- = cos a, •
a ,
- = tan a.
The corresponding values of the functions of the angle y8 result
from Art. 10.
17. Area of right triangles. The formulas for 'the area of a
right triangle follow from tiie familiar geometric theorem
Area = |^ x base x altitude,
or expressed symbolically,
A = \ab. (1)
The substitution for a of its value from the preceding article gives
^ = 1 6c sin a. (2)
Again, introducing the values of both a and 5,
A = I c2 sin a • cos a. (2)
Other formulas for the area may also be obtained.
18. Method of solution. The solution of any problem consists
of four parts : the analysis, the algebraic solution, the arithmetical
computation, and the interpretation of the results.
(1) The student should read and analyze the problem, noting
which parts are known and which are desired. The construction
of a neat and sufficiently accurate figure is helpful and advisable.
(2) The student should select from the five formulas of Art.
16 those containing a single unknown part each, in addition to the
known parts, and should solve them for these unknown parts while
still in the literal form.
Experience has led to the adoption of the following two rules
of procedure :
SOLUTION OF EIGHT TRIANGLES 21
(tI) The use of the Pythagorean formula, a^ -f 5^ = c^, is to be
avoided save when the data are very simple or a table of squares
and square roots is at hand.
(^) So far as is consistent with rule A, each unknown part
should be found in terms of those parts originally given in the
problem, in order to avoid accumulation of errors.
In conformity with these rules, the angle relation a-\- 13 = 90°,
and two of the three trigonometric formulas serve to effect the
solution, while the remaining trigonometric formula affords a
check on the work.
(3) The solution is now effected by introducing the numerical
data and performing the necessary computations. The correctness
and accuracy of the results are greatly enhanced by extreme order-
liness of arrangement and neatness of detail.
The use of the trigonometric tables and the employment of
suitable checks will be discussed in subsequent articles.
(4) The geometric or physical significance of the results ob-
tained should be fully considered and interpreted.
Example i. Given c == 254, a = 30°, to find a, 5, /3.
In this instance the analysis and construction are obvious.
The three appropriate formulas yield at once the forms
yS=90°-a,
a= c sin ct,
h = c cos a.
The formula b = a tan /3 affords the check.
On introducing the numerical data, we obtain
^ = 90° - 30° = 60°,
a = 25ixi =254 X. 5 = 127,
^> = 254 X 1V3 = 254 X .86605 = 219.976T.
The check formula gives
6 = 127x1.7321 = 219.9767.
Example 2. Given « = 39.00, 6 = 80.00, to find c, a, /9, and
the area.
As before, we may pass immediately to the second stage. Now
c is given directly in terms of a and b by the formula c?= Va^ + h^
22 RIGHT TRIANGLES
If we are to avoid the use of this formula, we must first find a and
/8, and then get c by means of one of these angles. We use the
forms :
I a =
a
"V
/9 =
90°-
C =
a
sm a
A =
\ah.
c =
h
sin l3'
and for the check
We obtain, then tan « = 39 -^ 80 == .4875,
a = 25° 59^ as found from Table III,
/3= 90° -25° 59' =64° 01',
^ = 39 _j. .4381 = 89.01,
^ = 1 X 39 X 80 = 1560,
and for the check c=SO-^ .8989 = 89.00,
showing a difference of .01.
On account of the simplicity of the numbers, we may, by using
the formula c^ = a^-]- P, find, exactly, c = 89.
Explain the accumulation of errors and, hence, the reason for
rule of procedure (jB).
Examples, l. Given c = 42, ^ = arcsin .28 ; find a and h,
2. Given 5 = 27, « = tan~i .75 ; find a and c.
3. Given a = 300, a = cos"^ .45 ; find c and b.
4. Given c= 200, a— arccot 1.12; find a and h.
19. Trigonometric tables. In the first example worked in the
preceding article, the functions of 30° had been determined in
Art. 11. In the second example, however, the value of tan a was
not one of those previously ascertained, and the value of a was not
recognizable from its tangent. For convenience of reference the
numerical values of the sines, cosines, tangents, and cotangents of
all angles differing by intervals of one minute from 0° to 90° have
been collected in Table III, on pages 71-89. The arrangement is
simple and plain. The degree numbers from 0° to 44° occur at the
top of the page, with the minutes running down the left margin.
TABLES. ERRORS AND CHECKS 23
The numerical values of the functions, computed to four decimal
places, are placed in columns under the names of the functions.
Since sin (90° — a)— cos a, and tan (90° — ce) = cot a, the space
required may be reduced one half. The degree numbers of angles
from 45° to 90° are printed at the bottom of the pages in reversed
order, the minutes run up the right margin, and the names of the
functions are in reversed order at the bottom.
For the present the student need not concern himself with
smaller divisions of the angle than the minute. Further refine-
ment is attained by a method to be described in Art. 26.
Table IV, on pages 91-93, contains the squares of numbers less
than 1000 and, by interpolation, of numbers up to 9999. The
first page gives directly the squares of numbers from 1 to 100.
On the second and third pages the tens and units digits of the
number to be squared are in the left margin, while the hundreds
digits are at the tops of the several columns. The last two figures
of the square are in the column at the right under U., opposite
the tens and units digits ; the first three, or four, figures of the
square are in the same line in the column under the hundreds
digit. In the right margin are the last two figures of the tabular
difference used in interpolation, to which must be prefixed the
remainder obtained by subtracting the first three, or four, figures
of the square from those in the same column immediately beneath,
or that remainder diminished by 1 when the asterisk (*) is present.
The use of the table is best shown by illustration.
Examples, i. 3282 = 107,584.
2. 475.3 = 4752 4- .3 X 951 = 225,625 + 285 = 225,910.*
3. 28.372 =28.32 + . 07 x567 = 800.89 + 3.97 = 80 1.56.
Square roots are extracted by reversing the process ; thus.
4. V27556 = 166.
5. V658,037 = V657,72l + 316 ^ 1623 = 811 + .2 = 811.2.
20. Errors and checks. The results obtained are not always,
nor even usually, exactly correct. The deviations from the true
values are of two sorts, mistakes and errors, and a sharp distinc-
tion must be made between them.
* This result is, of course, only approximately correct. The true result may be
obtained as follows :
475.32 = 4752 + .3 X (475.3 + 475) = 225,625 + 285.09 = 225,910.09.
24 RIGHT TRIANGLES
The data for problems arising in actual practice are derived
from observations made with instruments for measuring lengths,
angles, etc.
Mistakes may arise from a false reading of the observing instru-
ment, a misapprehension of the problem, the employment of the
wrong formula, faulty addition, etc. ^ They are never allowable or
excusable.
On the other hand, instruments are so constructed as to yield
results only to a certain degree of precision, which should be
ascertained for each instrument. Moreover, observation is per-
formed by the human apparatus, eyes, ears, etc., and a certain per-
sonal equation, an anticipation or lagging in sight or hearing, is
always present, varying with personal fitness and experience.
Methods of eliminating instrumental errors, so as to obtain the
maximum precision possible with the instruments used, are given
in standard works on engineering instruments. Again, the arith-
metical calculation involves the trigonometric ratios, which are, in
general, non-terminating decimal fractions, while their values in
the mathematical tables are computed only to a certain number of
decimal places. Errors, therefore, will always be present ; but
every precaution should be taken to keep the errors due to com-
putation well within the limits of error of the observed data and
desired results fixed by the nature of the problem.
In both observation and solution, certain additional processes
are employed, to avoid, or to reveal, mistakes. These processes
are known as checks and vary with the nature of the problem.
While no general rules for checks can be laid down, a frequent
practice in the solution of triangles is to make use of a formula
connecting the required parts, just found, noting if the results are
within the range of allowable error. The size of this allowable
error should be known for each table.
As a check to arithmetical computation, graphical construction
is well understood and strongly advised. As a means of avoiding
the grosser mistakes, a free-hand sketch will frequently suffice by
guiding the student to a reasonable interpretation of data, and
indicating possible results.
A drawing constructed to scale will further aid by yielding
values more or less approximate, approaching those obtained by
computation.
Carried a step farther as regards accuracy, by the use of pre-
cise instruments, the graphical construction often attains to the
PROBLEMS INVOLVING RIGHT TRIANGLES
25
dignity of an independent solution, with results falling within the
limits prescribed by the physical conditions of the problem.
There is no better evidence of careful work than the record-
ing of a reasonable error obtained by the comparison of two
methods. In practical work the allowable per cent of error
becomes an important consideration.
EXERCISE VIII
Find the missing parts of the following triangles, using the
natural trigonometric functions, Table IIL
a
/3
a
h
c
A
1.
2.5° 10'
34
2.
52° 20'
73
3.
61° 15'
243
4.
78° 35'
521
5.
21° 25'
235
6.
72° 45'
720
7.
80° 30'
1200
8.
17° 30'
1500
9.
240
360
10.
381
715
11.
521
630
12.
840
1400
13.
648
864
14.
595
600
15.
215
385
16.
2111
1234
17.
95
7980
18.
264
30360
19.
74° 20'
1225
20.
24° 50'
843
21. In the same vertical plane the distances shown in Fig. 15 were meas-
ured in feet along the surface of
the ground. The distances of the « ^ ^ ^ ^ f^R-
different points below the instru-
ment, as measured by a rod, are
given also in feet. The vertical
scale is exaggerated for clearness.
What is the horizontal distance
from B to Gl (Check by a table of squares and square roots.)
26 RIGHT TRIANGLES
22. A line surveyed across a ridge is 1500 feet in horizontal length.
Stakes are set 100 feet apart horizontally by level chaining. By leveling, the
elevations of the surface at the different stakes is obtained as follows : 730.2,
735.9, 739.7, 743.4, 750.1, 751.8, 760.7, 764.1, 764.3, 765.8, 765.0, 763.2, 758.3,
750.2, 743.1, 740.2. What length of wire will be required for fencing along
this line? (Check by a table of squares and square roots.)
23. If a gravel roof slopes one half inch to the horizontal foot, what angle
does it make with the horizon?
24. If the face of a wall has a batter or inclination of one inch in one ver-
tical foot, what is its angle with the vertical ?
25. What is the angle of ascent of a railway built on a 2 per cent grade
(i.e. 2 vertical feet to 100 horizontal feet) ?
C 26. The pitch of a roof is the ratio
— . (See Fig. 16.) What is the in-
clination to the horizon of a roof with
•^ pitch, I pitch, I pitch?
27. What is the pitch of a roof slop-
ing to the horizon at 15°, 30°, 45° ?
28. What is the inclination to the horizon of the corner or hip rafter of a
pyramidal roof whose pitches are ^?
29. What is the inclination from the vertical of the corner edge of a wall,
both of its faces having a batter of ^^^ ?
30. At what angle does a railway slope if it has a grade of 0.25%, 0.5%,
2.5%?
31. At what angle must a cog railway ascend in order to rise 2640 feet in
one horizontal mile ?
32. A battleship known to be 341 feet long subtends an angle of 3° 20'
when presenting its broadside to a fort on shore. For what distance should
guns be sighted when trained upon it ? (Note that the isosceles triangle hav-
ing the length of the ship for its base is separable into two right triangles.)
33. In planning the stairway for a house it is desired that the riser, or
vertical distance between steps, shall be 7 inches, and the treads, or horizontal
distances between faces, 11 inches. What will be the angle of inclination of
the hand rail?
34. Taking the data of the preceding problem, what will be the length of
the hand rail if straight, provided the height between floors is 11 feet 8 inches?
35. A cylindrical water tower whose external diameter is 25 feet subtends
a horizontal angle of 5° 30' as viewed from a distance. How far is its center
from the instrument?
(Note that we have a triangle that is right-angled when the line of sight
is tangent. The base is the radius of the tower and the opposite angle is half
of the one observed.)
PROBLEMS INVOLVING RIGHT TRIANGLES
27
36. What horizontal angle would be subtended, at a distance of 2 miles^
by a vertical cylindrical gas receiver 60 feet in diameter ?
(See note to problem 35.)
37. The end of a pendulum 34 inches long swings through an arc of 3| inches.
Find the angle through which the pendulum swings.
38. When vertically over a village, a balloon's angle of inclination, as
viewed from 9 miles distant, was 15° 20'. Assuming the surface of the country
to be fairly level, what was the height of the balloon ?
39. A flagstaff 110 feet high is covered by a vertical angle of 12° 30' at a
point approximately on a level with its center. How far is the observer from
the staff?
40. The data of a preliminary survey are as follows:
AB = 240.9 feet.
BC = 310.7 feet.
CZ> = 611.5 feet.
DE = 237.2 feet.
J5:i^= 528.0 feet.
Considering A, Fig. 17, as the
origin of coordinates and AB a,s
the axis of abscissas, it is required
to compute coordinates for all
points given, thus providing for
the accurate mapping of the
survey.
41. Find the missing parts
and area of the following isos-
celes triangles (see Fig. 18 for
lettering)
Angle at 5 = 62° 11' left.
Angle at C = 55° 50' left.
Angle at D = 43° 42' right.
Angle a.tE = 51° 23' right.
Fig. 17.
35°, a = 42;
« = 72°, & = 12o;
350, & = 180;
/3 = 54°, a = 360 ;
51° 26', & = 480;
a = 640, b = 840.
42. Find the lengths of the chords of the follow-
ing arcs in terms of the radius: 30°, 36°, 40°, 45°, 60°,
75°, 90°, 120°. Compute, given R = 100.
43. Express in terms of the sine and radius the relation between the chord
of an arc and the chord of half the arc.
44. Express in trigonometric form the most important relations between
the radius R of the circumscribed circle, the radius r of the inscribed circle, the
side s, and the number of sides n of a regular polygon.
28 RIGHT TRIANGLES
45. Compute and tabulate the perimeter and the circumferences of the
circum- and in-circles of a regular polygon of n sides for n = 4, 8, 16, 32, given
72 = 10.
46. Compute and tabulate the area of a regular polygon of n sides and of
its circum- and in-circles for n = 4, 8, 16, 32, given R = 10.
47. Repeat example 45 for n == 6, 12, 24, 48.
48. Repeat example 46 for n = 6, 12, 24, 48.
49. A body is acted upon by three forces of magnitudes 20, 40, 60, parallel
to the sides of an equilateral triangle. Resolve these forces along two perpen-
dicular axes, then combine, and thus find the magnitude and direction of the
resultant.
50. A body situated at one vertex of a regular hexagon is acted upon by
five forces represented in magnitude and direction by the vectors drawn to the
five other vertices. Resolve along and perpendicular to the diameter through
the point and find the magnitude and direction of the resultant.
51. A point describes a circle with uniform speed. Determine the position
of its projection upon a diameter in terms of its angular displacement from that
diameter.
52. A point describes a circle of radius 30 feet at a rate of 4 revolutions
per minute. Find the position of its projection upon a diameter at the end of
5 seconds after passing the extremity of that diameter.
53. Determine the components of the vertical acceleration g along and
perpendicular to a plane inclined at an angle a to the horizon.
54. If ^r = 32, find the acceleration along and perpendicular to a plane
whose inclination to the horizontal is 30°, 15°, 10°, 5°.
55. A man weighing 150 pounds stands midway on a 30-foot ladder whose
foot is 10 feet horizontally from the vertical wall against which it leans.
Find the normal (perpendicular) pressure on the ladder and the force tending
to cause him to slide along the ladder.
56. Find the components along the X- and F-axes of a force of 65 pounds
making an angle of 28° 13' with the Z-axis.
57. A steamer is sailing in such a way that its speed due east is 12 miles
per hour and its speed due south is 14 miles per hour. Find the direction of
the steamer's course and the speed in that course.
58. In an oblique triangle, angle B = 45°, angle C = 32°, and side b = 16.
Find side c. (Suggestion. Draw the perpendicular from the vertex A upon
the opposite side.) Attempt to deduce a general relation between the func-
tions of the acute angles of au oblique triangle and the opposite sides.
CHAPTER IV
LOGARITHMS
21. Definition of a logarithm. If we have given
we can find the product of 5Q and 79 without performing the
operation of multiplication, provided we know in advance the
powers of 10. For, we have from the general laws governing
exponents,
56 X 79 = 10i-^4«i^ X W'^''^'
__ -j^Ql.74819+1.89763
=:103-«4'^82^4424.
It will be seen that the process of multiplication has been replaced
by the simpler one of addition.
Many other processes in computation can be simplified in a
similar manner ; for example, if we wish to find the cube root of
a number, say 89.1, we have
89.1 = 101-94988^
and consequently •
V89.1 = (10i-94988y = 100-64996 ^ 4.466+.
In this case the extraction of a root has been accomplished by the
simple process of division. In order to extend this method we
must know all of the powers of some convenient number. The
exponents involved are called logarithms, and the number raised
to a power is referred to as the base of the logarithmic system.
We may define a logarithm more exactly as follows :
If a is any number and x and n are so related that «^ = n^ then
X is^,«^lled the logarithm of n to the base a ; that is, a logarithm is
the index of the power to which the base must be raised to obtain
the given number.
This relation is denoted symbolically by writing
X = log„ n,
and is read ^'•x is equal to the logarithm of n to the base a."
29
30 LOGARITHMS
Thus 3 is the logarithm of 8 to the base 2, since 2^ = 8 ; and
in the illustrations given above, 1.74819 is the logarithm of 66 to
the base 10, etc.
The two statements
a^ = n, x = logo n
are inverse to each other, just as are the relations sin x and
arcsin x^ etc., of Art. 14.
Exercise. Find by inspection log3 27, log5.625, log8 32,
log,. 04. . . . ,
The logarithm of a number to itself as base is unity, since 71^ = 71.
The logarithm of 1 to any base other than zero is zero, since
a^ = 1.
In conformity with the definition just laid down, it follows that,
if two numbers are equal, their logarithms to the same base are
equal. It is also true conversely, that if the logarithms of two
numbers to the same base are equal, the numbers are equal.*
If the base is real and positive, real logarithms produce only
positive numbers. If the base is real and negative, even loga-
rithms produce positive numbers ; odd logarithms, negative num-
bers. For this reason only real positive bases are chosen in prac-
tice, and only positive numbers are combined by the aid of their
logarithms. The sign of the result is ascertained entirely apart
from the numerical computation.
22. Laws of combination. Logarithms are important in trigo-
nometry and elsewhere as labor-saving devices in calculations with
numbers containing many digits. Only so much of the theory of
logarithms as is necessary for this purpose will be developed in the
present chapter.
The laws of combination of numbers by the aid of their loga-
rithms follow at once from the definition of the preceding
article.
I. The logarithm of the product of two factors is equal to the sum
of their logarithms, all to the same hase.
For, if a; = log„ n and y = log^ m we may write •
n = a^ and m = a^.
* In the theory of analytic functions a broader definition of the logarithm is laid
down, and the statement just made requires modification.
LAWS OF COMBINATION 31
Multiplying, we have, by the exponential law,
nm = a^+^,
whence, loga nm ^x-\-y= logc* ^ + lo&a ^- (1 )
This law may evidently be extended to any finite number of
factors.
II. The logarithm of the quotient is equal to the logaritJim of the
dividend minus the logarithm of the divisor^ all to the same base.
For, if a; = log^ri and y = log^ m, we may write as before,
n = a*, m = a^.
n
Dividing, we have — = a^"^,
m
whence, log„ ^— j =x — y = log« n— log„ m. (2)
Manifestly log« f — j = — log^ m.
III. The logarithm of the power of a number is equal to the loga-
rithm of the number multiplied by the index of the power.
For, if a; = log^ n, then n = a^.
Hence, n^ = {a^y = a^"^
or, log« (nP) =px = p loga n- (3)
IV. The logarithm of the root of a number is equal to the loga-
rithm of the number divided by the index of the root.
For, if :r = log^ n, then n = a^. Extracting the ^'th root of both
members, we get
v n = aQ,
whence, log^^^ = - = i log„ n. (4)
q <1
23. Common logarithms. Any number may be used as a base
of a system of logarithms. For certain purposes the so-called
natural system of logarithms, which has for its base the number
e = 2.71828183 •••, has advantages. For the purposes of ordinary
numerical computation, however, it is most convenient to employ
for the base of the system of logarithms, 10, the base of the
universally adopted system of numeration.
32 LOGARITHMS
The common logarithms of all exact integral powers of 10 are
positive integers ; for instance
logio (1000000) = logio(W)
= 6 log,, 10
= 6.
The logarithms of reciprocals of integral powers of 10 are
negative integers ; thus
logio (.00001) =logi„ (10-0
= -51og,„10
= -5.
The losrarithms of numbers situated between two consecutive
integral powers of 10, say between 10^ and 10*+^, lie between k
and k + 1, where k is any integer, positive or negative. Thus
103 < 2417 < 104,
whence, , 3 < log^^ 2417 < 4,
or, logjQ 2417 = 3 + a number lying between and 1.
The logarithms of numbers greater than the base consist of an
integer plus a proper fraction. The fractional part is written
decimally, calculated to a number of decimal places, depending
on the degree of accuracy desired in the use of the table. The
integral part of the logarithm is called the characteristic; the
decimal fraction, its mantissa.
Hereafter, in this book, except in Chapter IX, we shall have
to do only with common logarithms and, unless otherwise expressly
stated, log n will denote logjQ n.
24. Characteristic. If one number is equal to another number
multiplied by a factor which is a power of 10, the logarithms of
the two numbers differ by an integer. For
log (10* xn')= log (10^ + log n
= k + log n.
Example. log 34000 = 3 + log 34
= 4 + log 3.4, etc.
Every number containing one digit at the left of the decimal
point lies between 10^ and 10^. The characteristic of its logarithm
CHARACTERISTIC. MANTISSA 33
is therefore 0. The cipher should always be written to indicate
that the characteristic has not been overlooked.
Every number containing k digits at the left of the decimal
point is 10*"^ times a number with one digit at the left. The
characteristic is therefore k — \. We have then the following
rule for the characteristic :
The characteristic of the logarithm of any number greater than
unity is one less tlian the number of digits at the left of the decimal
point.
Should the number be less than unity, move the decimal point
ten places to the right (thus multiplying by 10^^) and apply the
same rule as before, then write — 10 after the logarithm for
correction. Thus
log 7.12 = 0.85248,
log 71200 = log (10* X 7.12)
= 4.85248,
log .00712 = log (10-10 X 71200000)
= log (10-10 X 107 X 7.12)
= 7.85248-10.
The positive part of the last characteristic is seen to be the
difference found by subtracting from 9 the number of ciphers
immediately following the decimal point in the number.
The characteristic of the logarithm of any number less than unity
is found by subtracting from 9 the number of ciphers between the
decimal point and the first significant digit, then affixing —10.
25. Mantissa. We have seen that moving the decimal point
in the number merely changes the characteristic of the logarithm,
leaving its mantissa unaltered. The mantissa depends solely upon
the sequence of significant digits.
In the tables given, the logarithms are computed to five deci-
mal places (see pp. 1-21), and the mantissas alone for all numbers
from 100 to 9999 are given, arranged in the following manner :
Running down the left margin, under iV, are to be found the
first three digits of the number. In the next, (open) column
occur the first two figures of the mantissa. In the next ten
columns are the remaining three figures of the mantissa arranged
under the fourth digit of the number at the top of the columns.
34 • LOGARITHMS
Thus to find the mantissa of log 3814, we select the row having
381 in the left margin. The first two figures of the mantissa, 58,
are found in the first column. The three remaining figures, 138,
are found in the column headed 4, the fourth digit of the number,
giving the mantissa .58138.
To avoid repetition, the first two figures, 58, are not printed in
every line, but are to be used from 3802 to 3890, inclusive. The
prefixed asterisk, *006, denotes that the mantissa of 3891 is .59006,
not .58006.
EXERCISE IX
1. Find by inspection logg 16, log,, 27, log^ ^^.
2. Find by inspection logg 81, logg 32, logo; 9.
3. What numbers correspond to the following logarithms to base 4 : 0, 1,
2,2.5,3, -2,-3?
4. What numbers correspond to the following logarithms to base 8 : 0,
1,H, -I, -2?
5. Find by logarithms: («)^; (P) '^^^f^iM.
6. Find («) VtW; (b) \/W7 ; (c) \/9l.
^ Find^
18 X V240 X 753
72 X Vim X 200
3/
, Find (^xV720xl5Y^
V2x V480x 248/
9. Find { "^^ ] * , where k = 1.41.
\ 65 /
10. Solve for x: ^^ = 24.
11. Solve for x : 6* = 25.
The amount A attained by a principal P at interest at the rate r com-
pounded annually for n years is
A =P(1 +r)~.
12. Find the amount of $ 3680 at 4 per cent in 6 years.
13. Find the principal which, in 7 years at 5 per cent, amounts to ^ 5820.
14. At what rate will ^ 5000 amount to $7500 in 8 years ?
15. In how many years will $ 86,500 amount to $ 129,600 at 3^ per cent ?
16. If a city increases its population I each year, in how many years will
it double its size ?
INTERPOLATION 35
26. Interpolation. It will be shown in Art. 79 that the differ-
ence in the logarithms of two numbers is approximately propor-
tional to the difference in the numbers provided these differ-
ences are small. Thus, approximately,
log 51473 - log 51470 ^ 51473 - 51470 ^ 3
log 51480 - log 51470 51480-51470 10*
We have, then,
log 51473 = log 51470 + ^\ (log 51480 - log 51470).
Introducing the values from Table I,
log 51473 = 4.71155 + ^^^ (4. 71164 - 4.71155)
= 4.71155 4-. 3 X .00009
= 4. 71155 +.00003
= 4.71158.
The difference .00009, or omitting the denominator, the 9 is
called the tabular difference corresponding to the logarithm of
5147. Note that the added difference is computed to the nearest
fifth decimal place.
This process is called interpolation by the principle of pro-
portional parts. To facilitate interpolation, tables of proportional
parts are inserted in the logarithmic tables in the column headed
P.P. At the top of each of the P.P. tables is the tabular differ-
ence and under this is the number to be added corresponding to
the digit at the left. For example
log 38.25 = 1.58263
log 38.26 = 1.58274.
The difference is .00011 and in the P.P. column is a table
headed 11. Suppose now that log 38.257 is required. Opposite
7 under 11 is found 7.7 ; hence 8 is to be added in the fifth deci-
mal place, giving
log 38.257 = 1.58271.
27. Numbers from given logarithms. The inverse process of
finding the number corresponding to a given logarithm is best
explained by an illustration. Given the logarithm 3.84235. Only
the mantissa need be considered at first, as the characteristic
merely determines the position of the decimal point in the number.
36 LOGARITHMS
Looking for 84 in the first column after the margin, we find it
corresponding to numbers from 692 to 707. The nearest tabular
number (mantissa) smaller than 235 is 230, corresponding to the
number 6955. The difference is 5, while the tabular difference,
found by subtracting 230 from 236, is 6. We have now the pro-
portion for the next digit,
n _5 ^
10~6'
so that the next digit is found by dividing 50 by 6. It is inad-
visable to carry the interpolation beyond one additional digit.
Since 50 -^ 6 = 8 • + • • •, we have found the desired number to be
6955.8. The decimal point is placed after the fourth digit accord-
ing to the rule for the characteristic, the given characteristic
being 3. Should the logarithm be followed by — 10, the decimal
point must finally be moved ten places to the left.
28. Cologarithms. The logarithms of divisors have to be sub-
tracted. Subtraction, however, can be avoided and the logarith-
mic computation of a succession of multiplications and divisions
effected by a single addition process. There is no advantage in
using cologarithms when but two factors are involved. When,
however, more than two are involved, instead of dividing by the
denominator or divisor factors, we may multiply by their recipro-
cals, obviously a legitimate substitution. Now
log— = — log m = (10 — log m) — 10.
m
This logarithm, (10 — log m) — 10, is called the cologarithm
of m, written cologm. It may be written down immediately from
the table by beginning at the left and subtracting each figure from
9, until the last figure, which must be subtracted from 10. Thus
log 28.24 = 1.45086
and colog 28.24 = 8.54914 -10.
29. Logarithms of trigonometric functions. Logarithms of the
trigonometric functions are arranged in Table II in the same
manner as are the natural functions, or true numerical values of
the functions. Logarithmic secants and cosecants need not be
printed, since they are the cologarithms of the cosines and sines.
The sines and cosines of angles and the tangents of angles less
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS 37
than 45° are numerically less than unity. In conformity with
Art. 24, therefore, their logarithms are written in the augmented
^^^^^' log sin 6d° 21^ = 9. 95850 - 10.
The — 10 is not printed in the table but it is always understood.
The positive portion of the characteristic is printed in the table.
Usage differs with respect to printing the logarithmic tangents
of angles greater than 45°. Engineering and physical instruments
are usually graduated to minutes or larger divisions of the angle,
so that it is not feasible to carry the interpolation farther than to
tenths of minutes. The tables of functions and of proportional
parts printed in connection with this book are arranged with this
in view.
Astronomic observations justify carrying the interpolation to
seconds, and astronomers use for this purpose tables computed to
seven or more decimal places.
For example,
log sin 29° 37' = 9.69890 - 10,
log sin 29° 38' = 9. 69412 - 10.
The difference is .00022, and in the P.P. column is a table headed
22. Suppose now that log sin 29° 37.4' is required. Opposite 4
under 22 is found 8.8 ; hence 9 is to be added in the fifth decimal
place, giving
log sin 39° 37.4' = 9.69399 - 10.
EXERCISE X
1. Find from the table the logarithms of 72484, 619.25, 695 x 10^
.00064375, 3 x lO^i.
2. Find from the table the logarithms of 91386, 14.295, 321 x 10^,
.000078541, 2 x lO^*.
3. Find the numbers whose logarithms are 3.71295, 12.61242, 8.21312 - 10.
4. Find the numbers whose logarithms are 4.21382, 11.75153, 6.13579 - 10.
5. Find Young's modulus of elasticity from the formula Y= — ^, if
m = 4932.5, g = 980, I = 110.5, tt = 3.1416, r = .25, s = .3. '^'' ^
6. Find the radius of the sun if its mass is 2.03 x 10^^ grams, and its
average density is 1.41, knowing that mass = volume x density.
7. The radius r of each of two equal, tangent, iron spheres which attract
each other with a force of 1 gram's weight, is given by the formula
4r2 i22'
38 LOGARITHMS
in which the density of iron p = 7.5, the mass of the earth ikf = 6.14 x 10*^
grams, and the radius of the earth R — 6.37 x 10^ cm., while ir = 3.1416. Solve
for r and compute by logarithms.
8. Solve example 7 for spheres of lead with density p = 11.3.
9. The population of a county increases each year by 12.5 per cent of the
number at the beginning of the year. If its population Jan, 1, 1776, was
2.5 X 106, what will it be Dec. 31, 1926?
10. If the number of births and deaths per annum are 3.5 per cent and
1.2 per cent respectively of the population at the beginning of each year, and*
the population on Jan. 1, 1830, was 5 x 10^ find the population Jan. 1, 1905.
11. Find from the tables log sin 25° 32.3', log cot 71° 18.6', colog cos 16°
29.2'.
12. Find from the tables log cos 19° 25.7', log tan 31° 16.2', colog sin
65° 12.8'.
13. Find the angles corresponding to log cos a = 9.31723, log cot y8 = 9.16251,
log tan y = 0.61253.
14. Find the angles corresponding to log sin a = 9.63152, log tati
(3 = 9.71728, log cot y = 0.15382.
15. Francis deduces the following formula for the discharge over a weir,
q = 3.01 bH^-^, in which q is the discharge in cubic feet per second, b the breadth
of the crest, and H the head of water. Find by logarithms the discharge when
6 = 3.5 and// =1.2.
16. A common formula for finding the diameter of a water pipe is
m
d = 0.479
h
in which /is a friction factor, I the length of the pipe, q the discharge, and h
the head. Find d when /= 0.02, I = 500, ^ = 5, ^ = 10.
17. The discharge from a triangular weir is given as ^ = c I'V V2 g H^, in
which c is a constant, g the acceleration of gravity, and H the head. Find q
when g = 32.2, H = 1.2, c = 0.592.
18. The formula for velocity head is h = 0.01555 V^. Find ?i when V = b.
19. The elevation of the outer rail on what is known as a one-degree railwav
curve to resist centrifugal force is sometimes given by the formula e = 0.00066 V^,
e being in inches and V the speed of the train in miles per hour. When F = 45,
comjmte e.
20. Another expression for the relation of the preceding problem is
qY-2
c = '^ • Here e is in feet, g is the gauge of the track, V is the speed in feet
per second, and R is the radius of the curve. Given g = 4.71, V = 66, R = 5730,
compute e.
THE SLIDE RULE 39
21. The difference between the base and hypotenuse of a right triangle is
given by c — a = , and when a and c are nearly equal, approximately by
yi c -\- a
c — a = — .
2c
Find the per cent of error introduced by the second method when the angle
between a and c is 15°.
22. If a = length of a short circular arc and c = its chord, then approxi-
mately a — c = . Given a = ^ and R — 100, compute the value of this
difference.
23. The relation between the pressure and volume of air expanding under
certain conditions is pj??j ^-^^ — pv^*\ where p^ and v^ are initial values. If p^ = 40,
v^ = 5.5, find V when j9 = 24 ; also when p = 16.
24. The relation between the initial and final temperatures and pressures
is given by the equation
With ^j = 60 and the other data as in Ex. 23, find the final temperatures
for p = 24: and j9 = 16, respectively.
30. The slide rule. The principles of logarithmic computation
are conveniently illustrated by means of the slide rule, now widely
used in performing mechanically such operations as admit of the
use of logarithms. A brief description of this instrument will be
found profitable at th^s stage, and its use by the student as a
ready check upon the numerical solution of problems is strongly
recommended. As will be seen by an inspection of the simplified
diagram of Fig. 19, the rule is essentially a device for adding and
A B Rule C
1 2 3 4 5 6 7
a Slide h
1 1 — I — I — I —
3 4 5 6789 10
Fig. 19.
subtracting logarithms, thereby giving a wide range of computa-
tions. In the figure the point 6X on the "slide" is set opposite
the point B on the "rule." If both scales, which are alike, are
so divided that AB is equal, or proportional, to log 2 and ab to
log 3, then on the rule opposite b on the slide gives the distance
^(7 equal, or proportional, to log 6. That is, log 2 + log 3 = log
(2x3) = log 6.
Similarly by subtraction, AC — ab = AB ;
that is log 6 — log 3 = log 2.
40
LOGARITHMS
The point a of the slide is called the index^ hence we have the
following rules for simple operations.
1. To multiply two numbers, set the index opposite one num-
ber on the rule and opposite the other number on the slide read
the product on the rule.
2. To divide one number by another, set the divisor on the
slide opposite the dividend on the rule and read the quotient on
the rule opposite the index.
In the instrument as actually constructed, * Fig. 20, there are four scales
denoted respectively by A, B, C, and D, of which scales B and C are on the
Fig. 20.
slide. For convenience in compound operations the rule is provided with a
runner r by means of which a setting of the slide may be preserved while the
slide is moved to a new position. The following example will illustrate the
manipulation of slide and runner.
Example 1. Find 6^^^115x27.
14.6 X 342
Set 14.6 on C scale opposite 63 on D scale ; move runner to 115 on C scale ;
move 342 on C scale to runner, and opposite 27 on C scale read result on D scale.
In this, as in all slide-rule computations, the decimal point must be
located by inspection.
On the lower side of the slide are three scales, the outer of which are marked
S and Irrespectively. The following examples illustrate the use of these scales.
Example 2. Find 36 sin 22^
Set 22 on the S scale opposite the mark on the slot in the right-end of the
rule ; then opposite the end of the A scale can be read on the B scale the natu-
ral sine of 22°. Now opposite 36 on the A scale read the result on the B scale.
Example 3. Find 26.5 tan 13° 15'.
Reverse the slide and set 13° 15' on the T" scale opposite the mark on the
slot ; then opposite the end of the B scale can be read on the D scale the natu-
ral tangent of 13° 15'. Set the runner at this point and replace the slide with
the index at the runner. Opposite 26.5 on the C scale read the required prod-
uct on the D scale.
Example 4. Find 56i''.
Set the index of C scale opposite 56 on D scale and opposite the mark on
the under side of the right-hand end of the rule read 748 on the middle scale
* A more detailed description of the slide rule is not within the scope of this
book. A manual describing fully the use of tlie instrument can be had of any firm
selling slide rules.
LOGARITHMIC SOLUTION OF RIGHT TRIANGLES 41
of the lower side of the slide. This reading is the mantissa of the logarithm
of 56. The characteristic 1 must be supplied as usual. Now in the usual way-
find 1.3 X 1.748; that is, put index to 1.748 on D scale and opposite 1.3 on C
scale read the product 2.272. This is the logarithm of SG^-^. Set the mantissa
272 on the logarithm scale opposite the mark on the rule and read 118.7 on the
D scale opposite the index.
EXERCISE XI
1 17- /I ^ \ 64 X 37 ,,. 193 . . 0.05 x 137 x 62
1. liind (a) (b) — ; (c) •
^ 163 ^ ^ 67 X 2.1 ^ ^ 14 X 28 X 6.5
2. Find (a) 127 sin 24°, (6) 0.32 sin 72°, (c) 16.5 cos 35°.
3. Find (a) 37 tan 8° 20', {h) 1.35 tan 40° 10'.
4. Find («) 11^2^32:, (/.) 35.5 ?H^^
^ ^ 64 ^ ^ sin 47°
5. Find (a) 28^ {h) y/^^, (c) 7.311-27.
31. Right triangles solved by logarithms. — It is now possible,
with the aid of the logarithmic tables, to solve right triangles the
numerical values of whose parts contain more digits than those
given in Chapter III, without entailing laborious multiplications
and divisions.
Example 1. Given a = 51.72, jS = 73° 46^
Solving the proper formulas for the unknown parts, we have
a
^ = B'
cos p
h=a tan /3,
A = \a^ tan ^,
h — c cos a, check.
Sum of angles =90° 00^
^= 73° 46^
«=16°14^
log «= 1.71366
log cos /3= 9.44646 -10
log (? = 2.26720
.•.c= 185.01
42 LOGARITHMS
log a = 1.71366
log tan 13 = 0.53587
log 6 = 2.24953
.-. 5 = 177.64
21og«= 3.42732
log tan /3= 0.53587
colog 2 = 9.69897-10
log ^ = 13.66216 -10
.-. J. = 4593.67
Check
logc= 2.26720
log cos a = 9.98233-10
log 5 = 12.24953 -10
.-.5 = 177.64
Note that log a^=2 log a. In solving, first write all the forms
needed for the complete solution ; secondly, look up and write in
all the needed logarithms of the data from the tables ; thirdly, per-
form the additions and subtractions ; lastly, from the logarithmic
results find the numbers. Then log cos /3, log tan /3, and log
cos a ( = log sin )S) can all be found from once turning to the
angle 73° 46'.
A form of computation sometimes used is given below. It has
the advantage of being more compact than the usual form, and
furthermore the logarithms of the data stand close to the data,
thus permitting easy verification of results or correction of
mistakes. ^^^^^
a=51.72 log 1.71366 logl. 71366
/3 = 73° 46' log cos 9.44646-10 log tan 0.53587
c = 185.01
log 2.26720
2
^ = 4593.7
log 2.26720
5 = 177.64
« = 16°14'
5 = 177.64
log 2.24953
log cos 9.98233- 10
log 2.24953
log 3.42732
log tan 0.53587
colog 0.69897 -10
log 3.66216
LOGARITHMIC SOLUTION OF RIGHT TRIANGLES 43
Example 2. Given 5 = 7124.5, c = 9365.4.
We have,
cos a =
1
c
^ =
: 90° - a.
a =
■ e sin a,
A =
: ^ he sin a,
a =
•■ h tan a, check.
log 5 =
: 3.85275
logc =
: 3.97153
log cos a =
: 9.88122-
10
a =
: 40° 28.4^
=
: 49° 31.6'
loge^
: 3.97153
log sin a =
: 9.81231-
-10
loga =
= 13.78384-
-10
a =
: 6079.2
Check
log 5 =
: 3.85275
log tan a =
: 9.93109-
-10
log a = 13.78384 -10
a =6079.2
The following is the compact arrangement of the computation :
Check
b = 7124.5 log 3.85275 log 3.85275
c = 9365.4 log 3.97153 log 3.97153
a = 40° 28.4^ log cos 9.88122-10 log sin 9.81231- 10 log tan 9.93109- 10
13 = 49° 31.6^
a = 6079.2 log 3.78384
a = 6079.2 log 3.78384
It appears that the Pythagorean proposition, a^ + 5^ = c^, is
not used because it is not adapted to the use of logarithms. It
might be used in this case, however, in the form
44
LOGARITHMS
EXERCISE XII
Find the missing parts of the following triangles, using loga-
rithms. (The work may be checked with a slide rule.)
a
fi
a
b
c
A
1.
63°
2584
2.
7531
8642
3.
75° 15.2'
965.24
4.
47.193
3972.6
5.
7.3298
6.1032
6.
18° 25.5'
32.96
7.
132.97
985.27
8.
53.215
13.712
9.
65983
72916
10.
29° 50.2'
10.207
11.
25° 17.4'
382.97
12.
.00020
.00037
13.
63° 12.7'
7.1436
14.
.07154
.09127
15.
35° 16.4'
.62961
16.
35° 16.8'
41658
17.
.00615
.00415
18.
80° 12.5'
5.2108
19.
.00729
.01625
20.
25° 18.2'
1729.3
The examples 1-20 of Exercise VIII may also be solved by
logarithms and the results compared with those there obtained.
21. Find the radius of the circle inscribed in a regular pentagon whose
side is 12 feet.
22. Find the side of a regular pentagon inscribed in a circle whose radius
is 15 feet 7 inches.
23. Find the area of a regular octagon whose circumscribed circle has a
diameter of 10 feet.
24. A tower 120 feet high throws a shadow 69.2 feet long upon the plane
of its base. What is the angle of inclination of the sun?
25. The top of a certain lighthouse is known to be 73 feet above the
water. From a boat the angle between the top and its reflection is measured
as 6° 45'. How far is the boat from the light ?
26. Two trains leave a station at the same time, one going north at the
rate of 30 miles per hour, and the other east at the rate of 40 miles per hour.
PROBLEMS INVOLVING RIGHT TRIANGLES
45
How far apart will they be in 20 minutes, and what is the direction of the line
joining them ?
27. Show that if a is the side of a regular polygon of n sides, the area of
1 1 S()°
the polygon is given by ^4 = - d^n cot
28. Show that if r is the radius of a circle, then the area of a regular cir-
cumscribed polygon of n sides is A = rhi tan
n
29. Find a value for the area of an inscribed polygon corresponding to
that given above.
30. Taking the moon's diameter as 31' 20" and its distance from the earth
as 239,000 miles, what is its diameter in miles ?
31. At what distance may a mountain 4 miles high be seen across a plain,
the earth being taken as a sphere of 4000 miles radius?
32. If the sun's diameter is taken at 866,000 miles and its distance from
the earth as 93,000,000 miles, what angle should it subtend at the center of the
earth.
33. An approximate formula for the distance from the midpoint of a cir-
cular arc to the midpoint of its chord is m = - — — — ■,
4 100
in which I is the length of the chord in feet and a the
deflection or circumferential angle subtended by a base
or chord of 100 feet. Find m for Z = 30, a = 2°.
34. If / is the angle of intersection between two
tangents to a circle of radius i2, the distance T from
a point of tangency to the point of intersection is given
hj T = Rcod. Find T ior R = 3000 feet, and / = 22° 52'.
35. The length of a chord is given by 2 72 sin i 7, in which / is the central
angle. Find the chord length for R = 2000, / = 12° 13'.
36. A river which obstructs chaining on a survey is passed by tri-
angulation. The line ^J5,'Fig. 22, is measured 200
feet perpendicular to AC, and the angle ADC found
to be 35° 27'. AVhat is the distance AC 2
37. With an instru-
ment at A, Fig. 23, a
level line of sight passes
6 ft. above the top of a wall as measured
on a rod. The angles of depression * to
the top and bottom of the vertical face
are respectively, 2° 31' and 42° 16'.
What is the height of the wall?
* The angles of elevation and depression of an object measure respectively its
angular distance above or below the horizon of the observer.
Fig
Fig. 22.
Fig. 23.
46
LOGARITHMS
AC
DAB
Fig. 24.
14° 41' find AD and DB.
38. In order to obtain both the horizontal
and vertical distances to an inaccessible point,
the solution of two triangles may be necessary.
Fig. 24 represents two views of the problem.
Wishing the distances AD and BD, first lay out
the base line A C of any convenient length per-
pendicular to AB. Measure the angle A CD and
compute AD.
Next from AD and the angle DAB^ the
angles of elevation, compute DB.
Having ^C = 300 ft., ACB = Ql°d4:', and
CHAPTER V
THE OBTUSE ANGLE
32. Definitions of the trigonometric functions of obtuse angles.
If an obtuse angle (^.e. an angle greater than 90° and less than
180°) is placed on the axes of coordinates in the same manner as
was the acute angle in Art. 6, the terminal line will extend
into the second quadrant. The trigonometric functions of such
angles are defined exactly as in Art. 6. Thus in Fig. 25,
sin a
I
cos a
tan a
, etc.
4F
33. Signs and limitations in value. The abscissas of all points
in OA (Fig. 25) are negative, while their ordinates and radii
vectores are positive. It is evident, therefore, that some of the
defining ratios are negative. In
accordance with the law of signs
in algebraic division, we find
that the sines and cosecants of all
obtuse angles are positive, while
their cosines, secants, tangents,
and cotangents are negative.
The student should verify
each of these statements in de-
tail and become unhesitatingly
familiar with these fundamental
facts.
P\irthermore, the sine and cosine cannot be numerically greater
than unity and the secant and cosecant cannot be numerically
less than unity.
47
^X
Fig. 25.
48
THE OBTUSE ANGLE
Query. What are the limitations in value of the tangent and co-
tangent ?
34. Fundamental relations. If the effects of the law of signs
are traced, it will be seen that all the relations of Art. 9 hold also
for functions of an obtuse angle without any modifications.
35. Variation. As the angle 6 varies from 90° to 180°, while
V remains constant, x is always negative and varies from to — v,
and y is positive and varies from v to 0. Consequently, as 6
increases from 90° to 180°, sin 6 decreases from 1 to 0, cos 6 decreases
(algebraically) from to — 1, tan 6 increases from — oo to 0, cot 6
decreases from to — oo, sec 6 increases from — oo to — 1, esc 6
increases from 1 to oo.
The terms positive infinity and negative infinity require careful
consideration. If 6 varies continuously from 89° to 90°, tan
varies in such a way as to exceed in magnitude any previously
assigned definite value, however large. As it is positive for all
values of 9 in the first quadrant, it is consequently said to become
positively infinite (+oo). If 6 varies continuously from 91° to
90°, tan 6 varies so as to exceed numerically any previously
assigned definite value. As it is, however, always negative for
values of 6 in the second quadrant, it is said to become negatively
infinite (-co). The plus or minus sign written before the symbol
00 merely indicates whether the trigonometric function increases
numerically without limit through a positive or a negative set of
values.
36. Functions of i8o°. As 6 approaches 180°, v remaining
constant, x approaches — v and y approaches 0. We have,
\Y then,
sin 180° = 0,
cos 180° = - 1,
tan 180° = 0,
cot 180° = 00,
sec 180° = - 1,
FIG. 26. cscl80°=oc.
37. Functions of supplementary angles. Two angles are called
supplementary if their sum is 180°. Thus, in Fig. 27, a and fi are
FUNCTIONS OF (180^^ - a) AND (90°+ a)
49
supplementary, and yS=180 — «, a being acute. The triangles
OMP and ONQ are similar, but ON is negative. The pairs of
corresponding sides are v and v\ x and r^:^ y and yK Hence we have
sin (180° - a) = sin /3
cos (180° - a) = cos yS = ^
v' V
v'
sm a.
= — cos a,
tan (180° - a) = tan ^
Similarly :
cot (180° - a) = - cot a,
sec (180° — a) = — sec a,
esc (180° — a) = CSC a.
As a consequence of the
relation sin (180° — «) = sin a,
two values exist for arcsin m,
the one acute, the other obtuse,
and supplemental to each other.
Y
y _ y
— tan a.
Y
P^^
"^-^^
V ^
y'
^' J
f3 3^
y
I
ST
X'
X
M
Fig. 27.
In case m = 1, the two values are
identical.
Fig. 28.
QdERY. Is this also true of arccos m,
arctan tw, etc. ?
38. Functions of (90° + a). In
y Fig. 28, ^ = 90° + a, a being acute.
-^f-^^ The triangles OMP and ONQ are
similar, but the pairs of homologous
sides are v^and v\ x and y, y and a;',
while 2:' is negative. We thus obtain
sin (90° 4- a) = sin /8 = ^ = - = cos a,
cos (90° + a) = cos /8
_^_ ^_
sma,
tan (90° + a) = tan /3
- = — cot a.
50 THE OBTUSE ANGLE
In like manner,
cot (90° + a) = - tan a,
sec (90° + a) = — CSC a,
esc (90° + a)= sec a.
EXERCISE XIII
1. Find the values of the functions of 135°. (See Art. 11.)
2. Find the values of the functions of 150°. (See Art. 11.)
3. Find the value of sin [cos-^(— |f)], tan (csc-if ^), cos [arctan (— :^)],
the angles being of the second quadrant.
4. Find the value of cos (arccos — ^j), sin [tan-i ( — j^^)], cot (arcsin ff),
the angles being of the second quadrant.
5. Express in terms of an angle less than 45°, cos 160°, tan 130°, sec 150°.
6. Express in terms of an angle less than 45°, sin 170°, esc 95^ cot 140°.
7. Verify for a = 60°, the ftquatiiuiig - ^c/& ^^^c/^S
sin 2 a = 2 sin a cos a, cos 2 a = 2 cos^ a — 1.
8. Verify for a = 45°, the equations
sin 3 a = 3 sin ct — 4 sin^ a,
cos 3 a = 4 cos^ a — 3 cos a. •
9. Verify for a = 120°, the equations
^^^ - ^ -. ,- + cos a
cos -'—'»'
tanl«:=V^-"Q^^^^-^»^^.
2 ^ 1 + cos a sin a
10. Verify for a = 120°, the equations
i«=4
cot- ^--^1- + cos«_l + cosa
cos a sm a
11. Verify for a = 120°, (3 = 30°, the equations
sin (« + ^) = sin a cos /3 + cos a sin ^,
cos (a- 13) = cos a cos )8 + sin a sin /S.
12. Verify for a = 120°, /3 = 60°, the equations
sin (a — )8) = sin a cos /3 — cos a sin y8,
cos (a -\- f3) = cos a cos /3 — sin a sin ^.
FUNCTIONS OF OBTUSE ANGLES 51
13. Fill in the proper values in the following table for handy reference :"
a
sin a
cos a
tan a
cot a
sec a
CSC a
0°
30=^
i
45°
lV2
60°
W3
90°
1
120°
iV3
135°
iV2
150°
i
180°
CHAPTER VI
OBLIQUE TRIANGLES
39. Formulas for solution. In the oblique triangle ABC,
Fig. 29, let the angles be denoted by a, ^, 7, and the lengths of
the opposite sides by a, 5, c?, as in the figure.
The relation a 4- p + 7 = 180° al-
ways exists, and consequently when
two of the angles are known, the
third is determined. Five of the six
parts of the triangle still remain to
be found ; namely, the three sides
and two angles. It has been shown
in elementary geometry that if any
three independent parts are given, the triangle is determined and
the remaining parts can be found. Then two formulas, in addi-
tion to the one just stated, are sufficient for the complete solution.
It is, nevertheless, convenient to express the relations between
the sides and angles in a variety of forms. Those given in the
following pages are selected on the score of utility. They fall
into sets of three each. From any one of each set the other two
may be written by cyclic advance of the letters involved ; i.e. by
changing a into 6, h into c, c into a, and at the same time a into
^, /3 into 7, 7 into a. The legitimacy of this process and the
truth of the resulting formulas appear from the consideration that
no distinction is made as to any one side or any one angle. Any
side and its opposite angle can be exchanged for any other pair.
The cyclic advance affords a convenient systematic method of
writing all possible forms.
From any one of these sets, as for instance that of Art. 40,
or that of Art. 42, all the other sets may be derived by purely
analytical processes. An independent geometric proof is given of
each, however. The derivation by the analytic method suggested
will afford a valuable review exercise after Chapter VIII has been
studied.
62
LAWS FOR OBLIQUE TRIANGLES
53
40. Law of projections. If, in
Fig. 30, the perpendicular CD is
drawn from to AB^ the portions
AB and BB are respectively the
projections on the side AB of the
other two sides AC and CB. Con-
sequently, by Art. 15, we have
Fig. 30.
AB ==ACco^a-\-CB cos /3,
or c = & cos a + a cos p.
By drawing the perpendicular from A and B in turn, we get
a zz c cos p + 6 cos "y,
6 = a cos "Y + c cos a.
By cyclic advance of the letters the first formula is transformed
into the second, the second into the third, and the third into
the first.
41. Law of sines. Connect the circumcenter K in Fig. 31
with the vertices, A^ B, C, and the midpoints, X, M, iV, of the sides.
Then is Z BK0 = 2 «, Z CKA = 2 /3,
Z.AKB=2y. (Why?) In the
right triangle KLO^ /. LKO = a,
and LC=^a. Denoting the cir-
cumradius by R^ Art. 16 gives
R sin a.
Fig. 31.
J- a
The other right triangles give
likewise
^b = R sin yS,
^ c = R sin y.
Equating the values of 2 R, we obtain the law of sines ; namely,
a _ b _ c
sin a sin p sin 'y
The cyclic symmetry is apparent.
The student should draw the figure and give the proof in case
one angle of the triangle is obtuse.
42. Law of cosines. In Fig. 32 the perpendicular jt? drawn from
(7 divides the opposite side c into two portions m and w, and the
54
OBLIQUE TRIANGLES
whole triangle into two right triangles ADC and BDO, In
the latter triangles, we have, by
Art. 16,
a^ = n^ -\-p^
= ((? — 771)2 -{-p^
Fig. 32. or a^ — ^2 ^ ^2 _ 2 hc COS a.
Proper changes in the figure yield
62 = c'^ + a^ -2ac cos p,
c2 = a2 _f. ^2 -2 ah cos 7.
These again may be written by cyclic advance of the letters.
Useful forms for writinsr these laws are :
cos a =
52+^2-
a^
2 be
cos /3 =
c2-fa2-
h'^
2 ao
i->r»o i\/ —
a'^+P-
-6'2
lab
43. Law of tangents. lu Fig. 33 draw AE the bisector of the
angle at J., and BF and CD perpendicular to it from the other
vertices.
Then
ABAF=^DAC=^a,
while
Z DCE= Z. EBF= 90° - Z BEF
= 90° - (Z ABE + Z ^^ JS')
= K« + ^ + 7)-(^ + -|«)
= K7-/3).
Again,
i)^= ^^+ DE=AF- AD.
From the right triangles in the figure we get
AF~AD
Fig. 33.
(e-
b) cos J a
^^^2^7 Z:^; ^^ ^^^ FB + CD FB^-CD ((? + *) sin i«
or
taiil(v-p)
c + 6
cot ^ a.
LAWS FOR OBLIQUE TRIANGLES
55
The forms
tani (a-v) = ^-^cotlp,
a
tani(p-a)=— --cot|Y,
& + «
may be obtained from suitably altered figures or by cyclic advance.
The formula may be written symmetrically
tanK7-y3) ^g-^
tan i (7 + yg) c-^h
Ji b>c, the first formula will stand
tan i (/? - ry) = -Zl cot I a.
Similar changes may occur in the other two.
44. Angles in terms of the sides. Construct the inscribed
circle, Fig. 3-4, and denote its
radius by r. Denoting the perim-
eter a-{-b-{- c by 2 s, we have
AE=AF=s-a,
BD = BF=s-h,
CD=CE = s- c.
Consequently, by Art. 16,
V V
tan J a = , tan \ p = -, tan \ y =
The value of r in terms of the three sides is derived in the
corollary of Art. 45, thus completing this theorem.
45. Area of oblique triangles.
(1) By elementary geometry, we
have (see Fig. 35)
Introducing the value of p found by
Art. 16, we get the formula
^ = 1 6c sin a.
56 OBLIQUE TRIANGLES
with the cognate forms
A= ^ca sin p, ^ = | ab sin -y.
(2) Squaring both members of the formula just derived, we
obtain, with the aid of readily justifiable transformations and sub-
stitutions.
= i5V(l-cos2
a)
,
= — (l + coscc) •
|(1-
- cos a)
her. ^h^ + e^-
->
!(■
b'^^c^-
2 5c
.■)
2bc-hb^ + c^-
aP' 2bc-
52_,.2 4.^2
4
4
b-\-c + a b +
c— a
a-
-b-{- c a-\- b — c
2 2
= s(s — «)(s — 5)(s — c).
Whence we have the desired formula
A = Vs(s — a)(s — h) (s — c).
(3) If r is the radius of the inscribed circle, we have, by
elementary geometry,
A = rs.
Corollary. Equating the values of A found in (2) and (3),
and solving for r, we get
^ s
the result needed to complete the theorem of Art. 44.
46. Numerical solution. The formulas of Arts. 40 and 42 are
not adapted to the employment of logarithms. They are useful,
however, in case the numerical values of the sides contain few
digits.
The solution of oblique triangles falls into four well-defined
cases, according as the three given parts consist of
I. Two angles and one side.
II. Two sides and an angle opposite one of them.
III. Two sides and the included angle.
IV. Three sides.
NUMERICAL SOLUTION. CASE I 57
Each of these three cases with a model solution is discussed in
detail in the following articles.
47. Case I. Given two angles and one side. Let the given
parts be a, j3, a.
The solution is effected by means of the formulas of Arts. 39
and 41. Solving for the unknown parts, we have
7 = 180°-(« + /3),
, a sin y8
Example.
sin a '
a sin 7
c=—. -,
sin a
t, , b sin 7
formula c = —. — ~ .
sm l3
Given a = 47° 13.2'
^=65° 24.5'
a = 43.176
sum of angles = 180°
« 4-/5 = 112° 37.7'
.-. 7= 67° 22.3'
loga= 1.63524
log sin /3= 9.95871-
-10
cologsina= 0.13433
log 5 = 11.72828 -
-10
.-. 5 = 53.491
loga= 1.63524
log sin 7= 9.96522-
-10
cologsina= 0.13433
log c= 11.73479-
n^
.-. c = 54.299
Check
log 5= 1.72828
log sin 7= 9.96522-
10
cologsiny8= 0.04129
log (? = 11.73479-
-10
/. (?= 54.299
58 ^ OBLIQUE TRIANGLES
The compact form of computation is as follows ;
log 1.63524
a = 43.176
^ ^ 65° 24.5'
a = 47° 13.2'
b = 53.491
y = 67° 22.3
c = 54.299
log 1.63524
log sin 9.95871 - 10
colog sin 0.13433
log b 1.72828
coiog sin 0.13433
log sin 9.9652g - 10
Check
colog sin 0.04129
log 1.72828
log sin 9.96522
log c 1.73479
Examples
Find the remaining three parts, given
1. ;8 = 65°15.5', y = 81° 24.6',
2. /? = 38°37.4', 7 = 75° 32.8',
3. a= 48° 29.2', y= 115° 33.8',
^^— 4. a = 68° 41.5', y = 110° 16.5',
10
log c 1.73479
b = 724.32.
c = 129.63.
a = 14.829.
c = 9.4326.
48. Case II. Given two sides and an angle opposite one.
Let the given parts be a, 5, a.
The solution is effected by the formulas of Arts. 39 and 41.
Solving, we have
. ^ b sin a
sin/3= — ,
a sin 7
c =
with the check formula
(? =
sm a
5 sin 7
sin y8
An ambiguity arises in this case, however, since to any value
of the sine correspond two supplementary angles, one acute, the
other obtuse. Thus we also have
/3'
= 180°-/3,
i
= 180°- (a
+ /30.
c'
a sin 7'
sin a '
e'
h sin 7'
siuiS'
CASE II
59
The nature of this ambiguity will appear from the construction
of the triangles with the given parts. If the given angle a is
acute, there will be no solution, one solution, or two solutions,
according as the free end of a (see Fig. 36), swinging about
.>— .
A\
T^L
Fig. 36.
(7, meets the line AL in no points, one point, or two points ;
i.e. as a is shorter than (72), the perpendicular from upon AL,
longer than AO^ or intermediate between CD and AC For
a = CD there is a single right triangle ; and for a = AC, a single
isosceles triangle.
When a is right or obtuse, there is no solution or one solution,
according as a is shorter or longer than AC.
These results may be tabulated for reference.
«<90'
'a<h sin a,
b sin a<a<h^
a = b sin a,j
no solution,
two solutions,
one solution.
>
90^
a :^ 6, no solution,
> 5, one solution.
If we proceed with the numerical work, without previously
testing the number of solutions possible, the case of a single
solution will appear from the fact that a -\- jS^ > 180°. (Whence
a + (180° - /9) > 180°, or a - yS > 0, or ^ < a.) When there is no
solution, we shall get log sin fi>0 ; i.e. its augmented character-
istic will be 10 or greater. A preliminary free-hand sketch will
ordinarily serve to determine the number of possible solutions.
Example i. Given
a = 3541,
b = 4017,
a = 61° 27'.
60
OBLIQUE TRIANGLES
By careful arrangement of the work, we can determine the
number of solutions by inspection.
Check
log
&=4017
az=61°27'
h sin a
a = 3541
^= 85° ir
a-f;8=146°38'
7 = 33° 22'
c=2217.16
c = 2217.16
3.60390
log sin 9.94369-10
log 3.54759
log 3.54913
log sin 9.99846
colog sin 0.05631
log 3.54913
log sin 9.74036-10
log 3.34580
log
3.60390
colog sin 0.00154
log sin 9.74036-10
log 3.34580
From the logarithms of 6, a, and h sin a it is seen that h sin a
<a<h, whence there are two solutions. For the second solution
we have :
a= 61° 27'
)8'= 94° 49'
« + /?' = 156° 16'
y' = 23°44'
a = 3541
6 = 4017
c' = 1622.52
c' = 1622.52
colog sin 0.05631
log sin 9.60474 - 10
log 3.54913
log
3.21018
Check
colog sin 0.00154
log sin 9.60474 - 10
log 3.60390
log 3.21018
Example 2. How many triangles are determined by the
given parts a = 30°, h = 24, « = 10, 12, 20, 24, 30 ?
Here 6 sin a = 24 x J = 12. Accordingly, we have, for a = 10,
no triangle ; for « = 12, one right triangle ; for a = 20, two triangles ;
for a = 24, one isosceles triangle ; and for a = 30, one triangle.
Examples
1. How many triangles are determined by the given parts (3 = 43°, c = 120,
and h = 63, 81.884, 95, 120, 150?
2. How many triangles are determined by the given parts y = 54°, a = 75,
and c = 51, 60, 67.5, 70, 75, 100?
Find the remaining parts of all possible triangles, given
3. a
62.518,
4. a= 429.15,
5. 6 = 3912.7,
6. 6 = 129680,
6 = 72.932,
c= 328.12,
c = 3526.5,
c = 152960,
/3= 98° 23.5'.
a = 130° 33.7'.
y= 35° 25.8'.
13= 38° 28.8'.
CASE III
61
49. Case III. Given two sides and the included angle. Let
the given parts be «, 6, 7, with a>h. The solution is effected by
the formulas of Arts. 43, 39, and 41. Solving, we have
tan 1 (« — /3) =
I(« + ^) = 90°-i7,
a sin 7
cot J 7,
with the check formula
Example. Given
y = 78° 15'
a = .745
6 = .231
a-/; =.514
log 9.71096-
-10
a + 6=. 976
colog 0.01055
^=39° 7.5'
2
log cot 0.08969
^^^=32° 55.3'
2
^^±^=50° 52.5'
2
log tan 9.81120 -
-10
a = 83° 47.8'
)8 = 17°57.2'
c = . 73368
c = . 73367
sm a
h sin 7
smy3
a =.745,
5 = .231,
7 = 78°15^
log sin 9.99080- 10
log 9.87216-10
colog sin 0.00255
log 19.86551-20
Examples
Find the unknown parts, given
1. 6 = 284.12, c = 361.26, a = 125° 32'.
2. c = 395.71, a = 482.33, ^ = 137° 21'.
3. a = .06351, c = .10329, /8 = 83° 29.4.'
4. c = .00397, h = .00513, a = 68° 21.8^
log sin 9.99080-10
log 9.36361 - 10
colog sin 0.51109
19.86550-20
50. Case IV. Given the three sides. The given parts are
a, 5, G.
62
OBLIQUE TRIANGLES
The solution is effected by the formulas of Art. 44, with the
formula for r from Art. 45. We have at once
s = 1 (a -f 6 + 0'
^_J(«-«)(«-*)(«
-0
tan -« = , etc.
2 s — a
« + /3 + 7 = 180°, serves as a check formula.
Example i. Given
a =.05341,
5 =.06217,
tf=. 03482.
Then 2 s = .15040
s= .07520
colog
1.12378
s-a=. 02179
log
8.33826-10
s- 5 = .01303
log
8.11494-10
«_ ^ = .04038
log
8.60617-10
r2
log
16.18315-20
r
log
8.09157-10
^=29° 32.3'
log tan
9.75331-10
1 = 43° 27.6'
log tan
9.97663-10
^=17° 0.1'
log tan
9.48540-10
a= 59° 4.6'
• y8= 86° 55.2'
7= 34° 0.2'
sum of angles =180° 0'
When the three sides are given and only one angle is required,
say /S, the two appropriate formulas may be combined into one, as
tan -^
_^l(8 — a)(8 — c)
sCs-h)
CASE IV. COMPOSITION OF FORCES 63
Example 2. Given
a= 35,
h= 64,
c= 73.
Then 2 s = 172
s= 86 colog 8.06550-10
s-a= 51 log 1.70757
8-h= 22 colog 8.65758-10
8-e= 13 log 1.11394
2)19.54459-20
i;8=30°37.4' log tan 9.7723U-10
;e=61°14.8'
Examples
Find the angles of the following triangles :
1. a = 6123, ^> = 7148, c = 6815.
2. a = 12,545, 5=8612, ^=10,353.
3. a = .05431, 5 = .03714, 6'=. 06513.
-- — 4. a = .006152, b = .008174, c = .007534.
5. ^. = 72,584,
5 = 125,217, c?= 36,925.
6. a = 13,579,
6 = 35,791, (? = 24,680; find /3.
7. a = 80,812,
h = 37,194, e = 43,618.
8. « = 36,925,
5 = 25,814, c= 14,703; find 7.
Find the areas in examples 1 and 2.
51. Composition and resolution of forces. Equilibrium. In
mechanics the solution of oblique triangles is frequently required
in problems relating to the composition and resolution of forces,
velocities, and other directed quantities.
In this article will be stated, without proof, some of the laws
governing the combination of such quantities, showing the appli-
cation of trigonometry to certain of the problems involved.
Suppose the line segments AB and JL(7, P'ig. 37, to represent
in magnitude and direction two forces acting at a point J., and in-
cluding between their lines of action the angle <^.
64 OBLIQUE TRIANGLES
Complete the parallelogram ABBQ. The diagonal AB^ drawn
from the point A^ is the line segment representing the resultant
of the two given forces, i.e. the sin-
gle force that will produce the same
effect as the two given forces. The
process of finding the resultant of
two or more given forces is called
the composition of forces.
Conversely, the two line segments
AB and AC may be taken as the
components of AB. Thus the two
Fig. 37. ^ forces AB and AC, acting together
at A, produce the same effect as the single force AB. The pro-
cess of finding two or more forces equivalent to a given force is
called the resolution of the force into its components.
Since the segment BB is equal and parallel to AC, it follows
that the resultant and the two components form a closed triangle
ABB, and the relation between the forces may be obtained by
solving this triangle. Note that the angle ABB is the supple-
ment of the angle <^, so that by Art 37,
cos ABB = — cos (/).
Example 1. Find the resultant of two forces of 320 dynes
and 400 dynes, respectively, acting on a common point, at an angle
of 54° 28^
In the triangle ABB, Fig. 37, we have given two sides and
the included angle. If only the magnitude of the resultant is
desired, it may be obtained by the law of cosines. Art. 42. Thus
we obtain „ «
AB = ^\A^ -f ^(7+2 AB AC ■ cos c^j.
If the angle formed by the resultant with its components is also
required, the logarithmic computation may be effected as in Case
III, Art. 49.
Example 2. Resolve a force of 40 pounds into components
making angles of 32° and 74° 20 with its line of action.
Referring to Fig. 37, we have
^D = 40, Z ^^2> = 32°, and Z i>^ (7= Z ^i)^ = 74° 20' .
Denoting the sides opposite the angles A, B, B, respectively, by
a, h, d, we have from the law of sines,
,sin^ J ysini)
a — b -, d = o— — — •
sin B sin ^
EQUILIBRIUM OF FORCES
65
Hence the components may be computed.
Three forces are in equilibrium when the resultant of any two
forces is equal and opposite to the third. Thus in Fig. 37, if
the direction of the force AD is reversed, it and the forces AB
and AQ will be in equilibrium. The necessary conditions that
three forces shall be in equilibrium are :
1. Their lines of action shall lie in the same plane.
2. Their lines of action shall meet in a point.
3. The line segments representing the three forces when laid
off in order shall form a triangle.
In Fig. 38 the forces a, 5, and c applied at a common point are
in equilibrium. The angles between the lines of action are de-
noted by J., B^ C^ as indicated. When the forces are laid off to
form the triangle, the angles of the triangle are seen to be the
supplements of the corresponding angles A, B^ O.
That is,
a = 180° — J., whence sin a = sin A,
/3 = 180° - B, whence sin /S = sin B.
etc. etc.
From the law of sines.
Therefore,
a h
c
sin a sin ^
sin 7
a h
c
sin A sin B sin O
66
OBLIQUE TRIANGLES
EXERCISE XIV
Find the unknown parts of the following triangles :
a
18
y
a
b
c
1.
62° 35'
82916
59278
2.
75290
92841
69289
3.
25° 36.2'
68° 13.5'
3.9168
4.
55° 55.4'
.25317
.36291
5.
69° 17.5'
329.12
689.12
6.
100° 10'
62198
29322
7.
.0000713
.0000987
.0001255
8.
61° 15.2'
49° 16.3'
58.291
9.
120° 50.2'
2.8315
4.1217
10.
38° 17.2'
21.992
50.715
11.
150° 24.2'
.038251
.047319
12.
58° 06.5'
57.15
67.31
13.
75° 19.3'
70° 29.2'
658.42
14.
100.05
200.07
150.08
15.
126° 26.4'
.0021868
.0032292
16.
10° 32.8'
25.317
37.293
17.
50010
70020
90030
18.
48° 25.3'
56° 34.5'
7219.2
19.
120° 15'
62158
75292
20.
90° 00'
725.63
617.25
Solve the following triangles, given
21. a = 2500, c = 2125, A = 208,690.
22.. ft = 103.5, c = 90, A = 4586.7.
23. a = 73° 10', b = 753, A = 74,803.
24. ^ = 57° 25', c = 57.65, A = 3055.7.
25. Find the areas in examples 1, 9, 17.
26. Find the areas in examples 2, 4, 14.
27. Determine the magnitude and direction of the resultant of two forces
of magnitudes a and h, if their lines of action include an angle <f>.
28. Carry out the computation of example 27 in the following cases :
a = 20, & = 36, <^ = 45° ; a = 300, b = 540, <^ = 64°;
a = 75, 6 =3 60, <^ = 145° ; a = 250, b = 320, <^ = 120°.
29. Find the directions of three forces in equilibrium if a = 7, 6 = 10,
c = 15; also if a = 24, 6 == 36, c = 42.
EXERCISES
67
30. Referring to Figure i
a = 695, 6 = 483, = 155°: a
' solve completely and interpret physically when
720, b = 840, B = 100°.
= 135,
31. Solve and interpret when a = 1200, 5 = 135°, C = 150°
h = 142, c = 95.
32. Resolve a force of magnitude 84 into two equal components making an
angle of 60° with each other.
33. Resolve a force of magnitude 240 into two components of 120 and 180
each and find the directions of the components.
34. Determine the formula for one side of a quadrilateral in terms of the
other three sides and their included angles. Compute for a = 10,b = 12, c = 15,
^6 = 135°, 6c = 60°.
Query. How many given parts serve to determine the remaining parts
of a quadrilateral?
35. Given the four sides and one angle of a quadrilateral, determine the
other angles and the diagonals. Compute for a = QO, b = 72, c = 90, d = 100,
Q = 120°.
36. Given three angles and two sides of a quadrilateral, determine the
remaining sides. Compute for a = 630, b = 500, ab = 100°, be = 80°, cd= 60°.
37. Find the angles and the lengths of the sides of a regular pentagram,
or five-pointed star, inscribed in a circle of radius 8.
38. Compute the volume for each foot in depth of a
horizontal cylindrical tank of length 30 feet and radius
6 feet.
39. Having measured the
following data, ^A = 80° 30',
B = 72° 15', and c = 232.5
feet, compute the inaccessi-
ble distance b (Fig. 39).
40. Compute the dis-
tance a across a lake. Fig.
40, having measured A,
B, and c, which are respectively 51° 20', 72° 40' and ,
3420.5 feet.
Fig. 41.
41. A being invisible
from C, find the distance b
through a forest, having
measured a = 1037 feet, c = 1208 feet, B = 69° 25'.
42. In Fig. 42, BC, the distance of the foot of
a wall below the instrument is 12.3 feet, 6 and a,
the angles of elevation and depression, are 15° 20'
and 21° 15', respectively. Find the height of the wall and its distance
from the instrument.
Fig. 42.
68
OBLIQUE TRIANGLES
Fig. 43.
Fig. 44.
43. A pole BC, Fig. 43, is 12 feet long and leans two feet from a vertical
toward the instrument at ^ . If the angles of elevation of the top and bottom
are respectively 37° 15' and 11° 50', what are the horizontal and
vertical distances from the instrument to the foot of the pole?
44. It is desired to find the
horizontal distance and eleva-
tion of the inaccessible
point B, Fig. 44,
with reference to
an instrument at
A. Having laid
out a base line
A C, 250 feet long,
the angles at A and C are found to be 87° 10'
and 73° 51', respectively, and from A the angular elevation of B is 11° 32'.
,(7 45. Given 5 = 110° 05', ^ J5: = .4 7) = 200 f eet,
DE = 125 feet, and ^5 = 632 feet; find the distance
AC to be laid off, and the
inaccessible distance BC
(Fig. 45).
46. From measure-
ments we have (Fig. 46)
AB = Qm feet, BAC = 10° 40', BAD = Q2° 30',
ABD = 65° 32', ABC = 89° 25'. Find the inacces-
sible distances AD F^«- ^^^
and DC, and the angle between DC and AB.
47. From the instrument at A (Fig. 47)
the angles of elevation to the top and base of
the vertical wall are 15° 12' and 1° 23', respec-
tively. A base line AB \& measured 75 feet
toward the wall down a plane inclined 8° 16',
and from B the angle of elevation to the top
of the wall is 37° 46'. Compute the height of
the wall and its horizontal distance from A.
Fig. 47.
48. It is required to prolong the line AB (Fig. 48) beyond an obstacle.
At B is made an angle 52° 20' to the right
and at C an angle of 110° 00' to the left, BC
being 210 feet. Compute the proper distance
CD and angle to the right at Z), also the
inaccessible distance BD. Note that by mak-
ing B = D = 60° and C = 120°, then BC=CD
= BD and all computations are avoided.
49. Having but one point C (Fig. 49) from which both inaccessible points
A and B are visible, we are required to find the inaccessible distances AC
EXERCISES
69
and AD and the angle between AB and DC.
ADC = 87° 42', DC A = 60° 32', DCE = 170° 05',
BCE = 41° 20', CEB = 111° 35', DC = 365.2 feet,
C^^ = 410.7 feet.
50. It is required to ascertain the length and j)
position of an in-
A_, ,B
Fig. 50.
accessible line AB
(Fig. 50), its ex-
tremities not being visible from a common
point beyond the obstacles. By chaining
we have CD = 210.7 feet, DE = 390.4 feet,
EF = 173.5 feet.
Then the follow-
ing angles are measured : A CD = 83° 41', CDE =
19° 12' left (180°-19° 12'), CDA = 79° 49', FEB =
53° 20', DEF = 42° 03' left, EFB = 115° 27'.
In order to locate points suitably upon a map,
find lengths AB, AD, and BE.
/77777my
51. A tower 115 feet high casts a shadow 157
feet long upon a walk which slopes downward Fig- 51.
from its base at the rate of 1 in 10. What is the elevation of the sun above
the horizon?
CHAPTER YII
THE GENERAL ANGLE
Only those parts of trigonometry that are necessary for the solution of triangles
have been developed thus far. In this and the following chapters are considered
some of the more important topics of another phase of trigonometry that is no less
essential for the further study of pure and applied mathematics.
52. General definition of an angle. If a straight line rotates
about one of its points, remaining always in the same plane, it
generates an angle. The angle is measured by the amount of ro-
tation by which the line is brought from its original position into
its terminal position. For the small rotation leading to acute and
obtuse angles this definition agrees with the customary elementary
definition, the knowledge of which has been presupposed in the
foregoing chapters.
As in Art. 3, counterclockwise rotation generates positive
angles ; clockwise rotation, negative.
In the sexagesimal system of angle measurement the standard
unit is the angle produced by one complete rotation of the
generating line. This angle is divided into 360 equal parts
called degrees^ the degree into 60 minutes, and the minute into
60 seconds.
In the circular system the standard unit is the radian^ the
angle produced by such a rotation that each point in the generat-
ing line describes an arc equal in length to its radius. Angu-
lar magnitudes are stated in radians and decimal fractions
thereof.
Instruments are graduated and tables printed in accordance
with the sexagesimal system, which is used in practical numerical
calculations. Astronomers, however, employ decimal fractions of
seconds, while engineers make use of tenths of minutes and deci-
mal divisions of degrees. In theoretical discussions the radian
system is commonly employed. Hereafter, in this book, the two
systems will be used interchangeably.
70
DEFINITIONS OF TRIGONOMETRIC FUNCTIONS 71
.^^-^
Since the circumference of a circle is equal to 2 tt times its
radius, where 7r= 3.14159---, we may write the following relations
between the two systems :
2 IT radians = 360°
1 radian = 57.29578°. .
= 57° 17' 44.8''
and, in general, the number of degrees in any angle is equal to
180
the number of radians multiplied by • , while the number of
TT
radians is equal to the number of degrees multiplied by -^.
Thus the straight angle is tt radians ; the right angle, — radians.
A
If the radius of the circle is represented by r, the arc by a,
and the angle, in radians, by a, we have the important relation
a — vQ^.
53. Axes, quadrants, etc. Let the two axes of coordinates be
assumed as in Art. 4 ; and, as in Art. 6, let the angle be placed
upon the axis, its vertex at the origin, and its initial line
extending along the X-axis toward the right. The sign and
magnitude of the angle will determine the position of the terminal
line, causing it to coincide with one of the axes or to fall in one
of the quadrants. An angle is said to be of the first, second,
third, or fourth quadrant according as its terminal line falls in
that quadrant.
While the acute angle is of the first quadrant, the converse
is by no means necessarily true. The ter-
minal line of every angle, however large,
must coincide with the terminal line of
some positive angle less than 360° (see Fig.
52). For the purpose of trigonometry as
developed in the present chapter, for every
angle, positive or negative, and of any mag-
nitude, may be substituted a positive angle
less than 360°.
54. Definitions of the trigonometric functions. The trigono-
metric functions of angles of any size are defined identically as in
72
THE GENERAL ANGLE
Art. 6. Thus for all positions of the terminal line, Fig. 53,
y . X
- = sm a, - = cos a,
V V
y X
— = tan a, — = cot a,
X
y
V V
- = sec a, - = CSC a.
X y
W
O X
M
p\a
{a)
Q>) (c)
Fig. 53.
(d)
55. Signs and limitations in value. The abscissas are positive
for all points in the first and fourth quadrants, negative for those
in the second and third. Ordinates are positive for all points in
the first and second quadrants, negative for those in the third and
fourth. The radius vector is, by agreement, considered positive
for all points.
In conformity with the sign law of algebra, the functions of
angles of the different quadrants will have signs as displayed in
the following table :
Quad.
Sine
Cosine
Tangent
Cotangent
Secant
Cosecant
I
+
+
+
+
+
+
II
+
-
—
—
-
+
III
-
—
+
+
—
—
IV
-
+
—
—
+
-
It will be noticed that for angles of the first quadrant all six
functions are positive. In each of the other quadrants one pair of
mutually reciprocal functions are positive, the other two pairs are
negative. These positive pairs run as follows : second quadrant,
sine and cosecant : third quadrant, tangent and cotangent : fourth
quadrant, cosine and secant.
SIGNS AND LIMITATIONS IN VALUE 73
The student should establish these statements regarding the
signs of the functions and memorize them.
Since the lengths of the abscissa and ordinate can never exceed
that of the radius vector, it follows that the sine and cosine can
never be numerically greater than unity, and the secant and
cosecant can never be numerically less than unity. The tangent
and cotangent can have numerical values either greater or less
than unity.
EXERCISE XV
1. Express in degrees, minutes, and seconds the angles — , — , '-~^,
o,r« o« 3^ ■ 4 3 6
2. Express in radians the angles 30°, 15°, 45°, 120°, 240°, 300°, 450°.
/-^r' In a circle of radius 60 cm., what is the length of the arc which sub-
tends at the center the angle 30°, 60°, ^, ^ ?
^ ' ' 3 ' 4
4. In a circle of radius 10 inches, what is the circular measure of the angle
subtended by an arc whose length is 10, 5, 20, 5 ir inches?
5. A friction gear consists of two tangent wheels, whose radii are 8 and
12 inches, respectively. The smaller wheel makes 4 revolutions per second. Find
the number of revolutions per second made by the larger, the angular velocity
of each, and the linear velocity of a point on the circumference of each. If
the larger wheel is attached to the rear axle of an automobile whose rear wheel
has a diameter of 30 inches, find the speed of progress of the machine.
6. The diameters of the front and rear sprocket wheels of a bicycle are
10 inches and 4 inches, respectively, and the diameter of the rear wheel is 28
inches. Find the rate of pedaling when the bicycle is traveling 12 miles per
hour, the corresponding angular velocities of the two sprocket wheels, and the
linear velocity of the chain.
7. Determine the quadrant to which each of the following angles belongs :
210°, 465°, 745°, - 830°, ^, i^, -^.
3 ' 4 ' 3
8. Determine the signs of the functions of the following angles: 240°,
330°, 400°, ^, -'Le, 6^.
3 4'
9. Show that the quadrant to which an angle belongs is determined if the
signs of any two non-reciprocal functions are given.
10. To what quadrant does an angle belong if its sine and tangent are
negative ; its secant and cotangent positive ; sine and secant negative ; tangent
and cosine positive ?
74
THE GENERAL ANGLE
11. Determine the quadrants of the following angles:
sin"i|; arccos — j\; arctanf; cot-^ — j\.
12. Determine the quadrants of the following angles :
sin-i I = cot-i — f ; arccos — j% = arccsc i|.
13. For what values of a is sin a — cos a positive ?
14. For what values of cc is tan a — cot a negative ?
15-20. Find the missing values in the following table :
z
sin
cos
tan
cot
sec
CSC
QlFAD.
a
7
8
it
-II
--A
— 15
II
III
III
IV
¥
IV
^
-¥
III
56. Variation of the trigonometric functions. A change in
the angle will produce a corresponding change in the values of the
coordinates and in their ratios. If, for convenience, the chosen
point in the terminal line of the angle is maintained at a constant
distance from the vertex, the radius vector will retain the constant
value + V.
As the angle increases continuously from 0° to 360°, the
abscissa and ordinate vary continuously between the limits — v
and 4- v. As 6 increases from 0° to 90°, x is positive and decreases
from V to ; as 6 increases from 90° to 180°, x is negative and
decreases (algebraically) from to — v ; as ^ increases from 180°
to 270°, X is negative and increases from —v to ; and as 6
increases from 270° to 360°, x is positive and increases from to v.
As 6 increases from 0° to 90°, i/ is positive and increases from to
y ; as ^ increases from 90° to 180°, ^ is positive and decreases from
V to ; as ^ increases from 180° to 270°, y is negative and decreases
from to —V ; as increases from 270° to 360°, i/ is negative and
increases from — v to 0. Upon introducing these varying values
into the ratio definitions, we are enabled to trace the variation of
the trigonometric functions.
We see, for example, that as increases from 0° to 360°,
tan 6 continually increases algebraically, changing sign from
negative to positive through the value as ^ passes through 0°,
GRAPHS OF THE TRIGONOMETRIC FUNCTIONS
75
180°, and 360°, and from positive to negative by becoming infinite
as 6 passes through 90° and 270°. There is an infinite discon-
tinuity in tan 6>, for 6 = 90° and d = 270°.
Query. Which of the trigonometric functions other than the tangent
become infinite and therefore discontinuous ?
The student should trace the variation of each function in detail, stating
the narrative verbally.
57, Graphs of the trigonometric functions. The whole behavior
of each function can be conveniently represented by means of the
graphical method already introduced in Art. 4. Assume a pair
Fig. 54. Graph of sin 6.
of axes of coordinates, as in Art. 4, and along the JT-axis to the
right lay off equal spaces corresponding to the number of degrees
in the angle 6. At each point in the JT-axis erect a perpendicular
whose length is proportional to the value of the sine of that angle.
Each point thus determined has the property that its abscissa
represents the angle 6 and its ordinate the corresponding value
of sin 6. Now having located a sufficient number of points, draw
through them a smooth curve. It will be seen that the value,
sign, and variation of the sign at each instant is fully exhibited
by the ordinate, position, and inclination of the curve or graph.
The same may be done for each of the functions.
The graphs of the different functions are here presented.
The student should trace carefully the intimate and exact cor-
respondence of the graphical and the verbal narratives.
76
THE GENERAL ANGLE
^T
Fig. 55. Graph of cos d.
-h.
O
Fig. 56. Graph of tan d.
GRAPHS
77
^Y
X
Fig. 57. Graph of cot d.
/^Y
O
Fig. 58. Graph of sec 6.
78
THE GENERAL ANGLE
AY
_7C_
-7t 2
2
371
2
^
2
X
Fig. 59. Graph of esc d.
58. Functions of 270° and 360°. By the method of limits em-
ployed in Art. 12, we get the following sets of values :
sin 270° = - 1, cos 270° = 0,
tan 270° = oo, cot 270° = 0, ■
sec 270° = 00. CSC 270° = - 1.
sin 360° = 0, cos 360° = 1,
tan 360° = 0, cot 360°= oo,
sec 360° = 1, CSC 360° = oo.
Here oo is used as before to denote the value of a fraction whose
numerator remains finite while its denominator approaches zero.
The sign + or — is prefixed to the symbol oo according as tlie
variable becomes oo through a positive or a negative sequence of
values. In the light of this discussion the values of the functions
oih y. —(k any integer) may be tabulated, the upper of the pair
of double signs arising when the angle approaches the critical
value from below.
e
sin 9
COS
tan 9
cot 6
sec B
esc B
TO
+ 1
TO
Too
+ 1
Too
f
-fl
±0
±cc
±0
±co
+ 1
TT
±0
-1
TO
Too
- 1
±GO
It
-1
TO
±cc
±0
Too
-1
27r
±0
+ 1
TO
T 00
+ 1
Too
FUNDAMENTAL RELATIONS 79
EXERCISE XVI
n
1. Trace the variation, as d varies, (a) of sin 2 6\ (b) of cot -•
n
2. Trace the variation, as ^ varies, (a) of tan 2^; (h) of cos - •
id
3. Draw the graph of cos 2 B.
4. Draw the graph of sin 3 B.
5. In what points will a horizontal line \ unit above the JT-axis intersect
the graph of sin Q ? Explain the significance of the result.
6. In what points will a horizontal line 1 unit above the X-axis intersect
the graph of tan B1 Explain.
7. If the graphs of tan B and cot B are drawn on the same axes to the same
scale, where will they intersect? What is the significance?
8. If the graphs of sin B and cos B are drawn on the same axes to the same
scale, where will they intersect? What is the significance ?
9. Construct the graph of logj^x, taking values of the number x as abscis-
sas and the corresponding logarithms as ordinates.
59. Fundamental relations. Just as in Art. 9 we find, by in-
spection,
(1)
(2)
(3)
by division, tan a = ^*L^^ (^4)
(5)
sm a
by virtue of the Pythagorean proposition,
sin^ a + cos^ a = 1, (6)
tan^ a + 1 = sec^ a, (T)
cot^ a -f- 1 = csc^ a. (8)
The student should prove that all these formulas conform, for
angles in all quadrants, to the algebraic law of signs.
CSC a
1
sina^
sec a
1
cos a
cot a
1 .
tan a'
tana
sin a
cos a
nnt CL
_ cos a .
80 THE GENERAL ANGLE
60. Line representations of the trigonometric functions. As the
names tangent and secant indicate, the trigonometric functions
were originally defined as certain lines measured in terms of a
standard unit line. The adoption of the abstract ratios, as in
this book, is of comparatively recent date. It is both interesting
and advantageous to know the line representations and sliow that
they lead to the same science of trigonometry as do the ratio deti-
nitions.
The line representations most frequently used involve the use
of a unit circle, i.e. a circle of radius unity. It is evident that
we may replace each of the defining ratios of Art. 54 by an equal
ratio so chosen that its denominator is positive unity. The value
of the ratio will be equal to that of the numerator. In other
words, if a positive unit radius is taken as the denominator, the
length and sign of the numerator will represent the function in
magnitude and sign. We have, then, simply to select six lines
whose ratios to the radius agree with the definitions of Art. 54.
The ratio of the subtended arc to the radius is, by Art. 52, the
circular measure of the angle.
Suppose, then, a circle of unit radius drawn with its center at
the origin of coordinates.
The angle is placed upon the axes just as in Art. 6, and from
the point P of intersection of the terminal line with the circle,
perpendiculars MP and NP are drawn to the two axes. From
the two points Jl and ^ where the positive axes cut the circle,
tangents ^2^ and BS are drawn meeting the terminal line (pro-
duced if necessary) in the points T and S.
Since P, T, jS, Figs. 60-63, lie in the terminal line, we have, at
once, in accordance with Art. 54 (or Art. 6) :
MP NP
AT , BS
tan a = -—- , cot « = -— ,
OA OB
OT OS
sec«=^, csc« = -^.
But by construction,
OP=OA=OB = 1.
LINE REPRESENTATIONS
81
These denominators may then be suppressed and the functions
represented graphically as indicated below :
t
\^
B
^
\
\
A ,
1 M
1
\ 1
^•\r
Fig. 60
Fig. 61.
Fig. 62. Fia. 63.
sin a — MP, cos a = NP,
tan a = AT, cot a = BS,
sec a= OT, CSC a = OS.
Moreover, the angle, in radians, is represented as follows
arc AP
OP
arc AP.
According to the modern view, the line is not the function, but
by its length and direction represents the function in magnitude
and sign.
Note that the line representing the tangent is always drawn
from the point A and that representing the cotangent from B.
All the lines are read from the axes to the terminal line. Hori-
zontal lines are positive toward the right, negative toward the left.
Vertical lines are positive upward, negative downward.
82 THE GENERAL ANGLE
By means of the Pythagorean proposition, and the theorems
concerning similar triangles, the fundamental relations given in
the preceding article, as also the limitations of value stated in Art.
55, are readily established. So, also, the subsequent theorems of
trigonometry may be interpreted by means of the line represen-
tation of the trigonometric functions. This graphic interpretation
frequently presents special advantages. This is the case, for ex-
ample, in the investigation of the variation of the functions con-
sidered in Art. 5Q. So, too, the construction of the graphs of the
functions as treated in Art. 57 is facilitated, since the lengths of
the defining lines may be transferred by the use of dividers.
EXERCISE XVil
Find the values of the following expressions :
1. cos^ a — sin^ a, when a = arctan ( — |), in the 2d quadrant.
2. • H , when a = sec-i(— 3), in the 3d quadrant.
1 — tan a 1 — cot ct
3. '- -, when a = arcsin(— 14), in the 4th- quadrant.
tana - sec a + 1 \ o^/ ^
4.
1 , when a = cos-^M, in the 4th quadrant.
CSC a — cot a esc a + cot a
Solve the following equations, finding all the angles less than
2 TT that satisfy each equation :
5. cos/? =^.
6. tan y8 = - Vs.
7. sin 2 a = - ^V^.
8. cot 3 a =: 1.
9. 4 sin2 a — 4 cos a — 1 = 0.
10. 3tan2y8-l = 0.
11. 2 sin ^ cos ^ - sin ^ = 0.
12. 2 sin a + \/3 tan a =0.
In exercises 13-24, verify the given identities by transforming
the first member into the second.
13. (sin a + cos a) (cot a + tan a) = sec « + esc a.
14. (sec a — cos a) (esc a — sin a) = sin a cos a.
15. ^^"^ + ^^^^ ^tan«cot)8.
cot a + tan ft '
PERIODICITY 83
16. (r cos Oy -{- (r sin cos <^)2 + (r sin ^ sin <f>y = r\
^_ tan a — tan y8 ^ cot a cot ^ + 1 _ ^
1 + tan a tan ^ cot y8 — cot a
18. CSC a (sec a — 1) — cot a (1 — cos a) = tan a — sin a.
19. (sin a cos 13 — cos a sin ^)^ + (cos a cos ^ + sin a sin ^)2 = 1.
20. sec ct CSC a (1 — 2 cos'^ a) + cot a = tan a.
21. (sin a cos /? + cos « sin f^)'^ + (cos ot cos /? — sin a sin I3y = l.
oo o 2 (l-tan2a)2 . »
22. sec- a csc^ ct — ^^ ^ = 4.
tan-^ a
23. (cos a + V— 1 sin a) (cos a — V — 1 sin a) = 1.
24. (cos a + V— 1 sin a)^ + (cos a — V— 1 sin «)2 z= 4 cos^ a — 2.
25. By means of Fig. 60 show that, when is acute and measured in
radians, sec > tan 0>0> sin 0.
26. By means of Fig. 60 show that, when 6 is acute and measured in
radians, esc ^ > cot ^ > ( '^ — ^ j > cos ^.
61. Periodicity of the trigonometric functions. It was pointed
out, in Art. 53, that if two angles differing by an integral multiple
of 360° are placed on the axes, their terminal lines coincide. As
an immediate consequence, it follows that corresponding functions
of the two angles are identical. Thus we may write
sin (2 kir -^ a) = sin a,
and, in general,
F(2kiT -\- a) = jP(a),
where F denotes the same function in both members of the equa-
tion, and k is an integer.
62. Functions of
k^±a . Precisely as in Art. 10, 37, and
38, we may express the functions of the angles ± a, 90° ± a,
180° ± a, 270° ± a, 360° ± a, and other similarly compounded angles
in terms of the functions of cc, no matter what the quadrant of the
angle a. Because of the periodicity brought out in the preced-
ing article, it is not necessary to carry the investigation beyond
the five multiples of the right angle mentioned ; indeed, the fifth
reduces to the first. On account of the double signs and the pos-
sibility^ of a belonging to any one of the four quadrants, there
exist thirty-two distinct cases. The demonstration is tlie same
84
THE GENERAL ANGLE
for all cases, involving the same proportionality of sides of similar
triangles and the same question of agreement or opposition of signs.
The working out of the proof in three characteristic instances
should be sufficient to enable the student to do the same for any
and all cases. The theorem is, however, somewhat elusive, and
the student can completely master it and render it an infallible
instrument only by actual careful construction and proof of most
of the cases. Upon first study it may be well to limit considera-
tion to the cases in which a is of the first quadrant.
Let it be required first to express the functions of (180° -f- a)
in terms of functions of a, when a is an angle of the first quad-
rant. If, in Fig. 64, Z XOA = a,
thenZXOB=/3=lS0°-\-a. The
two triangles OMP and OJSfQ are
similar, the pairs of correspond-
ing sides being v and v', x and x\
and y and y. Notice also that
x^ and ?/' are negative, all the
other sides being positive. Giv-
ing due attention to signs, we
Fig 64. may write :
sin (180° + a) = sin /3 =
y
sin a,
cos (180° + a) = cos /5 =
= — cos OJ,
V
tan (180°+ a) =tan /5 = ^ = ^ = tan a,
cot (180° + a) = cot /3 = - = - = cot «,
y y
sec (180°+ a) = sec /?
sec a,
esc (180° + «) = CSC /3 = ^
y
- = — CSC a.
y
Again, let it be required to express the functions of (270® — a)
in terms of functions of «, when a is of the first quadrant. In
Fig. Qb, Z XOA = «. Z XOB = /3 = 270° - a. The two triangles
OMP and ONQ are similar, the pairs of corresponding sides now
FUNCTIONS OF
[..|±«]
being v and v\ x and y\ and y and x\ The sides x^ and
negative, all the others positive. We may then write;
sin(270°-«) = sinyS =
cos(270°-a)=cos)8 =
tan(270°-«)=tan^ =
cot(270°-«)=coty8 =
sec(270°-«)=secy8 =
csc(270°-a) = csc,/S =
As a third and especially important
instance, let us find the functions of — a,
when a is of the second quadrant. In
Fig. m, XOA = «, XOB= /3= - a.
The two triangles OMP and ONQ
are similar, the pairs of correspond-
ing sides being v and v', x and x\
y and y\ while a;, a;', and ^' are ^
negative.
We then have, as before : p^^ ^g
sin ( — a ) = sin ,8 = ^ = — ^= — sin a,
85
are
y
v'
-
X
V ~
- COS a.
x'
v'
-
V
- sin a.
y'
x'
x
y
= cot
a.
x'
y'
=
y
X
= tan
a.
v[
_
V _
- CSC a.
x'
y
v'
y'
=
—
V _
X
• sec a.
X X
cos ( — a) = COS ^ = — = - = cos a,
v' V
tan ( — a) = tan /9 = '-^
cot ( — a) = cot yS = — =
tana.
— cot a,
sec ( — a ) = sec ytj = — = - = sec a,
CSC ( — a) = CSC /3 = -r = = — esc a.
v' X
86 THE GENERAL ANGLE
It will be noticed that whenever the number of right angles
involved is even the pairs of corresponding sides are v and v\x and
x' ^ y and y^ ; while whenever the number of right angles is odd
the pairs of corresponding sides are v and v\ x and y\ y and x' .
Thus we have the theorem : Any function of an even number of
right angles plus or minus a is
numerically equal to the same func-
tion of a; any function of an odd
number of right angles plus or minus
a is numerically equal to the cor-
responding co-function of a; the
agreement or opposition of signs is
to be determined from the quadrants
of a and of the compound angle. It
may easily be verified that in all
cases this agreement or opposition
of signs is the same as when a is of first quadrant.
The general theorem may also be stated as follows : If the sum
or difference of two angles is an even number of right angles^ the
functions of the one are numerically equal to the same functions of
the other. If the sum or difference of two angles is an odd number
of right angles^ the functions of the one are numerically equal to the
corresponding co-functio7is of the other. The agreement or opposition
of signs is to be determined from the quadrants of the two angles.
The significance of the theorem is made clear by application
to an example: Required to find the value of cos (810° + a).
Here 810° = 9 x 90°, an odd number of right angles. When a is
considered as of the first quadrant (and its functions consequently
positive), the compound angle (810° + a) is of the second quadrant
and hence its cosine is negative. The required relation is, there-
fore,
cos (810° + a) = — sin «,
which holds for all values of a.
Again, to find the value of tan 1230°. We have
1230° = 14 X 90° - 30°, and is of second quadrant.
Then tan 1230° = _ tan 30° = - — .
V3
The student may, if he prefers, construct the figure and proceed
as in the demonstration just given.
FUNCTIONS OF Fa; • | ± a"] 87
As a consequence of these relations, it follows that to every
inverse function correspond two angles, lying between and 2 tt.
Thus arc sin a = a and ir — a^
arc cos h = a and 2 tt — a,
arc tan c= a and tt + a,
arc cot d= a and tt + «,
arc sec e = a and 2 tt — «,
arc esc/ = a and tt — a.
These statements should be verified by the student.
EXERCISE XVIII
Express in terms of a positive angle less than 45° :
1. sin 700^ 4. cot - 35°.
2. cos 260°. 5. CSC 930".
3. tan436^ 6. sec 1400°.
Find the value of cos a -}- sin a and of tan a — cot a when a has
the value
7. ?. 10. '^'^.
6 (j
8. -2j-. 11 iijr
3 ■ 3 *
9. 1^. 12. -^.
4 4
Find all the values between 0° and 360° of
13. arctaii V3. 16. arcsec 2.
14. csc-i(- V2). 17. arccot(-l).
15. arccos (- .5). 18. sin-i (- ^ V3).
Find the value of
19. sin 480° sin 690° + cos ( - 420°) cos 600°.
20. tan 840° cot 420° + tan (- 300°) cot (- 120°).
21. tan llr tan ii5 + cot ( - 11^) cot ( - t'^) .
22. sin 1|^ cos ( - ^) - sin If cos ( - ^
88 THE GENERAL ANGLE
23. If sin 200° 30' = - .35, find cos 830° 30'.
24. If tan 558° 26' = i, find cot 468° 26'.
25. If cot 520° = - a, find sin 160°.
26. If cos 590° = - m, find tan 850°.
27. Express cos (a — 90°) as a function of a.
28. Express sin (a — 180°) as a function of a.
29. Express tan (a — 360°) as a function of a.
30. Express cot (a — 270)° as a function of a.
CHAPTER VIII
FUNCTIONS OF TWO ANGLES
63. Formulas for sin (a + p) and cos (a4- p). Suppose a and
y8 to be acute angles. In Fig. 67 (ct + /3) is acute ; in Fig. 68
(a + yS) is obtuse. The following demonstration applies to both
figures.
Let ZXOA = a, ZAOB = 0', then ZX0B = a + /3. From
P, a point in 0J5, draw Pil[f perpendicular to OX, PQ perpendic-
M N
Fig. 67.
>X
Fig. 68.
ular to OA^ and from Q draw §iV perpendicular to OX, and QR
perpendicular to MP. The angle RPQ == a and PP = QP cos a,
PQ =z QP sin a, by Art. J6. By the same article.
MP=OP s'm (a + /3).
Also MP = MR + RP = NQ-\-RP
= 0§ sin a + ^jP cos a
= OP sin a cos yS + OP cos a sin y8.
Equating the two values of MP and dividing through by the
common factor OP, we have the theorem
sin (a + p) = sin a cos P + cos a sin p.
80
0)
90 FUNCTIONS OF TWO ANGLES
In like manner
Oi)^f= OP cos («+)S),
and also OM=^ON-MN= ON- RQ
= OQ cos a — QP sin a
, = OP cos a cos /S — OP sin a sin p.
Hence the companion theorem
cos (a + p) = cos a cos p — sin a sin p. (2)
These are called the addition formulas and are fundamental
in trigonometry.
64. Extension of addition formulas. The two formulas of the
last article were proved only for angles both of the first quadrant.
It remains to be shown that they hold when a and /S denote any
angles.
First, let a be an angle of the second quadrant. Then
(^ (= a — 90°) is an angle of the first quadrant. Now a = 90° + <^,
so tliat, by Art. 38,
sin a = cos <^, cos a = — sin <f).
Since <^ is of the first quadrant, the formulas of Art. 63 apply
and we have
sin (a H- yS) = sin (90° +(/)+yS)
=:COS(</> + /S)
= cos (^ COS /3 — sin <f) sin /3,
= sin a cos /3 + cos a sin /3.
Likewise
cos (« + /8) = cos (90° + (^ + /3)
= -sin(^ + /3)
= — sin </) cos /3 — cos <f> sin /S,
= cos a cos yS — sin a sin y8.
The formulas are therefore true when one angle is of the first
and the other of the second quadrant. By adding 90° succes-
sively to each of the angles, the formulas are established for two
positive angles of all quadrants. If one of the angles is negative,
it can be augmented by such an integral multiple of 360° as to
produce a positive angle possessing the same functions.
The addition formulas are, therefore, true for angles of any
size.
ADDITION AND SUBTRACTION FORMULAS 91
EXERCISE XIX
Evaluate the addition formulas for
1. a = 60°, /? = 30°. 3. a = 240°, 13 = 150°.
2. a = 45°, /? = 90°. 4. a = 300°, ^ = 150°. .
5. « = arctanf, ^ = arccos (— -^j), a first quadrant, ft second.
6. a = sin-i (— j^^), /3 = cot-^ 5^, a fourth quadrant, /? third.
Find the value of
7. cos (^ + a) cos (1 + /«) - sill (| + «) sin (| + )»).
8. sin (7 + a) cos(^+/8) + cos (| + a) sin (| + ysV
9. sin (1 + n) a cos (1 - n) « + cos (1 + n) « sin (1 — n) a.
10. cos (1 + n) a cos (1 — n) a — sin (1 + n) a sin (1 — n) a.
11. sin (^ + <^) cos {6 - <f>) + cos (^ + <^) sin (^ - <^).
12. cos (^ — <^) cos ^ — sin (^ — <^) sin <^.
13. Evaluate the addition formulas for a = 60°, ^ = 45°, and thus find
sin 105°, cos 105°, sin 15°, cos 15°.
14. Evaluate the addition formulas for a = 45°, ^ = 30°, and thus find
sin 75°, cos 75°, sin 15°, cos 15°.
65. Subtraction formulas. In the addition formulas replace /3
by — fi. We have
sin (a — yg) = sin a cos (—/?) + cos a sin (— yS).
But by Art. 62,
sin (— y8) = — sin y5, cos (— jS') = cos y5.
Making this substitution, we have
sin (a — P) = sin a cos p — cos a sin p. (1)
In like manner
cos (a — yS) = cos a cos (— yS) — sin a sin (— yS),
or, by the same substitution,
cos (a — p) = cos a cos p + sin a sin p. (2)
92 FUNCTIONS OF TWO ANGLES
66. Formulas for tan (a ± p), cot (a ± p). From Arts. 59 and
63 we have
. . , a\ sin (a 4- yS)
tan(a + /3)= ) ^ ^(
cos (« + /3)
_ sin a cos /8 + cos a sin /3
cos a cos yS — sin a sin yS
sin a cos /3 cos « sin ^
cos a cos /8 cos a cos /S
cos a cos yS _ sin a sin /S
cos ct cos ^ cos ct cos ff
or, finally,
-I ^ . ON tan a + tan B ^^ ^
tan (a + P) = ;; ^ • (1)
^ ^^ 1-tanatanP ^ ^
In like manner we may derive
X / o\ tan a — tan B ^^;.
tan (a — P) = -^- (2)
^^ 1 + tanatanP ^ ^
cot(« + ^)=22^fi±|l,
Sin (ct + yS)
_ COS a cos /6 — sin a sin /3
sin a cos fi + cos a sin yS'
cos ct cos /3 sin cc sin y8
__ sin a sin y8 sin a sin ^
sin a cos ^5 cos a sin yS'
sin a sin jS sin a sin /3
Again,
or
Likewise
^ ^^ cot p + cot a ^ ^
J. r ox cot a cot p + 1 ^ . ^
cot (a — P) = ~ ^ — - • (4)
^ cot p - cot a ^ ^
EXERCISE XX
1. Demonstrate geometrically the formula for sin (« — y8), when a > (3^
both acute.
2. Demonstrate geometrically the formula for cos (a — (3), when a > /?,
both acute.
FUNCTIONS OF 2 a AND ^ 93
Evaluate the formulas of Art. ^^ for
3. a = 60°, y8 = 120°. 5. a = arcsin ^V, ^ = arctan - ^f .
4. a = 240°, y8 = 150°. 6. a = cot-i |^, ^ = cos-i ||.
Evaluate the formulas of Art. ^^ for
7. a = 330^ ^ = 150°. 8. a = 210°, )8 = 300°.
9. a = cos-i y5_, ^ :^ tan-i ( - f ).
10. a - arccot \\, /? = arctan f Q.
11. Find the functions of 15° by putting a = 45°, (3 = 30°.
12. Find the functions of 15° by putting a = 60°, /S = 45°.
Show that
13. sin (a + (3) sin (a- /3) = sin-^ a — sin^ (3 = cos^ /? — cos^ a.
14. cos (a + /3) cos (a - yS) = cos^ a — sin^ ^ =: cos^ (3 — sin^ a.
Expand by successive applications of the formulas :
15. sin(« + /?+y). 17. tan(« + ^ + y).
16. cos(a + j8 + y). 18. cot(a+^ + y).
Show that
19. sin(^+ « J - sin f "^ - a j = sina.
( - — a y = \/3 cos a.
6 ■ / ■ \6 /
67. Functions of twice an angle. In the addition formulas of
Arts. 63 and 6Q, place jS = a. We then obtain
sin 2 a = 2 sin a cos a, (1)
cos 2 a = cos^ a — sin^ a, (2)
= l-2sin2a, (2 a)
= 2cos2a-l. (2?0
. n 2 tan a ^ox
tan2a = - —- (3)
1 — tan^ a
. o cot^ a — 1 ^ , X
cot 2 a = ~ (4)
2 cot a
68. Functions of half an angle. From Art. 67 we may write
cos 2 /3 = 1 - 2 sin2 /8,
94 FUNCTIONS OF TWO ANGLES
and solving for sin/3,
sin ^ = Vi(l-cos2yS).
Now placing 2 /S = a, so that y^ = ^, we obtain
sin I a = Vi (1 - cos a). (1)
Similarly
cos 2 yS = 2 cos2 y8 - 1 ;
so that
cos y8 = Vi(l + COs2y8),
and, with the same substitution,
cos I a = Vi(l + cosa) . (2)
Dividing the first formula by the second, we get
. 1 ^ /I — cos a >-Q\
taii-a = ^— , (d)
2 ^ 1 + cos a
and inverting,
.1 ^ /I + COS a ^^N
cot a = X|- (4)
a ^ 1 — cos a
Rationalizing the numerators of the last two formulas, we get
other useful forms,
^-^^ 1 1 — cos a ,r>,
tan -a = -. , (5)
2 since
^^i. 1 1 4- cos a y^«>,
cot - a = — : (6)
2 sin a
Query. — Why are formulas (1) to (4) ambiguous in sign, while (5) and ((>)
are apparently not ?
EXERCISE XXI
Find the values of
1. The functions of 60° from those of 30°.
2. The functions of 120° from those of 60°.
3. The functions of 75° from those of 150°.
« 4. The functions of 15° from those of 30°.
Find the values of the functions of
5. 2 arctan :f%. 8. I arctan \\l,
6. 2 cos-i (— j\). 9. arcsin ^^ + 2arccot f.
7. J sin-i ( — M)' ^0- arctan ^^ — 2 arccos f .
CONVERSION FORMULAS 95
Transform the first member into the second :
11. l+sin^-cos2^ ^^^^^^
cos ^ + sin 2 ^
12. l+cos^ + cos2^^^^^^
sin 6 + sin 2 6
13. (VT+sina+ Vl — sin a)2z=4cos2ia.
14. ( Vl + sin a — Vl —sin a) 2 = 4 sin^ i a.
15. tan f - + a J - tan f - - a J = 2 tan 2 a.
16. cot [- + a^ - cot [ - - a] = - 2 tan 2 a.
Find the values of a which satisfy the following equations
17. (2 + V3)(l-sin2a) -2cos22a = 0.
18. sin 2 a + 2 cos 2 a = 1.
19. 4 sec2 2 a + tan 2a = l.
20. CSC 2 a + cot 2 a = 2.
Show that
21. tan-i ^^^ = cos-i ^
3 Va;2 - 4 a: + 13
2 a: +
— — ^i^:^^::^^::::^ — arccsc — ---
y/x'^ + 2 x - 3 2
9 a: + 1
22. arctan — " — arccsc -
23. Find sin (^- 2 tan-i
'\--
\-x \
l+xj'
24. Find sin fsin-i m 4- tan"!^^-^^ — ^V
\ ml
25. Find sin f arccos (1 — a) — 2 arctan -^ — ^^ — | •
26. Find cos (arccos (1 — 2 a) — 2 arcsin Va).
69. Conversion formulas for products. Adding the two first
formulas of Arts. 63 and 65, we have
sin (« + ,5) + sin (« — yS) = 2 sin a cos /3,
or, reversing and dividing by 2,
sin a cos p = 1 [sin (a + p) + sin (a - p)] . (1)
If we subtract, instead of adding, we get
sin (a + yS) — sin (a — /S) = 2 cos a sin /3,
or
cos a sin p = i [sin (a + P) - sin (a - p)]. (2)
96 FUNCTIONS OF TWO ANGLES
Treating the two second formulas in like manner, we obtain
cos a COS p =1 [cos (a + p) + cos (a — p)], (3)
and
smasmp = -l [cos(a + p) -cos(a- P)]. (4)
By means of these formulas, products of sines and cosines are
expressed as sums or differences. By successive applications
higher powers and products are reducible to expressions linear
in sines and cosines. The same transformations may often be
effected by application of the formulas of Art. 67, written in the
form
sin a cos a = | sin 2 a, (5)
sin^ a = J (1 — cos 2 a), (6)
cos2a= I (l + cos2a). (7)
EXERCISE XXII
Reduce the following products to linear expressions :
1. sin 5 a cos S (^. 6. sin a cos^ u-
2. cos 6 a sin 4 cc. 7. cos^ a.
3. sin 7 a sin 3 a. 8. sin^ a.
4. cos 2 a cos 5 a. 9. cos^ a sin^ a.
5. sin^ a cos a. 10. sin^ a cos^ a.
Show that
11. cos a sin (/? — y) + cos /S sin (y — a) + cos y sin (« — ^) = 0.
12. sin(^ - y) sin (« - 8) + sin(y - a) sin (/3 - 8) + sin (a-/3) sin (y- 8) = 0.
13. sin — cos^+ sin^cosi^ = 0.
5 5 5 5
14. 2 cos^cos^ + sin^ + cos^ = 0.
8 4 8 8
Solve for a, making use of Art. 10.
15. COS (50'^ + a) sin (50° -a)- cos (40° - a) sin (40° + a) = 0.
16. sin (70° + a) sin (70° - a) + sin (20° + a) sin (20° - a) = 0.
Solve for a, making use of Art. 69.
17. cos 3 a + cos 9 a = 0.
18. sin 5 a — sin 10 a = 0.
CONVERSION FORMULAS 97
19. cos (a + 0) cos (a- 6) + cos (3 a + 6) cos (3 a - 6) = cos 2 0.
20. sin (a + 6) cos (a - ^) + sin (3 a + ^) cos (3 ct - ^) = sin 2 0.
70. Conversion formulas for sums and differences. In the
process of deriving the formulas of the last article, before revers-
ing and dividing by 2, substitute a + /3 = ^, a— (3 = 6^ so that
We then obtain the following formulas :
sincj) + sin 6 = 2 sin ^ ^^^ T '> (^)
sine))- siii6 =2cos^ sin ^ , (2)
coscf) + cos6 =2 cos^-— — cos^— — , (3)
cos<)) — cosG = - 2sin^-- — ^i^^^^ — (^)
These formulas serve to effect transformations converse to
those mentioned in Art. 69.
71. Multiple angles. In the formula for sin (a + /3) put y8 = 2 a.
Then
sin 3 a = sin a cos 2 a + cos a sin 2 a
= sin a — 2 sin^ ce + 2 sin a cos^ a
= 3 sin a — 4 sin"^ a.
Again, cos 3 a = cos a cos 2 a — sin a sin 2 a
= 2 cos^ a — cos a — 2 sin^ « cos a
= 4 cos^a — 3 cos a.
In like manner the other functions of 3 ct and, by repeating the
process, the functions of any integral multiple of a may be ex-
pressed in terms of functions of a.
EXERCISE XXIII
Show that
1. ?i5A«±^iEij? = tan5a.
cos 6 a + cos 4 a
2 co^3^t-_cos^^^^^^^
sin 3 a + sin 5 a
98 FUNCTIONS OF TWO ANGLES
« sin 7 « — sin 5 a ,
3. = tan a.
cos 7 a + cos 5 a
. cos 4 a — cos 2 « j. o
4. - — : — -— = - tan 3 a.
sin 4 « — sm 2 cj
5 sina-sin^^^^^^gM:^^^^a-^^
sin ct + sin |8 2 2
6 cos« + co^^_^^^«+^^^^«-^
cos a — cos )8 2 2
^ cos 3 ^ + 2 cos 5 ^ + cos 7 _ , ^ ^
sin 3 ^ + 2 sin 5 ^ + sin 7 ^ ~
8 sin ^ - 2 sin 4 ^ + sin 7 6 _. .q
' cos ^ - 2 cos 4 ^ + cos 7 ^ ~
Solve the following equations :
9. cos 6 + cos 5 ^ = cos 3 ^.
10. sin ^ + sin 5^ = sin 3^.
11. sin 2 ^ + 2 sin 4 ^ + sin 6 ^ = 0.
12. cos3^ + 2cos4^+cos5^ = 0.
Derive the formulas for :
13. cot 3 a. (In terms of cot a.)
14. tan 3 a. (In terms of tan a.)
15. sin 4 a.
16. cos 4 a.
Solve the equations :
17. sin 3 a = V2 sin 2 a.
18. V3cos3a + 2sin2a = 0.
19. cos 3 a = cos a cos 2 a.
20. sin 3 a = sin a cos 2 a.
CHAPTER IX
ANALYTIC TRIGONOMETRY
The foregoing chapters constitute an introduction to the elementary principles
of trigonometry. The student ought now to be prepared for a more advanced
study of the theory of the trigonometric functions, which may be entitled analytic
trigonometry. It is beyond the scope of this book to consider more than a few of
the most important topics which might be discussed under this head. For a more
extended treatment the student is referred to the treatises by Henrici and Treut-
lein, Hobson, Lock, Loney, Todhunter, and others, and, of course, to articles in
the various mathematical journals.
72. Limits of 6/sin 6 and 6/tan 9 as
6 approaches zero. Let 6 be an acute
angle measured in radians. Con-
struct, as in Fig. 69, the angle
XOP = ^, repeated symmetrically
as XOQ. Draw through P the arc
PAQ with center 0, the chord PMQ,
and the broken or double tangent
PTQ, Then
AP
OP
= 6,
MP
OP
= sin 6^
TP
OP
= tan 6,
By elementary geometry,
PMQ < PAQ < PTQ.
Whence, dividing by 2 and by OP,
sin S < e < tan 6.
Dividing equation (1) through by sin 6, wo have
e
(1)
1 <
sin
e
< sec 6.
Now in Art. 12 it was proved that as 6 approaches the limit 0,
cos B and its reciprocal sec 6 approach the limit 1. Thus, the
value of ^/sin 6 is always intermediate between 1 and a number
that approaches the limit 1, as approaches 0. The ratio ^/sin 6
99
100 ANALYTIC TRIGONOMETRY
must, therefore, approach the limit 1 at the same time. This is
expressed symbolically by writing
Again, dividing equation (1) through by tan 0, we get
cos e < — ^ < 1.
tan u
Now as approaches 0, cos 6 approaches 1, and hence, as before,
O/tand approaches the limit 1 at the same time. Symbolically,
0=0 Vtan 0/
Note. — Since sin 6 and tan d both approach along with 6, it might seem that
they therefore approach equality, and then the theorems would follow. The fallacy
of assuming that the limiting form - has the value 1 will appear on considering the
following instances. The circumference and area of a circle approach zero simul-
taneously with the radius. We have, however, the general relations
Circumference 2 Trr ^ ^ ^ 6.28318 ••.,
Radius r
Area _ irr^
Radius r
= Trr = 3.14159 ...r.
Now when r approaches the limit 0, the limit of the first ratio is the constant 2 tt,
and the limit of the*second ratio is 0.
The limiting form - will be discussed at length in calculus. (See Townsend
and Goodenough's ''First Course in Calculus," Art. 13.)
Example. If is increased by an angle S, let it be required
to determine the limit of the ratio of the consequent increase in
sin 6 to the increment 3 of 6, as that increment 8 approaches zero.
By Art. 70, we have
2cosf(9 + |^sin I
s in ((9 + 3) -sin (9 _ V 27 2
= cos (6 +
sin -
S\ 2
2/ 8
2
De moivrp:'S theorem lOi
Now when B approaches 0, cos ( ^ + ^ ) approaches cos 6 and
. 8
sm-
— -— approaches 1.
o
2
Hence
lim sin ((9 + 3) - sin 3 ^ ^^^ ^
6 = g
It will be noticed that the numerator and denominator approach
simultaneously, but that the limit of the value of their ratio is
a number somewhere between — 1 and + 1, and depending upon
the value of 0.
Examples
In like manner find the limits as 3=0, of
, cos (6 + 8) — cos
8
n, sec (0 -\- 8) — sec ^c< -r« • j. r • \
2. ^^ — ^^—^ • (Suggestion. Express in terms of cosme.)
« CSC (0 -\-S) - CSC $
3. U
4 tan (0 + 8) — tan (Suggestion. Express in terms of sine and
8 cosine.)
5 cot (^ + 8) -cot^
73. De Moivre's theorem. If we adopt the customary nota-
tion z = V— 1, so that P = —1, we have, on performing the mul-
tiplication,
(cos a+ i sin a) (cos /3 -f ^ sin /3) = cos a cos /3 — sin a sin ^
+ { (sin a cos /3 4- cos a sin /3)
= cos (« -1- /3) + « sin ((X + /3), (1)
a relation which holds for all values of a and /3, whether positive
or negative.
Putting y8 = a, we get
(cos a-\-i sin a)^ = cos 2a + i sin 2 a.
Again, putting /3 = 2 a in equation (1) and making use of the
relation just established, we get
(cos a+ i sin a)^ = (cos a-{-i sin cc) (cos 2a + i sin 2 a)
= cos 3 a -f i sin 3 a.
102 ANALYTIC TRIGONOMETRY
Repetition of this process proves the relation
(cos a H- ^ sin a)" = cos na + i sin na (2)
for all positive, integral values of n.
It, is evident^ ypon multiplying, that
''"'' rcr.«. '(^QQs ^_|_ ^- gii^ ^^(^cosa — z since) = 1,
whence
(cos a 4- i sin a)~^ = cos a — i sin a.
Suppose 71 to be a negative integer. Let n = — m^ where m is
a positive integer. Now
(cos P — i sin /3)""' = (cos yS + ^ sin yS)""
= cos myS + ^ sin m^.
Substituting m = — n and yS = — a, we get
(cos a + z sin «)" = cos Tia + i sin Tia,
true also for negative integral values of n.
Suppose 71 to be a fraction, either positive or negative. Let
71 = -, where r and s are integers. Now
r 1
(cos yS + z sin y8) * = (cos ryS + ^ sin rp~) ' .
Raising both members to the sth power,
r
(cos Sy8 + i sin s/3) * = cos r/3 + i sin ry8.
Introducing - = ti, and putting sfS = a, so that r^= - - s^=na,
s ' s .
we get (cos a + ^ sin a)" = cos 7i« + i sin 7i«.
This relation, therefore, holds for all rational values of n.
By an argument involving the method of limits it can be proved
also for all irrational values of n. This is De Moivre's theorem, an
instrument of great importance in some branches of mathematics.
Example. An illustration of its use is afforded by applying
it to tlie derivation of the formulas for the sines and cosines of
multiple angles. Thus
cos 3 a -h ^ sin 3 a = (cos a -f- 1 sin a)^
= cos^ a 4- 3 ^ cos^ « sin « — 3 cos a sin^ a— i sin^ a.
COMPLEX NUMBERS 103
On equating the real terms on each side, and also the imagi-
nary terms, separately, we have at once
cos 3 a = cos^ a — 3 cos a sin^ a
= 4 cos^ ct — 3 cos a.
sin 3 a = 3 cos^ « sin a — sin^ a
= 3 sin a — 4 sin^ a.
The functions of 4 « and of higher multiples of a are as readily
found. The simplicity and beauty of the method appears on
comparison with that of Art. 71.
Examples
1. Show that cos«+^'s;"« ^ cos («-/?) + / sin (« - S).
cos (3 +isin^
2. Show that ( cos "^ ^ 1- i sin ^ — ) = cos cc + i sin a.
\ n n J
3. Show that f cos ^ ^^ "^ " + f sin - ^'^ + ^ V = cos a + « sin a, where /[: is
\ n n J
any integer.
2 y^TT 4- C£
4. Show that the angle — — ~ — has n different values as k takes the suc-
n
cessive values, 0, 1, 2, • • • n — 1 (n being a positive integer). Show also that
for all integral values of k outside these limits, the terminal sides of the angles
coincide with those of the n angles already found.
5. Since cos -\- i sin = 1, find the ?i different nth roots of 1, of which all
but one are imaginary. Making use of the tables of natural sines and cosines
compute for n = 2, 3, 4, 6.
6. Since cos tt + « sin tt = — 1, find the n different nth roots of — 1, of
which all but one are imaginary when n is odd, and all imaginary when n is
even. Compute for n = 2, 3, 4, 6.
74. Graphical representation of complex numbers. An interest-
ing application of De Moivre's theorem is found in the graphical
representation of complex numbers, devised by Wessel, a Danisli
mathematician, and published by Argand in 1608. The treatment
of this topic belongs rather to the courses in algebra and function
theory. (See Rietz and Crathorne's "Algebra.") Only so much
of the rudiments of the method will be developed here as possess a
trigonometric interest.
A pure imaginary is an indicated square root of a negative
number. A complex number is an indicated sum of a real number
104 ANALYTIC TRIGONOMETRY
and a pure imaginary. All pure imaginaries can be expressed in
the form «/^, and all complex numbers in the form x + yi. Here
^ = V— 1, so that z^ = — 1 ; while x and y are real numbers, either
rational or irrational.
Argand's method makes use of a pair of mutually perpendicu-
lar axes. The Argand diagram must not, however, be confused
with the Cartesian scheme of coordinates.
All real numbers, rational or irrational, are represented by dis-
tances from the origin to points in the horizontal axis, called now
the axes of reals, positive to the right, negative to the left. To
every real number corresponds a point in this axis, and conversely,
to every point in this axis corresponds a real number. Thus there
is said to be a one-to-one correspondence between the totality of
real numbers and the totality of points in the line.
All pure imaginaries are represented by distances from the
origin to points in the vertical axis, now called the axis of imagi-
naries, points above and below the origin giving, respectively,
positive and negative coefficients for the imaginary unit factor
i = V— 1. Here again there exists a one-to-one correspondence
between the totality of pure imaginaries and the totality of points
in the vertical axis.
Notice that the origin alone, of all points in the plane, is on
both axes. The number zero belongs to both systems. With this
single exception, no pure imaginary can equal a real number, since
the directions of the two axes are essentially different.
In order to represent the complex number x + yi recourse must
be had to the method of adding coplanar but non-collinear directed
line segments employed in the graphical composition and resolution
of forces in physics. Since directed line segments may undergo
translation, the segment yi may be placed with its initial point
upon the terminus of the segment x. The complex number is
therefore represented by the right line segment (radius vector) v
from the origin to the resulting terminus of the segment yi. For
y—^ we have real numbers, for a: = we have pure imaginaries.
As the lengths of the horizontal segment x and the vertical
segment ^^ measure respectively the magnitudes of the reals and the
pure imaginaries, so the length of the radius vector v may be said
to measure the absolute magnitude of the complex number
v — x-\- yi. This is called the absolute or numerical value of v,
and is denoted by the letter r. Evidently all points on the unit
circle about the origin possess the absolute value 1.
COMPLEX NUMBERS
105
The directed line segment, or radius vector, v makes in general
an oblique angle with the axis of reals, and its direction is deter-
mined by the angle it forms with the positive axis of reals. This
angle is denoted by ^, and is called the amplitude of the complex
number. All points lying on the same radius have a common
amplitude, while radii vectores extending from the origin in
opposite directions have amplitudes differing by tt. All positive
real numbers have the amplitude ; negative reals, ir ; pure
TT 3 TT
imagmaries, — or -— -.
The right triangle formed by a;, ^, and v yields the relations
6 = arctan-1
X
\Y
x = r cos ^, 3/ = ^ sin 6,
We may write interchangeably,
v^ ov x + yi^ or r (cos ^ + ^ sin ^).
The expression cos 6 -\-i sin d consequently denotes a unit segment
(complex unit) with the amplitude ^, while r is a purely arith-
metical factor.
Conjugate complex numbers, x -f yi and x — y% evidently have
the same absolute value and amplitudes which are negatives of
each other.
Addition is effected graphically by placing the initial point of
the second segment upon the
terminus of the first and con-
necting the initial point of the
first to the terminus of the
second. Thus in Fig. 70,
= ^1 + %i + ^2 + %
= (^1 + ^2) +^'(^1 + ^2)-
The values of r and d in terms of /-j, r^^ 6^ and 0^ are readily deter-
mined, but exhibit little of present interest. Suffice it to point
out that
r < rj + 9-2,
e^e^ + e^.
Subtraction reduces at once to addition on reversing the sub-
trahend segment.
Fig. 70.
106 ANALYTIC TRIGONOMETRY
On attacking the problem of multiplication, we must define
the product of a directed rectilinear segment by the imaginary
unit { as a segment of equal length turned through a positive
right angle. Thus v = x-{-iy =r (cos 6 -\-i sin ^) multiplied by i
gives
= — y -\-ix=r cos ( ^ + ^ ) + 2 sin ( ^ + ^ j •
2 7 V2
The absolute value is unchanged, while the amplitude is in-
ir
creased by — • This is consistent with the original scheme of rep-
A
resentation, since reals multiplied by i give pure imaginaries, and
these multiplied by i give — 1 times the original, i.e. the original
radius vector reversed.
Multiplying a directed segment by a positive real number
simply stretches it, multiplying its length and leaving its direc-
tion unchanged. Multiplying
V =x-\-iy = r (cos -\-i sin ^) by k^ we get
v' = hv = kx -\- iky = kr (cos -\-i sin ^).
The absolute value is multiplied by the factor k, while the ampli-
tude is unchanged.
Multiplication of one complex number by another is effected
by combining the two processes just described, applying the asso-
ciative and distributive laws. Thus
v = v^'V^= (^1 H- iyi) ' (a^2 + ^^2)
= ^1 (^2 + ^>2) + *>i (^2 + %)
= (.V2 - ViVi) + ^^1^2 + ^2^1) •
Using the other notation and applying De Moivre's theorem,
v = v^ • V2 = r^ (cos ^j + i sin ^j) • r^ (cos 0^ + i sin 0^^
= r^r^ ' [cos ((9i + 0^-) + i sin ((9^ -hO^)-].
Figure 71 illustrates the multiplication of 5 — 2 ^ by 2 4- 3 ^.
The product is shown to be 16 + 11 i. We have then the law that
the absolute value of the product of two complex numbers equals
the product of their absolute values, while the amplitude of the
product equals the sum of their amplitudes.
COMPLEX NUMBERS
107
The inverse process of division is readily performed, with the
result
or
V = -1 =
^2
X, + ly^ _ x^x^^ + y^y^ . x^y^ - x^y^
^2 + ^Vl ^2 + ^2
r^(cos 0^ + ^ sin Q^
r^ (cos ^2 + ^ sin ^2) '
+ ^
+ ^2^
i; = ^ [cos (6'i - 6>2) + ^ sin ((9i - 6>2)] •
^2
The absolute value of the quotient is equal to the quotient of
the absolute values, while the amplitude of the quotient is equal
to the difference of the amplitudes.
We have further,
vz=v{' = r-^ (cos nO^ + i sin nO^.
The absolute value of
the power is equal to
the power of the abso-
lute value, while the
amplitude of the power
is equal to the ampli-
tude of the number
multiplied by the index
of the power. Here
" power " is used to de-
note the result of affect-
ing the number by the
exponent n^ whatever
the value of n. This
includes both involu-
tion and evolution. In
particular let n be the
reciprocal of a positive integer m.
Fig. 71.
V- m/
Then
cos—i +
m
mj
But Vj is just as well and exactly represented by
r [cos (2 ^TT + ^) + i sin (2 kir + ^)],
where h is any integer. Thus the mth root just found is only one
of an infinite number, all given by the form
m/-r 2kir + 0.
Vr. cos =^— i
I
+ i sin
^kir
m
±6,1
^ J
108 ANALYTIC TRIGONOMETRY
in which k assumes all integral values. This form gives m dif-
ferent values for the root, corresponding to A; = 0, 1, 2, ••• m — 1.
All the others are repetitions of these m roots, since the terminal
sides of all the other amplitude angles will coincide with the ter-
minal sides of the m amplitudes specified.
Hence every complex number has m different mth roots, whose
common absolute value is the arithmetical mth root of the absolute
value of the number, while their amplitudes have the m different
values,
e. ^TT + e^ 4 7r + 6>^ 2fm-l)7r + (9^
m m m m
all less than 2 tt.
In the special case of any positive real number a^j, whose am-
plitude is therefore zero, we obtain m different mth roots with the
common absolute value Vr^, which is called the principal value of
Va^j, and the m different amplitudes,
/^ 2 TT 4 7r Gtt 2(m — l)7r
u, , , , ••• •
m m m m
Only one of these is real, the first, and it is called the principal
mth root of the positive real number.
The student should construct figures to illustrate the foregoing
theorems. Still another analytic notation for complex numbers
will be brought out in Art. 75.
Examples
1. Represent by Argand's diagrams^the numbers 2, —3, 3i, — 4i, 3 + 5t,
4 - 3 1, - 2 + i, - 5 - 3 i, 4 + VZ^ Vs - VIT?.
2. Write the numbers the termini of whose radii vectores have the Carte-
sian coordinates (3,4), (-3,2)^ (7,-3), (-5,-2), (6,0), (0,5),
(-2,0), (0, -6), (0,0), (V^, V5).
3. Find the absolute values and the amplitudes (expressed in degrees and
minutes) of the numbers in examples 1 and 2.
4. Describe the situation of the number points which have : (1) the
common absolute value 3 ; (2) the common amplitude 30° ; (3) the amplitudes
45° and 225°.
5. Perform graphically: (3 + 40 + (7 - 2 ; (-3 + 2/) + (6 - 3 {) ;
(7 - 3 - (4 + 2 i) ; (3 - 2 - (- 6 - 3 i) ; (5 + 2 i) + (3 - 4 i) - (6 - 3 {).
6. Perform graphically, taking the first factor in each case as the multi-
plier : 3.(5 + 2 0; I -(3 + 50; 2 t- (6-30; -4.(2+50; -6 i . (3+ 2 0;
(4+ 2 • (3 + 4 ; (3 + 4 -(4 + 20.
EULER'S EXPONENTIAL VALUES 109
21 + ; 6 - 17 1
7. Construct the quotient of
3 + 2f 4-3t
8. Construct: (3 + 202; l-l^i^V-, (l-iV'dy;i\
9. Find by construction: V7-24i; \/- 119 + 120 i; (-5 + 12i)^'
^/ITl; ^; ^16.
10, Write the general solution of the binomial equation : x" — a" = 0.
11. Find all the roots of the equations a:^ — 1 = ; x'^ -{- 1 — 0; x^ — 1 = ;
a;3 - 8 = 0.
75. Exponential values of the trigonometric functions. The
first form of De Moivre's theorem, Art. 73, Eq. (1), may be written
symbolically,
which is read, function of a times (the same) function of /9 equals
(the same) function of (a + /3) ; or, the product of the (same)
functions of two numbers equals the (same) function of the sum
of the two nuifibers. Now this is identically the characteristic
relation or law governing the exponential function, that is, a
function of the form a^ ; thus.
For reasons discussed in Art. 77, it is found that instead of the
more general function a% we must place
cos a-\-i sin a = e'% (1)
where e = 2.71828183 ••• is the base of the Naperian, or natural,
system of logarithms given in Art. 23.
Note that the law of exponents, derived for positive integral
exponents, and assumed to hold also for negative, fractional, and
irrational exponents, is still further assumed for exponents which
are pure imaginaries and complex numbers. As in the former
cases, the significance must be determined in conformity to the
action of the assumed law. Indeed, the law defines the function.
Since cos a — i sin a = ;— : , we have also
cos a + ^ sin a
cos a—i sin a = er^°-. (2)
Adding and dividing by 2, we obtain
cosa = ; Ko)
110 ANALYTIC TRIGONOMETRY
again, subtracting and dividing by 2 ^^
sm a = — ^j— . (4)
These values were first given by Euler in 1743. Starting from
these two exponential values as fundamental definitions, and de-
fining further
, sin rt , 1 1 1
tana = , cota = , seca = , csca =
cos a tan a cos a sm a
it is possible to develop all the laws and formulas of trigonometry
as contained in Arts. 59 and 63-71, quite apart from any geo-
metric meaning attached to the functions or their argument a.
The analogous derivation of those trignometric theorems de-
pendent on the periodicity of the trigonometric functions involves
the periodicity of the logarithm, and is therefore postponed until
the later mathematical study of the student.
A third notation for complex numbers now becomes manifest.;
for
v = x + iy = r (cos 6 -\- i sin ^) = re^^.
The consequent theorems regarding the absolute values and am-
plitudes of products, quotients, powers, and roots follow readily,
and should be worked out by the student.
Examples
1. Find the exponential values of tan a, cot a, sec a, esc a.
2. Derive from the exponential values the laws sin^ a + cos^ a = 1, etc., of
Art. 59.
3. Derive from the exponential values the formulas of Arts. 63-71.
4. Derive from the exponential notation the laws for the absolute values
and amplitudes of products, quotients, powers, and roots of complex numbers.
76. Hyperbolic functions. Closely allied to Euler's forms of the
last article are the two interesting and important forms.
2 2
They are called, by analogy, the hyperbolic cosine and hyperbolic
sine. Thus, employing the customary notation,
cosh a = , sinha= —
HYPERBOLIC FUNCTIONS 111
The remaining hyperbolic functions are defined from these, just as
in Art. 75 :
tanh a = ^IBIL^, coth a = —1—, sech a = -—, csch a = ___.
cosh a tanli a cosh a sinh a
A very simple relation exists between the hyperbolic and the
circular (^i.e. ordinary trigonometric) functions. Evidently
cosh a = cos ^a,
sinh a = — i sin m,
tanh a = — ^ tan ia ;
and conversely,
cos a — cosh za,
sin a = — i sinh m,
tan a = — { tanh za.
To each formula of Chapter YIII corresponds a formula for the
hyperbolic functions, which may be deduced either directly from
the exponential definitions, or by substituting the values just
given in the formulas for the circular functions. The student
should derive these formulas by both methods.
The analogue to De Moivre's theorem is
(cosh a -h sinh a)" = cosh na -\- sinh na.
Cosh a and sinli a possess an imaginary period 2 7^^, since
gM _ ^u+2kni^ jf. being any integer. (See treatises on the theory of
functions.)
77. Exponential and trigonometric series. In the present
article values in the form of infinite series will be derived for
certain exponential, logarithmic, and trigonometric functions.
In the proof, however, the use of the binomial formula and the
manipulation of the series introduce a lack of rigor requiring ex-
tended consideration in the subsequent courses in algebra, the
calculus, and the theory of functions.
(1) Exponential series.
Expanding by the binomial formula,*
n) ~ ll'^^ 2! ^ 3! ' n^
*The symbol \k, or k !, is used to denote the product 1 • 2 • 3 ••■ A:, where k is
any positive integer, and is read " factorial A;".
112
ANALYTIC TRIGONOMETRY
.2
(2)
Now as n becomes infinite, the binomial factors [ 1 ), f 1 — — ),
etc., all approach the common limit 1, and we shall have, in the
limit,
,- = limrA+^Y1=l + ^ + ^ + |^+.... (1)
n=^[\ nJ J 1! 2! 3!
This series is convergent for all finite values of x. (See Rietz and
Crathorne's "Algebra.")
For a; = 1 we get
1,1,1.1,1,
^=^ + l! + 2!-^3! + T!-^-
The terms diminish rapidly in value and, when expressed deci-
mally, the value of e is found to be 2.71828183 •••.
The series for e^ is valid also for negative and imaginary values
of X ; thus, substituting successively —x, ix, and —ix for a;, we have
* =^-T! + 2!-8T+-'
/y>^ /"Y*^ /^O /y% /y»t> /y*0 /y»7 I
^ =^-2! + 4!-6! + - + ln-^ + 6!-f!+-}
_,> -, x^ , x^ x^ , .r X a^ , x^ x'^ , ~]
^ ='-2! + 4!-0l+--iT!-^ + 5!-7!+-J-
(2) Logarithmic series.
From the expansion just obtained for e"^ can be derived a series
for log, (1 + ^).
Since* ^t^ = e^l«ge«,
we have ^^ = 1 + ^ (log, ^) + |y (loge ^)^ + ^y (log, uy + -- -.
Placing u=l + t/f
(l+y)-=l + ^log,(l+y) + ^[log,(l + y)P
+ ^Doge(l+y)?+--
* Let w = u*. Taking logarithms to base e, we have loge w = x log« u. Now
taking exponentials to base e, lo = w* = e* log*".
TRIGONOMETRIC SERIES 113
Expanding the first member by the binomial formula,
(1+^)^ = 1 + — y+ ^ ^^ ^ f + -^ ^y^ -f+ ••••
Picking out and equating the coefficients of x in the two expres-
sions, the required expansion is obtained,
log«(l+2/) = f-f + |-J+-. (8)
This series is convergent for — 1 < ?/ ^ + 1.
If w = loga u^ we have a"' = u; wlience, taking logarithms to
base g, w log^ a = logg u. Therefore w = log« u = • log. u,
log, a
Substituting from (3) we see that
(3) Trigonometric series.
From De Moivre's theorem
cos mO -\-i sin mO = (cos 6 -\-i sin ^)"*.
Expanding b}^ the binomial formula and- separating the real terms
from the imaginary,
cos mO + i sin md = cos"^ 6 - ^^^ ~ ^^ cos'^-s ^ sin2
. w(m— l)(m — 2)(m— 3) m-A a - 4. a
+ — ^^ — ^ cos"^ ^ a sin^ a •"
4!
-f ^ f — - cos"* 1 6 sm 6 — — ^ ^ ^ cos"*-3 gm^ j^ ... \
Equating separately the real and the imaginary parts,
cos mO = cos"* - ^^^"-^^ cos"*-^ sin2 (9 4- •••,
sinmu==—j cos"* ^^sm^ ^^ /y^ ^cos"* ^osin^6-{- •••.
Place now
m$ = a, so that ^ = — ;
co.„ = cos"Q-^(^cos-Qsi,.Q+...,
sm
1 ! \wy Vm/
m(m-l)(m-r2)
3!
-<^)^''<3^--
114 ANALYTIC TRIGONOMETRY
Now let approach the limit zero and m become infinite, while
still obeying the condition that mO = a, where a remains finite.
By Art. 12, cos—, cos^— , etc., approach the limit 1 as m becomes
mm . .
infinite, and in the calculus the same is shown for cos"*f — j,
cos^^-if-Y etc.
Again, m sm — = •,
\mj u
■ m(m-l)sin2('"'\=a(«-6>)./'^Y,
m{rn - 1) (m - 2) sin^/"-^ = «(« - ^)(« - 2 l9) • f^}^\\ etc.
Since li^^^ ^ = 1, the limits approached by these expressions, as
^ = 0, are a, a\ a^, etc.
Making these substitutions, we obtain, in the limit,
^ «2 «4 ^6 .
cos« = l-- + jj-^j+..., (6)
'^"" = r!-3!+6T-^+-- <^^>
These series are convergent for all values of a.
Tan a may also be expanded into the series
*'^"" = i + F + i5 + W+-- ('>
It will be noticed that the series for cos a contains only even
powers of a, while those for sin a and tan a contain only the odd
powers of a. (See the third example worked out in Art. 62.)
The assumption of Art. 75 may now be justified. For, on sub-
stituting for e'^, e"'^ sin a:, and cos a; their expansions in series, we
obtain
cos x-\-i sin x = e'^,
cos ic — ^ sin a; = e~"^,
and cos x —
^rx
4- e-''
2
^ix
- e-'^
sma:
2i
COMPUTATION OF TABLES • 115
(4) Hyperbolic series.
From the analogous relations the expansions for the hyperbolic
functions are readily obtained.
ri^i 1 e'^ + e~'^ 1 , ^2 ^4 ^6
Ihus cosha = -^^=l + - + - + -+ .-, (8)
Examples
1. By substituting — a for a, find the series for sin(— a), cos(— a),
tan (— a), and by comparison, verify the corresponding relations of Art. 62.
2. By substituting ia for a, verify the relations of Art. 76, cosh a — cos la,
etc.
3. Using two terms of the expansions for sin a and cos«, and retaining
only powers of a below the fifth, obtain an approximate verification of the fol-
lowing formulas :
sin^a + cos^a = 1, sin (a + /?) = •••, cos (a + y8)= •••, sin 2 a= ••-, cos 2 a = ••••
4. Do as required in example 3 for the hyperbolic functions.
5. Repeat examples 3 and 4, using three terms of the series and retaining
powers of a below the seventh, thus arriving at a closer approximation.
78. Computation of trigonometric tables. The numerical values
of the sine, cosine, and otlier trigonometric functions of angles
from 0° to 90°, as tabulated in Table III, may be calculated by
means of various trigonometric formulas, or better, by the use of
the series derived in Art. 77.
Euler gave the following series, carrying the computation to
28 decimal places and a corresponding number of terms : Place
a = m ' -^ in the series for sin a and cos a ; whence
sin ^w. 1^ = 1.570 796m -0.615 964^3
+ 0.079 693 m^ - 0.004 682 m'^
-f- 0.000 160 m^- 0.000 004m"
+ ,
cos C^ • I) = 1-000 000 - 1.233 700 m^
+ 0.253 699 m^ - 0.020 863 m^
-{- 0.000 919 m^ - 0.000 025 m^^
+
116 * ANALYTIC TRIGONOMETRY
We need to calculate the sines and cosines of angles up to
45° only, so that m is a fraction and always less than |-. The
terms of the series converge rapidly and a few terms suffice to give
values correct to a small number of decimal places.
More extended discussion of this topic may be found in Hob-
son's "Trigonometry," Chap. IX; Todhunter's "Plane Trigonome-
try," Chap. X ; and other advanced treatises on trigonometry.
The series for log (1 + ^) converges too slowly for convenient
calculation, but a modified form is easily obtained. ^ Manifestly
/2
~ 8
log (1 - ^) = - ^ - |- ^
and
log l^f = 1^'g (! + «/)- log (1 - ^)
— 9
I + -8+5 +
}
Place y = , Avhence ^ = ; then
2i; + l 1 — y ^
log !i±i = 2^-1-+ 1 + 1 : + ...),
or
log (v + 1) = lo^ V -h 2r — - — H h — + •••T
This series converges rapidly and by it log 2 can be computed
from log 1 = 0, log 3 from log 2, etc. Logarithms of composite
numbers can be checked by adding the logarithms of their factors.
79. Proportional parts. In using the logarithmic and trigo-
nometric tables it Avas assumed, as stated in Art. 2^, that for
small differences in the number, the differences in the logarithm
are proportional to the differences in the number, and that like-
wise, for small differences in the angle, the differences in the sine
(or other trigonometric function, or logarithmic function) are pro-
portional to the differences in the angle.
We have
log (a; + 8) - log a; = log^±^ = log f 1 + -^
X ' \ xj
^S_j5^ J3 ^
X 2X^ SX^ 4:X^
__ 8 (Approximately for small
X values of 3.)
PROPORTIONAL PARTS. INVERSE FUNCTIONS 117
Therefore, we have approximately
log ix-\-8^)-lo^x ^ S^ /^ = ^1 , (13
log (x + 82) - log X hjx S2'
for small differences.
Again,
sin (3 -}- |9) - sin (9 = 2 cos [o + -") sin |
= cos Q • 3. (Approximately for small
values of 3.)
Hence, approximately
sin (0 4- 3j) — sin \ cos ^ \
sin (^ + ^2) — sin Q h^ cos d h^
(2)
For the other functions, the proof follows exactly similar lines,
and can easily be supplied by the student.
Full discussion along this line may be found in Loney's " Plane
Trigonometry," Chap. XXX ; Lock's " Higher Trigonometry,"
Chap. VIII ; Hobson's " Trigonometry," Chap. IX ; etc.
80. General inverse functions. In Art. 14 only acute angles
were under consideration, so that the relations
m = sin a, a = arcsin m,
expressed a one-to-one correspondence. In other words, under
the condition that
0°<«<90°, 05m<l,
to each value of « there corresponds one and only one value of
m and conversely.
On considering the general angle, it became evident that to
any one angle there corresponds one and only one value of the
sine (or other function), but that to one value of the sine (or
other function) correspond many angles. We define arcsin m,
arccos m, arctan ?w, etc., as the numerically smallest angle having
the given sine, cosine, tangent, etc. It follows that arcsin m^
arctan w, arccot m, arccsc m always lie between — — and +— ,
while arccos m and arcsec m always lie between and + tt.
These are called the principal values of the general inverse func-
118 ANALYTIC TRIGONOMETRY
tions Arcsin w, Arccos w, etc. As results of Arts. 61, 62, we
may write, if k is any integer,
Arcsin m = 2 A^tt + arcsin m,
Arccos m = 2 kir -{- arccos m,
Arctan m = Jctt -{- arctan w,
Arccot m — Jctt -\- arccot m,
Arcsec m = 2 kir + arcsec m,
Arccsc m = 2k7r -{- arccsc m.
Similar relations exist for the inverse hyperbolic functions, the
periods being imaginary, 2 kiri and kiri.
From the relations of Art. 76 may be derived the following :
arccos m = ( ± ) ^ inv cosh m^
arcsin m = — i inv sinli im^
arctan m = — z inv tanh iyn ;
and
inv cosh m = ( ± ) ^ arccos w,
inv sinh m = — i arcsin im^
inv tanh m = — i arctan im.
81. Logarithmic values of inverse functions. Since the circular
and hyperbolic functions are expressible as exponential functions,
it would seem that the inverse functions should be expressible as
logarithmic functions. Such is, indeed, the case, and the desired
values may be found by solving for a the forms given in Arts. 75
and 76.
( 1 ) Circular functions .
If, for example,
z — Sin a = -— — ,
2 I
we have the quadratic equation in e'%
g2.« _ 2 {ze^<^ -1 = 0,
whose roots are
e'" = iz ± Vl — z^.
Choosing the upper sign, and taking logarithms to base e, we
^^^ ia = log {iz + vr^^),
whence ^ ^ ^^^^.^ ^ _ _ . ^^^ ^ .^ _^ ^^^^ )
gives the principal value of the arcsine.
INVERSE HYPERBOLIC FUNCTIONS 119
(2) Hyperbolic functions.
As may be expected, the values of the inverse hyperbolic
functions are real in form. Thus from
z = smh cf = ,
we get, on solving,
a — inv sinh z = log {z + Vs^ + 1 )•
Examples
Obtain
1. arccos z = — i log {z + Vz^ — 1 )•
2. arc tan s = - i log ^ + ^^
2 " 1 - I.
iz
3. inv cosh z — log 2 {z + va;^ — 1 ).
1 1-4-2
4. invtanhs = - log =--^.
2 ^ \-z
REVIEW EXERCISES
1. To what quadrant do the following angles belong : 560°, 653°, 1030°,
425°, -1260°?
2. To what quadrant do the following angles belong: — ^, r!L_^ 8^,
5 o
13 TT^'
3 '
25^, ■
4 ' -
3
3.
Reduce to radians
: 75°
300°, -250°, 2000°, 465° 20'.
4. Reduce to the degree system : 4^, - 6^, i^, ^, _ I^.
o o 2
5. Find the lengths of the arcs subtended by the following angles at the
center of a circle of radius 6 : 45°, 120°, 270°, ^^, ^^, 5^-
'4 8 3.
6. A polygon of n sides is inscribed in a circle of radius r. Find the
length of the arc subtended by one side. Compute the numerical values if
r = 10 and w = 3, 4, 5, 6, 8.
7. Taking the radius of the earth to be 4000 miles, find the difference in
latitude of two points on the same meridian 300 miles apart.
8. Find the difference in longitude of two points on the equator 1200
miles apart.
9. Find the distance in degrees between two points, one of which is 800
miles due north of the other.
10. A city is surrounded by a circular belt line 5 miles in radius. How
long will a train require to go at a speed of 20 miles an hour from a station due
east of the center to one due northwest, if the motion is clockwise ; if counter-
clockwise ?
11. Find with the protractor the angles formed successively by the radii
vectoresof the points (3, 0), (2, 4), (-3, 5), (0, 6), (-4, 2), (- 2, 1), (5, -3),
(8, 0).
12. Find with the protractor the angles of the triangles formed by the
abscissa, ordinate, and radius vector of each of the following points : (4, 4),
(1,3), (3, -3), (-2,2), (-4, -8).
13. Find by measurement the coordinates of the point whose radius vector
is 4 and makes an angle of 30° with the positive x-axis ; 5 and 120° ; 8 and 225°.
14. Find by measurement the length and inclination angle of the radii
vectores of the points whose coordinates are (2, 5), (—5, 12), (— 8, — 15).
120
REVIEW EXERCISES
121
15. If the earth were assumed to be a plane, and one degree of latitude or
longitude were 60 miles, what would be the distance and direction from a point
in 20° N. lat., 60° E. long., to one in 50° N. lat., 30° E. long. ; from a point in
30° S. lat., 15° E. long., to one in 45° N. lat., 40° W. long. ?
Note. This assumption is made by navigators as a basis for what is known as
Plane Sailing. In Great Circle Sailing the earth is considered a sphere. Let the
student devise a system of coordinates for the latter.
16. Find by measurement the six trigonometric functions of 36°, 155°,
285°, - 130°.
17. Find by measurement the following angles: arccot | and of 1st quad-
rant ; arcsin f and of 2d quadrant ; arccos ( — 0.3) and of 3d quadrant ; arcsec
0.6 and of 4th quadrant.
18. Find the lacking functions in the following table :
Angle
Sink
Cosine
Tangent
Cotangent
Secant
Cosecant
Quadrant
a
-tV
III
ft
f
IV
y
-i
ir
a
Y-
III
e
_ 2
TIT
<t>
3
T
19. Find the value of ^^^ " + ^"^ ^ ii a = arcsin - and of 2d
quadrant. 1 - tan « l-cot« 5
20. Find the value of tan /? + sec y8 - 1 if ^ ^ arccsc f - — ) and of 3d
quadrant. tan ^ - sec /? + 1 ^ V 6 J
21. Find the value of tan^g - sin^ct .^ ^ ^ arccot f - I] and of 4th quad-
rant. '^^'^ ^ 2;
22. Find the value of 1 + cosy -2 secy -^ arctan-^ and of 1st quad-
rant. 3 + cosy + 2 secy 40
23. Express ^^^^ " + ^^^^ ^ in terms of tan a.
sec2 a + csc2 a
24. Express 4-sin^-3csc^ j^ ^^^^^ ^^ ^^^^^
sin /8 - 1 - 6 CSC /^ ^
25. Express Q - cos «) (1 -f sec «) -^ ^^^^^ ^^ ^^^ ^
(1 -sin a)(l -fcsca)
26. In the following identity transform the first member into the second,
(1 + tan y) (cosy- cot y) ^ _ ^^^
(1 + cot y) (sin y — tan y)
122 REVIEW EXERCISES
27. Show that (l-tana)(l + cot(x) ^ _ ^^
(1 + tan a)(l - cotw)
28. Show that sin^ /g + cos^ /S siii« ^ - cos^ ^ ^ _ ^ ^ .^^, ^ ^^^, ^^
Solve the following equations and find the angle in degrees :
29. 4sin2y- tan^ynzO.
30. 2 tan2 a - sec a = 4:.
31. 4 CSC ;8 + cot2 y8 = 5.
32. tan2y + 3cot2y = 4.
33. sin2 a + sin^ /3 = I, cos"^ a + cos^ (3 = 0.
34. 2 cos2 a + sin2 y8 = 2, sin a + cos^ y8 = 0.
35. For what range of values of a between and 2 tt is sin a + cos a posi-
tive ; negative ?
36. For what range of values of (3 between and 2 rr is tan jS — cot/3 posi-
tive; negative?
37. Show that tan y + cot y must always be numerically greater than unity.
38. Trace the variation of sin^ ^ as ^ varies from to 2 tt..
39. Trace the variation of cos^ ^ as ^ varies from to 2 tt.
40. Trace the variation of 1 — sin ^ as ^ varies from to 2 tt.
41. Trace the variation of 1 — cos ^ as ^ varies from to 2 tt.
42. Find by inspection logg .625, log8i27, loggg .008.
43. What numbers correspond to the following logarithms to base 9 : — 3,
-2, -1.5, -1,0, .5, 1,2,3?
44. In the formula W=^^^p^v^ [( ~)"^ ~ "^T ^^^^^ ^ives the work
of an air compressor, find W when n = 1.3, p^ = 14.7, p2 = 72, vi = 6.
45. Work the following with the slide rule :
,. .72x137x14 . .J. fl20y-^ . ., 42 sin 27° .
^^^ 372x778 =' ^'M42J =^ ^'^-13^- = ^
46. Solve for a: : 52^ = 6; 8»=-i = 7.
Query. Does the result depend on the base of the system of logarithms
used?
47. Solve for x: 32« - 4 • 3'' + 3 = 0.
48. Find the amount of $2000 in 5 years at 4% compound interest.
49. At what rate, compound interest, will $45,000 amount in 8 years to
$60,000?
50. In how many years will a city become three times its original size if
it increases | each year ?
REVIEW EXERCISES 123
51. Derive the formulas of Art. 42 from those of Art. 40. [Suggestion.
Multiply respectively by a, b, and - c and add.]
52. Derive the formulas of Art. 40 from those of Art. 42. [Suggestion.
Solve for a cos /8 and b cos a and add.]
53. From the law of sines, Art. 41, show that ^^^-^ = sm^-smy ^
b + c sin 13 + sin y
54. By applying the formulas of Art. 70 to the result obtained in example
53, derive the law of tangents of Art. 48.
55. From the formulas of Arts. 42 and 68 derive the results
sin ^^ = J5ZS5ZI) ; COS « = JiSZ3 ; tan ^ = JKEMlZs) .
2 ^ be ' 2 > 6c 2 > s(s-a)
56. Draw the graph of sin ^ + cos ^ and thus trace its variation. What
values of cause the given expression to assume maximum values ; minimum?
57. Draw the graph of tan + cot and thus trace its variation. What
values of make the given expression a maximum ; a minimum ?
58. Draw the graph of arcsin u and trace its variation. [Suggestion.
Lay off the values of u as abcissas, of arcsin u as ordinates.]
59. Draw the graph of arccos u and trace its variation.
60. Draw the graph of arctan u and trace its variation.
61. Draw the graph of arccot u and trace its variation. What discon-
tinuities are exhibited by the functions of examples 58-61 ?
62. Find from the table the values of cos 625° 12' ; of sin 238° 25'; of
tan 324° 6'; of cot 921° 32'.
63. Find without reference to the table the value of
cos 285° cos 345° + sin 195° sin 465°.
64. Find without reference to the table the value of
tan 205° cot 335° + tan 295° cot 115°.
65. Find all the values between and 2 tt of
arcsin ( — ) ; arccos ( ] : arctan - .
V 2/' V V2/ 3
66. Find all the values of a between and 2 tt if
sin 3 a Z3 A ; if tan 2 a= - \/3 ; if cos - = — ; if cot - = - 1.
V2 '2 V2 3
67. Find the value of sin (2 a + /3) if a = arcsin | and of 2d quadrant,
ft = arctan | and of 1st quadrant.
68. Find the value of tan (3 a + 2 /?) if a = arcsin | and of 2d quadrant,
P = arccos y\ and of 4th quadrant.
69. Derive the formula for sin (« + /8 + y) in terms of sine and cosine.
70. Derive the formula for cos (^ + (S -\- y) in terms of sine and cosine.
124 REVIEW EXERCISES
71. Derive the formula for tan (« + ^8 + y) in terras of tangent.
Discuss the results of examples 69-71 in case a + ft + y = ir.
72. In the results of examples 69-71 put ft = y — a, and thus obtain the
formulas for sin 3 a, cos 3 a, and tan 3 a.
73. Find the value of cos (3 a — 2 /3) if « = arccos ( — jV) and of 3d quad-
rant and ^ = arctan | and of 1st quadrant.
74. Find the value of cot {^ a — ft) \i a — arcsin f and of 1st quadrant
and ft = arccos (- ^f) and of 2d quadrant.
75. Find the value of sec (a + /8) if a = arctan ^^ and ft = arcsin ^V? both
of 1st quadrant.
2 tan a
76. Show that sin 2 =
77. Show that cos 2 a --
1 + tan'-^ a
1 - tan2 a
1 + tan2 a
78. Show that tan ( 45° — ^ ) =: esc a + cot a.
79. Show that cot (45° — - J =csc a — cot a.
Transform into products or quotients the following expres-
sions (80-84) :
80. cot a + tan a.
81. cot a — tan a.
82. 1 + tan a tan ft.
83. cot a — tan ft.
g^ cot a + cot ft
tan a + tan ft
If a + /3 + 7 = TT, show that (85-87)
85. sin a + sin /? + sin y = 4 cos - cos ^ cos ^. [Suggestion. Apply
A Ji Ji
Art. 70 to the first two terms and Art. 67 to the third term.]
86. cos a + cos ft + cos y = 1 + 4 sin ^ sin ^ sin ^ •
Jd Z 2i
87. tan a + tan ft + tan y = tan a tan y8 tan y. (See example 71.)
88. How does it appear from example 87 that either all three angles of a
triangle are acute or else two are acute and one obtuse? (Consider the signs.)
89. How does it appear from example 87 that if one angle of a triangle
is obtuse, it is numerically nearer 90° than either of the acute angles?
In the following equations find the angle :
90. tan 2 a tan « = 1.
91. sin (60° - ft) - sin (60° + yg) = f .
SECONDARY TRIGONOMETRIC FUNCTIONS 125
92. cos 6 y — cos 2 y = 0.
93. r sin ^ = 8, r cos ^ = 15.
94. r sin 6 cos <f> = S, r sin 6 sin <^ = 4, r cos 6 = 12.
95. Show that sin ^ sin ^ sin ^ sin i^ = A.
5 5 5 5 16
96. Show that cos — cos |^ cos ^ cos i^ = -i; •
15 15 lo 15 lb
97. Show that 2 sin ^ = ± VI + sin a ± VI - sin a.
98. Show that 2 cos ^ = ± Vl + sin a =f VI - sin a.
99. The formula for the horizontal range of a projectile fired at an eleva-
tion a with a muzzle speed u, is — sin 2 a. Show that the maximum range
9
is attained for an elevation of 45*^.
100. A triangle is formed by two given sides of constant length b and c,
including a variable angle a. For what value of a is the third side a maxi-
mum ; the area a maximum?
SECONDARY TRIGONOMETRIC FUNCTIONS
In addition to the trigonometric functions defined in Art. 6,
32, and 54, there are certain other expressions which are also
functions of the angle. While of less importance than the six
primary functions, an investigation of their properties will be
valuable, not only for the results obtained, but as a review of the
fundamental principles of trigonometry.
We may define, then,
versed sine a = vers a = 1 — cos a,
coversed sine a = covers a = 1 — sin a,
exsecant a = exsec a = sec a — 1,
excosecant a = excsc a = esc a— 1,
101. By reference to
Fig.
53,
show that
vers a = 1 ,
V
covers « = 1 — -
V
exsec a = - — 1 ,
excsc a = ' — 1 .
V
102. By reference to Figs. 60-63, show that, in line representations,
vers a = MA, covers a = NB,
exsec a = PT, excsc a = PS.
103. Determine the signs and limitations in value of each of the four
secondary functions in the different quadrants.
126 REVIEW EXERCISES
104. Show that if the quadrant of the angle and the value of any one of
its ten functions are given, the values of the other nine can be found.
105. Find the values of the four secondary functions, given :
a = arcsin ( — ^%"j-) and of '3d quadrant ; jS = arccos |§ and of 4th quadrant ;
y = arctan (— -^^j) and of 2d quadrant; 8 = arccot ff and of 1st quadrant.
106. Find all ten functions, given:
a = arcvers |° and of 4th quadrant; ^ = arccovers t| and of 2d quadrant;
y = arcexsec ^ and of 1st quadrant ; 8 — arcexcsc 2^^ and of 3d quadrant.
107. Trace the variation of each of the secondary functions as the angle
varies from to 2 tt.
108. Draw the graph of each of the secondary functions. What discon-
tinuities, if any, are present.
109. Find the secondarv functions oi k •-, for ^ = 1, 2, ••• 8.
4
110. Find the secondary functions of ^' .--, for ^ = 1, 2, ••• 12.
111. Verify the relations of Art. 61 for the secondary functions.
112. Determine the relations analogous to those of Art. C2 affecting the
secondary functions. [Suggestion. Use Art. 60.]
113. By means of the cosine series. Art. 77, Eq. (.5), show that lim Y^^— — 0.
PXSPO
114. From the preceding example, show that lim — '-^ — = 0.
115. Show that
vers a + covers a exsec a + excsc a 2 vers a covers a
vers cc — covers a exsec a — excsc a vers a — covers a
116. Show that exsec^ /? + 2 exsec fi = tan^ p.
117. Solve and find y in degrees : 2 vers y (2 — vers y) = 1.
118. Solve and find 8 in degrees : tan^ 8 + exsec 8 = 4.
119. Show that, if a is the angle at the center of a circle of radius r, the
ordinate at the middle of the chord is given by the formula m = r vers -. Find
m for r = 1433, a = 11° 32'. ^
120. If T is the intersection angle of two tangents to a circle of radius r,
the shortest distance of their point of intersection from the arc is given by
the formula d = r exsec -. Find d for r = 5730, t = 5° 32'.
121. Reduce the first member to the second in the identity
(exsec a + vers a) (excsc a + covers a) = sin a cos a.
122. Show that vers 2 a = 2 sin^ a.
123. Show that exsec 2 «= _2_sm^-a__^
1-2 sin2 a
124. Show that excsc 2 a cos 2 a = tan a.
REVIEW EXERCISES
127
125. At points in a straight line ordinates are erected siich that for each
point (x, y), x = vers (arcsin y). Show that the graph thus determined is a
circle tangent to the F-axis at the origin.
126. At each point in a circular arc the radius is extended an amount equal
to the exsecant (in line values) of the arc measured from a fixed point in it.
What is the graph thus determined?
127. Two equal circles have their centers in the same horizontal line. Show
that the horizontal distance between two points in the neighboring arcs is equal
to twice the versed sine (in line values) of the arc, measured from the point of
tangency.
128. Two tangents to a circle intersect at an angle t. Show that the dis-
tance of the point of intersection from the midpoint of the chord of contact
equals exsec t + vers t (in line values).
129. Show how the versed sine is of practical use in staking out a circular
railroad track passing through three given points.
130. Show how the exsecant is of practical use in staking out a circular
railroad spur of given radius branching tangentially from a straight track.
Compute the missing parts of the following triangles, distinguishing right
from oblique :
"A
'3
a
b
c
131.
37° 42.8'
90°
6244.8
132.
72° 25.6'
90°
64.863
133.
90°
375.84
296.57
- 134.
54° 36.9'
24.465
42.850
^135.
136° 36.8'
36902
37490
136.
68° 51.5'
90°
7532.8
137.
90°
396.45
531.53
138.
90°
.005428
.006395
139.
148° 24'
7.4536
5.3648
140.
.038456
.028638
.051524
141.
125° 34.6'
35° 25.3'
2584.6
142.
24° 36.8'
2.4657
3.6542
143.
80° 04.5'
90°
30.007
144.
94° 46.8'
34.086
52.475
145.
.93274
.40586
.63208
146.
76° 46.3'
85° 38.7'
8.4637
147.
90°
29846
53857
148.
29° 57.4'
43° 52.6'
64.475
149.
17° 46.8'
.39475
.29478
— 150.
36875
28467
48542
128 REVIEW EXERCISES
151. In a given triangle a = 280, c = 420, y = 38° ; find the radius of the
circumscribed circle.
152. In a given circle a = 63, & = 81, y = 54°; find the lengths of the bisec-
tors of the interior angles.
153. The sides of a given triangle are 220, 350, 440 ; find the lengths of the
three medians.
154. In a given triangle b = 340, a = 48°, y = 63°; find the lengths of the
radii of the inscribed and of the three escribed circles.
155. A boat drifts in a stream whose current runs 4 miles an hour due east
under a breeze of 10 miles an hour from the southwest. Determine the motion
during 35 minutes, if the resistance reduces the effect of the wind 30 %.
156. Three forces of 1800, 2200, and 2700 dynes are in equilibrium ; find
the angles they make with one another.
157. A helical spring is fastened to the door 16 inches from the axis of the
hinges, and to the jamb 4 inches from the same line in the same horizontal
plane. Find the length of the spring when the door is closed, open at 30°, 45°,
70°, 90°, 120°. Neglect the thickness of the door.
158. A cable 30 feet long is suspended from the tops of two vertical poles
20 feet apart and 15 and 18 feet high, and bears a load of 200 pounds hanging
from it by a trolley. Find the position of the trolley when at rest, and the
lengths, inclinations to the horizon, and (common) tensions of the segments of
the cable. Neglect the weight of the cable.
159. Let the data be as in the preceding example, save that the load hangs
from a ring knotted at the center of the cable ; find the inclinations to the hori-
zon and the (unequal) tensions of the segments. Solve when the ring is
knotted at a point 12 feet from the lower end of the cable.
160. The eye is 40 inches in front of a mirror and an object appears
to be 35 inches back of it, while the line of sight makes an angle of 48° with
the mirror. Find the distance and direction of the object from the eye.
(Note. The angles of incidence and reflection are equal.)
161. The line from the eye to the object recedes from the mirror at an
angle of 32°, while the object is 36 inches from the eye and 12 inches from the
mirror. Find the angles of incidence and reflection, and the point of reflection.
162. Two railway tracks intersect at an angle of 75°, and are connected by
a circular " Y " of 800 feet radius lying in the obtuse angle and tangent to the
two tracks. Find the distances of the points of tangency from the crossing and
the length of the " Y ".
163. Two railway tracks, intersecting at an angle of 62°, are joined by a
circular " Y " in the acute angle and tangent to the two tracks at points 900
feet from the crossing. Find the radius of the " Y " and its length.
REVIEW EXERCISES
129
Fig. 72.
164. In setting a door frame 6 feet wide and 8 feet high, the vertical side is
found to be 2 inches (horizontally) out of plumb. Find the angles of the paral-
lelogram and the lengths of the diagonals. Is the diagonal of the true rectan-
gle the average (arithmetic mean) of these two ?
165. The staking out of a certain building requires the setting of stakes at
the vertices of a rectangle 32 x 48 feet. A test of the trial setting shows the
figure to be a parallelogram whose sides are as given above, but whose diagonals
dilfer by 9 inches. Find the angle through which the longer sides must be
swung to correct distortion, and the
chord of the arc through which the
back corners must be moved.
166. A triangular roof truss is 80
feet long and divided into 8 equal seg-
ments, by vertical members, as shown
in Fig. 72. The height being 25 feet, what are the lengths and inclinations
to the horizon of the various members?
167. The triangular roof truss shown in Fig. 73 is 60 feet long and 20
feet high. The bottom chord and rafters are divided into equal segments.
Find the lengths and inclinations of the members.
168. In order to determine the
exact location of the point G (Fig.
74), a base line AB \s laid oif due
north and south, measuring precisely
130 rods. Convenient intermediate
stations are chosen, and angles meas-
ured as follows: ^^0=^36° 35',
BA C = 61° 10', BAD = 43° 54', CAD = 17° 16', DCE = 66° 36', CDE = 47° 41',
EDF=55°4:H',DEF =
73° 12', FEG = 5S°32\ B
EFG = 10° 28'. Com-
pute the distances com-
posing the sides of the
triangles in the figure.
169. By projecting
the distances AC^ CE,
EG (Fig. 74), perpen-
dicular and parallel to
A B, compute the east-
erly and southerly dis-
tances of G from A;
find thence the direct Fig. 74.
distance and direction of G from A.
Fig. 73.
FORMULAS
GENERAL TRIGONOMETRY
1
CSC a —
sec « =
cot a —
tan a =
cot a =
sill a
sin^a + cos^ a = 1.
tan^ a + I = sec^ a.
cot^H- 1 = csc2 a.
2 TT^ = 360°.
F(2k7r + a) = F(a), k an integer.
sm a
1
cos a
1
tan a
sin «
cos a
cos a
Ffk
± a] = ±F (a), k an even integer.
Fik'-± a]= ± co-F (a), ^ an odd integer,
sin (a ± /3) = sin a cos yQ ± cos a sin yS.
. cos (a ± y6) = cos a cos /3 =F sin a sin /3.
tan(«±^)= tan«±tanff ^
1 =F tan at'dn 8
cot(«±ff)=«"t«C"t/3Tl.
cot y8 ± cot a
130
GENERAL TRIGONOMETRY 131
sin 2 a = 2 sin a cos a.
cos 2 a = cos^ a — sin^ a = 2 cos^ a — 1 = 1 — 2 sin^ «,
2 tan a
tan 2 a =
cot 2 « =
1 — tan^ a
cot^ ce — 1
2 cot a
sin \a= V| (1 — cos a).
cos I a = V| {\ 4- cos a).
.1 /l — cos a 1 — cos a
tan - a = \— = — :
2 ^ 1 H- cos a sm a
,1 /I + cos « 1 + cos a
cot - a=V— ^- =— ^^
2 ^ 1 — cos a sm a
sin a cos yS = | [sin (ct + y8) + sin (a — /8)].
cos « sin /3 = J [sin (a -f- ^) — sin (a ^ /S)] .
cos a cos yS = I [cos (« + /S) + cos (a — 6)] .
sin a sin /3 = — -| [cos (ot + yS) — cos (a — ^)].
sin a cos « = |^ sin 2 a.
cos^ a = 1 (1 + cos 2 a).
sin^ a = 1 (1 — cos 2 cj).
sin a -\- sin p = 2 sin — — ^ cos — --^ •
sin a — sm /3 = 2 cos — ^— ^ sm — --^»
cos a + COS p = 2 cos — —-^ cos — — ^ •
/o o- a-\- S . a— ^
cos a — cos p = — 2 sin — ^ sm — —^
RIGHT TRIANGLES
6^2 _|. ^2 ^ ^,
a + ^ = 90°.
a . ^
- = sin a = cos p.
132
FORMULAS
= cos a = sin y8.
a .
- = tan a
COtyS.
A= ^ ab = ^bcsin a= I c^ sin a cos a = ^ c^ gj^ 2 a.
OBLIQUE TRIANGLES
« + ye + 7=180°.
(? = 5 cos a-\- a cos yS, etc.
a _ h _ <?
sin ct sin yS sin 7
c^ = ^2 _j_ j2 _ 2 a5 cos 7, etc.
tan ^ ^ ^ = cot -, etc.
2 -\- c 2
tan
, etc.
2 s-a
r = ■% /('^ — ^)(g— ^)(g— g)
.A=j^ 5(? sin a = rs = Vs (s — a) (s — 6) (s — (?).
RIGHT SPHERICAL TRIANGLES
sin a = sin <? sin a.
sin 6 = sin c sin /3.
tan 6 = tan c cos a.
tan a = tan <? cos 0,
tan a = sin h tan a.
tan b — sin a tan yS.
cos c = cos a cos 5.'
cos /3 = cos b sin a.
cos a = cos a sin /?.
cos (? = cot a cot /3.
GENERAL TRIGONOMETRY
133
OBLIQUE SPHERICAL TRIANGLES
sin a sin h sin c
sin a sin /3 sin 7
cos (? = cos a cos J + sin a sin 5 cos 7, etc.
cos 7 = — cos a cos y8 + sin a sin y8 cos c, etc.
tan - =
tan r
2 sin (s — a)
, etc.
tan r
_ / sin (g — a) sin {s — h) sin (s — g)
^ sin s
tan - =
a _ cos (^S — a )
, etc.
2 cot Z^
^=l(« + ^ + 7).
cot i2 = J - c<>^ (>S^- «) cos (S-fi) cos (a^- 7)
^ cos aS'
tan 1 (/3 + 7) = ^^^i^l^cot 1 a, etc.
cos -^- (0 + c)
tan 1 (5 - ^) = ^"' 2 (f — ll tan 1 «, etc.
2^ sin 1(^4- 7)
tan 1 (5 + c?) = ^"^t ^^~^^ tan J a, etc.
- V cos 1 (^ - 7)
ANALYTIC TRIGONOMETRY
lim
0=0
lim
0=0
' e 1
_sin ^J
• e -
tan ^
= 1.
= 1.
(cos a-\-{ sin a) (cos ^+ i sin /3) = cos (a + /3) + i sin (a + /3).
(cos « + i sin a)*^ = cos na 4- ^ sin na.
cos a + ^ sin a = e**".
cos « — I sin a = e~**.
COS a = • .
134 ANALYTIC TRIGONOMETRY. CONSTANTS
sm a = — —
2^
cosh a + sinh a = g*.
cosh a — sinh a = e~"-.
cosh « =
sinh a =
2>
iog(i+.)=f-|Vf-|V-.
C0S«=l-- + jj-^+-.
tan « = - + — + — ^ +
1 8 15 315
cosh.= l + - + -+^4
CONSTANTS
7r = 3.14159265 •••.
7r^ = 180°.
l^ = 5T.295779e5° ••• =57° 17'44.8'^...
g = 2.7182818285 ••..
Mode 10 = — ^ = .4342944819 ....
CONSTANTS 135
1 inch = 2.54001 ••• centimeters.
1 foot = .3048 .-. meters.
1 mile= 1.60935 ••• kilometers.
1 centimeter = .3937 ••• inches.
1 meter = 3.28083 feet = 1.09361 yards.
1 kilometer = .62137 miles.
g= 32.086528 + .171293 sin^t^ feet per second per second.
= 9.779886 + .05221 sin^ (^ meters per second per second at
sea level for latitude ^.
ANSWERS TO EXERCISES
(Answers are omitted in case their knowledge would detract from the value of the
exercise.)
Exercise II
6. (0, 0), (a, 0), (a, a), (0, a); (^ V2, o), ("' | Vs), (-|V2,o),
(0,-|V2).
7. (5, 0), (0, - 5), (-4.33, -2.5), (-3.54, 3.54).
9. 5.6569; 7.6158.
11. Cross country distances, in miles : 5.099; 2.828; 2.236; 2.236; 6.325.
12. Distances saved, in yards : 773.2 ; 644.4 ; 1288.7 ; 128.9.
Exercise IV
9.
1681
11.
If.
13. cos
a.
15. Jl-siny
^l + sin y
1519
12.
If.
14. 2csc^.
10.
f.
Exercise V
16. 2(l+tan2y).
9.
h
13.
60°.
17.
(a)
60°; (b) 19.05 ft.
10.
W3.
14.
0° and 60°.
18.
8.08
; 16.17.
11.
i-
15.
60° and 90°.
19.
452.39.
12.
K3V3-
-2).
16.
45°.
20.
60°.
Exercise VI
11. 0° and .60°. 12. 30°. 13. 0°, 30°, and 45°. 14. 0°, 30°, and 45°.
Exercise VII
1. (a) 6.7,6.7; (b) 8.23,4.75. 4. 15.6; 9.
2. (a) 8; (6) 13.86. 5. (a) 2598.16; (b) 1500.
3. (a) 20, 34.64 ; (6) 28.28, 28.28. 6. 24 miles per hour, 30° east of north.
' Article 18
1. a = 40.32, & = 11.76. 3. 6 = 151.5, c = 381.6.
2. a = 20.25, c = 33.75. 4. a = 133.2, b = 149.2.
137
138 ANSWERS TO EXERCISES
Exercise VIII
(These results were ol)tained with four-place tables.)
1. ^ = 64° 50', a = 14.46, h = 30.77.
2. p = 37° 40', a = 57.79, b = 44.61.
3. a = 28° 45', a = 116.88, b = 213.04.
4. a = 11° 25', a = 103.11, 6 = 510.68.
5. (3 = 68° 35', b = 599.13, c = 643.66. •
6. /3 = 17° 15', b = 223.56, c = 753.93.
7. a = 9° 30', b = 7170.96, c = 7272.73.
8. a = 72° 30', a = 4757.40, c = 4988.36.
9. a = 41° 49', /3 = 48° 11', 6 = 268.33.
10. a = 32° 12', (3= 57° 48', 6 = 605.03.
11. a = 34° 13', 13 = 55° 47', a = 354.25.
12. a = 53° 8', ^ = 36° 52', a = 1120.
13. a = 36° 52', ^ = 53° 8', c = 1080.
14. « = 44°46', )8 = 45°14', c = 845.07. '
15. a = 29° 11', ^ = 60° 49', c = 440.94.
16. a = 59° 41', 13 = 30° 19', c = 2445.55.
17. a = 29° 29', (3 = 60° 31', b = 168.00, c = 193.00.
18. a = 41° 04', f3 = 48° 56', a = 230.00, c = 350.13.
19. (3 = 15° 40', a = 93.47, b = 26.21, c = 97.08-.
20. a = 65° 10', a = 60.35, 6 = 27.93, c = 66.50.
21. 200.1 ft. 23. 2° 23'. 25. 1° 9'.
22. 1501.73 ft. 24. 4° 46'. 26. 33° 41', 26° 34', 45°.
27. .134 pitch, .2887 pitch, | pitch.
28. 19° 28' inclination. 31. 26° 31'. 34. 21.73 ft. = 21 ft. 8| in.
29. 8° 3' inclination. 32. 5859.71 ft. 35. 260.4 ft.
30. 0° 9', 0° 17', 1° 26'. 33. 32° 28'. 36. 0° 20'.
37. 5° 54' ( = .1029 radians). Note that .1029 = sin 5° 52.5', an approxima-
tion. See Arts. 72 and 77 (3).
38. 2.468 miles. 39. 502.2 ft.
40. ^=(0,0), 5 = (240.9,0), C= (385.9, 274.8),
D = (98.7, 814.6), E = (162.8, 1043.0), F= (649.1, 1248.8).
41. 13 = 110°, b = 68.81, A = 828.81; ^ = 36°, a = 202.27, A = 12,641.88;
a = 75° 06', p = 29° 48', A = 30,441.6 ; a = 63°, b = 326.88, A = 52,425.01 ;
a = 64° 17', a = 553.12, A = 119,594.88; a = 41° 1', ^ = 97° 58', A = 202,809.6.
42. 51.76, 61.80, 68.40, 76.54, 1, 121.76, 141.42, 173.20.
43.
45.
5 = 2scos?.
4
44.
s = 2Rsin'''' = 2rt.u''''
n n
n
2'7rR
cos 180°.
n
e = 62.83, P4= 56.57, 0^= 44.43;
P8 = 61.23, C8 = 58.05;
P,e = 62.43, c,«= 61.62;
P32 = 62.72, C32 = 62.53.
Ap
^nW^&ui
180° 180° 1 po . 360°
cos = - nR^ sm
n n 2
n
Ai
= tt/^^ cos
n 2 \
360°\
, ^P4
= 200.00,
.4,4 = 157.08;
Ap^
= 282.84,
^,-8 = 268.15;
^P16
= 306.16,
A^Q = 302.21 ;
^P32
= 312.16,
^,32 = 311.14.
^6-
60.00, c,.
= 54.41 ;
^2 =
62.11, c,.
= 60.69 ;
^^24 =
62.64, c.„
= 62.29 ;
^48 =
62.78, c,«
= 62.69.
1, ^^6 = 259.80,
J,g= 235.62;
Aj,^^ = 300.00,
.4 ,-,2 = 293.11;
^,24 = 310.56,
^/ = 308.80;
A,^ = 313.20,
^,-^8 = 312.81.
ANSWERS TO EXERCISES 139
46. Ac=7rR%
^,. = 314.16,
47. C = 62.83,
48. ^. = 314.16
49. Z = -30, F=- 17.321, 22 = 34.641, 30° south of west.
50. A' = 6 r, F = 0, R = 6 r, due east.
51. Distance from center = r cos 0. 52. x= — 15.
53. Component along plane = g sin a, component perpendicular to plane
= g cos a.
54. 16, 27.71; 8.28, 30.91; 5.56, 31.51; 2.79, 31.88.
55. 50 pounds pressure, 141.42 pounds along ladder.
56. Z = 57.28, y= 30.73. 57. i2 = 18.44, ^ = 49° 24'. 58. c= 11.99.
Exercise IX
1. 4,1.5, -2.5. 4. 1,8, 16, i,^V
2. 4, if. 5. (a) .4724; (b) .01614.
3. 1, 4, 16, 32, 64, tV, ^?- 6. (a) 28.16; (b) .01913; (c) 2.465.
7. 17.978. 9. 524.9. 11. a: = 1.79. 13. $4136.09. 15. 11.6 years.
8. .76252. 10. a; = 2.29. 12. $4656.20. 14. 5.2%. 16. 3.8 years.
Exercise X
1. 4.86024, 2.79187, 9.84198, 5.80872 - 10, 21.47712.
2. 4.96088, 1.15518, 11.50651, 5.89510 - 10, 24.30103.
3. 516.35, 4.0966 x 10^2, .016335.
4. 16361, 5.64325 x 10", .00013671.
5. 9,067,800,000. 7. 88.594 cm. 9. 13,231 x lO^o.
6. 7.0048 X 1010 cm. 8. 71.68 cm. 10. 2,754,100.
11. 9.63459 - 10, 9.52928 - 10, 0.01824. 13. 78° 01.1', 81° 43.7', 76° 17.1'.
12. 9.97454 - 10, 9.78340 - 10, 0.04197. 14. 25° 20.7', 27° 32.6', 35° 3.6'.
15. 13.861. 16. .91186. 17. 3.9968. 18. .38875.
19. 1.3365 inches. 22. .074765.
20. .1111 foot (=1.3332 inches). 23. 6:711; 8.381.
21. 1.7% less than the true value. 24. - 11.85; - 61.38.
140
ANSWERS TO EXERCISES
Exercise XII
1. ^ = 27^ a = 2302.3, h = 1173.1.
2. cc = 60" 37.6', /3 = 29'^ 22.4', h = 4238.9.
3. ^ = 14° 44.8', b = 254.07, c = 998.12.
4. a = 15° 39.6', ^ = 74° 20.4', b = 168.36, c = 174.85.
5. a = 50° 13.1', (3 = 39° 46.9', c = 9.5378.
6. ^ z:. 71° 34.5', a = 10.417, ^> = 31.271.
7. a = 83° 38.4', y8 = 6° 21.6', 6 = 14.82, c = 133.79.
8. a = 75° 33', ^ = 14° 27', c = 54.953.
9. a = 64° 48.5', /S = 25° 11.5', 6 = 31,037.
10. ;8 rr 60° 9.8', a = 5.854, c = 11.766.
11. /? = 64° 42.6', a = 19.023, b = 40.264, c = 44.531.
12. a = 28° 23.6', (3 = 61° 36.4', c = .00042.
13. a = 26° 47.3', a = 3.2196, & = 6.3769.
14. a = 38° 23.3', ft = 51° 36.7', a = .056677.
15. a = 54° 43.6', b = .44535, c = .77120.
16. a = 54° 43.2', a =: 242.79, b = 343.16, c = 420.37.
17. a = 55° 59.3', 13 = 34° 0.7', c = .0074192.
18. a := 9° 47.5', a = .89928, c = 5.2878.
19. a = 63° 20.7', /3 = 26° 39.3', a = .014523.
20. a = 64° 41.8', a = 1563.4, 6 = 739.12.
21.
22.
8.2583 feet.
18 feet 3.8 inches.
23. 141.42 square feet.
24. 60° 1.8'.
25. 1237.8 feet.
26. 16| miles, 36° 52.2' north of west.
29. nR^ sin'-^cos'^-
n n
30. 2177.4.
31.
32.
33.
34.
35.
36.
37.
38.
178.8 miles.
0° 32'.
.078523 feet.
14834 feet.
425.64 feet.
142.4 feet.
118.1 feet.
554.06; 145.17.
Article 47
1. a = 33° 19.9', a = 438.23, c = 788.58.
2. a = 65° 49.8', a = 122.13, b = 885.60.
3. (3 = 15° 57.0', b = 5.442, c = 17.865.
— 4. ^ = 1° 02.0', a = 9.368, b = .18134.
Article 48
3. a = 57° .59.9', y = 23° 36.6', c = 29.526.
4. ^ = 13° 55.6', y = 35° 30.7', b = 135.96.
^ (a = 104° 31.3', 13 = 40° 2.9', a = 5889.9 ;
^ - 4° 37.1', ^' = 139° 57.1', a' = 489.8.
94° 17.9', y = 47° 13.3', a = 207,810;
6. ^
(a
[ a' = 47° 4.6', y' = 132° 46.7', a' = 152,600.
ANSWERS TO EXERCISES 141
Article 49
1. ^ = 23° 42.8', y =3o° 45.2', a = 450.35.
2. a = 23° 31.8', y = 19° 7.2', h = 818.54.
3. a = 33° 17.5', y = 63° 13.1', b = .11496.
4. ^ = 66° 27.0', y = 45° 11.2', a = .005202.
Article 50
66=
'49.4',
y =
z6r
*13.4';
A =
1.9181 X
101
42°
'51.8',
y =
= 54=
•51.8';
A =
4.4175 X
10^.
34°
45.4',
7 =
= 88^
'45.8'.
72=
' 33.2',
^y--
= QV
^33.4'.
r?
7.
Impossible.
Why?
8.
y = 8<
' 58.3'
'.
1. a = 51° 57.2', y8
2. a = 82° 16.4', /3
3. a = 56° 28.8', ^
4. « = 45° 53.4', 13
5. Impossible. Why?
6. IS= 136° 39.8'.
Exercise XIV
1. /8 = 74° 0.3', y = 43° 24.7', a = 76,568.
2. a = 52° 56.6', y8 = 79^^ 47.8', y = 47° 15.6'.
3. y = 86° 10.3', b = 8.4172, c = 9.0436.
4. a = 43° 29.3', y = 80° 35.3', 6 = .30470.
5. Impossible. Why?
6. a = 52° 11.2', y = 27° 38.8', a = 49,921.
7. a = 34° 32.1', ^ = 51° 41.8', y = 93° 46.1'.
8. y = 69° 28.5', a = 67,439, c = 72.037.
9. a = 23° 34.1', jS = 35° 35.7', c = 6.0804.
10. a = 15° 35.2', y = 126° 7.6', c = 66.113.
11. (3 = 13° 11.7', y = 16° 24.1', a = .082764.
I ^ = 32° 8.5', y = 89° 45', b = 34.993 ;
■ i /?' = 31° 38.5', y' = 90° 15', b' = 36.210.
13. a = 34° 11. .5', a = 382.48, c = 641.52.
14. a = 28° 57.0', /? =: 104° 28.6', y = 46° 34.4'.
15. a = 21° 13.9', y = 32° 19.7', b = .0048578.
16. a = 162° 18.9', y = 7° 08.3', a = 61.896.
17. a = 33° 33.1', /? = 50° 42.0', y = 95° 44.9'.
18. a = 75° 0.2', a = 8355.2, b = 6470.6.
19. a = 45° 29.5', /S = 14° 15.5', 6 = 2146.7.
20. a = 49° 36.8', ^ = 40° 23.2', c = 952.67.
21. a = 151° 56.6', /8 = 4° 30.4', y = 23° 38.0', b = 416.45.
22. a = 80° 0.0', ^ = 54° 45.2', y = 45° 14.8', a = 124.81.
23. /? =: 90° 50', y = 16° 0', a = 720.81, c = 207.58.
24. a = 95° 26.6', y = 27° 8.4', a = 125.81, b = 106.49.
25. 2.1815 X 109; 5.0105; 1,742,040,000.
26. 2.567 X 109; .038051; 7270.3.
27. Case III.
142
ANSWERS TO EXERCISES
28.
29.
30.
a = 15° 45', yS
a = 2r 52.6 , fi
a = 91° 54.7', 13
a = 47° 59.5', (3
r A = 156° 55.6', B = 145° 57.2', C = 57° 7.2' ;
t ^ = 145° 13.6', B = 121° 11.4', C = 93° 35.0'.
29° 15',
c = 52.1 ;
42° 7.4',
c = 723.6;
53° 5.3',
c = 43.042 ;
72° 0.5',
c = 291.38.
(A
31. ^
63° 25.9', B = 141° 34.1', c = 328.4 ;
122° 25.4', C = 137° 34.6', c = 575.41.
75°, & = 878.48, cz= 621.17;
L A = 113° 58.6', B = 106° 2.2', C = 139° 59.2'.
32. a = h = 48.5.
33. ^ = 151° 2.7', B = 133° 25.9', C = 75° 31.4'.
34. d^ = 0.2 + ^2 + c2 _ 2 a& cos ah -2 be cos kr + 2 ac cos (a6 + k-) ; d = 12.98
35. be = 84° 03.5', cS = 75° 53.0', (/a = 82° 03.5', ab - cd = 109.28,
6c - da = 114.47.
36. c = 1001.1, (/ = 568.6. 37. « rz 36°, s = 15.217.
„„ 6 -
38. V =
6
30 (6 - h) \/l2 A - A- : F^ = 135.0, V^ = 371.
F3 = 663.3, F4 = 990.0, F5 = 1338.0, F^ = 1696.4, F^ = 2054.8, Fg = 2402.8,
F9 = 2729.5, Fio = 3021.1, F„ = 3257.8, V^^ = 3392.8.
39. 6 = 483.4. 40. a = 3221.5. 41. b = 1286.
42. Distance = 31.63, total height = 20.97.
43. Distance = 24.24, height = 5.08.
44. AD = 738.2, DB = 150.6.
45. AC = 1075.1, BC = 679.5.
46. ^D = 1460, DC = 678, angle BXC at left = 17° 27.5'.
47. Distance = 135.74, height = 36.602.
48. D = 57° 40', CD = 196.73, BD = 233.55.
49. AD = 603.94, ^C = 693.12, BC = 838.82, 5^ = 595.76, AB = 867.48,
angle CXB at left = 4° 27.7'.
50. AC = 730.17, ^Z> = 737.37, BE = 805.40, BF = 715.52, ^5 = 841.67.
51. 39° 54'.
Exercise XV
1. 45°, 60°, 150°, 112° 30', 171° 53' 14.4", 42° 58' 18.6".
2.
77-^ TT^ TT^ 27r^ ilT^ 5^^ StT^
6'12'4'3'.3'3'2*
3. 31.416 cm., 62.832 cm., 125.664 cm., 47.124 cm.
4. 1^, l\ 2^ ^^
2 2
5. Smaller sprocket : 4 revolutions per second, angular velocity = 8 tt radi-
ans per second, linear velocity of circumference = 201.06 inches per second;
167r
larger sprocket: f revolutions per second, angular velocity
3
radians per
second, linear velocity of circumference = 201 .06 inches per second. Speed of
machine = 20.944 feet per second = 14.28 miles per hour.
ANSWERS TO EXERCISES 143
6. Linear velocity of chain and of circumferences of both sprockets = 75.43
inches per second ; angular velocity of larger sprocket = 15.09 radians per sec-
ond, of smaller sprocket = 37.76 radians per second ; smaller sprocket makes
5.1 revolutions per second.
13.
I<«<V^-
14.
0<«<^, '{<a
<
3 7r ^^^Stt 37r^^^'
Exercise XVI
4 *
5.
(i'l>- ■
- (!■)■ (V
, -1), etc.
6.
(-;,l).etc.
' fe f )• {
5 TT V2\ ,
Exercise XVII
1.
f.
7. 120°, 150°, 300°, 330°.
2.
l4-2\^
3
8. 15°, 75°, 135°,
9. 60°, 300°.
195°, 255°, 315°.
3.
5
10. 30°, 150°, 210<
^ 330°.
4.
-If-
11. 0°, 60°, 180°, 300°, 360°.
5.
60°, 300°.
12. 0°, 150°, 180°,
210°, 360°.
6.
120°, 300°.
Exercise XVIII
1.
- sin 20°.
9.
0,0.
22. 0.
2.
3.
- sin 10°.
cot 14°.
10.
1 + V3 2V3
2 ' 3 •
23. - .35.
24. -1.
4.
5.
6.
7.
- cot 35°.
- CSC 30°.
sec 40°.
1 + V3 2V3
2 ' 3 •
11.
12.
19.
V3 - 1 2V3
2 ' 3 •
0,0.
1 + V3
4
25. + VI - a\
26. ^l--^
m
27. sin«.
28. — sin a.
8.
1 + V3 2V3
2 ' 3 •
20.
21.
0.
0.
29. tan a.
30. - tan a.
Exercise XIX
5.
6.
7.
a + ^ = sin-i If = cos-i - |1,
cc + ft z= arcsin — ||f = arccos
-sin(a + y8). 9.
II quadrant.
— Iff, III quadrant,
sin 2 a.
11. sin 2 6.
8.
cos (a + y8).
10
. cos 2 a.
12. cos^.
13.
105°- arcsin ^" +
4
V6
1 \/2 - V6
= arccos ;
4.
15=
' =
105° 90" = arcsin^ 7^ =
4
\/6+V2
arccos J
144 ANSWERS TO EXERCISES
14. 75° = arcsm^ + ^^ = arccos^-^:
15° = 90° - 75° = arcsin^-^ = arccos^^ + ^
= arccos-
4 4
Exercise XX
11. tan 15° = 2 - V3, cot 15° == 2 + V3.
15. sin (a + /8 + y) = sin a cos y8 cos y + cos a sin y8 cos y + cos a cos jS sin y
— sin a sin y8 sin y.
16. cos (a+ /3 + y) = cos ct cos ^ cos y — sin a sin y8 cos y — sin a cos ^ sin y
— cos a sin /? sin y.
T „ , . , a , N tan a + tan B + tan v — tan « tan (3 tan v
17. tan (a-\-a-\-y)=: ^^ ? 1- ti _L_.
1 — tan ytJ tan y — tan y tan a — tan a tan /3
18. cot (a + B+y)= ^ot^^oty + cotycotg+cotctcoty ^
'"^ cot ct cot /? cot y — cot (/ — cot /5 — cot y
Note the symmetry in the last four formulas.
Exercise XXI
5. 2 a =: sin-i yV/o "= cos-i ifif.
6. 2 a = arcsin ± ilg = arccos — ^i|. Explain the signs.
7. sin ^ a = ± f and ± f , cos J =i ± f and ± f.
a s4«=.^^^ ana ±i^:eoa„..l^- and .^.
9. sin (a + 2 ;8) = f|f and - ||f, cos (a + 2 ;g; .. -^%% and - |04.
10. sin (a- 2 ft) =± |f | and T ff |, cos (a - 2 ^) = T^W and T |tf.
17. a = 30°, 45°, 60°, 210°, 225°, 240°. 18. a = 90^ 270°, and I arccos f .
19. a = 67° 30', 157° 30', 247° 30', 337° 30', and ^ arctan f .
20. a = 90°, 270°, and i cos-i f. 23. 2 x. 24. 1. 25. 0. 26. 1.
Exercise XXII
1. 1 [sin 8 a + sin 2 a]. 8. ^ [3 - 4 cos 2 a + cos 4 a].
2. ^ [sin 10 a - sin 2 a]. 9. -^ [3 sin 2 a - sin 6 a].
3. ^ [cos 4 a - cos 10 a]. 10. ^^ [1 - cos 4 a].
4. 1 [cos 7 « + cos 3 a]. ^5^ ;t . J; [/t = 0, 1, 2, 3, 4].
5. i [cos a - cos 3 a] . 4 > > » > j
6. K2sin2a + sin4a]. • royfc+n!!:. p/t - 1 9 31
7. i[3 + 4cos2a+cos4a]. ^^' ^"^ + ^^4' L^-«'1'-^J-
17. (2^+1)^; [/j=0, l,2,...ll]and(3A:+l)|; [A: = 0, 1, 2, 3].
18. (2A: + 1)^; [^ = 0, 1,2, ...29]and(2^-+l)^; [^ = 0, 1, 2, ...9].
19. A:.|; [A: = 0, 1, 2, ...7]. 20. ^.|; [^^ = 0, 1, 2, ... 7].
ANSWERS TO EXERCISES 145
Exercise XXIII
9. (2A: + 1)^; [/^ - 0, 1, 2, ...]. 13. ^0^3 a - 3 cot oc .
"^ ^6^ ' ' ' J 3cot2a-l
kir ^n kir -. ^ 3 tan a — tan^ a
10. — . 11. ^. 14.
3
12. TT and (2^ + 1)
3 4 1-3 tan'-^ a
15. 4 sin a cos^ a — 4 sin^ a cos a.
16. 8 cos* « - 8 cos2 a - 3.
17. 0°, 15°, 105°, 180°, 255°, 345^. (See Exercise XIX, examples 13 and 14.)
18. a = 60°, 90°, 120°, 270°, and arcsin ^. 19. — • 20. ^.
2V3 2 2
Article 72
1. - sin 0. 2. sec tan 0. 3. - esc cot ^. 4. sec^ 6. 5. - csc^ (9.
Article 73
5. cos^-+ I sin——; ±1; 1, ^ ; ±1, ±i; ±1, ^
_ (2;t + l)7r, . . (2^+l)7r , . . 1± V^^
6. cos ^^ — 1^ I sin -^^ ^— ; ±i: — 1, :
n n 2 '
± 1 ±« . ± V3±i
— Ti — ; ±h — 7^
Article 74
6. Products: 15 + 6i; -5 + 3t; 6 + 12 {; _8-20^; 12-18/; 4+22i;
4 + 22 i.
7. Quotients: 5-3*; 3 - 2 i. 8. Results: 4 + 12i; 1; - 8; 1.
9. Roots: 4 - 3 ^ 3 + 2 t; - 46 + 9 i ; ±21
10. ^/a [cos ?^ + i sin ?^] ; [>l- = 0, 1, 2, ... (n - 1)].
11. ±1; ±i; 1, ^\^ '^ 2, -l±V-3.
INDEX
[Eeferences are to articles, except where otherwise indicated.]
Addition formulas, 63, 64.
Ambiguous case of oblique triangles, 48.
Angle, general definition of, 52.
Angles, positive and negative, 3.
Answers to exercises, page 135.
Area, laws for
oblique triangles, 45.
right triangles, 17.
Checks, 20.
Common logarithms of numbers. Table I,
page 3.
Complementary angles, functions of, 10.
Complex numbers, graphical methods of
representation and combination, 74.
Composition of forces, 51.
Conversion formulas for products, 69.
Conversion formulas for sums and differ-
ences, 70.
Coordinates, 4.
Definitions of the trigonometric functions,
32, 54.
De Moivre's theorem, 73.
Directed rectilinear segments, 2.
Drawing instruments, 1.
Equilibrium of forces, 51.
Errors, 20.
Exponential values of the trigonometric
functions, 75.
Formulas for tan (a:t0), cot (a i |3), 66.
Formulas, list of, page 130.
Functions of 0°, 90°, 12.
of 180°, 36.
of 270°, 360°, 58.
of 30°, 45°, 60°, 11.
of (90° + a), 38.
of (^~±a],62.
of half an angle, 68.
of twice an angle, 67.
Fundamental relations between the func-
tions of a single angle, 9, 34, 59.
General inverse functions, 80.
Graphs of the trigonometric functions,
57.
Greek alphabet, page x.
Half an angle, functions of, 68.
Hyperbolic functions, 76.
Infinity, definition of, 12, 35, 58.
Inverse functions, logarithmic values of,
81.
Inverse trigonometric functions, 14.
Law for angles in terms of sides, 44.
Law of cosines, 42.
Law of projections, 40.
Laws of sines, 41.
Law of tangents, 43.
Laws of area,
oblique triangles, 45.
right triangles, 17.
Laws for solution of
oblique triangles, 39-44.
right triangles, 16.
Limitations in value of the trigonometric
functions, 8, 33, 55.
Limits of 6»/sin e and 0/ta.n e, 72.
Line representations of the trigonometric
functions, 60.
List of formulas, page 130.
Logarithms,
characteristic, 24.
cologarithms, 28.
common system, 23.
definition of, 21.
interpolation, 26.
laws of combination, 22.
mantissa, 25.
numbers from logarithms, 27.
of numbers, Table I, page 3.
of trigonometric functions. Table 1\
page 25.
Method of solution of triangles, 18.
Multiple angles, 71.
147
148
INDEX
Natural trigonometric functions, Table
III, page 71.
Oblique triangles,
area, 45.
laws for solution, 3f)-44.
Oblique triangles, solution of, 46-50.
Orthogonal projection, 4 (note), 15.
Periodicity of the trigonometric functions,
61.
Proportional parts, theory of, 79.
Purpose of trigonometry, 5.
Relation between the ratios and the
angle, 7.
Resolution of forces, 51.
Right triangles,
area of, 17.
laws for solution, 16.
solution by logarithms, 31.
solution by natural functions, 18.
Series, exponential, logarithmic, trigono-
metric, hyperbolic, 77.
Signs of the trigonometric functions, 8, 33,
55.
Slide rule, 30.
Solution of oblique triangles, 46-50.
Solution of right triangles,
by logarithmic functions, 31.
by natural functions, 18.
Squares of numbers. Table IV, page 91.
Subtraction formulas, 65.
Supplementary angles, functions of, 37.
Table I. Common Logarithms of Num-
bers, page 3.
Table II. Logarithms of the Trigonometric
Functions, page 25.
Table III. Natural Trigonometric Func-
tions, pagre 71.
Table IV. Squares of Numbers, page 91.
Trigonometric functions, definitions of,
for acute angles, 6.
for obtuse angles, 32.
for the general angle, 54.
logarithms of, Table II, page 25.
natural, Table III, page 71.
Trigonometric tables, pages 1-93.
computation of, 78.
description of, 19.
Trigonometry, purpose of, 5.
Twice an angle, functions of, 67.
Variations of the trigonometric functions,
13, 35, 56.
TABLES
-/
(-
TABLE I
COMMON LOGARITHMS
OF NUMBERS
N.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Lo^.
Infinity.
o.oo ooo
0.30 103
0.47 712
0.60 206
0.69 897
0.77 815
0.84 510
0.90 309
0.95424
1.04 139
1.07 918
I. II 394
1. 14 613
1. 17 609
1.20 412
1.23045
1.25 527
1.27875
1.30 103
I .32 222
I 34 242
1 36 173
1 . 38 02 1
1.39 794
1. 41 497
1.43 136
1.44 716
1 . 46 240
N.
1.47 712
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Log.
1.47 712
.49 136
.50515
.51851
•53 148
.54 4-'7
• 55630
. 56 820
. 57 978
.59 106
60 206
.61 278
.62 325
.63 347
.64345
.65 321
.66 276
.67 210
.68 124
.69 020
,69 897
.70757
.71 600
.72 428
.73239
.74036
.74819
•75 587
76343
.77085
1.77 815
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
Log.
1.77 815
78 533
.79239
.79934
.80618
.81 291
.81 954
. 82 607
.83251
.83885
.84 510
.85 126
.85733
.86332
.86923
.87 506
.88081
.88649
. 89 209
.89763
90309
90849
91 381
91 908
92 428
92942
93450
93952
94448
94 939
90 1.95424
3
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
Log.
1.95424
1.95 904
1.96 379
1 . 96 848
1.97 313
1.97 772
1.98 227
1.98677
1 .99 123
1.99 564
2 . 00 000
2.00 432
2 . 00 860
2.01 284
2.01 703
2.02 119
2.02 531
2.02 938
2.03 342
2.03743
2.04139
2.04 532
2.04 922
2.05 308
2.05 690
2.06070
2.06446
2.06 819
2.07 188
2.07 555
N.
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
120 2.07918 160 2.17609
Log.
2.07 918
2.08 279
2.08 636
2.08 991
2.09 342
2.09 691
2.10 037
2.10 380
2.10 721
2. 11 059
2. II 394
2. 1 1 727
2.12 057
2.12 385
2.12 710
2.13033
2.13354
2.13672
2.13988
2.14301
2. 14 613
2.14 922
2.15 229
2.15 534
2.15836
2.16 137
2.16435
2.16 732
2.17 026
2.17 319
TABLE I
N.
100
01
02
03
04
05
06
07
08
09
110
11
12
13
14
15
16
17
18
19
120
21
22
23
24
25
26
27
28
29
130
31
32
33
34
35
36
37
38
39
140
41
42
43
44
45
46
47
48
49
150
N.
O
oo ooo
432
860
01 284
703
02 119
531
938
03 342
743
043
04 139
532
922
05 308
690
06 070
446
819
07 188
555
918
08 279
636
991
09 342
691
10 037
380
721
11 059
394
727
12 057
385
710
13 033
354
672
988
I4_30i^
613
922
15 229
534
836
16 137
435
732
17 026
319
609
475
903
326
745
160
572
979
383
782
179
571
961
346
729
108
483
856
225
591
087
518
945
368
202
612
*oi9
423
822
218
954
314
672
*026
377
726
072
415
755
093
428
760
090
418
743
066
386
704
*oi9
333
644
953
259
564
866
167
465
761
056
348
638
610
999
385
767
145
521
893
262
628
990
350
707
*o6i
412
760
106
449
789
126
461
793
123
450
775
098
418
735
♦051
364
130
561
988
410
828
243
653
*o6o
463
862
258
650
♦038
423
805
183
558
930
298
664
*027
386
743
*o96
447
795
140
483
823
160
494
826
156
483
808
130
450
767
*o82
395
675 706
983 *oH
290
594
897
197
495
791
085
377
667
320
625
927
227
524
820
114
406
"696
73
604
030
452
870
284
694
*ioo
503
902
297
689
*077
461
843
221
595
967
335
700
063
422
*I32
482
830
175
517
857
193
528
860
189
516
840
162
481
799
*ii4
426
737
'045
351
655
957
256
554
850
143
435
725
217
647
♦072
494
912,
325
735
'141
543
941
689
*ii5
536
953
366
776
*i8i
583
981
336
727
500
881
258
633
'004
372
737
^099
458
814
*i67
517
864
209
551
890
227
561
893
222
548
872
194
513
830
*i45
457
768
♦076
381
685
987
286
584
879
173
464
754
260
732
*i57
578
995
407
816
*222
623
021
376
766
*I54
538
918
296
670
*04i
408
773
35
493
849
552
899
243
585
924
261
594
926
254
581
905
226
545
862
*I76
489
799
*io6
412
715
*oi7
316
613
909
202
493
782
303
415
805
*I92
576
956
333
707
*o78
445
809
*i7i
529
884
'237
587
934
278
619
958
294
628
959
287
613
937
258
577
893
^208
520
829
*I37
442
746
*o47
346
643
938
231
522
"sTT
S 9 Prop. Pts.
346
775
*I99
620
♦036
449
857
*262
663
*o6o
454
817
♦242
662
=078
490
898
*302
703
ICX)
493
844
♦231
614
994
371
744
= 115
482
846
*207
565
920
'272
621
968
312
653
992
327
661
992
320
646
969
290
609
925
*239
551
860
*i68
473
*o77
376
673
967
260
551
840
389
883
♦269
652
♦032
408
781
*i5i
518
882
=1=243
600
955
*307
656
*oo3
346
687
*02 5
361
694
*024
352
678
*OOI
322
640
956
*270
582
"89?
♦198
503
806
*io7
406
702
997
289
580
9
44
43
4*
4-4
4 3
4
8
8
8
6
8
13
2
12
9
12
17
6
17
2
16
22
21
5
21.
26
4
25
8
25-
30
8
30
I
29
35
2
34
4
33
39
6
38
7
37-
41 40 39
12 o
16 o
20 o
61240
7 28.0
8 32.0
9I36.0
37
3-7
7-4
II. I
14-8
18.5
22.2
25 9
29.6
33-3
3 9
7
II 7
15 6
19 5
23 4
27 3
31-2
35 »
36
3-6
7-
35
34
33
3-5
3 4
3
7
6.8
6.
10
5
10 2
9
14
136
13
17
5
17.0
16.
21
20.4
19
24
5
238
23
28
27 2
26.
31
5
30.5
29-
3a 31 30
16.0
19 2
22.4
25.6
28-8
3-0
6.C
9.0
12 o
15 5 »50
18 6 18 o
21 7
24 8
27 9
21 .0
24 o
27.0
Prop. Pts.
LOGARITHMS OF NUMBERS
N.
O
1
2
3
4
5
6
y
S 1 9
Prop. Pts.
150
51
17 609
898
638
926
667
955
696
984
725
*oi3
754
782
811
840
869
*04i
*o7o
*o99
*I27
*i56
29
28
52
iS 184
2n
241
270
298
327
355
3^4
412
441
53
469
498
526
554
583
611
639
667
696
724
1
2
2.9
5 8
2.8
5.6
54
752
780
808
«37
865
«93
921
949
977
*oo5
3
8.7
8.4
55
19033
061
089
117
145
173
201
229
257
285
4
n.6
11.2
56
312
340
368
396
424
451
479
507
535
562
5
14.5
14.0
57
590
618
645
673
700
728
756
783
811
838
6
17.4
16.8
58
893
921
948
976
*oo3
♦030
*o58
*o85
*II2
'/
20.3
19.6
59
160
61
20 140
167
194
222
249
276
303
330
602
358
629
385
656
8
9
23.2
26.1
27
22.4
25.2
26
412
683
439
466
493
520
548
575
710
737
763
790
817
844
871
898
925
62
952
978
*oo5
*032
*o59
*o85
*II2
*I39
*i65
*,92
1
2.7
5 4
2.6
5.2
63
21 219
245
272
299
325
352
378
405
431
458
64
484
511
537
564
590
617
643
669
696
722
3
8.1
7.8
65
748
11^
801
827
854
880
906
932
958
985
4
10.8
10.4
66
22 on
037
063
089
115
141
167
194
220
246
5
13.5
13.0
67
272
298
324
350
376
401
427
453
479
505
6
7
8
9
16.2
18.9
21.6
24 3
15.6
18.2
20.8
9.^ A.
68
531
557
5«3
608
634
660
686
712
737
.7^3
69
170
71
789
814
840
866
891
917
943
,968
994
*oi9
23 045
070
096
121
147
172 ' 198
223
249
274
300
32s
350
376
401
426
452
477
502
528
25
72
553
603
629
654
679
704
729
754
779
1
2 5
73
805
830
«55
880
905
930
955
989
*oo5
*o3o
2
5.0
74
24 055
080
los
130
155
180
204
229
254
279
3
7.5
75
304
329
35^
V^
403
428
452
477
502
527
4
10.0
76
551
576
601
625
650
674
699
724
748
773
5
6
7
12.5
15.0
17.5
77
797
822
846
871
895
920
944
969
993
*oi8
78
25 042
066
091
115
139
164
188
212
237
261
8
20.0
79
180
81
285
527
768
310
334
358
3«2
406
^648^
888~
431
672
912
455
479
503
9
22.5
551
575
600
624
864
696
720
744
983
. . 1
792
816
840
935
959
224:
23
82
26 007
031
055
079
102
126
150
174
198
221
1
2.4
2.3
83
245
269
293
316
340
3^4
3^7
411
435
458
2
4.8
4.6
84
482
505
529
553
576
600
623
647
670
694
3
7.2
6.9
9.2
85
717
741
764
788
811
8s4
858
881
90s
928
4
9.6
86
951
975
998
*02I
*o45
*o68
*o9i
Hl^
*i38
*i6i
5
6
12.0
14 4
11.5
13.8
87
27 184
207
231
254
277
300
323
346
370
393
7
16.8
16.1
88
416
439
462
485
S08
531
554
577
600
623
8
19.2
18.4
89
190
91
646
669
692
715
738
761
989
217
784
807
830
852
9
21.6
20.7
J71
28 103
898
126"
921
149
944
967
*OI2
*o35
*o58
*o8i
.. .. 1
171
194
240
262
285
307
22 Zi
21
92
330
353
375
398 421
443
466
488
511
533
1
2.2
2.1
93
556
578
601
623
646
668
691
713
735
758
2
4.4
4.2
94
780
803
825
847
870
892
914
937
959
981
3
4
b.b
8 8
6.3
ft 4
95
29 003
026
048
070
092
115
137
159
181
203
;=;
n fi
10.5
12.6
96
226
248
270
292
314
336
35«
380
403
425
6
13.2
97
447
469
491
513
535
557
579
601
623
64s
7
15.4
14.7
98
667
688
710
732
754
776
798
820
842
863
8
17.6
16.8
99
200
885
30 103
907
929
951
973
994
211
*oi6
*o38
*o6o
*o8i
9
19.8
18.9
125
146
168
190
233
255
276
298
1
2
3
4
5
6
7
« «|
Prop. Pts.
TABLE I
N.
1
2
3
4
5
6
7
8
9
Prop. Pt8.
200
01
02
03
30 103
320
535
750
12?
146
168
190
211
233
255
276
298
341
557
771
363
578
792
384
600
814
406
621
835
428
643
856
449
664
878
471
685
899
492
707
920
5'1
728
042
1
2
3
4
5
22
2.2
4.4
6.6
8.8
11.0
21
2.1
4.2
6.3
8,4
10.5
04
05
06
963
31 175
387
984
408
*oo6
218
429
*027
239
450
*048
260
471
*o69
281
492
♦091
302
513
*II2
323
534
♦133
345
555
*I54
366
576
07
08
09
210
11
12
13
597
32 015
618
827
035
639
848
056
"263-
660
869
077
284
681
890
098
305
702
118
723
931
139
744
952
160
765
973
181
785
994
201
6
7
8
9
13.2
15.4
17.6
19.8
12.6
14.7
16.8
18.9
222
243
325
346
366
387
408
'613
818
*02I
428
634
838
449
654
858
469
675
879
490
695
899
510
715
919
531
736
940
756
960
572
777
980
593
797
*OOI
1
20
2.0
4.0
14
15
16
33 041
244
445
062
264
465
082
284
486
102
304
506
122
526
143
546
163
365
566
183
385
586
203
224
425
626
3
4
5
6.0
8.0
10.0
17
18
19
220
21
22
23
646
846
34 044
666
866
064
686
885
084
706
90s
104
726
925
124
746
945
143
766
965
163
786
985
183
806
*oo5
203
826
*02 5
223
6
7
8
9
1
2
12.0
14.0
16.0
18.0
19
1.9
3.8
242
439
635
830
262
282
301
321
341
361
380
400
420
616
811
*oo5
459
850
479
674
869
498
518
908
537
733
928
557
753
947
577
772
967
596
986
24
25
26
35 025
218
411
044
238
430
064
257
449
083
276
468
102
488
122
315
507
141
334
526
160
353
545
180
372
564
199
392
583-
3
4
5
6
7
8
9
5.7
7.6
9.5
11.4
13.3
15.2
17.1
27
28
29
230
31
32
33
603
984
622
813
*oo3
641
832
*02I
660
851
*o4o
679
870
*o59
698
889
*o78
717
908
*o97
736
927
*ii6
755
946
*i35
774
965
*I54
36 173
192
211
229
248
267
286
305
324
342
.. 1
361
549
736
380
568
754
399
586
773
418
605
791
436
624
810
642
829
474
661
847
493
680
866
511
698
884
530
717
903
1
2
1»
1.8
3.6
34
35
36
922
37 107
291
940
125
310
959
144
328
977
162
346
996
181
365
*oi4
If.
*033
218
401
*o5i
236
420
♦070
254
438
*o88
273
457
3
4
5
6
5.4
7.2
9.0
10 8
37
38
H9
240
41
42
43
475
658
840
493
676
858
1"
694
876
530
712
894
548
731
912
566
749
931
767
949
603
967
621
803
985
639
822
*oo3
7
8
9
1
2
12.6
14.4
16.2
17
1.7
3.4
5.1
6.8
8.5
10.2
38 021
039
057
075
093
112
292
471
650
130
148
166
184
202
561
220
399
578
238
417
596
256
614
274
632
489
668
328
507
686
346
525
703
364
543
721
44
45
46
739
39 094
757
934
III
775
952
129
792
970
146
810
987
164
828
*oo5
182
846
*023
199
863
^^04 1
217
881
♦058
235
899
♦076
252
3
4
5
6
47
48
49
250
270
445
620
287
463
637
305
480
655
498
672
340
515
690
358
533
707
375
550
724
III
742
410
585
759
428
602
777
7
8
9
11.9
13.6
15.3
794
811
829
846
863
881
898
915
933
950
N.
1
2
3
4
5
6
7
8
9
Prop. Pts.
LOGARITHMS OF NUMBERS
7
N.
O
1
2
3
4
5
6
7
8
9
Prop. Pts.
250
51
52
52
39 794
811
829
846
863
881
898
915
933
950
967
40 140
312
985
157
329
♦002
175
346
♦019
192
364
*037
209
381
*o54
226
398
*o7i
243
415
*o88
261
432
*io6
278
449
*I23
295
466
1
2
3
4
5
18
1.8
3.6
5.4
7.2
9
54
55
56
483
654
824
500
671
841
518
688
858
535
705
875
552
722
892
569
739
909
586
756
926
603
773
943
620
790
960
637
807
976
57
58
59
260
61
62
63
993
41 162
330
497
*OIO
179
347
*027
196
363
*o44
212
380
l47
*o6i
229
397
564
*o78
246
414
581
*o95
263
430
*iii
280
447
*I28
296
464
*i45
313
481
647
6
7
8
9
1
2
10.8
12.6
14.4
16.2
17
1.7
3.4
514
531
597
614
631
664
830
996
681
847
*OI2
697
863
*029
714
880
*o45
731
896
♦062
747
913
*o78
764
929
*o95
780
946
*iii
797
963
*I27
814
979
*i44
64
65
66
42 160
325
488
177
341
504
193
357
521
210
374
537
226
390
553
406
570
259
423
586
275
439
602
292
455
619
308
472
635
3
4
5.1
6.8
S.5
67
68
69
270
71
72
73
651
813
975
667
830
991
684
846
♦008
700
862
*024
716
878
*040
732
894
♦056
217
749
911
*072
765
927
♦088
781
943
*io4
797
959
*I20
6
7
8
9
1
2
10.2
11.9
13.6
15.3
10
1.6
3.2
43 136
152
169
185
201
233
249
265
281
297
457
616
313
473
632
329
489
648
345
505
664
361
521
680
377
537
696
393
553
712
409
569
727
425
584
743
441
600
759
74
75
76
775
933
44 091
791
949
107
807
965
122
823
981
138
838
996
154
854
*OI2
170
870
*028
185
886
*o44
201
902
*059
217
917
*o75
232
3
4
5
6
7
8
9
4.8
6.4
8.0
9.6
11.2
12.8
14.4
77
78
79
280
81
82
83
248
404
560
264
420
576
279
43^^
592
295
451
607
467
623
326
483
638
342
498
654
358
514
669
373
529
685
389
545
700
716
731
747
762
778
793
809
824
840
855_
*OIO
163
317
.. 1
871
45 025
179
886
040
194
902
056
209
917
071
225
932
086
240
948
102
255
963
117
271
979
133
286
994
148
301
1
2
15
1.5
3.0
84
85
86
332
484
637
347
500
652
362
515
667
378
530
682
393
545
697
408
561
712
423
576
728
439
591
743
454
606
758
469
621
773
3
4
5
6
4.5
6.0
7.5
9
87
88
89
290
91
92
93
788
939
46 090
803
954
105
818
969
120
834
984
135
849
*ooo
150
864
*oi5
165
879
894
*o45
195
909
*o6o
210
924
*o75
225
7
8
9
10.5
12.0
13.5
240
"389"
687
255
404
553
702
270
285
300
315
330
345
359
374
.. 1
419
568
716
434
583
731
449
598
746
464
613
761
479
627
776
494
642
790
509
657
805
672
820
1
2
3
4
5
6
J.*
1.4
2.8
94
95
96
835
982
47 129
850
997
144
864
*OI2
159
879
♦026
173
894
*04i
188
909
*o56
202
923
*o7o
217
938
+085
232
953
*IOO
246
967
*ii4
261
4.2
5.6
70
8.4
97
98
99
300
276
422
567
290
436
582
305
451
596
319
465
611
334
480
625
770
349
494
640
784
363
509
654
799
378
524
669
I13
392
538
683
828
407
553
698
842
7
8
9
9.8
11.2
12.6
712
727
741
756
N.
1
2
3
4
5
6
7
8
9
Prop. Pts.
8
TABLE I
N.
1
2
9
4
5
6
7
8
9
Prop. Pts.
300
01
02
03
47 712
727
741
756
770
784
799
813
828
842
c^57
48 001
144
871
015
159
885
029
173
yoo
044
187
914
058
202
929
073
216
943
C87
230
958
lOI
244
972
116
259
986
130
273
15
04
Go
OG
287
430
572
302
444
586
316
458
601
330
473
615
344
487
629
359
643
373
657
387
401
544
686
416
558
700
1
2
3
1.5
3.0
4.5
07
08
09
310
11
12
13
996
728
869
*0I0
742
883
*024
756
897
*o38
770
911
♦052
785
926
*o66
799
940
*o8o
813
954
*094
827
968
*io8
841
982
*I22
4
5
6
7
8
9
6.0
7.5
9.0
10.5
12.0
13.6
49 136
150
164
178
192
206
220
234
248
262
402
541
679
276
415
554
290
429
568
304
443
582
318
457
596
332
471
610
346
485
624
3^
638
374
513
651
388
527
665
14
15
16
693
831
969
707
721
859
996
734
872
*OIO
748
886
*024
762
900
*o37
776
914
♦051
790
927
*o65
803
941
*o79
817
955
♦092
1
14
1.4
17
18
19
320
21
22
23
50 106
243
379
515
120
256
393
529
133
270
406
284
420
161
297
433
174
311
447
188
325
461
202
338
474
215
352
229
365
501
2
3
4
5
6
7
8
9
2.8
4.2
5.6
7.0
8.4
9.8
11.2
12.6
542
556
569
583
596
610
623
637
651
786
920
664
799
934
678
813
947
691
826
961
705
840
974
718
853
987
732
866
*OOI
745
880
*oi4
759
893
*028
772
907
*04i
24
25
26
51 055
188
322
068
202
335
081
215
348
095
228
362
108
242
375
121
255
388
135
268
402
148
282
415
162.
255
428
175
308
441-
27
28
29
830
31
32
33
455
587
720
468
601
733
481
614
746
495
627
759
508
640
772
654
786
534
667
799
548
680
812
693
825
574
706
838
1
2
3
4
5
6
7
13
1.3
2.6
3.9
5.2
6.5
7.8
9.1
851
865
878
891
904
917
930
943
957
970
983
52 114
244
996
127
257
*oo9
140
270
*022
III
*o35
166
297
*048
179
310
*o6i
192
323
*o75
205
336
*o88
218
349
*IOI
231
362
34
35
36
375
504
634
388
517
647
401
530
660
414
543
673
427
lit
440
569
699
453
582
711
466
595
724
479
608
737
492
621
750
8
9
10.4
11.7
37
38
39
340
41
42
43
763
892
53 020
148
776
905
033
789
917
046
802
930
058
815
943
071
827
956
084
840
969
097
853
982
no
866
994
122
879
*oo7
135
1
2
3
4
5
12
1.2
2.4
3.6
4.8
6
161
173
186
199
212
224
237
250
263
275
403
529
288
415
542
301
428
555
314
441
567
326
453
580
339
466
593
352
479
605
364
491
618
377
504
631
390
643
44
45
46
656
782
908
668
794
920
681
807
933
694
820
945
706
832
958
719
845
970
732
7s
744
870
995
757
882
*cx)8
769
895
*020
6
7
8
7.2
8.4
9.6
47
48
49
350
54 033
158
283
045
170
295
058
183
307
070
195
320
083
208
332
095
220
345
108
233
357
120
245
370
i
145
270
394
9
10.8
407
419
432
444
456
469
481
494
506
518
N.
1
2
3
4
5
6
7
s
9
Prop. Pts.
LOGARITHMS OF NUMBERS
9
N.
1
2
3
4
5
6
7
S
9
Prop. Pts.
350
61
52
53
54 407
419
432
444
456
469
481
494
506
518
531
654
777
543
667
790
555
679
802
568
691
814
580
704
827
716
839
605
728
851
617
741
864
630
753
876
642
765
888
13
54
55
56
900
55 023
145
913
035
157
925
047
169
937
060
182
949
072
194
962
084
206
974
096
218
986
108
230
998
121
242
*oii
133
255
1
2
3
1.3
2.6
3.9
57
58
59
360
61
62
63
267
388
509
279
400
522
291
413
534
303
425
546
315
437
558
328
449
570
691
340
461
582
352
473
594
364
376
497
618
4
5
6
7
8
9
5.2
6.5
7.8
9.1
10.4
11.7
630
871
991
642
654
666
678
703
715
727
739
763
883
*oo3
775
895
*oi5
787
907
*027
799
919
*o38
811
931
*o5o
823
943
*o62
835
955
*o74
847
967
♦086
859
979
♦098
64
65
66
56 no
229
348
122
241
360
134
253
372
146
265
384
158
277
.396
170
289
407
182
301
419
194
312
431
205
324
443
217
336
455
1
12
1.2
67
68
69
370
71
72
73
467
585
703
478
597
714
490
608
726
502
620
738
5H
632
750
526
644
761
656
773
549
667
785
561
679
797
573
691
808
2
3
4
5
6
7
8
9
2.4
3.6
4.8
6.0
7.2
8.4
9.6
10.8
820
832
844
855
867
879
891
902
914
+031
148
264
926
*043
159
276
937
57 054
171
949
066
183
961
078
194
972
089
206
984
lOI
217
996
113
229
*oo8
124
241
*oi9
136
252
74
75
76
287
403
519
299
415
530
310
426
542
322
438
553
334
449
565
461
576
357
473
588
368
484
600
380
496
611
392
507
623
77
78
79
380
81
82
83
634
749
864
978
646
761
875
657
772
887
669
784
898
680
795
910
692
807
921
703
818
933
830
944
726
841
955
852
967
*o8i
195
309
422
1
2
3
4
5
6
7
11
1.1
2.2
3.3
4.4
5.5
6.6
7.7
990
*OOI
*oi3
*024
*o35
*047
♦058
♦070
58 092
206
320
104
218
331
115
229
343
127
240
354
138
252
365
149
26^
377
161
274
388
172
286
399
184
297
410
84
85
86
433
546
659
444
557
670
456
569
681
467
580
692
478
591
704
490
602
715
501
614
726
512
625
737
524
636
749
647
760
8
9
8.8
9.9
87
88
89
390
91
92
93
771
883
995
59 106
782
894
*oo6
T18
794
906
*oi7
805
917
*028
816
928
♦040
827
939
♦051
838
950
*o62
850
961
♦073
861
973
♦084
872
984
♦095
1
2
3
4
5
10
1.0
2.0
3.0
4.0
5
129
140
151
162
173
184
195
207
318
428
539
218
329
439
229
340
450
240
461
362
472
262
373
483
273
384
494
284
395
506
7J>
517
306
417
528
94
95
96
550
660
770
671
780
572
682
791
693
802
594
704
813
605
824
616
726
835
627
737
846
638
748
857
649
6
7
8
6.0
7.0
8.0
97
98
99
400
N.
60 097
"206
890
?o1
901
*OIO
119
912
*02I
130
923
♦032
141
934
*o43
152
945
"054
163
956
*o65
173
966
+076
184
977
*o86
195
9
9.0
217
228
239
249
260
271
282
293
304
1
2
3
4
5
7
S
9
Prop. Pts.
10
TABLE I
N.
O
1
2
3
4
5
6
7
S
9
Prop. Pta.
400
01
02
03
6o 2o6
217
228
239
249
260
271
282
293
304
3H
423
531
325
433
541
336
444
552
347
455
563
358
466
574
369
477
584
379
487
595
390
498
606
401
509
617
412
520
627
04
05
06
638
746
853
649
756
863
660
767
874
670
778
885
681
788
895
692
799
906
703
810
917
713
82 F
927
724
83/
938
P5
842
949
11
07
08
09
410
11
12
13
959
61 066
172
970
077
183
981
087
194
991
098
204
*002
109
215
*oi3
119
225
♦023
130
236
*034
140
247
*o45
151
257
*o55
162
268
1
2
3
4
5
6
7
8
1.1
2.2
3.3
4.4
5.5
6.6
7.7
8.8
278
289
300
310
321
331
342
352
363
374
384
490
595
395
500
606
405
511
616
416
521
627
426
532
637
437
542
648
448
553
658
458
563
669
469
574
679
479
584
690
14
15
16
700
805
909
711
815
920
721
826
930
731
836
941
742
847
951
752
857
962
763
868
972
773
878
982
784
888
993
794
899
*cx)3
9
9.9
17
18
19
420
21
22
23
62 014
118
221
024
128
232
034
138
242
045
149
252
055
159
263
066
170
273
076
180
284
086
190
294
097
201
304
107
211
315
325
335
346
356
366
377
387
397
408
418
428
634
439
542
644
449
655
459
665
469
572
675
480
583
685
490
593
696
500
603
706
613
716
521
624
726
1
2
1.0
2.0
24
25
26
737
839
941
747
849
951
757
859
961
767
870
972
778
880
982
788
890
992
798
900
*002
808
910
*OI2
818
921
*022
829
931
*o33
3
4
5
6
3.0
4.0
5.0
6
27
28
29
430
31
32
33
63 043
144
246
053
155
256
063
165
266
073
175
276
083
185
286
094
195
296
104
205
306
114
215
317
124
225
327
134
236
337
7
8
9
7.0
8.0
9.0
347
357
367
377
387
397
407
417
428
438
639
739
448
548
649
458
659
468
568
669
478
579
679
488
589
689
498
599
699
508
609
709
619
719
629
729
34
35
36
749
849
949
759
859
959
769
869
969
779
879
979
789
889
988
799
899
998
809
909
*oo8
819
919
*oi8
829
929
*028
839
939
*038
9
37
38
39
440
41
42
43
64 048
147
246
058
157
256
068
167
266
078
177
276
088
187
286
098
197
296
108
207
306
118
217
316
128
227
326
137
237
335
1
2
3
4
5
6
7
8
0.9
1.8
2.7
3.6
4.5
5.4
6 3
7.2
345
355
365
375
385
395
404
414
424
434
444
542
640
454
650
464
562
660
473
572
670
483
582
680
493
591
689
503
601
699
611
709
621
719
532
631
729
44
45
46
738
836
933
748
846
943
758
856
953
768
865
963
777
875
972
787
885
982
797
895
992
807
904
*002
816
914
826
924
♦021
9
8.1
47
48
49
450
65 031
128
225
040
137
234
050
147
244
060
157
254
070
167
263
079
176
273
369
089
186
283
099
196
292
108
205
302
118
215
312
408
321
331
341
350
360
379
389
398
N.
i
2
3
4
5
6
7
§
Prop. Pts.
LOGARITHMS OF NUMBERS
11
N.
1
2
3
4
5
6
7
8
9
Prop. Pts.
450
51
52
53
65 321
331
341
350
360
369
379
389
398
408
418
514
610
427
523
619
437
533
629
447
543
639
456
552
648
466
562
658
475
571
667
485
581
677
495
504
600
696
54
55
56
706
801
896
811
906
725
820
916
734
830
925
744
839
935
753
849
944
763
858
954
772
868
963
782
877
973
792
887
982
10
57
58
59
4G0
61
62
63
992
66 087
181
276
*OOI
096
191
*OII
106
200
*020
115
210
*o3o
124
219
*039
134
229
*o49
143
238
♦058
153
247
*o68
162
257
*o77
172
266
1
2
3
4
5
6
7
8
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
285
295
304
3H
408
502
596
323
332
342
351
361
455
549
642
370
464
558
380
474
567
389
483
577
398
492
586
417
427
521
614
436
530
624
445
633
64
65
66
652
745
839
661
755
848
671
764
857
680
773
867
689
783
876
699
792
885
708
801
894
717
811
904
727
820
913
736
829
922
9
9.0
67
68
69
470
71
72
73
932
67 025
117
941
034
127
950
043
136
960
052
145
969
062
154
978
071
164
987
080
173
997
089
182
*oo6
099
191
201
210
219
228
237
247
256
265
274
284
293
1
2
3
4
6
6
9
0.9
1.8
2.7
3.6
4.5
5.4
302
486-
311
403
495
321
413
504
330
422
514
339
431
523
348
440
532
357
449
541
367
459
550
376
468
560
385
477
569
74
75
76
578
669
761
587
679
770
596
688
779
605
697
78S
614
706
797
624
715
806
633
724
815
642
733
825
651
742
834
660
752
843
77
78
79
480
81
82
83
852
68 034
861
952
043
870
961
052
879
970
061
888
979
070
897
988
079
906
997
088
916
*oo6
097
925
*oi5
106
934
*024
115
7
8
9
6.3
7.2
8.1
124
215
305
395
133
142
151
160
169
178
187
196
205
224
314
404
233
323
413
242
332
422
251
341
431
260
350
440
269
359
449
278
368
458
287
377
467
296
386
476
84
85
86
485
574
664
673
502
592
681
511
601
690
520
610
699
529
538
628
717
547
637
726
Pi
646
735
655
744
8
87
88
89
490
91
92
93
753
842
931
762
851
940
771
860
949
780
869
958
878
966
797
886
975
806
895
984
815
904
993
824
913
*002
833
922
*9ii
1
2
3
4
5
6
7
8
0.8
1.6
2.4
3.2
4.0
4.8
5.6
6.4
69 020
028
037
046
055
064
073
082
090
099
108
197
285
117
205
294
126
214
302
135
223
311
144
232
320
152
241
329
161
249
338
170
258
346
179
267
355
188
276
364
94
95
96
373
461
548
469
557
390
478
566
399
487
574
408
417
504
592
425
513
601
434
522
609
443
452
627
9
7.2
97
98
99
500
636
723
810
644
732
819
653
740
827
662
749
836
671
758
845
932
679
767
854
688
775
862
697
784
871
705
III
714
801
888
897
906
914
923
940
949
958
966
975
N.
1
2
3
4
5
6
7
8
9
Prop. Pts.
12
TABLE I
N.
O
1
2
3
4
5
6
7
8
9
Prop. Pts.
600
01
02
03
69 897
984
70 070
157
906
914
923
932
940
949
958
966
975
992
079
165
*OOI
088
174
*OIO
096
183
*oi8
105
191
*027
114
200
♦036
122
209
*044
131
217
*o53
140
226
*o62
148
234
04
05
06
243
329
415
252
338
424
260
346
432
269
355
441
278
364
449
286
458
295
467
303
389
475
312
321
406
492
9
07
08
09
510
11
12
13
501
586
672
757
842
927
71 012
509
595
680
518
603
689
612
697
P5
621
706
791
876
961
046
544
629
714
800
552
638
723
808
561
646
731
569
655
740
578
663
749
1
2
3
4
5
3
7
8
0.9
1.8
2.7
3 6
4.5
0.4
6.3
7.2
766
851
935
020
774
859
944
029
783
868
952
037
817
825
834
885
969
054
893
978
063
902
986
071
910
995
079
919
*oo3
088
U
15
16
265
105
189
273
"3
282
122
206
290
130
214
299
139
223
307
147
231
315
i9'5
240
324
248
332
172
257
341
9
8.1
17
18
19
520
21
22
23
349
433
517
357
441
525
366
450
533
374
458
542
383
466
550
391
475
559
399
483
567
408
492
575
416
500
584
592
8
10.8
21.6
600
684
767
850
609
617
625
634
642
650
659
667
675
692
775
858
700
784
867
709
792
875
717
800
883
725
809
892
734
817
900
742
825
908
750
834
917
759
842
925
24
25
26
933
72 016
099
941
024
107
950
032
115
958
041
123
966
049
132
975
057
140
983
066
148
991
074
156
999
082
165
*oo8
090
173
4
5
6
3.2
4.0
4.8
27
28
29
530
31
32
33
181
263
346
428
189
272
354
198
280
362
206
288
370
214
296
378
222
304
387
230
313
395
239
321
403
247
329
411
255
337
419
7
8
9
5.6
6.4
7.2
436
444
452
460
469
477
485
493
501
509
591
673
518
599
681
607
689
534
616
697
1^^
624
705
550
632
713
558
640
722
567
648
730
575
656
738
583
665
746
34
35
36
^54
835
916
762
843
925
770
852
933
779
860
941
787
868
949
795
876
957
803
884
965
811
892
973
819
900
981
827
908
989
7
37
38
39
540
41
42
43
997
73 078
159
239
*oo6
086
107
*oi4
094
175
*022
102
183
♦030
III
191
*o38
119
199
*o46
127
207
*o54
135
215
*o62
143
223
♦070
151
231
1
2
3
4
5
6
7
8
0.7
1.4
2.1
2.8
3.5
4.2
4.9
5 6
247
255
263
272
280
288
296
304
312
320
400
480
328
408
488
336
416
496
344
424
504
352
432
512
360
440
520
368
448
528
376
456
536
384
464
544
392
472
552
44
45
46
560
719
568
648
727
576
656
735
584
664
743
592
672
751
600
679
759
608
687
767
616
695
775
624
703
783
632
711
791
9
6.3
47
48
49
650
957
807
886
965
815
894
973
823
902
981
830
910
989
838
918
997
846
926
*cx>5
854
933
*oi3
862
941
*020
870
949
*028
74 036
044
052
060
068
076
084
092
099
107
N.
1
2 3
4
5
6
7
S
9
Prop. Pts.
LOGARITHMS OF NUMBERS
13
N.
O
1
2
3
4
5
6
7
S
9
Prop. Pt8.
51
52
53
74 036
044
052
060
068
076
084
092
099
107
115
194
273
123
202
280
131
210
288
139
218
296
147
225
304
155
233
312
162
241
320
170
249
327
178
257
335
186
265
343
54
55
56
351
429
507
359
437
515
367
445
523
374
453
531
382
4.61
539
390
468
547
398
476
554
406
484
562
414
492
570
421
500
578
57
58
59
560
61
62
63
IS
741
593
671
749
601
679
757
609
687
764
617
695
772
624
702
780
632
710
788
640
718
796
648
726
803
656
733
811
889
819
827
834
842
850
858
865
873
881
896
974
75 051
904
981
059
912
989
066
920
997
074
927
*oo5
082
935
*OI2
089
943
*020
097
950
*028
105
958
*o35
113
966
*043
120
1
2
0.8
1.6
64
65
66
128
205
282
136
213
289
143
220
297
151
228
305
159
236
312
166
243
320
174
328
182
259
335
189
266
343
197
274
351
3
4
5
6
2.4
3.2
4.0
4.8
67
68
69
670
71
72
73
358
435
5"
587
366
442
519
595
374
450
526
458
534
389
465
542
618
397
473
549
626
404
481
557
633
412
488
565
420
496
572
427
504
580
7
8
9
5.6
6.4
7.2
603
610
641
648
656
664
740
815
671
747
823
679
831
686
762
838
694
770
846
702
778
853
709
785
861
717
868
724
800
876
732
808
884
74
75
76
891
967
76 042
899
974
050
906
982
057
914
989
065
921
997
072
929
*oo5
080
937
*OI2
087
944
*020
095
952
*027
103
959
*o35
no
77
78
79
580
81
82
83
118
193
268
125
200
275
III
283
140
215
290
148
223
298
155
230
305
1%
313
170
245
320
178
253
328
185
260
335
343
350
358
365
373
380
388
395
403
410
418
492
567
425
500
574
433
507
582
440
515
589
448
522
597
455
530
604
462
537
612
470
545
619
477
III
486
559
634
1
2
7
0.7
1.4
84
85
86
87
88
89
590
91
92
93
641
716
790
864
938
77 012
649
723
797
871
945
019
656
730
805
879
026
664
If.
886
960
034
671
819
893
967
041
678
753
827
901
975
048
686
760
834
908
982
056
842
916
989
063
701
775
849
923
997
070
708
782
856
930
*oo4
078
3
4
5
6
7
8
9
2.1
2.8
3.5
4.2
4.9
5.6
6.3
085
093
100
107
115
122
129
137
144
151
159
232
305
166
240
313
173
247
320
181
254
327
188
262
335
269
342
203
276
349
210
283
357
217
291
364
225
298
371
94
95
96
379
452
525
386
459
532
393
466
539
401
474
546
408
481
554
415
488
561
422
430
503
576
437
583
444
517
590
97
98
99
670
743
605
677
750
612
685
757
619
692
764
627
699
772
634
706
779
641
714
786
648
721
793
866
656
728
801
663
735
GOO
815
822
830
837
844
851
859
873
880
N.
1
2
3
4
5
6
7
8
9
Prop. Pts.
14
TABLE I
N.
O
1
2
3
4
5
6
7
8
9
Prop. Pts.
600
01
02
03
n 815
887
960
78 032
822
830
837
844
851
859
866
873
880
895
967
039
902
974
046
909
981
053
916
988
061
924
996
068
931
*oo3
075
938
*OIO
082
945
*oi7
089
952
*025
097
04
05
06
104
176
247
III
183
254
118
190
262
125
197
269
132
204
276
140
211
283
147
219
290
226
297
161
233
305
168
240
312
8
07
08
09
GIO
11
12
13
319
390
462
326
398
469
333
405
476
340
412
483
347
419
490
355
426
497
362
433
504
369
440
512
376
447
519
383
455
526
1
2
3
4
6
6
7
8
0.8
1.6
2.4
3.2
4.0
4.8
5.6
6.4
533
540
547
554
561
569
576
583
590
597
604
675
746
611
682
753
618
689
760
625
696
767
633
704
774
640
711
781
647
718
789
654
725
796
661
732
803
668
739
810
14
15
16
817
888
958
824
895
965
831
902
972
838
909
979
845
916
986
852
923
993
859
930
*ooo
866
937
*oo7
873
944
*oi4
880
951
*02I
9
7.2
17
18
19
620
21
22
23
79 029
099
169
036
106
176
043
113
183
050
120
190
057
127
197
064
134
204
274
071
141
211
078
148
218
085
155
225
092
162
232
239
246
253
260
267
281
288
295
302
309
379
449
316
386
456
323
393
463
330
400
470
337
407
477
344
414
484
351
421
491
358
428
498
365
435
505
372
442
511
1
2
7
0.7
1.4
24
25
26
518
588
657
525
595
664
532
602
671
539
609
678
546
616
685
553
623
692
560
630
699
637
706
574
644
713
650
720
3
4
5
2.1
2.8
3.5
4 9
27
28
29
630
31
32
33
727
796
865
734
803
872
810
879
748
817
886
754
824
893
761
831
900
768
837
906
775
844
913
782
851
920
789
858
927
7
8
9
4.9
5.6
6.3
934
941
948
955
962
030
168
969
975
982
989
996
80 003
072
140
010
079
147
017
085
154
024
092
161
037
106
175
044
051
120
188
058
127
195
06s
134
202
34
35
36
209
277
346
216
284
353
223
291
359
298
366
236
305
373
243
312
380
250
387
257
325
393
264
332
400
271
339
407
6
37
38
39
640
41
42
43
414
482
550
421
489
557
428
496
564
434
502
570
441
509
577
448
584
652
455
523
659
462
530
598
468
536
604
475
543
611
1
2
3
4
5
6
7
8
0.6
1.2
1.8
2.4
3.0
3.6
4.2
4 8
618
625
632
638
645
665
672
679
686
754
821
693
760
828
699
767
835
706
774
841
713
781
848
720
787
855
726
794
862
733
801
868
740
808
875
747
814
882
44
45
46
889
956
81 023
895
963
030
902
969
037
909
976
043
916
983
050
922
990
057
929
996
064
936
*oo3
070
943
*oio
077
949
♦017
084
9
5.4
47
48
49
650
090
158
224
097
164
231
104
171
238
III
178
245
117
184
251
124
258
131
198
265
137
204
271
144
211
278
218
285
291
298
305
3"
318
325
331
338
345
351
N.
1
2
3
4
5
6
7
8
9
Prop Pts.
LOGAEITHMS OF NUMBERS
15
N.
O
1
2
3
4
5
6
7
8
9
Prop. Pts.
650
51
62
53
8 1 291
298
305
311
318
325
331
338
345
351
358
425
491
365
498
438
505
378
445
511
385
518
391
458
525
398
465
531
405
471
538
411
478
544
418
485
551
54
55
56
|5»
624
690
564
631
697
637
704
578
644
710
584
651
717
591
657
723
598
664
730
604
671
737
611
677
743
617
684
750
57
58
59
660
61
62
63
823
889
763
829
895
770
836
902
776
842
908
783
849
915
790
856
921
796
862
928
803
869
935
809
875
941
816
882
948
954
82 020
086
151
961
968
974
981
987
994
*CXXD
*oo7
*oi4
027
092
158
033
099
164
040
105
171
046
112
178
053
184
060
125
191
066
132
197
073
138
204
079
145
210
1
2
0.7
1.4
64
65
66
217
282
347
223
289
354
230
295
360
236
302
367
Itl
373
249
315
380
256
387
263
328
393
269
334
400
276
406
3
4
5
6
2.1
2.8
3.5
4.2
67
68
69
670
71
72
73
413
478
543
419
484
549
426
491
556
432
497
562
439
504
569
445
510
575
452
582
458
Pi
465
530
595
471
536
601
7
8
9
4.9
5.6
6 3
607
672
802
614
620
627
633
640
646
653
659
666
679
III
685
750
814
692
756
821
698
763
827
705
769
834
711
776
840
718
782
847
724
789
853
730
795
860
74
75
76
866
930
995
872
937
*OOI
879
943
*cx)8
885
950
*oi4
892
956
*020
898
963
*027
905
969
*033
911
975
*04o
918
982
*o46
924
988
♦052
77
78
79
680
81
82
83
83 059
123
187
065
129
193
072
136
200
078
142
206
085
149
213
091
155
219
097
161
225
104
168
232
no
174
238
117
181
245
251
257
264
270
276
283
289
296
302
308
315
378
442
321
385
448
327
391
455
334
398
461
340
404
467
347
410
474
353
417
480
359
423
487
366
429
493
372
436
499
1
2
0.6
1.2
84
85
86
506
569
632
512
575
639
518
645
525
588
651
531
594
658
537
601
664
544
607
670
|5°
613
677
556
620
683
563
626
689
3
4
5
6
1.8
2.4
3.0
3.6
87
88
89
690
91
92
93
696
759
822
702
765
828
708
771
835
778
841
721
784
^47
727
790
853
734
797
860
740
803
866
746
809
872
879
7
8
9
4.2
4.8
5.4
885
891
897
904
910
916
923
929
935
942
*oo4
067
130
948
84 on
073
954
017
080
960
023
086
967
029
092
973
036
098
979
042
105
985
048
III
992
055
117
061
123
94
95
96
198
261
142
205
267
148
211
273
155
28c
161
223
167
230
292
298
180
242
305
186
248
311
192
255
317
97
98
99
700
N.
323
448
330
392
454
336
398
460
342
404
466
348
410
473
354
417
479
361
485
367
429
491
373
435
497
379
4.^2
504
510
516
522
528
535
541
547
553
559
566
1
2
3
4
5
6
7
8
9
Prop. Pts.
16
TABLE 1
N.
O
1
«
3
4
5
6
7
S
9
Prop. Pts.
700
01
02
03
84 510
572
634
696
516
522
528
535
541
547
553
559
566
628
689
751
640
702
584
646
708
590
652
714
597
658
720
603
665
726
609
671
733
677
739
621
683
745
04
05
06
^57
819
880
763
825
887
770
831
893
776
837
899
782
844
905
788
850
911
794
856
917
800
862
924
807
868
930
813
874
936
7
07
08
09
710
11
12
13
942
85003
065
948
009
071
954
016
077
960
022
083
967
028
089
973
034
095
979
040
lOI
985
046
107
991
052
114
997
058
120
1
2
3
4
5
6
7
8
0.7
1.4
2.1
2.8
3.5
4.2
4.9
5.6
126
132
138
144
150
156
217
278
339
163
169
175
181
248
309
193
254
315
199
260
321
Tel
327
211
272
333
224
285
345
230
291
352
236
297
358
242
303
364
14
15
16
370
431
491
376
437
497
382
443
503
388
449
509
394
455
516
400
461
522
406
467
528
412
473
534
418
479
540
425
485
546
9
6.3
17
18
19
720
21
22
23
552
612
673
558
618
679
564
625
685
570
631
691
576
697
582
643
703
588
649
709
594
655
715
600
661
721
606
667
727
733
739
745
751
757
763
769
775
781
788
794
854
914
800
860
920
806
866
926
812
872
932
818
878
938
824
884
944
830
890
950
836
896
956
842
902
962
848
908
968
1
2
6
0.6
1.2
24
25
26
974
86 034
094
980
040
100
986
046
106
992
052
112
998
058
118
*oo4
064
124
*OIO
070
130
*oi6
076
136
*022
082
141
*028
088
147
3
4
5
1.8
2.4
3.0
27
28
29
730
31
32
33
153
213
273
332
159
219
279
338
165
225
285
171
231
291
177
237
297
183
243
303
189
249
308
195
255
3H
201
261
320
207
267
326
6
7
8
9
3.0
4.2
4.8
5.4
344
350
356
362
368
374
380
386
445
504
564
392
451
510
390
457
516
404
463
522
410
469
528
415
475
534
421
481
540
427
487
546
433
493
552
439
499
558
34
35
36
570
629
688
576
635
694
581
641
700
587
646
705
593
652
711
658
717
605
664
723
611
670
729
617
676
735
623
682
741
5
37
38
39
740
41
42
43
747
806
864
923
753
812
870
759
817
876
764
823
882
770
829
888
776
894
782
841
900
788
847
906
794
853
911
800
859
917
1
2
3
t
6
6
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
929
935
941
947
953
958
964
970
976
982
87 04.0
099
988
046
105
994
052
III
058
116
*oo5
064
122
*OII
070
128
*oi7
075
134
*023
081
140
*029
087
146
*o35
093
151
44
45
46
'57
216
274
163
221
280
169
227
286
175
233
291
181
239
297
186
245
303
192
251
309
198
256
315
204
262
320
210
268
326
9
4.5
47
48
49
750
332
390
448
396
454
344
402
460
408
466
355
413
471
361
419
477
367_
483
373
489
379
437
495
384
442
500
506
512
518
523
529
535
541
547
552
558
N.
1
2
3
4
5
6
7
S
9
Prop. Pts.
LOGAEITHMS OF NUMBERS
17
N.
1
2
3
4
5
535
593
651
708
6
7
8
9
Prop. Pts.
750
51
52
53
87 506
512
518
523
529
541
547
552
558
564
622
679
570
628
685
633
691
581
639
697
587
645
703
599
656
714
604
662
720
610
668
726
616
674
731
54
55
56
737
795
852
800
858
806
864
812
869
760
818
875
766
823
881
772
829
887
777
f35
892
783
841
898
789
846
904
57
58
59
760
61
62
63
910
967
88 024
915
973
030
978
036
927
984
041
933
990
047
938
996
053
944
*OOI
058
950
♦007
064
955
*oi3
070
961
*oi8
076
081
087
093
098
104
no
116
121
127
133
190
247
304
138
195
252
144
201
258
150
207
264
156
213
270
161
218
275
167
224
281
173
230
287
178
235
292
184
241
298
1
2
0.6
1.2
64
65
66
3??
366
423
315
372
429
321
377
434
326
383
440
389
446
338
395
451
343
400
457
349
406
463
355
412
468
360
417
474
3
4
5
6
1.8
2.4
3.0
3 6
67
68
69
770
71
72
73
480
536
593
485
542
598
491
604
497
553
610
502
|59
615
508
564
621
513
570
627
519
632
5^5
581
638
F
643
7
8
9
4.2
4.8
5.4
649
655
660
666
672
(>77
683
689
694
700
705
762
818
711
767
824
717
773
829
722
779
835
728
784
840
734
790
846
739
795
852
801
857
750
807
863
756
812
868
74
75
76
874
880
936
992
885
941
997
891
947
*oo3
897
953
*oc9
902
958
*oi4
908
964
*020
913
969
*025
919
975
♦031
925
98.
*037
77
78
79
780
81
82
83
89 042
098
154
048
104
159
053
109
165
059
115
170
064
120
176
070
126
182
076
131
081
137
193
087
143
198
092
148
204
260
209
265
?76
215
271
326
382
221
226
232
237
243
248
254
276
332
387
282
337
393
287
343
398
293
404
298
354
409
304
360
415
310
365
421
315
426
1
2
5
0.5
1.0
84
85
86
542
437
492
548
443
498
553
448
504
559
454
509
564
459
515
570
465
520
575
470
526
581
476
We
481
537
592
3
4
5
p
1.5
2.0
2.5
3
87
88
89
790
91
92
93
597
708
603
658
713
609
664
719
614
669
724
620
675
730
625
680
735
631
686
741
636
691
746
642
697
752
647
702
757
7
8
9
3.5
4.0
4.5
763
768
774
779
785
790
796
801
807
812
818
873
927
823
878
933
829
883
938
834
889
944
840
894
949
845
900
955
851
905
960
856
911
966
862
916
971
867
922
977
94
95
96
982
90 037
091
988
042
097
993
048
102
998
"III
*oo4
059
i'3
*oo9
064
119
*oi5
069
124
*020
075
129
*026
080
135
*o3i
086
140
97
98
99
800
146
200
255
'53
206
260
157
211
266
162
217
271
168
222
276
173
227
282
179
233
287
184
238
293
189
244
298
195
249
304
309
314
320
325
331
336
342
347
352
358
N.
1
2
3
4
5
6
7
8
9
Prop. Pts.
18
TABLE I
N.
O
1
2
3
4
5
6
7
8
9
Prop. Pts.
800
01
02
03
90 309
363
417
472
314
320
325
331
336
342
347
352
358
369
423
477
374
428
482
380
434
488
385
439
493
390
445
499
396
450
504
401
455
509
407
461
515
412
466
520
04
05
06
526
580
634
531
585
639
536
590
644
596
650
601
655
553
607
660
|5^
612
666
f3
617
671
569
623
677
574
628
682
07
08
09
810
11
12
13
687
741
795
693
747
800
698
752
806
703
811
709
763
816
714
768
822
875
720
773
827
881
725
779
832
886
730
784
838
-89T
736
789
843
897
849
854
859
865
870
902
956
91 009
907
961
014
^6^
020
918
972
025
924
977
030
929
982
036
041
940
993
046
945
998
052
950
*oo4
057
1
2
0.6
1.2
14
15
16
062
116
169
068
121
174
180
078
185
084
137
190
089
142
196
094
148
201
100
153
206
105
158
212
no
164
217
3
4
5
6
1.8
2.4
3.0
3 6
17
18
19
820
21
22
23
222
275
328
228
281
334
233
286
339
238
291
344
243
297
350
249
302
355
254
307
360
259
312
365
265
318
371
270
323
376
7
8
9
4.2
4.8
5.4
381
387
392
397
403
408
413
418
424
429
434
487
540
440
492
545
445
498
551
450
503
556
455
508
561
461
514
566
466
519
572
471
524
577
477
582
482
535
587
24
25
26
593
645
698
598
651
703
603
656
709
609
661
714
614
666
719
619
672
724
624
677
730
630
682
735
635
687
740
640
693
745
27
28
29
830
31
32
33
751
803
855
756
808
861
761
814
866
766
819
871
772
824
876
777
829
882
782
834
887
7^7
840
892
793
845
897
798
850
903
908
913
918
924
929
934
939
944
950
955
960
92 012
065
965
018
070
971
023
075
976
028
080
981
033
085
986
038
091
991
044
096
997
049
lOI
*002
054
106
*oo7
059
III
1
2
6
0.5
1.0
34
35
36
117
169
221
122
174
226
127
179
231
132
184
236
137
189
241
143
195
247
148
200
252
153
205
257
158
210
262
163
215
267
3
4
5
1.5
2.0
2.5
3
37
38
39
840
41
42
43
273
324
376
278
330
381
283
335
387
288
340
392
293
345
397
298
350
402
304
355
407
309
361
412
3H
366
418
319
371
423
7
8
9
3.5
4.0
4.5
428
433
438
443
449
454
459
464
469
474
480
531
583
485
536
588
490
542
593
495
547
598
500
603
505
557
609
562
614
516
567
619
521
572
624
526
578
629
44:
45
46
634
686
737
639
691
742
7A7
650
701
752
655
706
758
660
711
763
665
716
768
670
722
773
675
727
77S
681
732
783
47
48
19
850
N.
788
840
891
793
845
896
799
850
901
804
855
906
809
860
911
814
865,
916
819
870
921
824
875
927
829
881
932
834
886
937
942
947
952
957
962
967
973
978
983
988
1
2
3
4
5
6
7
8
9
Prop. VU.
LOG
ARIl
HMS
OF NUMBERS
19
N.
1
2
3
4
5
6
7
§
9
Prop. ris.
850
51
52
53
92 942
947
952
957
962
967
973
978
983
988
993
93 044
095
998
049
100
*oo3
054
105
*oo8
059
no
*oi3
064
115
*oi8
069
120
*024
075
125
*029
080
131
*oi!
136
*039
090
141
54
55
5G
146
197
247
151
202
252
156
207
258
161
212
263
166
217
268
171
222
273
176
227
278
181
232
283
186
192
242
293
6
57
58
59
860
61
62
63
298
349
399
303
354
404
308
359
409
313
364
414
369
420
323
374
425
328
379
430
334
384
435
339
389
44.0
344
394
445
1
2
3
4
5
6
7
8
0.6
1.2
1.8
2.4
3.0
3.6
4.2
4.8
450
500
III
455
460
465
470
475
526
576
626
480
485
490
495
505
556
606
510
611
515
566
616
520
571
621
531
636
541
591
641
546
596
646
64
65
66
651
702
752
656
707
757
661
712
762
666
717
767
671
722
772
676
727
777
682
732
782
687
787
692
742
792
697
747
797
9
5.4
67
68
69
870
71
72
73
802
852
902
807
857
907
812
862
912
817
867
917
822
872
922
827
877
927
832
882
932
837
887
937
842
892
942
847
897
947
952
957
962
967
972
977
982
987
992
997
1
2
3
4
5
6
5
0.5
1.0
1.5
2.0
2.5
3.0
94 002
052
lOI
007
106
012
062
III
017
067
116
022
072
121
027
077
126
131
086
136
042
091
141
047
096
146
74
75
76
151
201
250
156
206
255
161
211
260
166
216
265
171
221
270
176
226
275
181
186
285
191
240
290
196
245
295
77
78
79
880
81
82
83
300
349
399
448
305
354
404
310
359
409
364
414
320
369
419
325
374
424
330
379
429
335
384
433
340
438
345
394
443
7
8
9
3.5
4.0
4.5
453
458
463
468
473
478
483
488
493
498
547
596
503
552
601
507
557
606
512
611
567
616
522
621
527
576
626
532
630
537
586
635
r42
591
640
84
85
86
645
694
743
650
699
748
655
704
753
660
709
758
665
714
763
670
719
768
675
724
773
680
729
778
685
734
783
689
738
787
4
87
88
89
890
91
92
93
792
841
890
797
846
895
802
851
900
807
856
905
812
861
910
817
866
915
822
871
919
827
876
924
880
929
836
885
934
1
2
3
4
5
6
7
8
4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
939
988
95 036
085
944
993
041
090
949
954
959
*oo7
056
105
963
*OI2
061
109
968
973
978
983
998
046
095
*002
051
100
♦017
066
114
*022
071
119
*027
075
124
♦032
080
129
94
95
96
231
139
187
236
143
192
240
148
197
245
153
202
250
158
207
255
163
211
260
168
216
265
173
221
270
177
226
274
9
3.6
97
98
99
900
279
^;6
284
332
381
289
337
386
294
342
390
299
347
395
303
352
400
308
357
405
361
410
318
366
415
323
371
419
424
429
434
439
444
448
453
458
463
468
N.
1
2
3
4
5
6
7
8
9
Prop. Pts.
20
TABLE I
■,
N.
1
2
3
4
5
6
7
8
9
Prop. Pts.
900
01
02
03
95 424
429
434
439
444
448
453
458
463
468
472
569
477
525
574
482
530
578
487
535
583
492
540
588
497
545
593
501
598
506
554
602
511
559
607
564
612
04
05
06
617
665
713
622
670
718
626
674
722
631
679
727
636
684
732
641
6S9
737
646
694
742
650
698
746
655
703
751
660
708
756
07
08
09
910
11
12
13
761
809
856
904
766
813
861
818
866
823
871
780
828
875
785
880
789
885
794
842
890
799
847
895
804
852
899
909
914
918
923
928
933
938
942
947
952
999
96 047
957
*oo4
052
961
*oo9
057
966
*oi4
061
971
♦019
066
976
*023
071
980
*028
076
985
*o33
080
,995
*042
090
1
2
0.5
1.0
14
15
IG
095
142
190
099
147
194
104
152
199
109
156
204
114
161
209
118
166
213
123
218
128
175
223
133
180
227
137
185
232
3
4
5
6
1.5
2.0
2.5
3.0
17
18
19
920
21
22
23
237
284
332
379
426
473
520
1%
336
246
294
341
251
298
346
256
303
350
261
308
355
402
450
497
544
265
360
270
317
365
275
322
369
280
327
374
7
8
9
3.5
4.0
4.5
384
388
393
398
407
412
417
421
478
525
435
483
530
440
487
534
445
492
539
454
548
459
506
553
464
558
468
562
24
25
26
567
614
661
572
619
666
577
624
670
581
628
675
586
633
680
591
638
685
P5
642
689
600
647
694
605
652
699
609
656
703
27
28
29
930
31
32
33
708
755
802
848
713
759
806
717
764
811
722
^^?
816
727
774
820
825
736
783
830
741
788
834
745
792
839
750
844
853
858
862
^7
872
876
881
886
890
895
942
988
900
946
993
904
951
997
909
956
*002
914
*oo7
918
965
*OII
923
970
*oi6
928
974
*02I
932
979
*025
♦030
1
2
4
0.4
0.8
34
35
36
97 035
081
128
132
044
090
137
049
095
142
053
100
146
058
104
151
063
109
155
067
114
160
072
118
165
077
123
169
3
4
5
6
1.2
1.6
2.0
2.4
37
38
39
940
41
42
43
174
220
267
.79
225
271
183
230
276
188
192
197
243
290
202
248
294
206
253
299
211
257
304
216
262
308
7
8
9
2.8
3.2
3.6
313
359
405
451
317
322
327
331
336
340
345
350
354
400
447
493
364
410
456
368
414
460
373
419
465
377
424
470
382
428
474
387
433
479
391
437
483
396
442
488
44
45
46
497
589
502
548
594
506
552
598
511
603
562
607
•520
566
612
525
571
617
529
575
621
534
580
626
630
47
48
49
950
635
681
727
640
685
731
644
690
736
649
695
740
653
699
745
658
704
749
663
708
754
667
713
759
672
717
763
676
722
768
772
m
782
786
791
795
800
804
809
813
N.
1
2
«|4
5
6
7
8
9
Prop. PtR.
LOGARITHMS OF NUMBERS
21
N.
o
1
2
3
4
5
6
7
8
9
Prop. Pts.
050
51
52
53
97 772
777
782 786
791
795
800
804
809
813
8i8
864
909
823
868
914
827
873
918
832
^77
923
836
882
928
841
886
932
845
891
937
850
896
941
855
900
946
859
905
950
54
55
56
0^55
98 000
046
959
005
050
964
009
055
968
014
059
973
019
064
978
023
068
982
028
073
987
032
078
991
21
996
041
087
57
58
59
960
61
62
63
091
096
141
186
100
146
191
105
150
195
109
155
200
114
159
204
118
164
209
123
168
214
127
173
218
132
177
223
227
232
236
241
245
250
254
259
263
268
272
363
277
322
367
281
327
372
286
331
376
290
336
381
295
340
385
299
345
390
304
349
394
308
354
399
313
358
403
1
2
5
0.5
1.0
64
65
66
408
453
498
412
457
502
417
462
507
421
466
511
426
471
516
430
475
520
480
525
439
484
529
444
489
534
448
.493
538
3
4
5
6
1.5
2.0
2.5
3.0
67
68
69
970
71
72
73
543
588
632
547
592
^57
552
597
641
601
646
561
605
650
610
655
570
614
659
619
664
579
668
583
628
673
7
8
9
3.5
4.0
4.5
677
682
686
691
695
700
744
789
834
704
709
713
717
722
767
811
726
771
816
731
820
825
740
784
829
749
III
798
843
802
847
762
807
851
74
75
76
856
900
945
860
905
949
865
909
954
869
914
958
874
918
963
878
923
967
883
927
972
^2>7
932
976
892
936
981
896
941
985
77
78
79
080
81
82
83
989
99 034
078
994
038
083
998
043
087
*oo3
047
092
*oo7
052
096
*OI2
056
100
145
189
233
277
^=016
061
105
*02I
065
109
*025
069
114
*029
074
118
123
127
131
136
140
149
154
158
162
167
, 211
255
171
216
260
176
220
264
180
224
269
185
229
273
193
282
198
242
286
202
247
291
207
251
295
1
2
4
0.4
0.8
84
85
86
300
344
388
304
348
392
308
396
313
357
401
317
361
405
322
366
410
326
370
414
330
374
419
335
379
423
339
383
427
3
4
5
1.2
1.6
2.0
2 4
87
88
89
090
91
92
93
432
476
520
436
480
524
441
484
528
445
489
533
449
493
537
454
498
542
458
502
546
590
463
506
550
467
511
555
471
515
559
603
7
8
9
2.8
3.2
3.6
564
568
572
577
581
585
629
673
717
594
599
607
695
612
656
699
616
660
704
621
664
708
625
669
712
614
677
721
it
726
642
686
730
647
691
734
94
95
96
739
782
826
743
787
830
747
791
835
752
795
839
756
800
843
760
804
848
765
808
852
769
813
856
774
817
861
822
865
97
98
99
1000
870
9'3
957
874
917
961
878
922
965
883
926
970
887
930
974
891
935
978
896
939
983
900
944
987
904
948
991
909
952
996
00 coo
004
009
013
017
022
026
030
035
039
N.
1
2
3
4
5
6
7
§
9
Prop. Pts.
22
TABLE I
N
1
2
3
4
5
6
7
S
9
Prop. Pts.
1000
1001
1002
1003
ooo ooo
~434
868
ooi 301
043
477
911
344
087
130
174
217
260
304
347
391
521
388
564
998
431
608
♦041
474
651
♦084
517
694
♦128
561
738
*i7i
604
781
♦214
647
824
♦258
690
44
1004
1005
1006
734
002 166
598
777
209
641
820
252
684
863
296
727
907
339
771
950
382
814
993
857
♦036
468
900
*o8o
512
943
*I23
555
986
1
2
3
4.4
8.8
13.2
1007
1008
1009
1010
1011
1012
1013
003 029
461
891
073
504
934
116
547
977
159
590
*020
202
633
*o63
245
676
*io6
536
288
719
*i49
331
762
*I92
374
805
*235
417
848
*278
4
5
6
7
8
9
17.6
22.0
26.4
30.8
35.2
39.6
004 321
364
407
450
493
579
622
665
708
751
005 180
609
794
223
652
837
266
695
880
309
738
923
352
966
395
824
*oo9
438
867
*052
481
909
*o95
524
952
*i38
567
995
1014
1015
1016
006 038
466
894
081
509
936
124
552
979
166
594
*022
209
637
*o65
252
680
*io7
295
723
*i5o
338
*i93
808
♦236
f3
851
*278
1
43
4.3
1017
1018
1019
1020
1021
1022
1023
007 321
748
008 174
600
oop 026
451
S76
364
790
217
406
833
259
449
876
302
492
918
345
961
387
577
*oo4
430
620
*o46
472
662
*o89
515
705
*I32
558
2
3
4
5
6
7
8
9
8.6
12.9
17.2
21.5
25.8
30.1
34.4
38.7
643
685
728
770
813
856
898
941
983
068
493
918
III
536
961
578
*oo3
196
621
*o45
238
663
*o88
281
706
*i30
323
748
*i73
366
791
*2I5
408
833
*258
1024
1025
1026
010 300
724
on 147
342
766
190
232
427
851
274
470
893
317
936
359
978
401
597
*020
444
639
*o63
486
681
*io5
528
1027
1028
1029
1030
1031
1032
1033
570
993
012 415
*035
458
655
''078
5cx>
697
*I20
542
740
*l62
584
782
*204
626
824
*247
669
866
*289
711
909
*33i
753
951
*373
795
1
2
I
6
7
42
4.2
8.4
12.6
16.8
21.0
25.2
29.4
837
879
922
964
*oo6
♦048
♦090
*I32
*I74
*2I7
013 259
680
014 ICX)
301
722
142
343
764
184
226
427
848
268
469
890
310
511
932
352
553
974
395
596
*oi6
437
638
♦058
479
1034
1035
1036
521
940
015 360
563
982
402
605
*024
444
647
*o66
485
689
*io8
527
730
*i5o
569
772
*I92
611
814
*234
653
856
♦276
695
898
*3i8
737
8
9
33.6
37.8
1037
1038
1039
1040
1041
1042
1043
779
016 197
616
017 033
821
239
657
863
281
699
904
323
741
946
365
783
988
407
824
+030
448
866
♦072
490
908
*ii4
532
950
♦156
574
992
1
2
3
4
5
41
4.1
8.2
12.3
16.4
20 5
075
117
159
200
242
284
326 ! 367
409
451
86S
018 284
492
909
326
534
368
576
993
409
618
"034
451
659
♦076
492
701
*ii8
534
743
*I59
576
784
*20I
617
826
*243
659
1044
1045
1046
700
019 116
532
742
158
573
784
199
615
825
241
656
867
282
698
908
324
739
950
366
781
992
407
822
*o33
449
864
*o75
490
905
6
7
8
24.6
28.7
32.8
1047
10 i8
1019
1050
947
020 361
775
988
403
817
*030
444
858
*o7i
486
900
*ii3
527
941
*i54
568
982
"3^6
*I95
610
*024
*237
651
*o65
*278
693
*io7
♦320
734
*i48
561
9
36.9
021 189
231
272
313
355
437
479
520
N.
1
2
3
4
5
6
7
8
9
Prop. Pts.
LOGARITHMS OF NUMBERS
23
N.
1
2
3
4
5
6
7
S
9
Prop. i'U,
1050
1051
1052
1053
02I 189
231
272
313
355
396
437
479
520
561
974
387
799
603
022 016
428
644
057
470
685
098
511
727
140
552
768
181
593
809
222
635
851
263
676
892
305
717
933
346
758
42
1054
1055
1056
841
882
294
705
923
335
746
964
376
787
♦005
417
828
*047
458
870
*o88
499
911
*I29
541
952
*i7o
582
993
*2II
623
*o34
1
2
3
4.2
8.4
12.6
1057
1058
1059
1000
1061
1062
1063
024 075
486
896
025 306
116
527
937
157
568
978
198
609
♦019
239
650
*o6o
280
691
*IOI
321
732
*I42
363
773
*i83
404
814
*224
445
855
♦265
674
4
5
6
7
8
9
16.8
21,0
25.2
29.4
33.6
37.8
347
388
429
470
511
552
593
634
026 125
533
756
165
574
797
206
615
838
247
656
879
288
697
920
329
737
961
370
778
*002
819
*o43
452
860
♦084
492
901
1064
1065
1066
942
027 350
757
982
390
798
*023
431
839
*o64
472
879
*io5
513
920
*i46
961
♦186
594
*002
*227
635
*042
*268
676
*o83
*309
716
*I24
1
41
4.1
1067
1068
1069
1070
1071
1072
1073
028 164
_978
02Q 384
789
030 195
600
205
612
*oi8
246
653
*o59
287
693
*ICX)
327
734
♦140
368
775
*i8i
409
815
*22I
449
856
*262
490
896
♦303
531
937
*343
749
2
3
4
5
6
7
8
9
8.2
12.3
16.4
20.5
24.6
28.7
32.8
36.9
424
830
640
465
871"
276
681
506
546
587
627
668
708
316
721
952
357
762
992
397
802
*o33
438
843
*o73
478
883
*ii4
519
923
*i54
559
964
1074
1075
1076
031 004
408
812
045
449
853
085
489
893
126
530
933
166
570
974
206
610
*oi4
247
651
*o54
287
691
*o95
328
732
*i35
368
772
*i75
1077
1078
1079
1080
1081
1082
1083
032 216
619
033 021
256
659
062
296
699
102
337
740
142
377
780
182
4^7
820
223
458
860
263
498
901
J03
705
538
941
343
745
578
981
384
785
1
2
3
4
5
6
7
40
4.0
8.0
12.0
10.0
20.0
24.0
28.0
424
464
504
544
585
625
665
826
034 227
628
866
267
669
906
308
709
946
348
749
986
388
789
*027
428
829
*o67
468
869
*io7
50S
909
*I47
548
949
*i87
588
989
1084
1085
1086
035 029
430
830
069
470
870
109
510
910
149
550
950
190
590
990
230
630
♦030
270
670
♦070
310
710
*IIO
350
750
*i5o
390
790
♦190
8
9
32.0
36.0
1087
1088
1089
1000
1091
1092
1093
036 230
629
037 028
269
669
068
309
709
108
349
749
148
389
789
187
429
828
227
469
868
267
509
908
307
549
948
347
III
387
785
*i83
580
978
1
2
3
4
5
39
3.9
7.8
11.7
15.6
19 5
426
466
506
546
944
342
739
586
984
382
779
626
665
705
745
038 223
620
865
262
660
904
302
700
*024
421
819
*o64
461
859
*io3
501
898
*I43
541
938
1094
1095
1096
039 017
057
454
850
097
493
890
136
533
929
176
969
216
612
♦009
?55
652
♦048
295
692
*o88
335
731
*I27
374
771
*i67
6
7
8
23.4
27.3
31.2
1097
1098
1099
1100
040 207
602
998
246
642
*037
286
681
*o77
325
721
*ii6
761
♦156
405
800
*i95
590
444
840
*235
630
484
879
*274
669
523
919
*3i4
708
563
958
"353
748
9
35.1
041 393
432
472
511
551
N.
1
2
3
4
5
6
7
§
Prop. Pts.
TABLE II
LOGARITHMS
OF THE
TRIGONOMETRIC FUNCTIONS
FOR
EACH MINUTE
25
26
TABLE II
0^
2
3
_4_
5
6
7
8
_9_
10
II
12
13
;i
17
i8
19
20
21
22
23
24
26
27
28
29
30
31
32
33
34
36
37
38
39
40
41
42
43
44
II
49
60
51
52
53
il
ii
II
Jl
00
L. Sin.
6.46373
6.76476
6.94085
7.06 579
16 270
24 188
30882
36682
41 797
46373
50512
54291
57767
60985
63982
66 784
69417
71 900
74248
76475
78594
80615
82545
84393
86 166
87870
89509
91 088
92 612
94 084
95508
96 §87
98223
99520
8.00779
8 . 02 002
8.03 192
8.04350
8.05478
8.06578
8.07650
8.08696
8.09 718
8.10 717
II 693
12647
13 581
14495
15 391
8.16268
8.17 128
8.17971
8.18798
8.19 610
8 . 20 407
8.21 189
8.21 958
8.22 713
8.23456
8.24 186
L. Cos,
30103
17609
12494
9691
7918
6694
5800
5"5
4576
4139
3779
3476
3218
2997
2802
2633
2483
2348
2227
2119
2021
1930
1848
1773
1704
1639
1579
1524
1472
1424
1379
1336
X297
1259
1223
1190
1158
1128
IIOO
1072
1046
I022
999
976
954
934
914
896
877
860
843
827
812
797
782
769
755
743
.730
L. Tang, c. d. L, Cotg
46373
76476
94085
06579
16 270
24 188
30882
36682
41 797
46373
50512
54291
57767
60986
63982
66785
69418
71 900
74248
76476
78595
80615
82 546
84394
86167
87871
89 510
91 089
92613
94 086
95510
96889
98 225
99522
00 781
02 004
03 194
04353
05481
06581
07653
08 700
09 722
10 720
11 696
12 651
13585
14500
15395
16 273
17 133
17976
18804
19 616
20413
21 195
21 964
22 720
23462
8.24 192
L. Cotg.
30103
17609
12494
9691
7918
6694
5800
5"5
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
"59
1223
1x90
"59
Z128
ixoo
1072
X047
X022
998
976
955
934
915
895
878
860
843
828
812
797
782
769
756
742
730
c. d.
3 53627
3 23 524
3 05915
2.93421
2.83 730
2.75 812
2.69 118
2.63318
2 58 203
2.53627
2 . 49 488
2 45 709
2 42233
2.39014
2.36018
2 33215
2 30 582
2 28 100
2 25 752
2 23 524
2 21 405
2 19385
7 454
[5 606
13833
12 129
10 490
08 911
07387
05914
04 490
03 III
01 775
00478
99219
97996
96806
95647
94519
93419
92347
91 300
90278
89280
88304
87349
86415
85 500
84605
83727
82867
82 024
81 196
80384
79587
78805
78036
77 280
76538
75808
L. Tang,
89^
L. Cos.
000
000
000
000
000
000
000
000
000
oco
000
000
000
000
000
00 000
00 000
99 999
99 999
99 999
99 999
99 999
99 999
99 999
99 999
99 999
99 999
99 999
99 999
99998
99998
99998
99998
09998
99998
99998
99998
99997
99 997
99 997
99997
99 997
99 997
99997
99996
99996
99996
99996
99996
99996
99 995
99 995
99 995
99 995
99 995
99 994
99 994
99 994
99994
99 994
99 993
L. Sin.
35
34
33
32
3»
30
29
28
27
26
Prop. Pts.
3476
3218
.X
348
322
.2
695
644
•3
1043
965
•4
1390
1287
•5
1738
X609
2803
2633 1
.1
280
263
.2
560
527
•3
841
790
•4
1121
1053
•5
1401
X316
2227
203I
.1
223
202
.2
445
404
•3
668
606
•4
891
808
•5
IXX3
xoxo
1704
1579
.1
170
158
.2
341
316
•3
5"
474
■4
682
632
•5
852
789
1379
1297
.X
138
130
.2
276
259
•3
414
389
•4
552
519
•5
690
649
XX58
1x00
X16
110
232
220
347
330
463
440
579
550
999
954
.1
100
95
.2
200
191
•3
300
286
•4
400
382
•5
500
477
877
843
.1
88
84
.2
»75
169
•3
263
253
•4
351
337
•5
438
422
782
755
.1
78
75
.a
156
i5»
•3
235
226
• 4
313
302
•5
391
377
2997
300
599
899
1 199
1498
2483
24&
497
745
993
1242
1848
185
370
554
739
924
147a
H7
294
442
589
736
1223
122
245
367
489
612
X046
105
209
314
418
523
914
91
X83
274
366
457
8ia
81
162
244
325
406
730
73
146
219
29a
365
Prop. Pte.
LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS
27
L. Sill.
\
2
3
_4_
I
7
8
_9_
10
12
13
il.
;i
17
t8
20
21
22
23
24
26
27
28
29
31
32
33
34
36
37
38
ii
40
41
42
43
44
45"
46
47
48
60
51
52
53
il.
^i
II
59
60
24 186
24903
25 609
26304
26988
27 661
28 324
28977
29 621
30255
30879
31 495
32 103
32 702
33 292
33875
34450
35018
35578
36 131
36678
37217
37750
38276
38796
39310
39818
40320
40 816
41 307
41 792
42272
42 746
43216
43 680
44 139
44 594
45 044
45489
45930
46 366
46 799
47 226
47650
48 069
48485
48896
49304
49 708
50 108
50504
50897
51 287
51 673
52055
52434
52 810
53183
53552
53919
54282
L. Cos.
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
Tang.
24 192
24 910
25 616
26 312
26 996
27 669
28 332
28986
29 629
30263
30888
31 505
32 112
32 711
33302
33886
34461
35029
35590
36143
36689
37229
37762
38289
38809
39323
39832
40334
40830
41 321
41 S07
42 287
42 762
43232
43696
44 156
44 611
45 061
45507
45948
46385
46817
47245
47669
48089
48505
48917
49325
49 729
50 130
50527
50 920
51 310
51 696
52079
52459
52835
53208
53578
53 945
54308
L. Cotg.
c.d.
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
420
416
412
408
404
401
397
393
390
386
383
380
376
373
370
367
363
C.d.
L. Cotg.
75808
75090
74384
73688
73004
72331
71 668
71 014
70371
69 737
69 112
68495
67888
67 289
66698
66 114
65539
64971
64 410
63857
63 3"
62771
62238
61 711
61 191
60 677
60168
59 666
59 170
58679
58193
57713
57238
56768
56304
55844
55389
54 939
54 493
54052
53615
53183
52755
52331
51 9"
51495
51083
50675
50 271
49870
49 473
49 080
48 690
48304
47921
47541
47165
46 792
46 422
46055
.45692
L. Tang.
88°
L. Cos.
99 993
99 993
99 993
99 993
99992
99992
99992
99992
99992
99991
99991
99991
99990
99990
99990
99990
99989
99989
99989
99989
99988
99988
99988
99987
99987
99987
99 986
99 986
99 986
99985
99985
99985
99984
99984
99984
99983
99983
99983
99 982
99982
99982
99981
99981
99981
99 980
99 980
99 979
99 979
99979
99978
99978
99977
99 977
99977
99976
99976
99 975
99 975
99 974
99 974
9 99 974
L. Sin,
GO
59
58
57
_5i
55
54
53
52
_51
50
49
48
47
46
45
44
43
42
_iL
40
39
38
_3^
35
34
33
32
30
29
28
27
26
25
24
23
22
21
20"
19
18
17
16
15
14
13
12
II
To"
9
8
7
6
Prop. Pte.
717
695
.1
71.7
69.5
.2
M3-4
139.0
•3
215. 1
208.5
•4
286.8
278.0
• 5
358.5
347-5
653
634
.1
65.3
63-4
.2
130.6
126.8
•3
195-9
190.2
•4
261.2
253.6
•5
326.5
3170
599
583
59-9
58.3
119. 8
n6.6
179.7
174.9
239-6
233-2
299-5
291-5
553
539
.1
55-3
53-9
.2
no. 6
107.8
■3
165.9
161.7
•4
221.2
215.6
•5
276.5
269.5
514
503
51.4
50.2
102.8
100.4
154-2
150.6
205.6
200.8
257.0
251.0
480
470
.1
48
47
.2
96
94
•3
144
141
•4
192
188
•5
240
235
450
440
.X
45
44
.2
90
88
•3
135
132
•4
180
176
•5
225
220
430
410
.1
42
4»
.2
84
82
•3
126
123
•4
168
164
•5
210
205
390
380
39
38
78
76
117
114
156
152
>95
190
673
67.3
134.6
201 9
269.2
336.5
616
61.6
123.2
184.8
246.4
308.0
56.8
113.6
170.4
227.2
284.0
536
52.6
105.2
157.8
210.4
263.0
490
49
98
U7
196
245
460
46
92
138
184
230
430
43
86
129
172
215
400
40
80
120
160
300
37
74
III
148
Prop. Pts.
28
TABLE II
2'
3
±^
l
I
9_
10
II
12
13
:i
ii.
20
21
22
23
24
25
26
27
28
29
30
31
32
33
11
35
36
37
3«
39.
40
41
42
43
44
46
47
48
49
50
51
52
53
54
II
57
58
59
GO
L. Sin.
8 54 282
8 . 54 642
8.54999
8-55 354
8 . 56 054
8 . 56 400
8.56743
8.57084
8.57421
8.57757
8.58089
8.58419
8.58747
8.59072
8 59 395
8.59715
8 . 60 033
8.60349
8.60662
8.60973
8.61 282
8.61 589
8.61 894
8.62 196
8 . 62 49 7
8.62 795
8.63 091
8.6338c
8.6367
8.63968
8.64256
8.64543
8.64827
8.65 no
8.65391
8.65670
8.65947
8 . 66 223
8.66 497
8.66 769
8 ■ 67 039
8.67.308
8.67575
8.67841
8.68 104
8.68367
8.68627
8.68 886
8.69 144
8 . 69 400
8 . 69 654
8.69 907
8.70 159
8 . 70 409
8.70658
8 70 905
8.71 151
8.71 395
8.71638
8.71 880
L. Cos.
360
357
355
351
349
346
343
341
337
336
332
330
328
325
323
320
318
316
313
3"
309
307
305
303
301
298
296
294
293
290
288
287
284
283
281
279
277
276
274
272
270
269
267
266
263
263
260
259
258
256
254
253
252
250
249
247
246
244
243
242
L. Tang.
c.d.
8.54308
8 . 54 669
8.55027
8.55382
8 55 734
8 56~oS3"
8.56 429
8.56773
8.57 "4
8.57452
8.57788
8 58 121
8 58451
8 58779
8.59 105
8.59428
8 59 749
8.60068
8.60384
8.60698
8.61 009
8.61 319
8.61 626
8.61 931
8.62 234
8.62535
8.62834
8.63 131
8.63426
8.63718
8 . 64 009
8.64298
8.64585
8.64870
8.65 154
8.65435
8.65 715
8.65993
8.66269
8.66543
8.66816
8.67087
8.67356
8.67624
8 67890
8.68 154
8.68417
8.68678
8.68938
8.69 196
8.69453
8.69 708
8 . 69 962
8.70 214
8 . 70 465
8.70714
8 . 70 962
8.71 208
8.71453
8.71 697
8.71 940
L. Cotff.
361
358
355
352
349
346
344
341
338
336
333
330
328
326
323
321
319
316
3H
3"
310
307
305
303
301
299
297
295
292
291
289
287
285
284
281
280
278
276
274
273
271
269
268
266
264
263
261
260
258
257
255
254
252
251
249
248
246
245
244
243
L. Cotg.
1.45692
1-45 331
1.44 973
I .44 618
1 . 44 266
1-43917
I -43 571
1.43227
1.42886
1 . 42 548
I .42 212
1. 41 879
I -41 549
1 .41 221
1.40895
1.40572
1 .40 251
1-39932
1.39 616
1-39302
•38991
.38681
-38374
.38069
.37766
•27 ^§
.37 166
• 36869
-36574
. 36 282
1-35 991
1.35 702
1-35415
1-35 130
1.34846
1-34565
1-34285
1.34007
1-33 731
I 33 457
I 33 184
I 32913
1.32644
1.32376
I 32 no
C.d,
1. 31 846
I -31 583
1. 31 322
1 .31 062
I 30 804
I 30547
1 . 30 292
1 . 30 038
I . 29 786
1-29 535
1 . 29 286
1 . 29 038
I . 28 792
1.28547
1.28303
1.28060
L. Tang.
87^
L. Cos.
99 974
99 973
99 973
99972
99972
99971
99971
99970
99970
99969
99969
99 968
99 968
99967
99967
99967
99 966
99 966
99965
99964
99964
99963
99963
99 962
99 962
99961
99 961
99 960
99 960
99 959
99 959
99958
99958
99 957
99956
99956
99 955
9 99 955
9 99 954
9-99 954
99 953
99952
99952
99951
99951
99950
99 949
99 949
99948
99948
99 947
99946
99946
99 945
99 944
99 944
99 943
99942
99942
99941
99940
L. Sin.
GO
59
58
57
55
54
53
52
il
50
49
48
47
25
24
23
22
21
20
19
18
17
16
15
14
13
12
II
To
9
8
7
6
Prop. Pte.
360
350
36
35
72
70
108
105
144
140
180
175
216
210
• 7
252
245
.8
288
280
•9
324
315
330
320
1
33
32
3
66
64
3
99
96
4
132
128
5
165
160
6
108
192
7
231
224
8
264
256
9
297
288
300
290
385
30
29
28.
60
58
57
90
87
85-
120
116
114.
150
145
142.
.6
180
174
171.
•7
210
203
199.
.8
240
232
228.
•9
270
261
256
380
375
270
I
28.0
27-5
27-
.2
56.0
55-0
54-
•3
84.0
82.5
81.
•4
112.
IIO.O
108.
• 5
140.0
137-5
135-
.6
168.0
165.0
162.
-7
196.0
192.5
189.
.8
224.0
220.0
216.
•9
252.0
247 -5
243-
365
■ 26.5
•53-0
•79-5
106.0
132-S
159.0
1.85 5
212 o
260
.26.0
.52.0
.78.0
104.0
130.0
156.0
182.0
208.0
234.0
250
245
.25.0
•24.5
.50
-49
•75
•73
5
100
198
125
122
5
.6
150
»47
'75
171
5
200
196
•9
225
320
5
Prof). Pts.
LOGARITHMS OP THE TRIGONOMETRIC FUNCTIONS 29
3°
2
3
j4
I
7
8
9
10
II
12
13
J4_
;i
\i
Ji_
20
21
22
23
24
26
27
28
29
30
31
32
33
36
37
38
40
41
42
43
44
45
46
47
48
19.
50
51
52
53
54.
55
56
II
59
(JO
L. Sin.
8 74 226
8.74454
8.74680
8 . 74 906
■ 75 130
71 880
72 120
72359
72597
72834
73069
73303
73 535
73767
73 997
75 353
75 575
75 795
76015
76234
76451
76 667
76883
77097
77310
77522
77 733
77 943
78152
78360
78568
78774
78979
79183
79386
79588
79789
79990
80189
80388
80585
80782
80978
81 173
81367
8.B1 560
8.81 752
8.81 944
8.82 134
8.82324
82513
82 701
82888
83075
83261
83446
83630
83813
83996
84177
8.84358
L. Cos.
240
239
238
237
23s
234
232
233
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
20-i.
201
201
199
199
197
197
196
194
193
192
192
190
190
189
188
187
187
186
185
184
183
183
181
L. Tang.
c. d.
71 940
72 181
72 420
72659
72 896
73366
73 600
73832
74063
74292
74521
74748
74 974
75 199
75423
75645
75867
76087
76306
76525
76742
76958
77173
77387
77 600
77 811
78 022
78232
78441
78649
78855
79 061
8 79 266
8.79470
79673
79875
80076
80 277
80476
80 674
80872
81 068
81 264
81 459
8.81 653
8 81 846
8 82038
8 82 230
8 82 420
8 82610
8 82 799
8 82987
883175
8.83361
8 83547
8 83732
8 83 916
8 84 100
8 84282
8,84464
L. Cotg.
241
239
239
237
236
234
234
232
231
229
229
227
226
225
224
222
222
220
219
219
217
216
215
214
213
211
209
208
206
206
205
204
203
202
201
20I
199
198
I9&
196
196
194
193
192
192
190
190
189
186
186
185
184
184
182
L. Cotg.
1 . 28 060
1.27 819
1.27 580
I 27341
I .27 104
1.26868
1.26634
1 . 26 400
1.26 168
1-25937
I . 25 708
I 25 479
I .25 252
I .25 026
1 . 24 801
I 24577
1-24355
I 24 133
1-23913
1.23694
I 23 475
1.23258
1.23042
1.22 827
I .22 613
1 . 22 400
I .22 189
1. 21 978
1. 21 768
I-21 559
I 21 351
I 21 145
1.20939
1.20734
1.20530
c. d.
1 . 20 327
I .20 125
I 19 924
I 19 723
I 19 524
1. 19 326
1. 19 128
1 . 18 932
1. 18 736
1. 18 541
18347
18 154
17 962
17770
17580
1. 17 390
1.17 201
1.17013
1. 16 825
I 16 639
I 16453
1 . 16 268
1 . 16 084
1 . 1 5 900
1.15 718
15 536
L. Tang.
80°
L. Cos.
9 99940
9 99940
9 99 939
9 99938
9 99938
9 99 937
9 99936
9.99936
9 99 935
9 99 934
9 99 934
9 99 933
9.99932
9 99932
9 99931
9 99930
9 99 929
9.99929
9.99928
9.99927
9.99926
9.99926
9-99925
9-99924
9.99923
9.99923
9.99922
9 99921
9.99920
9.99920
9.99919
9.99918
9.99917
9 99917
9.99916
9 99915
9.99914
9 99913
9 99913
9.99912
9 99911
9 99910
9.99909
9.99909
9.99908
9,99907
9.99906
9 99905
9 99904
9 99 904
9.99903
9.99902
9.99901
9.99900
9.99899
9.99898
9.99898
9.99897
9 99896
9 99895
9.99894
L. Sin.
Prop. Pts.
238
334
sag
.1
23.8
23 -4
22
.2
47.6
46. «
45
•3
71.4
70.2
68
•4
95-2
93-6
91
•5
119.0
117.
114
.6
142.8
140.4
137
•7
166.6
163.8
160
.8
190.4
187.2
183
•9
214.2
210.6
206
225
220
.1
22.5
22.0
2
45.0
44.0
•3
67.5
66.0
•4
90.0
88.0
•5
112.5
IIO.O
.6
135 -o
132.0
■7
157-5
I54-0
.8
180.0
176.0
9
202.5
198.0
212
208
21.2
20.8
42.4
41.6
63.6
62.4
84.8
83.2
106.0
104.0
127.2
124.8
-7
148.4
145.6
.8
169.6
166.4
•9
190.8
187.2
201
197
.1
20.1
19.7
.2
40.2
39-4
•3
60.3
59-1
•4
80.4
78.8
•5
100.5
98.5
.6
120.6
118.2
•7
140.7
137-9
.8
160.8
157-6
•9
180.9
J77-3
i8g
18.9
37-8
56.7
75 6
94-5
"3 4
132.3
151.2
170.1
185
18.5
37-0
55-5
74 -o
92.5
III.O
129.5
148.0
166. 5
43a
1 0.4 0.3 0.2 c
2 0.8 0.6 0.4 o
3 1.2 0.9 0.6 o
4 1.6 1.2 o.'^ o
5 2.0 1.5 1.0 o
6 2.4 1.8 1.2 o
7 2.8 2.1 1.4 o
8 3.2 2.4 1.6 o
9 3.6 2.7 1.8 o
216
21.6
43-2
64.8
86.4
108.0
129.6
151.2
172.8
194.4
204
20.4
40.8
61.2
81.6
102.0
122.4
142.8
163.2
183 6
193
19-3
38.6
57-9
77.2
96.5
1158
135 I
154-4
173-7
181
18. 1
36.2
54-3
72.4
905
108.6
126.7
144.8
162.9
Prop. Pts.
30
TABLE II
_9
10
II
12
13
ii_
\i
18
20
21
22
23
24
25
26
27
28
29
30
31
32
33
il
36
37
3«
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
L. Sin.
8.84358
8 84539
8.84 718
8 84897
8 85 075
8 85 252
8.85429
8 85 605
8 85 780
8 85955
8 86 128
8 86 301
8 86 474
8 86 645
8 86816
8 86 987
8 87 156
8 87325
8 87494
8 87661
8 87 829
8 87 995
8 88 161
8 88 326
8 88 490
8 88 654
8 88817
8.88980
8 89 142
8 89 304
8 89 464
8 89 625
8 89 784
8.89943
8 90 102
8 90 260
8 90 417
8 90574
8 90 730
8 90 885
8 91 040
8 91 195
8 91 349
8 91 502
8 91 655
8 91 807
8 91.959
8 92 no
8 92 261
8 92 411
8 92 561
8 92 710
8 92 859
8 93 007
8 93 ^54
8 93 301
8 93448
8 93 594
8.93 740
8 93 885
8.94030
L. Cos.
179
179
178
177
177
176
>75
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
'55
155
154
153
153
152
152
151
151
150
150
149
149
148
147
147
147
146
146
H5
145
L. Tang.
8.84464
8.84646
8.84826
8.85006
8.85 185
c.d.
8 85 363
8.85 540
8.85717
8.85893
8.86 o6q
8 86 243
8.86417
8.86591
8.86 763
8.86935
8.87 106
8.87277
8.87447
8.87616
8 87 785
8.87953
8.88 120
8.88287
8.88453
8.88618
8.88783
8.88948
8.89 III
8.89274
8.89437
8 89598
8.89 760
8 . 89 920
8 . 90 080
8 . 90 240
8.90399
8 90557
8 90715
8.90872
8 91 029
8.91 185
8.91 340
8.91 495
8 91 650
8.91 803
91 957
o 92 no
8.92 262
8.92 414
8.92 565
8.92 716
8.92866
8.93 016
8.93 165
8 93 S13
8 93 462
8 9^ 609
8 93 756
8 93903
8 94 049
8 94 '95
182
180
180
179
178
177
177
176
176
174
^74
174
172
172
171
171
170
169
169
168
167
167
1 66
165
165
165
163
163
163
i6i
162
160
160
160
159
158
158
157
157
156
155
155
155
153
154
153
152
152
151
151
150
ISO
149
I J*8
149
I X47
147
147
I 146
j 146
L. Cotg.
I 15 536
I 15 354
I 15 174
1 . 14 994
I 14 815
1. 14 637
I 14 460
I 14 283
I 14 107
I 13 931
13757
13583
13409
13237
13065
1 . 12 894
I 12 723
I 12 553
I 12 384
1 . 12 215
12047
n 880
II 713
II 547
11382
I II 217
III 052
1 . 10 889
1 . 10 726
I 10 563
1 . 10 402
1 . 10 240
1 . 10 080
1 . 09 920
I . 09 760
I 09 601
1.09443
I 09 285
I .09 128
1.08 971
I 08815
1 .08 660
1.08505
I 08350
I .08 197
I 08 043
1.07 890
1.07738
1.07586
I 07 435
1 .07 284
I 07 134
I 06 984
1.06 835
1.06687
To6~538"
1.06 391
I 06 244
1 .06 097
I 059 51
I 05 805
L. Cos.
L. Cotg. ic. d.l L. Tang.
85^
99894
99893
99892
99891
99891
99 890
99889
99888
99887
99886
99885
99884
99883
99882
99881
99880
99879
99879
99878
99877
99876
99875
99874
99873
99872
99871
99 870
99 869
99868
99867
99866
99865
99 864
99863
99 862
. 99 861
9 99 860
. 99859
9 99 858
9 99 857
9 99 856
9 99855
99854
99853
99852
99851
99850
99848
99847
99 846
99845
99844
99843
99842
99841
99 840
99839
99838
99837
99836
99834
L. Sin.
GO
59
58
57
55
54
53
52
51
50
49
48
47
45
43
42
41
40
39
38
36
35
34
33
32
31
30
29
28
27
26
Prop. Pts.
181
179
I
18.1
17.9
.2
36.2
35-8
•3
54-3
53-7
•4
72.4
71.6
•5
90-5
89.5
.6
108.6
107.4
•7
126.7
125.3
.8
144.8
143-2
•9
162.9
161. 1
175
X73
.1
17-5
17-3
.2
35.0
34-6
•3
52-5
51-9
•4
70.0
69.2
•5
87.5
86.5
.6
105.0
103.8
■7
122.5
121. 1
.8
140.0
138.4
•9
157-5
155.7
168
166
I
.16.8
16.6
3
33-6
33-2
•3
50-4
49-8
•4
67.2
66.4
•5
84.0
83.0
.6
100.8
99-6
•7
117. 6
116.2
.8
134-4
132.8
•9
151-2
149.4
162
159
16.2
159
32-4
31-8
48.6
47-7
64.8
63.6
81.0
79-5
6
97-2
95-4
•7
"3-4
III. 3
.8
129.6
127.2
•9
145.8
I43-I
155
153
I
15 5
153
2
31-0
30.6
3
46.5
45-9
4
62.0
61.2
5
77-5
76.5
.6
93-0
91.8
•7
108.5
107.1
.8
124.0
122.4
•9
139-5
'37-7
149
147
14.9
M-7
29.8
29.4
44-7
59-6
44.1
58.8
.6
74-5
894
73-5
S8.a
•7
.8
104 3
119.2
102.9
1176
•9
134 -I
132 3
Prop. Pts.
LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS
31
2
3
_4
5'
6
7
8
9
10
II
12
13
14
15
i6
17
i8
20
21
22
23
26
27
28
29_
30
31
32
33
34
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
J9
60
L. Sin.
94030
94 174
94317
94461
94603
94 746
94887
95029
95 170
95310
95450
95589
95 728
95867
96 005
96 143
96 280
96417
96553
96 689
96825
96 960
97095
97229
97363
97496
97629
97762
97894
98 026
98157
98288
98419
98549
98679
98808
98937
99 066
99 194
99322
99450
99 577
99 704
99830
99956
00 082
00 207
00332
00 456
00 581
00 704
00828
00951
01 074
01 196
01 318
01 440
01 561
01 682
01 803
01 923
L. Cos.
L. Tang.
94195
94340
94485
94630
94 773
94917
95 060
95 202
95 344
954S6
95627
95 767
95 908
96 047
96 187
96325
96 464
96 602
96 739
96877
97013
97 150
97285
97421
97556
97691
97825
97 959
98 092
98 225
98358
98 490
98 622
98753
98884
99015
99 145
99275
99405
99 534
99 662
99 791
99919
00 046
00 174
00 301
00 427
00553
00 679
00 805
00 930
01 055
01 179
01 303
01 427
01 550
01 673
01 796
01 918
02 040
02 162
Cotg.
c.d.
c.d.
L. Cotg.
1.05 805
1 . 05 660
I 05 515
1.05 370
05 227
05 083
04 940
04 798
04 656
04514
04373
04233
04 092
03953
03813
03675
03536
03398
03 261
03 123
02 987
02 850
02 715
02579
02444
02 309
02 175
02 041
01 908
01 775
01 642
01 510
01378
01 247
01 116
1 . 00 985
00 855
I 00 725
I 00595
I 00 466
I 00338
1 . 00 209
I 00 081
o 99 954
o 99 826
0.99699
o 99 573
0.99447
0.99321
0.99 195
o . 99 070
o 98945
0.98821
0.98 697
0.98573
o . 95 450
0.98327
o . 98 204
o . 98 082
0.97 960
0.97
L. Tang.
84°
I
.Cos.
60
It
11
55
54
53
52
51
9
9
9
9
9
99834
99833
99832
99831
99830
9
9
9
9
9
99829
99828
99827
99825
99824
9
9
9
9
9
99823
99822
99821
99 820
99819
50
49
48
47
46
45
44
43
42
41
40
1
9
9
9
9
9
99817
99 816
99815
99814
99813
9
9
9
9
9
99812
99 810
99809
99808
99807
9
9
9
9
9
99806
99804
99803
99802
99801
35
34
33
32
31
30
29
28
11
9
9
9
9
9
99800
99798
99 797
99796
99 795
9
9
9
9
9
99 793
99792
99791
99790
99788
25
24
23
22
21
9
9
9
9
9
997S7
99786
99785
99783
99782
20
19
18
\l
IS
14
13
12
II
9
9
9
9
9
99781
99780
99778
99777
99776
9
9
9
9
9
99 775
99 773
99772
99771
99769
10
7
6
9
9
9
9
9
99768
99767
99765
99 764
99763
5
4
3
2
I
9
99761
I
.Sin.
f
Prop. Pts.
145
X43
141
14.5
14.3
14.
ap.o
28.6
28.
43-5
42.9
42
58.0
57-2
56
72-5
71-3
70
6
87.0
85.8
84
7
101.5
100. 1
98.
8
116.0
114.4
112.
9
130.5
128.7
126.
139
,^3-9
27
41
55
69
83
97
138
13 ■
27
41
55
69
82
96
no
.•24
135
133
.1
13.5
13.3
3
27.0
26.6
3
40.5
39-9
4
540
53-2
5
675
66.5
6
81.0
79.8
7
94-5
93-1
8
108.0
106.4
9
121.5
119.7
139
128
.1
12.9
12.8
.2
25.8
25.6
•3
38.7
38.4
•4
51.6
51.2
•5
645
64.0
.6
77-4
76.8
•7
90-3
89.6
.8
103.2
t02.4
•9
116. 1
115 2
135
133
.t
"•5
12.3
.2
25
24.6
3
37
5
36.9
4
50
49.2
•5
62
5
61.5
6
75
73-8
7
87
5
86.1
8
100
98.4
9
112
5
110.7
X3I
xao
.1
12. 1
12.0
.2
24.2
«4.o
•3
36.3
36.0
.4
48.4
48.0
•5
60.5
60.0
.6
72.6
72.0
7
847
84.0
.8
96.8
96.0
•9
108.9
108.0
136
13
27
40
54
131
13-1
26.2
39-3
52.4
65.5
78.6
91.7
104.8
117.9
136
12.6
25.2
37-8
50.4
63.0
75-6
88.2
100.8
"34
122
12.2
24.4
36.6
48.8
61.0
73-2
85.4
97.6
109.8
z
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Prop. Pts.
32
TABLE II
6^
__9_
10
II
12
13
14
15
i6
17
i8
19
20
21
22
23
il
26
27
28
29
31
32
33
34
36
37
38
39
40
41
42
43
44
46
47
48
49
50
51
52
53
il.
55
56
57
58
i9_
60
L. Sin.
01 923
02 043
02 163
02 283
02 402
02 520
02 639
02 757
02 874
02 992
03 109
03 226
03342
03458
03574
03 690
03 805
03 920
04034
04 149
04 262
04376
04 490
04 603
04 715
04828
04 940
05 052
05 164
05275
05386
05497
05 607
05 717
05 827
05937
06 046
06155
06 264
06372
06 481
06 589
06 696
06 804
06 911
07018
07 124
07231
07337
07442
07548
07653
07758
07863
07968
08 072
08 176
08280
08383
08486
08589
L. Cos.
d. L. Tang.
02 162
02 283
02 404
02 525
02 645
02 766
02885
03005
03 124
03 242
03 361
03479
03597
03 714
03832
03948
04 065
04 181
04297
04413
04 528
04 643
04758
04873
04987
05 lOI
05214
05 328
05441
05553
05 666
05 778
05 890
06 002
06 113
06 224
06335
06445
06 556
06666
06775
06885
06 994
07 103
07 211
07 320
07428
07536
07643
07751
07858
07964
08 071
08 177
08 283
08389
08495
08 600
08 705
08810
08 914
d« L. Cotg.
c. d.
c. d,
L. Cotg.
0.97838
0.97717
0.97 596
0.97475
0.97355
0.97234
0.97 115
0.96995
0.96 876
0.96 758
0.96639
0.96 521
o . 96 403
o . 96 286
0.96 168
0.96 052
0.95935
o 95 819
0.95 703
o 95587
0.95472
o 95 357
0.95 242
0.95 127
o 95013
0.94899
0.94 786
0.94 672
o 94 559
0.94447
0.94334
o 94 222
0.94 no
0.93998
0.93887
o 93 776
0.93665
o 93 555
o- 93 444
o 93 334
0.93 225
o 93 "5
o . 93 006
0.92 897
0.92 789
0.92 680
0.92 572
0.92 464
0.92357
0,92 249
0.92 142
o . 92 036
0.91 929
0.91 823
0.91 717
0.91 oil
o 91 505
0.91 400
0.91 295
0.91 190
0.91 086
L. Tang,
83°
L. Cos.
99 761
99760
99 759
99 757
99756
99 755
99 753
99 752
99 751
99 749
99 748
99 747
99 745
99 744
99 742
99741
99 740
99 738
99 737
99736
99 734
99 733
99731
99 730
99 728
99727
99 726
99 724
99 723
99721
99 720
99718
99717
99 716
99714
99 713
99 711
99710
99 708
99707
99705
99 704
99702
99 701
99699
99 698
99 696
99695
99693
99 692
9.99690
9.99689
9.99687
9 . 99 686
9 99684
99683
99 681
99 680
99678
99677
9 99675
L. Sin.
Prop. Pts.
lai
xao
I
12.1
12.0
.3
24.2
24.0
3
36.3
36.0
4
48.4
48.0
5
60.5
60.0
6
72.6
72.0
7
84.7
84.0
8
96.8
96.0
9
108.9
108.0
118
117
.1
II. 8
II. 7
.2
236
23
4
.3
35-4
35
I
•4
47.2
46
8
•5
59.0
58
5
.6
70.8
70
2
•7
82.6
81
9
.8
94-4
93
6
■9
106.2
los
3
"5
114
.1
"•5
II. 4
.2
33.0
22.8
•3
34-5
34-2
'4
46.0
45-6
.5
57-5
570
.6
69.0
68.4
• 7
80.5
79.8
.8
92.0
91.2
.9
103.5
102.6
iia
III 1
.1
II. 2
II. I
.2
22
4
22.2
•3
33
6
33-3
.4
44
8
44-4
•5
56
55-5
.6
67
2
66.6
•7
78
4
77-7
.8
89
6
88.8
•9
100
8
99.9
109
108
lOJ
.1
10.9
10.8
10
.3
31
8
21.6
21
•3
32
7
32.4
32
•4
43
6
43-2
42
•5
54
5
54 -o
53
.6
65
4
64.8
64
•7
76
3
75.6
74
.8
87
2
86.4
85
•9
98
I
97.2
96
X06
IC.5
I
10.6
10.5
.3
21.2
21.
•3
31-8
31 5
•4
42.4
42.0
•5
.6
53
63.6
52 5
63.0
.7
.8
74.2
84.8
73-5
84.0
•9
95-4
94-5
Prop. Pts.
LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS
33
L. Sin.
9_
10
II
12
13
\l
17
i8
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
36
37
38
39
40
41
42
43
44
46
47
48
49
50
51
52
53
II
II
59
GO
08589
08 692
08795
08897
08 999
09 lOI
09 202
09304
09405
09 506
09 606
09 707
09 807
09907
0006
o 106
o 205
0304
0402
o 501
0599
0697
0795
0893
o 990
I 087
1 184
1 281
1377
I 474
I 570
I 666
I 761
1857
1952
2047
2 142
2 236
2331
2425
2519
2 612
2 706
2 799
2 892
2985
3078
3 171
3263
3 355
3 447
3 539
3630
3 722
3813
3904
3 994
4085
4175
4 266
4356
L. Cos.
103
103
102
102
1 03
101
1 03
lOI
lOI
100
lOI
100
100
•99
100
99
99
98
99
98
98
98
98
97
97
97
97
96
97
96
96
95
96
95
95
95
94
95
94
94
93
94
93
93
93
93
93
92
92
92
92
91
92
91
91
90
91
90
91
90
L. Tang.
c. d.
08 914
09 019
09 123
09 227
09330
09434
09537
09 640
09 742
09845
09947
o 049
o 150
o 252
0353
0454
o 656
0756
0856
0956
I 056
1 155
1 254
I 353
452
649
747
845
1943
2 040
2138
2235
2332
2428
2525
2 621
2 717
2813
2 909
3004
3099
3 194
3289
3384
3478
3 573
3667
3 761
3854
3948
4041
4 134
4227
4320
4412
4504
4 597
4688
4780
L. Cotgr.
c. d.
L. Cotg.
0.91 086
0.90 981
0.90 877
0.90 773
0.90 670
o . 90 566
0.90463
o . 90 360
0.90 258
0.90155
0.90053
0.89 951
o . 89 850
0.89 748
o . 89 647
0.89 546
0.89445
0.89344
o . 89 244
0.89 144
o . 89 044
o . 88 944
0.88845
0.88 746
0.88647
0.88548
o . 88 449
0.88351
0.88253
0.88 155
0.88057
0.87960
c. 87 862
0.87 765
0.87668
0.87572
0.87475
0.87379
0.87283
0.87 187
0.87091
0.86 996
0.86 901
0.86806
0.86 711
0.86616
0.86 522
0.86 427
0.86333
o . 86 239
0.86 146
0.86 052
0.85959
0.85 866
0.85 773
0.85 680
0.85588
0.85 496
0.85403
0.85 312
0.85 220
L. Tang.
82°
L* Cos.
9.99667
9 . 99 666
9.99664
9.99663
9.99 661
99675
99674
99672
99670
99 669
99659
99658
99656
99655
99653
99651
99650
99 648
99647
99645
99643
99642
99 640
99638
99637
99635
99633
99632
99630
99 629
99627
99625
99624
99 622
99 620
99 618
99617
99615
99613
99 612
99 610
99 608
99607
99605
99603
9.99 601
9.99 600
9-99 598
99596
99 595
99 593
99591
99589
99588
99586
99584
99582
99581
99 579
99 577
9-99 575
L. Sin.
60
59
58
57
_5i
55
54
53
52
_5i
50
49
48
47
_46
45
44
43
42
41
40
39
38
_36
35
34
33
32
_3i
30
29
28
27
26
Prop. Pt8.
105
104
I
IO-5
10.4
.2
21.0
20.8
•3
31 5
31.2
•4
42.0
41.6
.5
52. s
52.0
.6
63.0
62.4
•7
73-5
72.8
.8
84.0
832
•9
94 5
93 6
102
lOI
I
10.2
10. I
2
20.4
20.2
3
•4
i
30.6
40.8
61 .2
303
40.4
I
l\:t
£i
■9
9.. 8
90.9
9S
)
98
I
9 9
9
?
19
8
19.
•3
29
7
29.
•4
39
6
39-
49
5
49-
.6
59
4
58.
7
69
3
68.
8
79
2
78.
9
89
I
88.
97
9 7
19
29
38
48
58
67
77
87
94
9-4
18.8
28.2
37 6
47.0
56.4
65.8
75-2
84.6
9e
' 1
9.6
19.2
28.8
38.4
48.0
57-6
67.2
76.8
86.4
93
9 3
18
6
27
9
37
2
46
55
1
65
I
74
4
83
7
103
10.3
20.6
309
41 .2
6i'8
72.1
82.4
92.7
100
10.0
20.0
30.0
40.0
50.0
60.0
70.0
80.0
90.0
4
2
95
9 5
19 o
285
380
47-5
91
90
.1
9 ^
9.0
.2
18
2
18.0
•3
27
3
27.0
•4
?6
4
36.0
.5
45
5
45
.6
54
b
.54
• 7
63
7
63.0
.8
r.
8
72.0
9
9
81.0
92
9.2
18.4
27.6
36.8
46.0
64.4
73 6
82.8
a
0.2
0.4
0.6
0.8
i.o
1.2
1-4
1.6
1.8
Prop. Pts.
34
TABLE II
8°
L. Sin,
10
II
12
13
14
15
i6
17
i8
i9_
20
21
22
23
24
26
27
28
30
31
32
33
34
36
37
38
39
40
41
42
43
44
46
47
48
j49
50
51
52
53
54
59
60
E
4356
4445
4535
4624
4 714
4803
4891
4 980
5069
5 J57
5245
5 333
5421
5508
5596
5683
5 770
5857
5 944
6030
6 116
6 203
6289
6374
6 460
6545
6631
6 716
6801
6 886
6970
7055
7 139
7223
7307
7391
7 474
7558
7641
7724
7807
7890
7 973
8055
8137
6 220
8 302
8383
8465
8547
8628
8 709
8790
8871
8952
9033
9 "3
9 193
9273
9 353
9 19433
L. Cos,
d.
89
L. Tan^.
4780
4872
4963
5054
5 145
5236
5327
5417
5 "^08
5598
5 688
5 777
5867
6 046
6135
6 224
6312
6 401
6489
6577
6665
6753
6841
6928
7 016
7103
7 190
7277
7363
7450
7536
7 622
7708
7 794
7880
7965
8051
8 136
8221
8306
8391
8475
8560
8644
8728
8812
8896
8979
9063
9 146
9 229
9312
9 395
9478
9561
9643
9725
9807
9889
9.19971
L. Cotg. c. d
c.d.
L. Cotg.
0.85 220
o 85 128
o 85037
o 84 946
0.84855
o 84 764
o . 84 673
084583
o . 84 492
o . 84 402
84312
84223
84133
84 044
83954
83865
83776
83688
83599
83 5"
0.83423
o 83335
0.83247
0.83 159
o . 83 072
0.82984
0.82897
0.82810
o 82 723
o 82 637
0.82 550
o . 82 464
0.82378
0.82 292
o 82 206
0.82 120
o 82 035
o 81 949
o 81 864
o 81 779
o 81 694
0.81 609
o 81 525
o 81 440
0.81 356
0.81 272
0.81 188
0.81 104
o 81 021
o 80 937
o 80 854
o 80 771
o 80688
o . 80 605
0.80 522
o . 80 439
0.80357
0.80275
0.80 193
0.80 III
. 80 029
L. T,aiig.
8r
L. Cos.
9 99 575
9 99 574
9 99 572
9 99 570
9 99 568
99566
99565
99563
99561
99 559
99 557
99556
99 554
99552
99550
99548
99 546
99 545
99 543
99541
99 539
99 537
99 535
99 533
99532
99530
99528
99526
99524
99522
99520
99518
99517
99515
99513
99 5"
99509
99507
99505
99503
99501
99 499
99 497
99 495
99 494
99492
99490
99488
99 486
99484
99482
99 480
99478
99476
99 474
99472
99470
99 468
99 466
99464
99462
L. Sin,
GO
59
58
57
_5i
55
54
53
52
_51
50
49
48
47
_46^
45
44
43
42
41
40
39
38
37
36
Prop. Pts.
9i
91
1
I
9.2
9.1I
2
18
4
2
3
27
6
27
3
4
36
8
36
4
<;
46
45
5
6
55
2
S4
6
7
64
4
63
7
8
73
6
72
8
9
82
8
81
9
90
9.0
18.0
27.0
36.0
45.0
54 o
63.0
72.0
81.0
89
87
17
26
34
43
52
60
69
78
85
17
25
34
42
51
59
68
76
83
8
16
24
33
41
49
58
66
74
8.8
17.6
26.4
35 2
44 o
52.8
61.6
70.4
79 2
86
i.6
17.2
25.8
34 4
43 o
51 6
60.2
68.8
77 4
84
8.4
16.8
25.2
33-6
42.0
50 4
58.8
67 2
75.6
16
24
32
41
49
57
65 6
738
81
80
I
8.1
8.0
2
16
2
16.0
3
24
3
24.0
4
32
4
32.0
s
40
5
40.0
6
48
6
48.0
7
S6
7
56.0
8
64
8
64.0
9
72
9
72.0
3
0.2
0.4
0.6
0.8
1 .0
1.2
1-4
1.6
1.8
Prop. Pts.
LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 35
9° 1
L. Sin.
d.
L. Tan^.
c.d.
L. Cot^.
L. Cos.
Prop. Pts.
9 19433
80
9 19 971
82
80 029
9 99462
(JO
I
9 19 513
9.20053
79 947
9.99460
59
83 81 80
2
9 19592
80
9 20 134
82
79 866
9 99 458
58 .1
8.2 8.1 8.0
3
9 19672
79
79
9 20 216
79 784
9 99 456
57 2
16.4 16.2 16.0
4
5
9 19 751
9 19 830
9 20 297
81
81
0.79703
9 99 454
56 .3
55 -4
24.6 24.3 24.0
32.8 32.4 32.0
9 20 378
. 79 622
9 99 452
6
9 19909
9 20 459
79541
9 99 450
54 -5
41 .0 40.5 40.0
7
9 19988
79
9 20 540
. 79 460
9 99 448
53 ^
49.2 48.6 48.0
8
9 20 067
79
78
78
9 20 621
80
81
0.79379
9 99446
52 •/
57.4 56.7 56.0
9
10
9 20 145
9 20 701
9 20 782
79 299
9 99 444
51 .^
50 5
65 6 64.8 64.0
73.8 72.9 72.0
9 20 223
79 218
9 99 442
II
9 20 302
78
9 20 862
80
79 138
9 99440
49
79
78
12
9 20 380
9 20 942
. 79 058
9 99438
48
.1 7.9
7-8
13
9 20 458
78
9 21 022
80
0.78978
9 99436
47
.2 15.8
15.6
14
9 20535
77
78
9.21 102
80
0.78898
9 99 434
46
45
.3 23.7
4 31 6
23 4
31.2
9 20 613
9
21 182
0.78818
9 99432
i6
9.20691
78
9
21 261
79
0.78739
9 99 429
44
5 39-5
2^2
17
9 20 768
77
9
21 341
80
0.78 659
9 99427
43
6 47.4
46.8
i8
9 ■ 20 845
77
9
21 420
79
0.78 580
9 99 425
42
l I^^
54 6
19
20
9 20 922
77
77
9
21499
79
79
0.78 501
9 99423
41
40
.8 63.2
•9l 71 I
62.4
70.2
9.20999
9
21578
0.78422
9 99 421
21
9.21 076
77
9
21 657
79
78343
9.99419
39
77
76
22
9 21 153
77
76
9
21 736
79
0.78264
9.99417
38
.1 7-7
7.6
23
9.21 229
9
21 814
78
0.78 186
9 99415
37
.2 15 4
^5-^
24
9.21 306
77
76
9
21893
79
78
0.78 107
9 99413
36
35
■3 23.1
.4 30.8
22.8
30.4
2S
9.21 382
9
21 971
. 78 029
9 99 411
26
9.21 458
76
9
22049
78
0.77951
9.99409
34
l ^l^
38.
45-6
27
9 21 534
76
9
22 127
78
0.77873
9.99407
33
.6 46.2
28
9.21 610
76
9
22 205
78
0.77795
9.99404
32
I in
9 693
^:8
68.4
29
30
9.21 685
75
76
9
22 283
78
78
0.77717
9 99402
31
80
9.21 761
9
22 361
0.77639
9.99400
31
9.21 836
75
9
22438
77
0.77562
9 99398
29
75
74
32
9.21 912
76
9
22 516
78
0.77484
9.99396
28
.1 75
7-4
33
9.21 987
75
9
22593
77
0.77407
9 99 394
27
.2 15.0
14 8
34
35
9 22 062
75
75
9
22 670
77
77
0.77330
9 99392
26
25
•3 22.5
•4 30
22.2
29.6
9.22 137
9
22747
0.77253
9 99 390
3^
9.22 211
74
9
22 824
77
0.77176
9 99 388
24
•5 37 5
.6 45.0
37
37
9 22 286
75
9
22 901
77
0.77099
9 99385
23
Tii
3«
9.22 361
75
9
22977
7b
0.77023
9 99383
22
.8 60.0
■ 9 67 5
39
40-
9 22435
74
74
9
23054
77
76
0.76946
9 99381
21
20
Iti
9 22509
9
23 130
0.76 870
9 99 379
41
9.22583
74
9
23 206
76
0.76 794
9 99 377
19
73
72
42
9 22657
74
9
23283
77
0.76 717
9 99 375
18
• I 7-3
7.2
43
9.22 731
74
9
23359
7b
0.76 641
9 99 372
17
.2 14.6
144
44
9.22805
74
73
9
23435
7b
75
0.76565
9 99370
16
15
.3 21.9
4 29.2
5 36.5
6 43 8
7 51 I
.8 58 4
•9 65.7
21.6
28.8
36.0
43-2
50 4
57 6
64.8
45
9 22878
9
23510
0.76490
9.99368
46
9.22952
74
9
23 586
7b
0.76414
9.99366
14
' 47
9 23025
73
9
23661
75
0.76339
9 99364
13
,4«
9 23 098
73
9
23737
7b
. 76 263
9 99362
12
49
60
9 23 171
73
73
9
23812
15
75
76 188
9 99 359
II
10
9 . 23 244
9
23887
0.76 113
999 357
51
9 23317
73
9
23962
75
0.76038
9 99 355
?> T
71
3 3
52
9 23390
73
9
24037
75
0.75963
0.75888
9 99 353
8 I
7.1 c
.6 0.4
.9 0.6
.2 0.8
5 '0
53
9.23462
72
9
24 112
75
9 99351
7 1
14.2 c
55
9 23535
73
72
9
24 186
74
75
0.75 814
9-99 348
6 3
5 1
4 6
21.3 c
28.4 I
35-5 I
42.6 I
49 7 2
56.8 2
63.9 2
9.23607
9
24 261
0.75739
9 -99 346
5t>
9 23679
72
9
24335
74
0.75665
9 99 344
.8 1 .2
H
9 23 752
73
9
24410
75
0.75590
9 99342
I -7
8
.1 1.4
.4 1.6
7 I 8
5^
9.23823
71
9
24484
74
75516
9 99340
59
60
9 23895
72
72
9 24 55«
74
74
0.75442
9 99 337
'
9.23967
9.24632
0.75368
9 99 335
^
L. Cos.
d.
L. Cotg.
c.d.
L. Tan^.
L. Sin.
t
Prop. Pts. 1
80° 1
36
TABLE II
10^
_9_
10
II
12
13
14
15
16
17
18
19
20
21
22
23
^_
'25
26
27
28
29
30
31
32
33
36
37
38
39
40
41
42
43
44
46
47
48
49_
50
51
52
53
51
55
56
57
58
19.
(io
]j. Siii<
23967
24039
24 no
24 181
24253
24324
24395
24 466
24536
24 607
24677
24 748
24818
24888
24958
25 028
25 098
25 168
25237
25 307
d* L. Tan^. c. d
25376
25445
25514
25583
25652
25 721
25 790
25858
25927
25995
26 063
26 131
26 r9'9
26 267
26335
26403
26 470
26538
26 605
26 672
26739
26806
26873
26 940
27 007
27073
27 140
27 206
27273
27 339
27405
27471
27537
27 602
27668
27 734
27 799
27864
27930
27995
28 060
L. Cos.
72
71
71
72
71
71
71
70
71
70
71
70
70
70
70
70
70
69
70
69
69
69
69
69
69
69
68
69
68
68
68
68
68
68
68
67
68
67
67
67
67
67
67
67
66
67
66
67
66
66
66
66
65
. 66
66
6S
65
66
65
65
24 632
24 706
,24779
24853
24 926
.25 000
25073
.25 146
25 219
25 292
25365
25437
25510
25582
25655
•25 727
25 799
.25 871
•25943
.26 015
26086
26 158
26 229
26 301
26372
26443
26514
26585
26 655
26 726
26797
26867
26937
27 008
27078
27 148
27 218
27288
27357
27427
9
9
9
9_
9
9
9
9
9.
9
9
9
9
9_
9
9
9
9
9_
9
9
9
9
_9
9
9
9
9
_9
9
9
9
9
J9
9
9
9
9
_9
L. Cot^.
2,7496
27 566
27635
27 704
27773
.27 842
.27911
27 980
28 049
28 117
,28 186
.28254
28323
28391
,28459
28 527
28 595
,28662
28 730
,28 798
,28865
74
73
74
73
74
73
73
73
73
73
72
73
72
73
72
72
72
72
72
71
72
71
72
71
71
71
71
70
71
71
70
70
71
70
70
70
70
69
70
69
70
69
69
69
69
69
69
69
68
69
68
69
68
68
68
68
67
68
68
67
c. d.
L. Cotg.
0.75368
0.75294
0.75 221
0.75 147
0.75074
o . 75 000
0.74927
0.74854
0.74781
o . 74 708
0.74635
0.74563
0.74490
0.74418
0.74345
0.74273
0.74 201
0.74 129
0.74057
0.73985
0.73914
0.73842
0.73 771
0.73699
0.73 628
0.73557
0.73486
0.73415
0.73345
0.73274
0.73203
0.73 133
0.73063
o . 72 992
0.72 922
L. Cos.
0.72 852
0.72 782
0.72 712
0.72643
0.72573
o . 72 504
0.72434
0.72 365
o . 72 296
0.72 227
0.72 155
o . 72 089
o . 72 020
0.71 951
0.71 883
0.71 814
0.71 746
0.71 677
0.71 609
0.71 541
0.71 473
0.71 405
0.71 338
0.71 270
0.71 202
0.71 135
L. Tang.
79^
9 99335
9 99333
9 99331
9.99328
9 99326
9 99
9 99
9 99
9 99
9 99
324
322
319
317
315
9 99
9.99
9 99
9 99
9 99
3^3
310
308
306
304
9.99301
9.99299
9.99297
9.99294
9,99292
9.99290
9 . 99 288
9.99285
9.99283
9.99 281
9.99278
9.99276
9.99274
9.99271
9.99269
9.99267
9.99264
9 . 99 262
9 . 99 260
9-99 257
9 99255
9.99252
9.99250
9.99248
9 99245
99243
99241
99238
99236
99233
9.99231
9.99229
9.99 226
9 99224
9.99 221
9.99219
9.99217
9,99214
9.99 212
9 99209
9.99207
9.99204
9 99 202
9 . 99 200
9 99 197
9 99 195
L. Sin.
60
59
58
56
45
44
43
42
_1L
40
39
38
_3i
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20^
^9
18
17
16
Prop. Pis.
74
73
I
^i
7.
2
14.8
14
•3
22.2
21.
4
29.6
29
.5
37.0
36.
.6
44.4
43
■7
51.8
51
.8
59.2
58
•9
66.6
165
7a
I
7.2
2
14.4
3
21,6
4
28.8
36. c
.6
43 2
.7
.50.4
.8
57-6
■9
64.8
70
7.0
14,0
21 .0
28.0
35 o
42.0
49 o
56,0
63.0
68
6.8
13-6
20.4
27.2
34 o
40.8
47.6
54-4
61.2
3
03
0.6
0.9
I .2
66
I
6.6
2
13.2
3
4
19.8
26.4
.7
.8
330
39 6
46.2
52.8
•9
59 4
4
7
71
71
14
21
28
35
42
49 7
56.8
63 9
69
69
13 '^
20,
27
34
41
48
55
62
67
6.
13
20.
26.
33
40.2
46.9
53 6
60.3
65
6.5
13.0
26.0
32 5
39 o
45 5
52.0
58.5
a
0.2
0.4
06
08
1 .0
12
14
16
1.8
Prop. Pts.
LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 37
11
7
8
_9_
10
12
13
\l
17
i8
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
36
37
38
39
40
41
42
43
44
46
47
48
49
60
51
52
53
ii
55
56
U
60
L. Sin.
9 28 060
9 28 125
9.28 190
9.28254
9,28319
9 • 28 384
9 2S 448
9.21 512
9.28577
9 28 641
9.28 705
9.28 769
9 28 833
9 . 28 896
9 28 960
9 29 024
9.29 087
9.29 150
9.29 214
9.29277
9 29340
9.29403
9 . 29 466
9.29529
9.29591
9.29654
9.29 716
9 29 779
9.29841
9.29903
9 . 29 966
9 . 30 028
9.30090
930 151
9.30213
9 30275
9 30336
9 30398
9 30459
9 30521
9.30 582
9 30643
9,30 704
9 30 765
9 . 30 826
9.30887
9 30947
9 31 008
9.31 068
9 31 ^29
9.31 189
9 31 250
9-31 310
9 31 370
9 31 430
9.31 490
9 31 549
9 31 609
9,31 669
9 31 728
9.31 788
L. Cos.
65
65
64
65
65
64
64
65
64
64
64
64
63
64
64
63
63
64
63
63
63
63
63
62
63
62
63
62
62
63
62
62
61
62
62
61
62
61
62
61
61
61
61
61
61
60
61
60
61
60
6i
60
60
60
60
59
60
60
59
60
L. Tang, c. d
28865
28933
29 000
29 067
29 134
29 201
29 268
29335
29 402
29 468
29535
29 601
29668
29734
29 800
29866
29 932
29998
30 064
30 130
30195
30 261
30326
30391
30457
30 522
30587
30652
30717
30 782
30 846
30 91 1
30975
31 040
31 104
31 168
31 233
31 297
31 361
31425
• 31 489
31 616
31 679
31 743
31 806
31 870
31933
31996
32059
32 122
32185
32248
323"
32373
32436
32498
32561
32623
32685
32747
68
67
67
67
67
67
67
67
66
67
66
67
66
66
66
66
66
66
66
65
66
65
65
66
65
65
65
65
65
64
65
64
65
64
64
65
64
64
64
64
63
64
63
64
63
64
63
63
63
63
63
63
63
62
63
62
61
62
62
62
L. Cotg.
0.71 135
0.71 067
0,71 000
o 70933
o . 70 866
0.70799
o . 70 732
o . 70 665
0.70 598
0.70532
o . 70 465
0.70399.
0.70332
o . 70 266
o . 70 200
0,70 134
o . 70 068
o . 70 002
0.69936
o 69 870
o . 69 805
0.69739
0.69 674
0.69 609
0.69543
0.69 478
0.69413
0.69348
o . 69 283
o 69 218
0.69 154
o . 69 089
o . 69 025
0.68 960
0.68896
L. Cos.
L. Cotg. Ic. d.
0.68832
0.68 767
0.68 703
o 68 639
0.68575
0.68 511
0.68448
0.68384
o 68321
0.68 257
0.68 194
0.68 130
0.68067
o . 68 004
o 67 941
0.67878
0.67815
o 67 752
o 67 689
0.67 627
0.67 564
0.67 502
o 67439
0.67377
o 67315
67253
L. Tang.
78^
9 99 195
9.99 192
9.99 190
9.99 187
9 99 185
9.99 182
9.99 180
9.99 177
9 99 175
9.99 172
9.99 170
9.99 167
9.99 165
9 99 162
9 99 160
9 99
9 99
9 99
9 99
9 99
157
155
152
150
147
9 99
9 99
9 99
9 99
9 99
145
142
140
137
135
9 99
9 99
9 99
9 99
9 99
132
130
127
124
122
9 99
9 99
9 99
9 99
9 99
9 99
9 99
9-99
9 99
9 99
106
104
lOI
099
096
9 99
9 99
9 99
9 99
9 99
093
091
088
086
083
9 99
9 99
9 99
9 99
9 99
080
078
075
072
076
9 99
9 99
9 99
9 99
9 99
067
064
062
059
056
9 99054
9,99051
9.99048
9.99046
9 99043
9 99 040
L. Sin.
60
59
58
55
54
53
52
_5L
50
49
48
47
46
25
24
23
22
21
20
19
18
17
16
Prop, rts.
68
6.8
13.6
20 4
27 2
34 o
40.8
47 6
54-4
61 2
66
6.6
13 2
10 8
26
33
39
46
52
59-4
64
6.4
12.8
19.2
25.6
32.0
38.4
44-8
51 2
576
63
62
12
24
31
37
43
49
55
60
6.0
12.0
18.0
24.0
30,0
36.0
42 o
48.0
54 oi
67
07
13 4
20 I
26 8
33 5
40.2
46 9
60.3
65
65
13
19
26
32
39
45
52
58
63
12.6
189
25 2
3^ 5
37 8
44 I
50 4
567
61
6
12
18
24
36
42
48
54
59
5 9
II
17
23
3
.1
03
.2
0.6
•3
0.9
4
1.2
I
'A
7
2. 1
8
2.4
9
2.7
a
0.2
0.4
0.6
0.8
1.0
1.2
14
16
1.8
Prop. Pts.
38
TABLE II
12° 1
/
L. Sin.
d.
L.Taug. c.d.|
L. Cotg.
L. Cos.
60
Prop. Pts.
9.31 788
9 32 747
63
0.67253
9.99040
9 31 847
9.32 810
0.67 190
9.99038
59
63 6a
2
9.31 907
9-32872
0.67 128
9 99035
58
I 63 6.2
3
9.31 966
59
9 32933
62
0.67067
9.99032
57
.2 12 6 12 4
_±
9.32025
59
9-32995
62
62
0.67005
9.99030
56
55
.3 18 9 18.6
.4 25 2 24.8
9.32084
9-33 057
0.66943
9 99027
6
9 32 143
59
9 33 "9
fi-W
0.66881
9 99 024
54
.5 31 5 31
7
9.32 202
59
9 33 180
62
61
0.66820
9.99022
S3
-6 37 8 37 2
8
9.32 261
59
58
59
9 33242
0.66 758
9.99019
52
-7 44-1 43.4
9
10
9 32319
9 33303
62
0.66697
9.99016
51
.8 50.4 49 6
•9 567 55-8
9 32378
9 33 365
0.66635
9.99013
50
II
9 32437
59
58
58
9 33426
61
0.66 574
9.99 on
49
61 60
12
9 32495
9 33487
61
0.66 513
9 99 008
48
.1 6.1 6.0
13
9 32553
9 33548
61
0.66452
9 99 005
47
.2 12.2 12
14
15
9 32 612
59
58
58
58
58
58
58
58
9.33609
61
61
61
61
60
61
0.66 391
9 99002
46
.3 18.3 18.0
.4 24.4 24.0
9 32 670
9-33670
0.66330
9 99000
45
16
9 32 728
9 33 731
0.66269
9 98 997
44
I ^2i ^2 °
17
932786
9-33 792
0.66208
9 98 994
43
.6 36.6 36.0
18
9.32844
9 33853
0.66 147
9 98991
42
.7 42.7 42.0
.8 48.8 48.0
•9 54-9 540
19
20
9.32902
9 33913
0.66087
998989
41
40
9.32960
9 33 974
. 66 026
9.98986
21
9.33018
9 34034
At
0.65 966
9-98983
39
59
22
9 33075
57
S8
9 34095
0.65905
9.98980
38
•' ^1
2S
9 33 133
9-34155
0.65 845
9.98978
37
.2 II. 8
24
25
9 33 190
57
58
9 34215
61
60
60
60
60
60
0.65 785
9 98975
36
35
.3 17 7
•4 23.6
•5 29-5
•6 35 4
9 33248
9.34276
0.65 724
9 98972
26
9 33305
57
9 34336
0.65 664
9 98969
34
27
9 33362
57
58
9 34396
0.65 604
9.98967
33
28
9-33 420
9 34456
0.65 544
9 98964
32
• 7 41 3
.8 47 2
9 53 I
58 57
1 5.8 5.7
2 II. 6 II. 4
29
30
9-33 477
57
57
9 34516
0.65 484
9.98 961
3^
30
9-33 534
9 34 576
0.65 424
9.98958
V
9 33591
57
9 34635
59
60
60
0.65 365
9 98955
29
32
9-33 647
56
9 34695
0.65 305
9 98953
28
33
9 33 704
57
9 34 755
0.65245
9.98950
27
34
9-33761
57
57
56
9 34814
59
60
0.65 186
9.98947
26
,3 17.4 171
.4 23.2 22.8
.5 29.0 28,5
.6 34.8 34 2
7 40 6 39 9
.8 46.4 45.6
.9 52 2 51.3
56 55
•I 56 55
3S
9 33818
9 34874
0.65 126
9 98944
25
3^
9 33874
9 34 933
59
0.65 067
9.98941
24
37
9 33931
57
56
56
57
9 34992
59
0.65 008
9.98938
23
3«
39
40
9 33987
9 -34 043
9 35051
9 35 "I
59
60
59
0.64949
0.64889
9-98936
9 98933
22
21
20
9 34 100
9 35 170
0.64830
9.98930
41
9 34156
56
9 35229
59
0.64 771
9.98927
19
42
9.34212
56
9 35288
59
0.64 712
9 98924
18
43
9 . 34 268
56
56
56
56
9 35 347
59
58
59
0.64653
9.98 921
17
3 16. 8i 16 5
.4 22 4 22.0
,5 28 27.5
.6 33.6 33.0
-7 39-2 38.5
8 44.8 44.0
44
45
9 34324
9 35 405
0.64595
9 98919
lb
15
9-34380
9 35 464
64 536
9 98 916
46
9-34436
9 35 523
59
58
0.64477
9 98913
14
47
9-34 491
55
9 35 581
0.64419
9 98 910
»3
48
9-34 547
56
9-35 640
59
58
59
58
58
58
58
S8
58
58
58
58
57
. 64 360
9 98907
12
49
50
9.34602
55
56
935698
. 64 302
9.98904
II
10
.9 50 4 49 5
3 3
9-34658
9 35 757
0.64243
9.98901
51
9-34713
55
9 35815
0.64 185
9.98898
9
I 0.3 0.2
.2 0.6 0.4
3 0.9 0.6
.4 1.2 08
•5 15 10
.6 1.8 1.2
52
9-34769
56
9 35873
0.64 127
9.98896
8
S3
9.34824
55
9 35931
. 64 069
9.98893
7
54
9 34879
55
55
9 35989
0.64 on
9.98890
6
5
9 34 934
9.36047
0.63953
9 98887
56
9.34989
55
9 36 105
63 895
9 98 884
4
57
9-35044
55
9.36 163
0.63837
9.98881
3
7 2.1 14
S«
9-35099
55
9.36221
0.63 779
9.98878
2
.8 24 16
59
or
9 35 154
55
55
9.36279
0.63 721
9.98875
I
.9 2.7 1.8
9 35209
9 36336
. 63 664
9.98872
L. Cos.
d. |l. Cotg.
c.d.
L. Taug.
L. Sin.
Prop. Pts.
77° 1
LOGARITHMS OF THE TKIGONOMETRIC FUNCTIONS
39
13° 1
L. Sin.
54
55
55
L. Tang.
c.d.
L. Cotg.
L. Cos.
60
Prop. Pts. 1
9 35209
9 36 336
58
58
57
0.63 664
9.98872
1
I
9
35263
9 36 394
0.63 606
998869
S9
58
57
2
9
35318
9 36452
63 548
9 98867
58
• I 5.^
? 5 7
3 II 4
\ 17 I
I 22 8
3
9
35 373
9 36 509
63 491
9.98864
57
.2 II. (
4
9
35427
54
9.36566
9.36624
58
63434
9 98861
56
S5
•3 17-
•4 23. i
9
35481
63 376
9 98858
6
9 35 536
9.36681
0.63319
9 98855
S4
•5 29.:
D 28 5
7
9 35 590
9 36738
63 262
9 98 852
SS
.6 34. {
^ 34 2
8
9 35644
9 36 795
63 205
9 98 849
S2
.7 40 6 39.9 1
9
10
9
"9
35 698
54
9.36852
57
63 148
9 . 98 846
51
50
.8 46 .
•9 52.^
\ 45-6
2 51 3
35752
9 36909
0.63091
9 98843
II
9 35 806
9 36966
0.63034
9 98 840
49
56
55
12
9.35860
9 37023
57
0.62977
9 98837
48
.1 5 <
^ 5 5
13
9
35914
9 37 080
0.62 920
9.98834
47
.2 II.:
2 II.
14
IS
9
35968
54
9 37 137
57
56
62863
9 98 831
46
45
.3 16.8 16.5
.4 22.4 22.0
9
36022
9 37 193
0.62 807
9.98828
i6
9
36075
53
9 37250
57
56
0.62 750
9 98825
44
•5 28.0 27 5
17
9
36 129
9 37306
. 62 694
9.98822
43
•6 33(
^ 330
i8
9
36 182
53
9 37363
56
57
56
56
56
56
56
0.62 637
9.98819
42
■9 50-^
\ 38.5
19
20
9
36236
53
9 37419
0.62 581
9.98816
41
40
5 44.0
^ 49-5
9
36289
9 37 476
0.62 524
9.98813
21
9
36342
9 37532
0.62468
9.98810
39
54
22
9
36395
937588
0.62 412
9.98807
38
.1
5 4
23
9
36449
54
9 37644
0.62 356
9.98804
37
.2 10.8 1
24
2S
9
36502
53
9.37700
0.62 300
9.98801
36
35
•3 I
•4 2
62
16
9 36555
9 37756
62 244
9.98798
26
9 36 608
53
9 37812
56
0.62 188
9 98 795
34
•5 27.0
27
9 ^6 660
52
9 37 868
56
0.62 132
9 98 792
33
t ^li
28
9 36713
53
9 37924
56
56
55
0.62 076
9.98789
32
■I V
29
9 36766
53
53
9 37980
. 62 020
9 98 786
31
30
.8 4
•9 4
3 2
8.6
9
36819
9 38035
0.61 965
9.98783
31
9
36871
52
9.38091
56
0.61 909
9 . 98 780
29
53
5a
32
9
36924
53
938 147
50
0.61 853
9.98 777
28
•I 5'
; 52
33
9
36976
52
9 38 202
55
0.61 798
9.98 774
27
.2 10. (
) 10.4
34
9
37028
52
53
9 38257
55
56
0.61 743
9.98771
26
•3 15 S
) 156
20.8
26.0
35
9
37081
9 38313
61 687
9.98768
25
.4 21.2
• 5 26. c
.6 31-^
3^
9
37 133
37185
52
9 38 368
55
61 632
9.98765
24
37
9
52
9 38 423
55
0.61 577
9.98 762
23
; 31.2
36 4
\ 416
' 46.8
3«
9
37 237
52
9 38 479
56
0.61 521
9 98 759
22
•7 371
.8 42.4
•9 47 y
39
40
9
37289
52
52
9 38534
55
55
0.61 466
9.98 756
21
9
37341
9 38 589
0.61 411
9 98 753
20
41
9
37 393
52
9 38644
55
61 356
9.98750
19
51
4
42
9
37 445
52
9 38 699
55
0.61 301
9.98 746
18
• I 51
0I
43
9 37 497
52
9 38 754
55
0.61 246
9 98 743
17
.2 10.2
44
4.S
9
37 549
52
51
9 38 808
54
55
0.61 192
9.98740
16
•3 15 3
.4 20.4
.6 30. e
9 45 5
3
1.2
1.6
2
37600
9 38 863
9.38918
0.61 137
9 98 737
15
46
9 37 652
52
55
0.61 082
9 98734
14
It
47
9 37703
51
9.38972
54
0.61 028
9.98 731
13
48
9 37 755
52
9 39 027
55
0.60973
9.98 728
12
11
49
60
9 37806
51
52
9.39082
55
54
0.60 918
9.98725
9.98 722
II
10
9
37858
9 39 136
. 60 864
51
9
37909
51
9 39 190
54
0.60 810
9 98 719
9
0.2
52
9
37960
51
9 39245
55
0.60 755
9 98 715
8
.2 0.6
53
9
38 on
^'
9.39299
54
0.60 701
9 98 712
7
0.6
08
I.O
54
55
9
38062
51
51
9 39 353
54
54
0.60 647
9 98 709
6
5
3 09
.4 12
9
38 113
9-39 407
0.60593
9 98 706
5^
9 38 164
SI
9.39461
54
0.60539
9 98 703
4
1.2
■■^^
9 38215
51
9 39515
54
0.60485
9.98 700
3
.7 2.1
8 2.4
\i
5«
9 38 266
51
9 39569
54
60 43 1
9.98697
2
59
60_
9 383^7
51
51
939623
54
54
0.60377
9.98694
I
9 2.7
1.8
9.38368
9.39677
0.60323
9.98690
L. Cos.
d.
L. Cotgr.
C.d.
L. Tangr.
L. Sin.
f
Prop. Pts. ' 1
76^ 1
40
TABLE II
14
9_
10
II
12
13
\l
18
i9_
20
21
22
23
24^
26
27
28
29
30
31
32
33
34
36
37
38
39
40
41
42
43
44
46
47
48
49_
50
51
52
53
il
II
60
L. Sin. d.
38368
38418
38469
38519
38570
38 620
38 670
38721
38771
38821
38871
38921
38971
39021
39071
39 121
39 170
39 220
39270
39319
39369
39418
39467
39566
39615
39664
39713
39 762
39 811
39860
39909
39958
40 006
40055
40 103
40 152
40 200
40249
40297
40346
40394
40 442
40 490
40538
40 586
40634
40 682
40730
40778
40 825
40873
40 921
4c 968
41 016
41 063
41 III
41 158
41 205
41 252
41 300
L. Cos. d.
L. Tang.
39677
39 731
39785
39838
39892
39 945
39 999
40 052
40 106
40159
40 212
40 266
40319
40372
40425
40478
40531
40584
40 636
40 689
40742
40795
40847
40 900
40952
41 005
41057
41 109
41 161
41 214
41 266
41 318
41 370
41 422
41 474
41 526
41 578
41 629
41 681
41 733
41 784
41 836
41 887
41939
41 990
42 041
42093
42 144
42 195
42 246
42297
42348
42399
42450
42501
42552
42 603
42653
42 704
42755
9 . 42 805
L. Cotg.
c.d.
c.d,
L. Cotg.
o . 60 323
o . 60 269
0.60 215
0.60 162
0.60 108
0.60 055
0.60 001
0.59948
0.59894
0.59841
0.59 788
0.59734
0.59 681
0.59 628
059 575
0.59 522
0.59469
0.59 416
0.59364
0.59311
0.59258
o . 59 205
0.59153
0.59 100
0.59048
0.58995
0.58943
0.58891
0.58839
0.58 786
0.58 734
0.58682
o . 58 630
0.58578
0.58 526
0.58474
o . 58 422
0.58371
0.58319
0.58 267
0.58 216
0.58 164
0.58 113
0.58061
0.58 010
0.57959
0.57907
0.57856
0.57805
o 57 754
0.57 703
0.57652
0.57 601
0.57550
o. 57 499
0.57448
0.57397
0.57347
0.57 296
0.57245
0.57195
L. Tang.
75°
L. Cos.
98 690
98687
98684
98681
98678
98675
98671
98668
98665
98662
98659
98656
98652
98 649
98646
98643
98 640
98636
98633
98630
98 627
98 623
98 620
98617
98614
98 610
98 607
98 604
98 601
98597
98594
98591
98588
98584
98581
98578
98574
98571
98568
98565
98561
98558
98555
98551
98548
98545
98541
98538
98535
98531
98528
98525
98521
98518
98515
98 511
98508
98505
98 501
98498
9.98494
L. Sin.
d.
60
59
58
55
54
53
52
IL
50
49
48
47
_46_
45
44
43
42
41
40
39
38
35
34
33
32
31
25
24
23
22
21
20
19
18
17
16
Prop. Pts.
54
53
I
5i
5-
2
10.8
10.
3
16.2
15-
4
21.6
21 .
27.0
26.
6
32.4
31
. 7
37.8
37-
.8
48 6
42.
•9
47
52
5-
10.
15-
20.
26.
41.6
46.8
50
50
10.0
15 o
20.0
25.0
30.0
350
40.0
45 o
4
0.4
o 8
1.2
1.6
2.0
2.4
2.8
36
48
47
I
4.8
4
2
9.6
9
3
14.4
14
4
19 2
18.
S
24.0
23
6
28.8
28.
7
336
32
8
384
37
9
43 2
42
Prop. Pts.
LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS
41
15^
L. Sin.
I
2
3
I
I
10
II
12
13
14
;i
17
i8
19
20
21
22
23
24
26
27
28
29
30
31
32
33
34
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
J.9
60
42 001
42047
42093
42 140
42 186
42 232
42 278
42324
42370
42 416
42 461
42507
42553
42599
42644
42 690
42735
42 781
42 826
42872
42917
42 962
43008
43053
43098
43 143
4318S
43233
43278
43323
43367
43412
43 457
43502
43 546
43591
43 635
43 680
43 724
43 769
43813
43857
43901
43946
43990
44034
47
47
47
47
47
47
46
47
47
46
47
46
47
46
47
46
46
47
46
46
46
46
46
46
45
46
46
46
45
46
45
46
45
46
45
45
46
45
45
45
45
45
45
45
44
45
45
45
44
45
44
45
44
45
44
44
44
45
44
44
L. Tang.
C. d.
42 805
42856
42 906
42957
43007
43 057
43 108
43 158
43 208
43258
43308
43358
43408
43458
43508
43558
43607
43657
43 707
43756
43 806
43855
43905
43 954
44 004
44053
44 102
44 151
44 201
44250
44299
44348
44 397
44446
44 495
44 544
44592
44641
44 690
44738
44787
44836
44884
44 933
44981
45029
45078
45 126
45 174
45 222
45 271
45319
45367
45415
45463
45 5"
45 559
45 606
45654
45 702
9 45 750
5»
50
51
50
50
51
50
50
50
50
50
50
50
50
50
49
50
50
49
50
49
50
49
50
49
49
49
50
49
49
49
49
49
49
49
48
49
49
48
49
49
48
49
48
48
49
48
48
48
49
48
48
48
48
48
48
47
48
48
48
L. Cotg.
0.57 195
0.57 144
0.57094
0.57043
0.56993
o 56943
o . 56 892
o . 56 842
0.56 792
0.56 742
o 56 692
o . 56 642
0.56 592
0.56542
0.56492
0.56 442
o 56393
0.56343
0.56 293
0.56 244
0.56 194
o 56 145
0.56 095
0.56 046
o 55996
55 947
55898
55849
55 799
55 750
o 55 701
o 55652
0.55603
o 55 554
o 55505
o 55456
0.55 408
o 55 359
o 55310
0.55 262
o 55213
0.55 164
o 55 116
0.55067
0.55019
0.54971
0.54922
0.54874
o . 54 826
o 54778
0.54 729
0.54 681
0.54633
o 54585
o 54 537
0.54489
0.54441
0.54394
0.54346
o . 54 298
0.54250
L. Cos. I d. I L. Cotg. c. d. L. Tang.
740
L. Cos.
9 98 4Q4
9.98491
9 98488
9 98484
9 98 481
98477
98474
98471
98467
98 464
98 460
98457
98453
98450
98447
98443
98 440
98436
98433
98429
98 426
98 422
98419
98415
98 412
98 409
98405
98 402
98398
98395
98391
98388
98384
98381
98377
98373
98370
98366
98363
98359
98356
98352
98349
98345
98342
98338
98334
98331
98327
98324
98317
98313
98309
98306
98 302
98299
98295
98 291
98288
9 . 98 284
L. Sin,
d.
60
59
58
55
54
53
52
51
50
49
48
47
_46_
45
44
43
42
41
40
39
38
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
Prop. Pts.
5t
.1
51
.2
10.2
3
15 3
4
20.4
•5
25 5
6
30 6
7
35-7
8
40 8
9
45 9
49
48
.1
4 9
4
.2
9.8
9
•3
14.7
14
•4
19,6
19
. 5
24 5
24
.6
29.4
28
7
34-3
33
.8
39 2
38
9
44 I
43
45
46
I
4 7
4-
2
9
4
9
3
4
Is
8
\l
5
11
5
23-
.6
2
27
7
32
9
32
.8
37
6
36.
9
42
3
41
43
1
I
4 51
2
9
3
13
5
4
18
5
22
5
6
27
■ 7
31
5
.8
36
9
40
5
Prop. Pts.
42
TABLE II
16
9_
10
II
12
13
14
15
i6
17
i8
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
il
55
56
II
-59.
60
L. Sin.
9.44472
9.44516
9 44 559
9.44 602
9.44646
45 120
45 163
45 206
45 249
45 292
44034
44078
44 122
44 166
44 210
44253
44297
44341
44385
44428
44 689
44 733
44776
44819
44 862
44905
44948
44992
45035
45077
45 334
45 377
45419
45 462
45504
45 547
45589
45632
45674
45 716
45 758
45 801
45843
45885
45927
45969
46 on
46053
46095
46 136
46 178
46 220
46 262
46303
46345
46 386
46 428
46 469
46 511
46552
9 46594
L. Cos.
d.
L. Tang:.
45 750
45 797
45845
45892
45 940
45987
46 035
46 082
46 130
46 177
46 224
46271
46319
46 366
46413
46 460
46507
46554
46 601
46648
46 694
46741
46 788
46835
46881
46 928
46975
47 021
47 068
47 114
47 160
47207
47253
47299
47346
47392
47438
47484
47530
47576
47 622
47668
47 714
47760
47 806
47852
47897
47 943
47989
48035
48080
48 126
48 171
48217
48262
48307
48353
48398
48443
48489
48534
L. Cotg.
c.d.
47
48
47
48
47
48
47
48
47
47
47
48
47
47
47
47
47
47
47
46
47
47
47
46
47
47
46
47
46
46
47
46
46
47
46
46
46
46
46
46
46
46
46
46
46
45
46
46
46
45
46
45
46
45
45
46
43
45
46
45
c. d,
L. Cotg.
0.54250
0.54203
o 54 155
0.54 108
o . 54 060
o 54013
0.53965
0.53918
0.53870
0.53823
0.53 776
0.53 729
0.53681
0.53634
o- 53 587
o 53540
0.53493
0.53446
0.53399
0.53352
0.53306
0.53259
0.53 212
0.53 165
0.53 "9
0.53 072
0.53025
0.52979
0.52 932
0.52886
0.52 840
0.52 793
0.52 747
0.52 701
0.52654
o . 52 608
0.52 562
0.52 516
0.52 470
0,52424
0.52378
0.52332
0.52 286
0.52 240
0.52 194
0.52 148
0.52 103
0.52057
0.52 on
o 51 965
0.51 920
0.51 874
0.51 829
0.51 783
0.51 738
o 51 693
0.51 647
0.51 602
o 51 557
05^ 5"
0.51 466
L. Tang.
73^
L. Cos.
98284
98281
98277
98273
98 270
98266
98 262
98259
9 98255
9 9825 1
9 98 248
98 244
98 240
98237
98233
98 229
98 226
98 222
98218
98215
98 211
98 207
98 204
98 200
98 196
98 192
98 189
98 185
98 i8i
98 177
98 174
98 170
98 166
98 162
98 159
98 155
98 151
98 147
08 144
98 140
98 136
98 132
98 129
98 125
98 121
98 117
98 113
98 no
98 106
98 102
98098
98 094
98 090
98087
9 8083
98079
98075
98071
98 067
98063
9 . 98 060
L. Sin."
(>0
It
1
55
54
53
52
11
50
49
48
47
i5_
45
44
43
42
41
40
39
38
37
Ji
35
34
33
32
31
30
29.
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
II
To
9
8
7
6
Prop. Pte.
48
47
I
4.8
4
2
9.6
9
3
14.4
H
4
19 2
18.
S
24.0
23
6
28.8
28.
7
,336
32
8
38 4
37
9
43 2
42.
At
45
I
4.6
4
2
9
2
9
3
13
8
IT,
4
18
4
18.
5
23
22.
.6
27
6
27
• 7
32
2
31
.8
36
8
36.
9
41
4
40
44
43
I
4 4
4
2
8.8
8.
3
13 2
12.
4
17.6
17
.q
22.0
21.
.6
26.4
25
• 7
30.8
30
.8
35-2
34-
9
39-6
38.
43 !
I
2
3
4
4.2
8.4
12.6
16.8
21 .0
6
25.2
7
8
9
29 4
336
37 ^
4
I
2
0.4
0.8
3
4
S
1.2
1.6
2.0
.6
.1
2.4
2.8
32
9
3-6
Prop. Pts.
LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 43
17'
L. Sin.
I
2
3
_4
i
7
8
_9_
10
I
12
13
:i
17
i8
ii.
20
21
22
23
24
25
26
27
28
29
31
32
33
36
37
38
39
40
41
42
43
44
46
47
48
50
51
52
53
il
60
46594
46635
46 676
46 717
46758
46 8cx)
46 841
46882
46923
46 964
47005
47045
47 086
47 127
47 168
47209
47249
47290
47330
47371
47 411
47452
47492
47 533
47 573
47613
47654
47694
47 734
47 774
47814
47854
47894
47 934
47 974
48 014
48054
48 094
48 133
48173
48213
48252
48 292
48332
48371
48 411
48450
48 490
48529
48568
48607
48647
48686
48 ,725
48 764
48803
48842
48881
48 920
48959
48998
41
41
41
41
42
41
41
41
41
41
40
41
41
41
41
40
41
40
41
40
41
40
41
40
40
41
40
40
40
40
40
40
40
40
40
40
40
39
40
40
39
40
40
39
40
39
40
39
39
39
40
39
39
39
39
39
39
39
39
39
L. Cos. I d.
L. Tang.
c. d.
48534
48579
48 624
48669
48714
48759
48804
48849
48894
48939
48984
49029
49073
49 118
49 163
49207
49252
49296
49341
49385
49430
49 474
49 519
49563
49607
49652
49 696
49 740
49 784
49828
49872
49916
49960
50 004
50 048
50 092
50136
50 180
50223
50 267
503"
50355
50398
50442
50485
9 50 529
9 50572
50616
50659
50703
50746
50789
50833
50876
509^9
50 962
51 005
51 048
51 092
51 135
51 178
45
45
45
45
45
45
45
45
45
45
45
44
45
45
44
45
44
45
44
45
44
45
44
44
45
44
44
44
44
44
44
44
44
44
44
44
44
43
44
44
44
43
44
43
44
43
44
43
44
43
43
44
43
43
43
43
43
44
43
43
L. Cotg.
0.51 466
0.51 421
0.51 376
0-51331
0.51 286
0.51 241
0.51 196
o 51 151
o 51 106
0.51 061
o 51 010
o 50971
0.50 927
0.50882
0.50837
0.50793
0.50748
0.50 704
0.50 659
0.50615
0.50 570
o 50 526
0.50481
o 50 437
0.50393
o 50 348
o . 50 304
o . 50 260
0.50 216
0.50 172
o. 50 128
o . 50 084
o . 50 040
0.49 996
0.49952
0.49 908
o . 49 864
o . 49 820
0.49 777
0.49 733
o 49 689
0.49 645
o 49 602
0.49 558
0.49515
0.49471
0.49 428
0.49384
0.49 341
0.49 297
0.49254
0.49 211
0.49 167
0.49 124
0.49 081
0.49038
o 48 995
o 48 952
0.48 908
o 48 865
0.48822
L. Cotg. c. d. L. Tang.
72°
L. Cos.
98060
98 056
98 052
98048
98 044
98 040
98 036
98 032
98 029
98025
98021
98017
98013
98 009
98 005
98 001
97997
97 993
97989
97986
97982
97978
97 974
97970
97966
97962
97958
97 954
97950
97946
97942
97938
97 934
97930
97 926
97 922
97918
97914
97910
97906
97902
97898
97894
97890
9.97886
9.97882
, 97 878
9.97874
9.97870
97866
97861
97857
97853
97849
97S45
97841
97837
97833
97829
97825
9782]
L. Sin.
d.
60
58
57
55
54
53
52
_5L
50
49
48
47
45
44
43
42
41
40
39
38
35
34
33
32
30
29
28
27
26
Prop. Pts.
45
I
4.5
2
9.0
3
13.5
4
18.0
22.5
.6
27.0
7
31-5
8
36.0
9
40.5
43
1
.1
4 3l
.2
8
6
•3
12
9
•4
17
2
I
21
25
I
:l
30
I
34
4
9
38
7
41
I
4-1
2
8.2
3
12.3
4
16.4
S
20. s
6
24.6
7
28.7
.8
32.8
9
36.9
39
I
39
2
7-8
3
II. 7
4
15.6
5
19 5
6
23 4
7
27 -3
8
31.2
9
35 I
4
.1
0.4
.2
0.8
■3
1.2
.4
1.6
•5
2.0
.6
2.4
7
2.8
8
3-2
9
36
4 4
8.8
13.2
17.6
22.0
26.4
30.8
35 2
39-6
42
4.2
8.4
12.6
16 8
21 .0
25.2
29.4
33-6
37-8
40
4.0
8.0
12.0
16.0
20.0
24.0
28.0
32.0
36.0
5
05
1 .0
15
2.0
2.5
30
3-5
4.0
4 5
3
0.6
0.9
1.2
2.1
2.4
2.7
Prop. Pts.
44
TABLE II
18^
I
2
3
4
I
7
8
_9_
10
II
12
13
14
15
16
17
18
i9_
20
21
22
23
24
25
26
27
28
29
30
31
32
35
31
36
37
38
J9_
40
41
42
43
44
46
47
48
49_
50
5^
52
53
■>!
]^
i9.
(JO
L. Sin.
9 48998
9 49037
49076
49 115
49 153
49 192
49231
49269
49308
49 347
9 49385
9.49424
9 49462
9 49 500
9 49 539
9 49 577
9.49615
9 49654
49692
49 730
49768
49 806
49844
49882
49920
49958
49996
50034
50072
50 no
50 148
9.50185
9.50 223
9.50 261
9 50 298
T 50336"
9 50 374
9 50 411
9 50449
50486
9 50523
9 50 561
50598
50635
50673
50 710
50747
50784
50 821
50858
50 896
50933
50970
51 007
51 043
9
9
9
9
_9_
9.51 080
9 SI "7
9 51 154
9.51 191
51 227
9.51 264
L. CoSu
39
39
39
38
39
39
38
39
39
38
39
38
38
39
38
38
39
38
38
38
38
38
38
38
38
38
38
38
38
38
37
38
3*?
37
38
38
37
38
37
37
38
37
37
38
37
37
37
37
37
38
37
37
37
36
37
37
37
37
36
37
L. Tang.
178
221
264
306
349
392
435
478
520
563
606
648
691
776
819
861
903
946
52031
52073
52 115
52157
52 200
52 242
52284
52 326
52368
52410
52452
52494
52536
52578
52 620
52 661
52703
52745
52 787
52 829
52 870
52 912
52953
52995
53037
53078
53 120
53 161
53 202
53244
53285
53327
53368
53409
53450
53492
53 533
53 574
53615
53656
9 53697
L. Cotg. c. d
c.d.
L. Cotg.
o . 48 822
0.48 779
0.48 736
0.48 694
0.48 651
0.48608
0.48 565
0.48 522
o . 48 480
0.48437
0.48394
0.48352
o 48 309
0.48 266
o 48 224
0.48 181
0.48 139
o 48 097
0.48 054
0.48 012
0.47969
0.47927
0.47885
0.47843
0.47 800
0.47 758
0.47716
0.47674
0.47 632
0.47590
0.47548
0.47 506
0.47464
0.47 422
0.47380
0.47 339
0.47297
0.47255
0.47213
0.47 171
0.47 130
0.47088
0.47047
0.47 005
0.46963
0.46 922
0.46880
o . 46 839
0.46 798
0.46 756
0.46715
0.46 673
o . 46 632
0.46 591
0.46 550
o . 46 508
0.46 467
o . 46 426
0.46 385
0.46344
0.46303
L. Tang.
71°
L. Cos.
d.
9.97821
60
9
97817
59
9
97812
58
9
97808
57
9
97804
56
55
9
97800
9
97796
S4
9
97792
53
9
97788
52
9 97 784
51
50
9 97 779
9-97 775
49
9
97771
48
9
97767
47
9
97763
46
45
9
97 759
9
97 754
44
9
97 750
43
9
97746
42
9
97742
^
41
40
9
97738
9
97 734
39
9
97729
38
9
97725
37
9
97721
36
3S
9
97717
9
97713
34
9
97708
33
9
97704
32
9
97700
31
30
9
97696
9
97691
29
9
97687
28
9
97683
27
9
97679
26
25
9
97674
9
97670
24
9
97666
23
9
97662
22
9
97657
21
20
9
97653
9
97649
19
9
97645
18
9
97640
17
9
97636
16
15
9
97632
9
97628
14
9
97623
13
9
97619
12
9
97615
II
10
9
97 610
9
97606
9
9
97602
8
9
97 597
7
9
97 593
b
S
9
97589
9
97584
4
9
97580
3
9
97576
2
9
97571
I
9
97567
L. Sin.
d.
f
Prop. Pte.
43
.1
4 3
2
8.6
3
12.9
4
17.2
1
l\i
7
30 I
8
34-4
9
387
4a
4.2
37-8
41
41
8.2
12.3
16.4
20.5
24.6
28.7
32.8
369
39
1
I
3 9
2
7.8
3
II
7
4
15
6
5
19
5
b
23
4
7
27
3
8
31
2
9
35
I
3^
' 1
I
3-7|
.2
7
4
3
II
I
4
14
8
•S
18
5
6
22
2
• 7
25
9
8
29
6
9
33
3
38
3-8
76
II 4
15 2
19 o
22.8
26.6
304
34-2
36
3-6
7.2
10.8
14.4
18.0
21.6
25.2
28.8
32 4
5
I
05
2
1.0
3
15
4
2.0
S
25
6
30
7
3-5
8
4.0
9
4-5
Prop. Pts.
LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS 45
19^
9_
10
12
13
\i
17
i8
19
20
21
22
23
24
26
27
28
29
30
31
32
33
34
36
37
38
39
40
41
42
43
44
46
47
48
49_
50
51
52
53
il
59_
60
L. Sin.
264
301
338
374
411
447
484
520
557
593
629
666
702
738
774
811
847
883
919
955
51 991
52 027
52063
52099
52 135
52 171
52 207
52 242
52 278
52314
52350
52385
52421
52456
52492
52527
52563
52598
52634
52 669
52 705
52 740
52 775
52 811
52846
52881
52 916
52951
52 986
53.0^
53056
53092
53 126
53 161
53 196
53231
53 266
53301
53336
53370
9 53405
L. Cos.
37
37
36
37
36
37
36
37
36
36
37
36
36
36
37
36
36
36
36
36
36
36
36
36
36
36
35
36
36
36
35
36
35
36
35
36
35
36
35
36
35
35
36
35
35
35
35
35
35
35
36
34
35
35
35
35
35
35
34
35
L. Tang.
53697
53738
53 779
53820
53861
c.d.
53902
53 943
53984
54025
54065
54 106
54147
54187
54228
54269
54 329
54350
54390
54431
54471
54512
54552
54 593
54633
54673
54714
54 754
54 794
54835
54875
54915
54 955
54 995
55035
55075
55 115
55 155
55 195
55235
55275
55315
55 355
55 395
55 434
55 474
55514
55 554
55 593
55633
55673
55712
55752
55 791
55831
55870
55910
55 949
55989
56028
56 067
56 107
41
41
41
41
41
41
41
41
40
41
41
40
41
41
40
41
40
41
40
41
40
41
40
40
41
40
40
41
40
40
40
40
40
40
40
40
40
40
40
40
40
40
39
40
40
40
39
40
40
39
40
39
40
39
40
39
40
39
39
40
L. Cotg.
0.46 303
o . 46 262
0.46 221
0,46 180
0.46 139
o . 46 098
0.46057
0.46 016
0.45 975
0.45 935
0.45 894
0.45 853
0.45 813
0.45 772
0.45 731
0.45 691
0.45 650
0.45 610
0.45 569
0.45 529
0.45 488
0.45 448
0.45 407
0.45 367
0.45327
0.45 286
0.45 246
0.45 206
0.45 165
0.45 125
0.45 085
0.45045
0.45 005
0.44965
0.44925
0.44885
0.44845
o . 44 805
0.44765
0.44 725
o . 44 685
0.44645
o . 44 605
0.44 566
0.44 526
o . 44 486
0.44446
0.44407
0.44367
0.44327
0.44 288
o 44 248
o . 44 209
0.44 169
0.44 130
o . 44 090
o . 44 05 1
0.44 on
o 43972
o 43 933
0.43 893
L. Cotg. Ic. d. L. Tang.
70°
L. Cos.
97567
97563
97558
97 554
97550
97 545
97541
97536
97532
97528
97523
97519
97515
97510
97506
97501
97 497
97492
97488
97484
97 479
97 475
97470
97466
97461
97 457
97 453
97448
97 444
97 439
97 435
97430
97426
97421
97417
97412
97408
97403
97 399
97 394
97390
97385
97381
97376
97372
97367
97363
97358
97 353
97349
97 344
97340
97 335
97331
97326
97322
97317
97312
97308
97303
9.97299
L. Sin.
d.
60
59
58
57
55
54
53
52
_51
50
49
48
47
45
44
43
42
41
40
39
38
,36
35
34
33
32
_3i
30
29
28
27
26
Prop. Pis.
4»
.1
4.1
.2
8.2
•3
12.3
4
16.4
•5
20.5
6
24 6
•7
28.7
.8
32.8
9
369
39
.1
3-
.2
7-
•3
II.
• 4
15
•5
19
.6
23
• 7
27.
.8
31
9
35-
37
I
3-7
2
7-4
•3
II .1
■4
14.8
18.5
.6
22.2
•7
25 -9
.8
29.6
9
33-3
3«
1
I
3 51
.2
7
•3
10
5
•4
14
. 5
17
5
.6
21
•7
24
5
.8
28 o|
9
31
51
5
.1
05
.2
1.0
•3
15
•4
2.0
•5
2.5
.6
30
• 7
3 5
.8
4.0
•9
4 5
40
4.0
8.0
12.0
16.0
20.0
24.0
28.0
32 o
36.0
36
36
7.2
10.8
14.4
18.0
21 .6
25.2
28,8
32 4
34
3 4
6 8
10.2
13 6
17.0
20 4
23 8
27 2
30.6
4
0.4
0.8
1 .2
1.6
20
2.4
2.8
36
Prop. Pts.
46
TABLE II
20^
I
2
3
I
7
8
10
II
12
13
'4
15
i6
17
i8
i2.
20
21
22
23
24
26
27
28
29
30
31
32
33
34
36
37
38
39
40
41
42
43
44
46
47
48
49
60
51
52
53
il
60
L. Sin.
9 53405
9 53440
9 53 475
9 53509
9 53 544
9 53 578
9 53613
9 53647
9 53 682
9 53 716
9 53 751
9 53 785
9 53819
9 53854
9 53 888
53922
53 957
53991
54025
54059
54093
54 127
54 161
54 195
54229
54263
54297
54331
54365
54 399
54 433
54466
54500
54 534
54567
54601
54635
54668
54702
54 735
54769
54 802
54836
54869
54903
54936
54969
55003
55036
55069
55 102
55 136
55 169
55 202
55235
55268
55301
55 334
55367
55400
55 433
L. Cos.
35
35
34
35
34
35
34
35
34
35
34
34
35
34
34
35
34
34
34
34
34
34
34
34
34
34
34
34
34
34
33
34
34
33
34
34
33
34
33
34
33
34
33
34
33
33
34
33
33
33
34
33
33
33
33
33
33
33
33
33
L. Tang.
56 107
56 146
56185
56 224
56 264
56303
56342
56381
56 420
56459
56498
56576
56615
56654
56693
56732
56771
56810
56849
56887
56 926
56965
57004
57042
57081
57 120
57158
57 197
57235
57274
57312
57351
57389
57428
57466
57504
57 543
57581
57619
57658
57696
57 734
57 772
57810
57849
57887
57925
57963
58 001
58039
58077
58 115
58153
58 191
58 229
58267
58304
58342
58380
58418
d. L. Cotg. c. d
c.d.
L. Cotg.
0.43893
0.43 854
0.43815
0.43 776
0.43 736
0.43 697
0.43 658
0.43 619
0.43 580
0.43 541
0.43 502
0.43463
0.43424
0.43 385
0.43346
0.43 307
0.43 268
o 43 229
o 43 190
o 43 151
0.43 "3
0.43074
o 43035
o 42 996
o 42 958
0.42 919
o 42 880
o 42 842
o . 42 803
o 42 765
0.42 726
o 42688
o . 42 649
o 42 611
0.42 572
0.42 534
0.42 496
0.42457
0.42 419
0.42381
0.42 342
0.42 304
o . 42 266
o 42 228
o 42 190
0.42 151
0.42 113
o 42075
o 42037
0,41 999
o 41 961
o 41 923
0.41 885
0.41 847
0.41 809
o 41 771
o 41 733
o 41 696
o 41 658
o 41 620
0.41 582
L. Tang.
69°
L. Cos.
9 97276
9.97271
9.97 266
9.97 262
9-97 257
97 206
97 201
97 196
97 192
97 187
97299
97294
97289
97285
97 280
97252
97248
97243
97238
97234
97229
97224
97 220
97215
97 210
97 182
97 178
97 173
97 168
97 163
97 159
97 154
97 149
97 145
97 140
97 135
97 130
97 126
97 121
97 116
97 III
97 107
97 102
97097
97092
9.97087
9.97083
9.97078
9 97073
9 97 068
97063
97059
97054
97049
97044
97039
97035
97030
97025
97 020
9.97015
L. Sin.
60
59
58
57
55
54
53
52
51
50
49
48
47
45
44
43
42
41
40
39
38
37
35
34
33
32
_31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
II
To
I
7
6
Prop. Pte.
40
.1
.2
Vo
3
12.0
4
16.0
c
20.0
^
24.0
28.0
9
32.0
36.0
39
3 9
78
II 7
156
19 5
38
I
38
2
7.6
3
II 4
4
15 2
5
19
6
22.8
7
26 6
8
30 4
9
34 2
37
3-7
7-4
II . I
14.8
18.5
22.2
25.9
29.6
33-3
35
3 5
7.0
10.5
14.0
17 5
21 .0
24 5
28.0
31 5
34
33
I
3-4
3.
2
6.8
6.
3
10.2
9
4
13 6
13.
5
17
16.
6
20 4
19.
7
23.8
23-
8
27 2
26.
9
30.6
29.
5
I
05
2
I.O
3
15
•4
2.0
.5
2 S
.6
30
•7
3 5
.8
4.0
9
4 5
4
0.4
0.8
1.2
1.6
2
2.4
28
^l
36
Prop, Pt8.
LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS
47
21
9_
10
12
13
;i
17
i8
19
20
21
22
23
24
26
27
28
29
30
31
32
33
34
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
L. Sin.
9 55 433
9 55466
9 55 499
9 55532
9 55 564
9 55 597
9 55 630
9 55663
9 55 695
9 55 728
55 761
55 793
55 858
55 891
55923
55 956
55988
56021
56053
56085
56 118
56150
56 182
56215
56247
56279
563"
56343
56375
56 408
56440
56472
56504
56536
56568
56599
56631
56663
56695
56727
56759
56 790
56822
56854
56886
56917
56949
56980
57012
57044
57075
57 107
57138
57 169
9.57201
9 57232
9 57264
9 57 295
9 57 326
9-57 358
L. Cos.
L. Tang«t€. i> L. Cotg
58418
58455
58493
58531
58569
58606
58644
58681
58719
58757
58794
58832
58869
58907
58944
58981
59019
59056
59094
59 131
59 168
59205
59243
59 280
59317
59 354
59391
59429
59466
59503
59540
59 577
59614
59651
59688
59725
59762
59 799
59835
59872
59909
59946
59983
60 019
60 056
60093
60 130
60 166
60 203
60 240
60 276
60 313
60349
60386
60422
60459
60495
60532
60568
60 605
60 641
d. L. Cotg.
37
38
38
38
37
38
37
38
38
37
38
37
38
37
37
38
37
38
37
37
37
38
37
37
37
37
38
37
37
37
37
37
37
37
37
37
37
36
37
37
37
37
36
37
37
37
36
37
37
36
37
36
37
36
37
36
37
36
37
36
41 582
41 545
41 507
41 469
41 431
41 394
41 356
41 319
41 281
41 243
41 206
41 168
41 131
41093
41 056
41 019
40 981
40944
40 906
. 40 869
40832
40795
40757
40 720
40683
40 646
40 609
40571
40534
40497
40 460
40423
40 386
40349
40312
40275
40238
40 201
40 165
40 128
40 091
40054
40 017
39981
39 944
L. Cos.
39907
39870
39834
39 797
39 760
39 724
39687
39651
39614
39578
39 541
39505
39468
39432
39 395
9.97015
9.97 010
9.97005
9 97 001
9 96 996
9 96 991
9 96 986
9 . q6 942
9.96937
96932
96927
96 922
39 359
c. d. L. Tang.
68°
96981
96 976
96971
96 966
96 962
96957
96952
96947
96917
96 912
96907
96 903
96898
96893
96888
96883
96878
96873
96868
96863
96858
96853
96848
d.
Q6843
96838
96833
96828
96823
96818
96813
96808
96803
96798
96 793
96788
96783
96778
96 772
96 767
96 762
96757
96 752
96 747
96 742
96737
96 732
96 727
96 722
9.96717
L. Sin.
60
59
58
55
54
53
52
_5i_
50
49
48
47
45
44
43
42
41
40
39
38
_36
35
34
33
32
31
Prop. Pts.
25
24
23
22
21
20^
19
18
17
16
10
I
7
6
38
I
3 8
2
76
3
11 4
4
15 2
5
19
6
22 8
7
26 6
8
30 4
9
342
37
3 7
1 \
II . I
14 8
185
22 2
25 9
29 6
33 3
36
33
I
36
3
2
72
6.
3
10 8
9
4
S
18
\l
6
21.6
19
•7
25.2
23
.8
28.8
26
9
32.4
29
33
64
96
12 8
16 o
19 2
22.4
25 6
28.8
31
.2
6.2
3
•4
i
9 3
12.4
• 7
.8
9
21.7
24.8
27.9
5
I
05
2
I.O
3
15
4
20
5
2.5
6
30
7
3-5
8
4.0
9
4 5
6
0.6
1.2
I
2.4
3-6
4.2
4.8
5-4
4
0.4
O.
I 2
1.6
2.0
24
2.8
36
Prop. Pts.
48
TABLE II
22'
I
2
3
_4
I
7
8
_9_
10
II
12
13
ii_
;i
17
i8
19
20
21
22
23
24
26
27
28
29
30
31
32
33
36
38
39
40
41
42
43
44
46
47
48
49
50
51
52
53
54.
II
II
59
60
L. Sin.
57358
57389
57420
57451
57482
57514
57 545
57576
57607
57638
57669
57700
57731
57762
57 793
57824
57855
57885
57916
57 947
57978
58008
58039
58 070
58 loi
58 131
58 162
58 192
58223
58253
58284
58314
58345
58375
58 406
58436
58467
58497
58527
58557
58588
58618
58648
58678
58709
58 739
58769
58799
58829
58859
58889
58919
58949
58979
59009
59039
59069
59098
59128
59 158
59188
L. Cos.
31
31
31
31
32
31
31
3»
31
31
31
31
31
31
31
31
30
31
31
31
30
31
31
31
30
31
30
31
30
31
30
31
30
31
30
35^
30
30
30
31
30
30
30
31
30
30
30
30
30
30
30
30
30
30
30
30
29
L. Tangr.
9.60 641
9.60 677
9.60 714
9 60 750
9 . 60 786
9 . 60 823
9 60 859
9 60 895
9.60 931
9 60 967
9.6
9.6
^i
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
9.6
96
9.6
9.6
004
040
076
112
148
184
220
256
292
328
364
400
436
472
508
544
579
615
651
687
722
758
794
830
865
901
936
972
9 , 62 008
9.62043
9 . 62 079
9.62 114
9.62 150
9.62 185
9.62 221
9.62 256.
9 . 62 292
9.62327
9 . 62 362
9.62 398
9 62 433
9 . 62 468
9.62 504
9 62539
9 62 574
c. d. L. Cotg.
9.62 609
9.62 645
9 . 62 680
9.62 715
9.62 750
9.62 785
L. Cotg.
36
37
36
36
37
36
36
36
36
37
36
36
36
36
36
36
36
36
36
36
36
36
36
36
36
35
36
36
36
35
36
36
36
35
36
35
36
36
35
36
35
36
35
36
35
36
35
35
36
35
35
36
35
35
35
36
35
35
35
35
c. d.
0.39359
0.39323
0.39 286
0.39 250
0.39214
0.39 177
0.39 141
0.39 105
o . 39 069
0.39033
o 38 996
o . 38 960
0.38924
0.38888
0.38852
0.38816
0.38 780
0.38 744
0.38 708
0.38672
0.38636
o . 38 600
o . 38 564
0.38528
0.38492
0.38456
0.38 421
0.38385
0.38349
0.38313
0.38278
0.38 242
o . 38 206
0.38 170
0.38135
0.38099
o . 38 064
0.38028
0.37992
0.37957
0.37921
0.37886
0.37850
0.37815
o 37 779
0.37 744
0.37708
0.37673
0.37 638
0.37 602
o 37567
0.37532
0.37496
0.37461
o 37 426
0.37391
o 37 355
o 37320
0.37285
o 37250
o 37215
li. Tang.
67°
L. Cos,
96717
96 711
96 706
96 701
96 696
96 691
96686
96681
96 676
96 670
96 665
96 660
96655
96 650
96645
96 640
96634
96 629
96 624
96 619
96 614
96608
96603
96598
96593
96588
96 582
96577
96572
96567
96 562
96556
96551
96546
96541
96535
96530
96525
96 520
96514
96509
96504
96 498
96493
96488
96483
96477
96472
96467
96 461
96456
96451
96445
96 440
9^35.
96 429
96424
96419
96413
96 408
9 96403
L. Sin.
60
59
58
1
55
54
53
52
S]_
50
49
48
47
45
44
43
42
41
40
39
38
_3i
35
34
33
32
31
Prop. Pts.
35
1
.1
3-71
.2
7
4
•3
II
I
4
14
8
.5
18
5
6
22
2
7
25
9
.8
29
6
9
33
3
36
36
7.2
10.8
14.4
18.0
21.6
25.2
28.8
32 4
35
3 5
7.0
10.5
14.0
17 5
21 o
24 5
28.0
31 5
32
I
32
2
6.4
3
9.6
4
12.8
5
16.0
.6
19.2
7
22.4
.8
25.6
28.8
9
30
29
I
2
3.0
6.0
2
3
4
9.0
12.0
II .
I
15
18.0
14
17
I
9
21.0
20
24.0
27
23
26
6
0.6
1.2
1.8
24
36
4.2
4.8
5 4
31
3^
6.2
9 3
12.4
\ll
21.7
24.8
27.9
5
05
I
I
2.0
2 5
30
3 5
4.0
4 5
Prop. Pts.
LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS
49
23° 1
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
Prop. Pts.
9 59 188
9.62785
0.37215
9.96403
6
■fiO"
I
9 59 218
9
62820
35
35
36
35
0.37 180
9
96397
59
2
9
59247
9
62855
0.37 145
9
96392
5
58
36
3
7
10
35
6 3-5
2 7.C
8 10.5
4. 14.0
3
9
59277
30
30
29
9
62890
0.37 no
9
96 387
57
_4
9
59307
9
62 926
0.37074
9
96381
5
6
56
55
2
3
9
59336
9
62 961
0.37039
9
96376
6
9
59366
9
62 996
37004
9
96370
54
\/\
7
9
59396
9
63031
0.36969
9
96365
53
5
t8
17.1;
8
9
59425
9
63 066
35
34
35
0.36934
9
96360
5
6
5
52
6
21
6 21
9
10
9
59 455
29
9
63 lOI
. 36 S99
9
96354
51
50
7
8
li.
2 24.5
8 28.0
9
59484
9
63135
0.36865
9
96349
II
9
59514
9
63 170
. 36 830
9
96343
96338
49
9
32.
4 31 5
12
9
59 543
9
63205
0.36 795
9
48
1
i.S
9
59 573
9
63 240
35
35
0.36 760
9
96333
6
5
6
47
1
14
15
9
59602
30
9
63275
0.36725
9
96327
46
45
.1
34
9
59632
9
63310
0.36690
9
96322
16
9
59 661
9
63345
■lA
0.36655
9
96316
44
.2
6.8
17
9
59690
29
9
63379
0.36 621
9
96 311
5
6
43
•3
[O 2
18
9
59 720
30
9
63414
0.36586
9
96305
42
4
[36
19
20
9
59 749
29
9
63449
35
0.36551
9
96300
5
6
41
40
i:
17.0
20.4
238
30.6
9
59778
9
63484
0.36516
9
96294
21
9
59808
9
63519
0.36481
9
96289
5
39
22
9
59^7
29
9
63 553
0.36447
9
96284
5
6
38
23
9
59866
29
9
35
0.36412
9
96278
37
■y 1 .
24
2S
9
59895
29
29
9
63623
34
0.36377
9
96273
5
6
36
35
1
9
59924
9
63657
0.36343
9
96267
30
29
2b
9
59 954
9
63692
0.36 308
9
96 262
5
34
27
9
59983
29
9
63 726
34
0.36274
9
96256
33
i.
5.8
8.7
II. 6
14. ■;
28
9 60012
29
9
63 761
0.36239
9
96251
5
32
29
30
9 60 041
29
29
9
63796
34
. 36 204
9
96245
5
6
31
30
i
4
9
12.
15 .
9 . 60 070
9
63830
0.36 170
9
96 240
31
9
60099
9
63 865
0.36 135
9
96234
29
18.
17 4
32
9
60128
9
63899
0.36 lOI
9
96 229
5
6
28
I
21 .
20 3
23 2
33
9
60157
29
9
63934
0.36066
9
96 223
96 218
27
?./[
34
35
9
60186
29
29
9
63968
35
0.36032
9
5
6
26
25
9
27.
26 I
9
60215
9
64 003
0.35997
9
96 212
3^^
9
60244
29
9
64037
0.35963
9
96 207
5
6
24
1
37
9
60273
29
9
64 072
35
35 928
9
96 201
23
38
3«
9
60302
29
9
64 106
34
0.35894
9
96 196
5
6
5
22
T
2 8
39
40
9
60331
29
28
9
64 140
35
0.35860
9
96 190
21
20
.2
3
56
8.4
9
60359
9
64 175
35825
9
96 185
41
9
60388
29
9
64 209
34
0.35 791
9
96179
19
.4
II .2
42
9
60417
29
9
64243
64278
0.35 757
9
96 174
5
18
s
14
43
9
60 446
29
28
29
9
0.35 722
9
96168
17
6
16 8
44
9
60474
9
64312
34
0.35 688
9
96 162
5
16
15
• 7
.8
196
22 4
45
9
60503
9 64 346
o- 35 654
9
96 157
46
9
60 532
29
9 64 381
0.35619
9
96 151
14
•9 -
25 2
47
9
60 561
29
28
9.64415
o- 35 585
9
96 146
5
6
13
1
48
9
60589
9
64449
34
0.35551
9
96 140
12
1
49
50
9
60618
29
28
9
64483
34
34
0.35517
9
96135
5
6
II
10
6
5
6 0.5
9
60 646
9
64517
35483
9
96 129
51
9
60675
29
9
64552
35
0.35448
9
96 123
9
2
I
2 I.O
52
9
60704
29
9
64586
34
0.35414
9
96 118
5
8
3
I
8 1.5
53
9
60732
9
64 620
34
0.35380
9
96 112
7
4
2
^ 2.0
54
55
9
60 761
29
28
9
64654
34
34
0.35 346
9
96 107
5
6
6
5
3 <
3
5 3.0
9
60789
9
64688
0.35312
9
96 lOI
56
9
60818
29
9
64 722
34
0.35 278
9
96095
4
7
4
I 3 5
i 40
57
9
60846
9
64756
34
0.35 244
9
96 090
5
3
8
4^
5«
9
60875
29
9
64790
34
0.35 210
9
96084
2
9
b-'
^ 4 5
59
60_
9
60903
28
9
64824
34
34
0.35 176
9
96079
5
6
I
9
60931
9
64858
0.35 142
9
96073
L. Cos.
d.
L. Cotgr.
c!T
L. Tang.
L. Sin.
d.
t
Prop. Pt8.
66^
50
TABLE II
24'
2
3
1
7
8
9
10
II
12
13
\l
17
i8
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
J4
35
3^
37
38
39
40
41
42
43
44
46
47
48
49
60
51
52
53
ii
II
57
58
Ji.
60
L. Sin.
6093:
60 960
60988
016
045.
073
lOI
129
158
186
214
242
270
298
326
354
382
411
438
466
494
522
578
606
634
662
689
717
745
773
800
828
856
883
911
939
966
994
62 021
62 049
62 076
62 104
62 131
62 159
d.
62 186
62 214
62 241
62 268
62 296
62323
62 350
62377
62 405
62432
62459
62486
62513
62541
62568
62595
L. Cos.
29
27
28
28
28
28
28
28
28
28
27
28
28
28
27
28
28
27
28
28
27
28
27
28
37
28
27
28
L. Tang.
c.d.
64858
. 64 892
. 64 926
.64 960
64 994
65 028
65 062
65 096
197
231
265
. 299
65333
65 366
9 . 65 400
9 65 434
9.65467
9 65 501
65535
65568
65 602
65 636
65 669
65 703
65 736
65 770
65 803
65837
130
164
65870
65 904
65937
65 971
66 004
66038
66071
66 104
66 138
66 171
66 204
66238
66 271
66 304
66337
9.66371
9 . 66 404
9.66437
9 . 66 470
9 66 503
9 66537
9 66 570
9 . 66 603
9 . 66 636
9 66 669
9.66 702
9 66735
9 66 768
9.66 801
9.66834
66867
L. Cotg.
L. Cotg.
0.35 142
0.35 108
o 35074
0.35 040
0.35 006
o 34 972
0.34938
0.34904
0.34870
o 34 836
o 34 803
0.34 769
0.34735
0.34 701
0.34667
0.34634
o . 34 600
0.34566
0.34533
0.34499
0.34465
0.34432
0.34398
0.34364
0.34331
0.34297
o . 34 264
0.34230
0.34 197
0.34 163
0.34 130
0.34096
o . 34 063
o . 34 029
0.33996
0.33 962
0.33929
0:33896
0.33 862
0.33829
0.33 796
0.33 762
0.33 729
0.33696
o 33663
0.33 629
0.33 596
0.33563
033 530
0.33497
0.33463
0.33430
0.33397
0.33364
0.33 331
c.d.
0.33 298
o 33 265
o 33 232
0.33 199
0.33 166
0.33 133
L. Tang.
65°
L. Cos,
9.96073
9 . 96 067
9 . 96 062
9.96 056
9 , 96 050
9 96 045
9 96 039
9 96 034
9 96 028
9 96 022
9.96 017
9.96 on
9 . 96 005
9.96 000
9 95 994
9.95988
9.95982
9 95 977
9 95971
9 95965
9 95 960
9 95 954
9 95948
9 95942
9 95 937
9 95931
9 95925
9.95920
9 95 914
9 95908
9 95 902
9 95897
9 95 891
9.95885
9 95879
9-95873
9 95 868
9.95 862
9 95856
9 95850
9-95 844
9 95839
9 95833
9.95827
9.95821
9 95815
9.95 810
9.95 804
9 95 798
9 95 792
9 95 786
9 95 780
9 95 775
9 95 769
9 95 763
9 95 757
9 95 751
9 95 745
9 95 739
9 95 733
9 95 728
L. Sin,
d.
60
59
58
1
55
54
53
52
iL
50
49
48
47
45
44
43
42
41
40
It
1
35
34
33
32
_3i
30
29
28
27
26
Prop. Pte.
34
33
I
^A
J.
2
6.8
6.
3
10 2
9
4
13 6
13
5
17.0
16
6
20,4
19
7
23.8
23
8
27.2
26
9
30.6
29
29
, I
2.9
.2
5.8
•3
8.7
.4
II. 6
•5
145
.6
17 4
•7
20.3
.8
23.2
•9
26.1
a8
.1
2.8
.2
5 6
.3
8.4
•4
II. 2
14.0
6
16.8
i
19.6
22.4
•9
25.2
37
.1
2.7
.2
5 4
•3
8. 1
•4
10.8
•S
13 5
.6
16.2
i
18.9
21 6
9
24 3
6
.1
0.6
.2
12
•3
1.8
4
2.4
. ^
3 c
6
3-6
7
4.2
8
4-8
9
5 4
5
05
1 .0
15
2.0
2.5
30
3 5
4.0
4-5
Prop. Pts,
LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS
51
25"
L. Siu.
_9_
10
II
12
13
!i
i8
19
20
21
22
23
24
26
27
28
29
30
31
32
33
34
36
37
3«
39
40
41
42
43
44
46
47
48
j49
50
51
52
53
H
55
56
II
GO
62595
62 622
62 649
62 676
62 703
62 730
62 757
62 784
62 811
62838
62865
62 892
62918
62945
62 972
62 999
63 026
63 052
63079
63 106
63 133
63213
63239
63 266
63 292
633^9
63345
63372
63398
63425
63451
63478
63 504
63531
63557
63 610
63636
63 662
63 689
63 715
63 741
63 767
63 794
63820
63 846
63872
63898
63924
63950
63976
64 002
64 028
64054
64 080
64 106
64 132
64 158
64 184
L. Cos.
d.
L. Tang.
66867
66 900
66933
66966
66 999
67032
67065
67098
67 131
67 163
67 196
67 229
67 262
67295
67327
67360
67393
67426
67458
67491
67524
67556
67589
67 622
67654
67687
67719
67752
67785
67817
67850
67882
67915
67947
67 980
68012
68 044
68077
68 109
68 142
68 174
68206
68239
68271
68303
68336
68368
68 400
68432
68465
68497
68529
68561
68593
68626
68658
68690
68 722
68754
68 786
68818
Cotgr.
L. Cotg.
0-33 133
0.33 100
0.33067
0.33034
0.33001
o 32 968
0.32935
0.32 902
o . 32 869
0.32837
o . 32 804
0.32 771
0.32 738
0.32 705
0.32673
o . 32 640
0.32 607
0.32 574
0.32 542
0.32 509
0.32476
0.32444
0.32 4n
0.32378
0.32346
0.32313
0.32 281
0.32 248
0.32 215
0.32 183
0.32 150
0.32 118
o . 32 085
0.32 053
o . 32 020
0.31 988
0.31 956
o 31 923
0.31 891
0.31 858
0.31 826
o 31 794
0.31 761
0.31 729
0.31 697
0.31 664
0.31 632
0.31 600
0.31 568
o. 31535
0.31 503
0.31 471
o 31 439
0.31 407
0.31 374
342
310
278
246
214
0.31 182
c. d. L. Tang.
64^
L. Cos.
9.95698
9.95692
9.95 686
9.95 680
9 95674
95 728
95 722
95 716
95 710
95 704
95668
95663
95657
95651
95645
95639
95633
95627
95 621
95615
95609
95603
95 597
95591
95585
95 579
95 573
95567
95561
95 555
95 549
95 543
95 537
95531
95525
95519
95513
95507
95500
95 494
95488
95482
95476
95470
95464
95458
95452
95446
95440
95 434
95427
95421
95415
95409
95403
95 397
95391
95384
95378
95372
9.95366
d.
L. Sin. I d.
60
59
58
57
55
54
53
52
_51
50
49
48
47
_46_
45
44
43
42
41
40
39
38
_3^
35
34
33
32
31
25
24
23
22
21
w
19
17
16
15
14
13
12
II
To
i
7
6
Prop. Pts.
33
I
3 3
2
6.6
3
9.9
4
13.2
16.5
6
19.8
7
8
26.4
9
29.7
33
6.4
9.6
12.8
16.0
19.2
22.4
25.6
28.8
27
2.7
i;t
10.8
16.2
.8.9
21.6
24-3
a6
2.6
7.8
10.4
lie
18 2
20 8
23 4
7
0.7
1-4
2.1
2.8
3 5
4 2
4-9
l^
6.3
.1
6
0.6
.2
12
•3
I 8
•4
2 4
it
i
t:i
•9
5-4
5
0.5
i.o
1-5
2 o
2.5
30
3-5
4.0
4-5
Prop. Pts.
52
TABLE II
26'
2
3
±
I
7
8
10
II
12
13
14_
IS
i6
17
i8
II.
20
21
22
23
26
27
28
30
31
32
33
Ji
39
40
41
42
43
44
46
47
48
49
50
51
52
53
54.
_59
GO
L. Sill.
64 184
64 210
64236
64 262
64288
64313
64339
64365
64391
64417
9 64442
9 64 468
9 64 494
9 64519
9 64545
964 571
9.64596
9 . 64 622
9.64647
9.64673
9 . 64 698
9.64724
9.64 749
9 64775
9 . 64 800
9 . 64 826
9.64 851
9,64877
9 . 64 902
9 64927
9 64953
9.64978
9.65003
9 . 65 029
9 65054
9.65 079
9 65 104
9.65 130
9 65 155
9.65 180
9 . 65 205
9.65 230
9 65255
9.65 281
9.65 306
9 65331
9 65356
9 65381
9 65 406
9 65431
d.
9 65456
9.65 481
9.65 506
9 65531
9 65556
9 65 580
9 65 605
9 65 630
9 65655
9 65 680
9 65705
L. Cos.
26
26
26
26
25
26
26
26
26
25
26
26
25
26
26
25
26
25
26
25
26
25
26
25
26
25
26
25
25
26
25
25
26
25
25
25
26
25
25
25
25
25
26
25
25
25
25
25
25
25
25
25
25
25
24
25
25
25
25
25
L. Tang.
c.d.
d.
9.68818
9.68850
9.68882
9.68 914
9 . 68 946
9.68978
9.69 010
9 . 69 042
9.69074
9.69 106
9.69 138
9.69 170
9 . 69 202
9 69234
9 . 69 266
9 . 69 298
9.69329
9.69361
9 69 393
9.69425
9 69 457
9 . 69 488
9.69 520
9 69 552
9.69584
9 69 615
9.69647
9.69679
9.69 710
9.69742
9.69 774
9 , 69 805
9.69837
9.69868
9 . 69 900
9.69932
9.69963
9 69995
9 70 026
9 . 70 058
9 . 70 089
9 70 121
9.70 152
9.70 184
9 70 2 1 5
9.70247
9.70 278
9.70309
9 70341
9.70372
9.70404
9 70435
9 70 466
9 70498
9 70529
9 . 70 560
9.70592
9 70 623
9.70654
9 . 70 685
9.70717
32
32
32
32
32
32
32
32
32
32
32
32
32
32
32
31
32
32
32
32
31
32
32
32
31
32
32
31
32
32
31
32
31
32
32
31
32
3»
32
31
32
31
32
31
32
31
31
32
31
32
31
31
32
31
31
32
31
31
31
32
L. Cotg.
0.31
0.31
[82
- %
0.31 118
o 31 086
o 3» 054
o 31 022
o . 30 990
0.30 958
o . 30 926
0.30894
o . 30 862
o . 30 830
0.30 798
0.30 766
o 30 734
0.30 702
0.30 671
o . 30 639
0.30 607
o 30575
0.30543
0.30 512
o . 30 480
0.30448
o 30 416
0.30 385
o. 30 353
0.30 321
o . 30 290
0.30258
o . 30 226
0.30 195
0.30 163
0.30 132
0.30 100
L. Cos.
o . 30 068
o 300^7
o . 30 005
0.29974
o . 29 942
0.29 911
0.29 879
o . 29 848
0.29 816
0.29 785
0.29753
0.29 722
0.29 691
0.29 659
0.29 628
o . 29 596
0.29 565
0.29534
0.29 502
0.29471
o . 29 440
o . 29 408
0.29377
0.29 346
0.29315
0.29 283
L. Cotg. led. I L. Tang.
68°
9 95 366
9 95360
9 95 354
9 95348
9 95341
9 95335
9 95 329
9 95323
9 95317
9 95 310
9 95304
9.95298
9.95292
9 95 286
9 95 279
9 95273
9.95 267
9 95 261
9 95254
9 95 248
9 95242
9 95236
9 95229
9 95223
9,95217
9 95 211
9.95 204
9.95 198
9 95 192
9 95 185
9 95 179
9 95 173
9 95 167
9 95 160
9 95 154
9.95 148
9 95 141
9 95 135
9.95 129
9 95 122
9 95 "6
9 95 "o
9 95 103
9,95097
9.95090
9.95084
9.95078
9.95071
9.95065
9 95059
9.95052
9.95046
9 95039
9 95033
9.95027
9.95 020
9.95014
9.95007
9 95001
9 94 995
9,94988
L. Sin,
d.
60
59
58
57
55
54
53
52
51
50
49
48
47
45
44
43
42
41
40
39
38
36
25
24
23
22
21
20
19
18
17
16
Prop. Pts.
3a
3
2
6
4
9
6
12
8
16
19
2
22
4
%
6
8
26
.1
2
.2
5
•3
7
■4
10
.5
13
.6
■7
W
.8
20
•9
23
3»
3.1
6,2
9 3
12 4
21 7
24 8
27.9
as
25
50
7-5
10.0
12 5
15 o
17 5
20.0
22.5
a4
24
8
2
6
o
4
8
2
6
7
I
07
2
14
•3
2 I
•4
28
i
3 5
42
I
S 6
9
6.3
6
2
8
4
o
6
2
8
54
Prop. Pts.
LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS
53
w)7o
9_
10
I
12
13
H
\l
\l
20
21
22
23
24
26
27
28
29
31
32
33
31
36
37
38
39L
40
41
42
43
44
46
47
48
49
60
51
52
53
ii
55
56
57
58
_59
60
L. Sin,
65 705
65 729
65 754
65 779
65 804
65828
65853
65878
65 902
65927
65952
65976
66 001
66 025
66 050
66075
66 099
66 124
66 148
66 173
9.66 197
9.66 221
9 66 246
9 66 270
9 . 66 295
66319
66343
66368
66392
66 416
66 441
66465
66489
66513
66 537,
66 562
66586
66 610
66 634
66658
66682
66 706
66 731
66755
66 779
66803
66827
66851
66875
66899
66 922
66 946
66 970
66 994
67018
67 042
67 066
67 090
67 "3
67137
67 i6i
L. Cos.
24
25
2S
24
25
24
25
25
24
25
24
25
24
24
25
24
25
24
24
25
24
24
25
24
24
24
24
25
24
24
24
24
24
24
25
24
24
24
24
24
24
24
23
24
24
24
24
24
24
24
23
24
24
L. Tang.
c. d,
70717
70748
70779
70 810
70 841
70873
70904
70935
70 966
70997
71 028
71 059
71 090
71 121
71 153
71 184
71 215
71 246
71 277
71 308
71 339
71 370
71 401
71 431
71 462
71 493
71 524
71 555
71586
71 617
71 648
71 679
71 709
71 740
71 771
71 802
71833
71863
71 894
71 925
72 017
72 048
72 078
72 109
72 140
72 170
72 201
72231
72 262
72293
72323
72354
72384
72415
72445
72476
72 506
72537
72567
Cotg.
3»
31
31
31
32
31
31
31
31
31
31
31
31
32
31
31
31
31
31
31
31
31
30
31
31
31
31
31
31
31
31
30
31
31
31
31
30
31
31
30
31
31
31
30
31
31
30
31
30
31
31
30
31
30
31
3?
31
30
31
30
L. Cotg.
0.29 283
0.29 252
0.29 221
0.29 790
0.29 159
0.29 127
o . 29 096
o . 29 065
0.29034
o . 29 003
0.28 972
0.28 941
0.28 910
0.28879
0.28847
0.28816
0.28 785
0.28754
0.28 723
o . 28 692
0.28661
o . 28 630
o . 28 599
0.28 569
0.28538
0.28 507
o 28 476
0.28 445
0.28 414
0.28383
0.28 352
0.28 321
0.28 291
0.28 260
0.28 229
0.28 198
0.28 167
0.28 137
0.28 106
0.28075
o . 28 045
0.28 014
0.27983
0.27 952
0.27 922
o 27 891
0.27 860
0.27 830
0.27 799
0.27 769
c.d,
0.27 738
o 27 707
o 27 677
o 27 646
0.27 616
0.27585
0.27555
0.27 524
0.27494
0.27463
0.27433
L. Tang.
62^
L. Cos.
94891
94885
94878
94871
94865
94988
94982
94 975
94969
94962
94956
94 949
94 943
94936
94930
94923
94917
94 91 1
94904
94898
94858
94852
94845
94839
94832
94 826
94819
94813
94 806
94 799
94 793
94786
94780
94 773
94767
94 760
94 753
94 747
94740
94 734
94727
94720
94714
94707
94700
94694
94687
94 680
94674
94667
94 660
94654
94647
94 640
94634
94627
94 620
94614
94607
94 600
9-94 593
L. Sin.
d.
60
59
58
57
55
54
53
52
11
60
It
47
J^
45
44
43
42
41
40
39
38
_3i
35
34
33
32
31
Prop. Pts.
2M
I
2
3
4
32
6.4
9.6
12.8
16
6
19 2
8^
22.4
25.6
9
28.8
SI
II
9-3
12.4
155
18.6
21 .7
24.8
27.9
30
30
6 o
90
12.0
15 o
18.0
21.0
24.0
27.0
as
I
2.5
2
50
3
7 5
4
10.
•5
12.5
.6
15
.7
17-5
.8
20.0
9
22.5
H
2.4
4.8
7.2
9.6
12.0
14.4
16.8
19 2
21.6
6.9
9.2
ii?
18.4
20 7
7
I
0.7
2
14
3
2. I
4
2.8
5
3 5
6
4.2
I
56
9
6.3
6
0.6
I .2
1.8
24
36
4.2
48
5 4
Prop. Pfs.
54
TABLE II
28^
9 67 i6i
I
2
3
i
7
8
9
10
II
12
13
II
;i
17
18
19
20
21
22
23
ii
26
27
28
80
31
L. Sin.
67185
67208
67 232
67256
67280
67303
67327
67350
67374
9.67398
9 67421
9 67445
9 67 468
9 67 492
9 67515
9 67539
0.67 562
9.67586
9 67 609
9 67 633
9 67 656
9 . 67 680
9.67 703
9 67 726
9 67 750
9 67 773
9.67 796
9 67 820
9 67 843
9 67866
^_ 9 67 890
32 967913
33 967936
34 9 67959
36
I 39
40
41
42
43
44
46
47
48
49
60
51
52
53
ii
59
60
9 67 982
9 68006
9 68 029
9 68 052
9 68 075
9.68098
9 68 121
9 68 144
9 68 167
9 68 190
9 68 213
9 68 237
9 . 68 260
9 68 283
9 68 305
9 68 328
9 68351
9 68 374
9 68 397
9 68 420
"9 68443
9.68466
9.68489
9.68 512
9-68 534
9 68 557
L. Cos.
24
23
24
24
24
23
24
23
24
24
23
24
23
24
23
24
23
24
23
24
23
24
23
23
24
23
23
24
23
23
24
23
23
23
23
24
23
23
23
23
23
23
23
23
23
24
23
23
23
23
23
23
23
23
23
23
23
23
23
23
L. Tang.
72567
72598
72 628
72659
72689
9 72 720
9 72 750
9 72 780
9 72 811
72 841
c.d.
72 872
72 902
72932
72 963
72993
73023
73 114
73 144
73 175
73205
73235
73265
73295
73326
73356
73386
73416
73446
73476
73507
73537
9 73567
9 12, 597
9 73 627
9 73 657
9.73687
73 717
73 747
73 777
73807
73837
73867
73897
73927
73 957
73987
74017
74047
9.74077
9 74 107
9 74 137
9 74 166
9 74 196
74 226
74256
74286
74316
74 345
9 74 375
31
30
3»
30
31
30
30
31
30
31
30
30
31
30
30
31
30
30
30
31
30
30
30
30
31
30
30
30
30
30
31
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
29
30
30
30
30
30
29
30
L. Cotg.
0.27433
0.27 402
0.27 372
o 27341
0.27 311
0.27 280
0.27 250
0.27 220
0.27 189
0.27 159
0.27 128
0.27 098
0.27 068
0.27 037
0.27 007
0.26 977
o . 26 946
0.26 916
0.26886
0.26856
0.26 825
0.26 795
0.26 765
0.26 735
0.26 705
0.26 674
o . 26 644
0.26 614
0.26 584
0.26554
0.26 524
o . 26 493
o . 26 463
0.26433
o . 26 403
L. Cos.
0.26373
0.26343
0.26 313
o . 26 283
0.26 253
o . 26 223
0.26 193
0.26 163
o 26 133
0.26 103
o 26 073
o 26 043
0.26 013
o 25983
o 25953
25923
25893
25863
25 834
25 804
0.25 774
o 25 744
0.25 714
o 25 684
0.25655
0.25 625
L. Cotg. c. d. L. Tang
61^
9 94 593
9 94 587
9 94 580
9 94 573
9 94567
9 94 560
9 94553
9-94 546
9 94540
9 94 533
9 94526
9 94519
9 94513
9 94506
9 94 499
9 94492
9 94485
9 94 479
9.94472
9 94465
9 94458
9-94 451
9 94 445
9-94 438
9 94431
9 94424
9 94417
9.94410
9.94404
9 94 397
d.
9 94390
9 94383
9 94376
9 94369
9.94362
9 94 355
9 94 349
9 94342
9-94 335
9.94328
9 94321
9 94314
9 94307
9.94300
9 94293
9 . 94 286
9.94279
9 94273
9 94 266
9 94259
9 94252
9 94245
9 94238
9 94 231
9 94224
9 94217
9 94 210
9 94 203
9 94 196
9 94 189
9.94 182
L. Sin.
60
1
55
54
53
52
SL
60
49
48
47
45
44
43
42
_4i_
40'
39
38
36
35
34
33
32
30
29
28
27
26
Prop. Pts.
31 1
I
.2
C
3
4
1
9 3
12.4
•7
.8
9
21.7
24.8
27.9
30
30
6 O
9 o
12 C
18 O
21 .0
24.0
27.0
29
29
58
8.7
II 6
14 5
17 4
20.3
23.2
26.1
34
I
2.4
2
4.8
3
7.2
4
9.6
• S
12
.6
14 4
.7
16.8
.8
19.2
9
21.6
23
6.9
92
II. 5
«3 8
16 I
18 4
20.7
7
I
0.7
2
14
3
2.1
4
2 8
5
3 5
6
42
I
S 6
9
6.3
sa
2.2
4 4
6.6
8 8
no
13 2
176
19.8
Prop. Pts.
LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS
55
29° 1
L. Sin.
d.
L. Tang.
c.d.
1. Cotg.
L. Cos.
d.
Prop. Pts.
9 68557
23
9 74 375
. 25 625
9.94 182
■fiO"
I
9
68580
9
74405
0.25595
9
94175
94168
S9
2
9
68603
9
74 435
0.25565
9
S8
30
3
9
68625
23
23
9
74465
0.25535
9
94 161
S7
_4
5
9
68648
9
74 494
30
25 506
9
94 154
56
SS
2
7
io
9
12.0
9
68671
9
74524
0.25476
9
94147
6
9
68694
9
74 554
. 25 446
9
94140
S4
4
7
9
68716
9 74 583
29
0.25417
9
94133
S3
5
15
18.0
8
9
68739
23
22
9 74613
30
0.25387
9
94126
S2
6
9
10
9
68 762
9 74 643
30
0.25357
9
94 119
51
50
I
21.0
24.0
9 68 784
9
74673
0.25327
9
94 112
II
9 68 807
9
74702
29
0.25 298
9
94 105 ! 1
49
9
27.0
12
9 68 829
9
74732
0.25 268
9
94098
48
1
13
9 68 852
23
22
9
74762
30
0.25 238
9
94090
47
14
IS
9 68 875
9
74791
30
. 25 209
9
94083
46
4S
1
29
2.9
9 68 897
9
74821
0.25 179
9
94076
I
i6
9 68 920
9
74851
30
0.25 149
9
94069
44
2
17
9
68942
9
74880
29
0.25 120
9
94062
43
3
8-7
II. 6
i8
9
68965
9
74910
30
0.25 090
9
94055
42
4
19
2r
9
68987
23
9
74 939
29
30
0.25 061
9
94048
41
40
•s^
145
17 4
20.3
'41
9
69 010
9
74969
0.25 031
9
94041
21
9
69032
9
74998
0.25 002
9
94034
7
39
22
9
69055
9
75028
0.24972
9
94027
38
23
9
69077
9
75058
30
0.24942
9
94020
37
24
25
9
69 100
22
9
75087
29
30
0.24913
9
94012
36
3S
1
9
69 122
9
75 117
0.24883
9
94005
23
26
9
69 144
9
75 146
29
0.24 854
9
93998
34
T
1:1
69
9.2
27
9
69 167
9
75 176
. 24 824
9
93991
33
28
9
69 189
9
75 205
29
0.24795
9
93984
32
•3
• 4
29
30
9
69 212
22
9
75235
30
29
0.24765
9
93 977
31
30
9
69234
9
75264
0.24736
9
93970
31
9 69256
9
75294
30
0.24 706
9
93963
29
6
32
9 69279
23
9
75323
29
0.24677
9
93 955
28
y
16. 1
33
9
69301
9
75 353
30
0.24647
9
93948
27
g
18.4
34
3S
9
69323
22
9
75382
29
29
0.24 618
9
93 941
26
25
.9
20.7
9
69345
69 368
9
75 411
0.24589
9
93 934
1
36
9
9
75441
30
0.24559
9
93927
24
1
37
9
69390
9
75470
29
0.24530
9
93920
23
aa
3«
9
69412
9
75 500
30
. 24 500
9
93912
22
. I
2.2
39
40
9
9
69434
22
9
75529
29
29
0.24471
9
93905
21
20
.2
.3
44
6.6
69456
9
•75558
0.24442
9
93898
41
9
69479
23
9
75588
30
0.24412
9
93?9i
19
• 4
8.8
42
9
69501
9
75617
29
0.24383
9
93884
18
II.
43
9
•69 523
9
75647
30
0.24353
9
93876
17
6
13.2
44
45
9
69545
22
9
• 75 676
29
29
0.24324
9
93869
16
15
:l
17.6
9
69567
9
•75 705
0.24295
9
93 862
46
9
.69589
9
• 75 735
30
0.24 265
9
93 855
14
9
19.8
47
9
69 611
9
■ 75 764
29
0.24236
9
93847
13
1
48
9
69633
9
•75 793
29
. 24 207
9
93840
12
1
49
50
_9
9
69655
22
9
.75822
29
30
0.24 178
9
93833
II
10
I
8
0.8
7
0.7
.69677
9
.75852
0.24 148
9
.93826
SI
9
.69699
9
.75881
29
0.24 119
9
93819
9
2
lb
14
S2
9
.69 721
9
75910
29
. 24 090
9
93 811
8
3
2.4
2.1
S3
9
69 743
9
75 939
29
0.24061
9
93804
7
4
32
2.8
54
SS
9
69 765
22
9
75 969
30
29
. 24 03 1
9
93 797
b
5
I
It
73
3 b
4.2
49
9
69787
9
75998
. 24 002
9
93789
S6
9
69809
9
.76027
29
0.23973
9
93 782
4
57
9
69831
9
76 056
29
0.23944
9
W
3
SB
9
69853
9
. 76 086
30
0.23914
9
8
7
2
y 1
59
9
.69875
22
9
.76115
29
29
0.23885
9
93760
I
9
.69897
9
76144
0.23856
9
93 753
L. Cos.
~
L. Cotg.
c.d.
L. Tangi
L. Sin.
d.
/
Prop. Pts.
60^ 1
56
TABLE II
30^
I
2
3
i
7
8
_9_
10
II
12
13
M
19
20
21
22
23
24
26
27
28
29
30
31
32
33
34
36
37
3«
39
40
41
42
43
44
46
47
48
49
60
51
52
53
il
i2.
00
L. Sin,
69897
69919
69941
69963
69 984
70 006
70028
70 050
70 072
70093
70 115
70137
70159
70 180
70 202
70 224
70245
70 267
76288
70310
70332
70353
70375
70396
70418
70439
70461
70 482
70504
70525
70547
70568
70590
70 611
70633
70654
70675
70697
70 718
70 739
70 761
70 782
70803
70 824
70 846
70867
70888
70 909
70931
70952
70973
70994
71 015
71 036
71058
71 079
71 100
71 121
71 142
71 163
71 184
L. Cos.
d.
L. Tally.
76 144
76 173
76 202
76231
76 261
76 290
76319
76348
76377
76 406
76435
76464
76493
76 522
76551
76580
76 609
76639
76668
76697
76725
76 754
76 1^2>
76812
76841
76870
76899
76928
76957
76986
77015
77044
77073
77 101
77 130
77 159
77188
77217
77246
77274
77303
77332
77361
7739c
77418
77 447
77476
77505
77 533
77562
77591
77619
77648
77677
77706
77 734
77 763
77791
77 820
77^^49
77877
d. L. Cotg.
c. d.
L. Cotg.
o 23 856
o 23 827
0.23 798
0.23 769
0.23 739
0.23 710
0.23 681
0.23 652
0.23 623
0.23 594
0.23565
0.23 536
0.23 507
0.23478
0.23449
o . 23 420
0.23391
0.23 361
0.23332
0.23303
0.23 275
0.23 246
0.23 217
0.23 188
o 23 159
0.23 130
0.23 lOI
0.23 072
0.23043
0.23014
. 22 985
O 22 956
O 22 927
O 22 899
0.22 870
c. d.
O 22 841
O 22 812
O 22 783
0.22 754
0.22 726
0.22 697
. 22 668
. 22 639
0.22 610
O 22 582
0.22553
0.22 524
. 22 495
O 22 467
. 22 438
0.22 409
O 22 381
O 22 352
. 22 323
O 22 294
0.22 266
0.22 237
. 22 209
O 22 180
0.22 151
0.22 123
L. Tang.
59^
L. Cos.
d.
60
9 93 753
9
93 746
59
9
93738
58
9
93 731
S7
9
93 724
7
56
9
93 717
55
9
93 709
54
9
93 702
S^
9
93695
52
9
93687
51
50
9 93 680
9 93 673
49
9 93665
48
9
93 658
47
9
93650
46
4S
9
93643
9
93636
44
9
93 628
4^
9
93621
42
9
93614
41
40
9
93606
9
93 599
39
9
93591
S8
9
93584
37
9
93 577
36
35
9
93569
9
93562
34
9
93 554
33
9
93 547
32
9
93 539
31
30
9
93532
9
93525
29
9
93517
28
9
93510
27
9
93502
26
25
9
93 495
9
93487
24
9
93480
23
9
93472
22
9
93465
21
20
9
93 457
9
93450
19
9
93442
18
9
93 435
17
9
93427
16
15
9
93420
9
93412
14
9
93405
13
9
93 397
12
9
93390
II
10
9
93382
9
93 375
9
9
93367
8
9
93360
7
9
93352
6
5
9
93 344
9
9Z2,i1
4
9
93329
3
9
93322
d.
2
9
93314
9
93307
L. Sin.
Prop. Pts.
30
I
3
2
6.0
3
9.0
4
12.0
15.0
6
18
7
21.0
8
24.0
9
27.0
2.9
8
7
6
5
4
5
8
II
14
17
20.3
23.2
26.1
28
28
8
.1
0.8
.2
r 6
3
2.4
4
32
1;
4.0
6
48
i
l\
9
7-2!
23
2 2
4 4
6 6
8.8
II. o
13.2
17 6
19.8
ai
2.1
42
8.4
12.6
14 7
16 8
18.9
Prop. Pts.
LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS
57
31° 1
L. Sin.
d.
L. Tang. c. d.
L. Cotg.
L. Cos.
d.
Prop. Pts.
9 71 184
977877 ,„
0.22 123
9 93307
g
60
I
9 71 205
9 77 906
29
28
. 22 094
9.93299
3
59
2
9 71 226
9 77 935
. 22 065
9 93291
58
33
^
9 71 247
9 77 963
0.22037
9 93 284
Q
57
T
2 9
9 71 268
21
9 77 992
28
29
28
22 008
9 93276
7
8
56
55
2
3
9 71 289
9 78 020
0,21 980
9.93269
6
9 71 310
21
9 78049
0.21 951
9.93261
8
54
4
II 6
7
9 71 331
9 78 077
21 923
9-93 253
53
14 5
8
9
10
9 71 352
9 71 373
21
20
9 78 106
9 78135
29
28
0.21 894
0.21 865
9.93246
9-93 238
8
8
52
51
60
7
8
17 4
20 3
23.2
9 71 393
9 78 163
21 837
9 93230
II
9 71 414
9.78 192
28
21 808
9.93223
8
49
9
26.1
12
9 71 435
9 78 220
31 780
9 93215
8
.48
1
n
9 71 456
9 78 249
21 751
9.93207
47
15
9 71 477
21
9.78277
29
28
21 723
9 93200
8
8
46
45
28
2 8
^.6
8.4
9 71 498
9.78306
0.21 694
9 93 192
I
16
9
71 519
9 78 334
0.21 666
9 93 184
44
2
•7
9
71 539
9 78363
29
28
28
29
28
21 637
9 93 177
8
8
7
8
8
43
3
18
9
71 560
^
9.78391
0.21 609
9 93 169
42
4
II .2
19
20
9
71 581
21
9.78419
0.21 581
9 93 161
41
40
I
14.0
16.8
19 6
22 4
25.2
9
71 602
9.78448
0.21 552
9 93 154
21
9
71 622
9.78476
0.21 524
9 93 146
39
22
9
71 643
9 78505
29
28
0.21 495
9 93 138
38
23
9
71664
9 78533
0.21 467
9 93 131
7
8
8
37
•y
24
2S
9
71685
20
9.78562
29
28
28
0.21 438
9 93 123
36
35
1
9
71 705
9 78 590
0.21 410
9 93 "5
26
9
71 726
9.78618
0.21 382
9.93 108
7
34
2 I
27
9
71 747
9 78 647
29
28
0.21 353
9.93 100
8
8
7
8
8
8
33
4.2
8.4
10 5
28
9
71 767
9.78675
0.21 325
9.93092
32
3
• 4
.5
29
30
9
71788
21
9 78 704
29
28
28
0.21 296
9.93084
31
30
9
71 809
9.78732
0.21 268
9.93077
31
9
71 829
9,78 760
0.21 240
9.93069
29
.6
12.6
32
9 71 850
9,78789
29
28
28
29
28
28
0.21 211
9.93061
28
.7
14 7
,1S
9 71 870
9.78817
0.21 183
9-93 053
27
.8
16.8
34
3S
9 71 891
20
9.78845
0.21 155
9.93046
8
26
25
9
18.9
9 71 911
9.78874
0.21 126
9-93 038
1
^^
9 71 932
9.78902
0.21 098
9.93030
24
1
37
9 71 952
9.78930
0.21 070
9.93022
8
23
• 20
3«
9
71 973
9 78959
29
28
28
28
0.21 041
9.93014
22
J
2.0
39
40
9
71 994
20
9 78987
0.21 013
9.93007
8
Q
21
20
.2
■ 3
40
6.0
9
72014
9.79015
. 20 985
9.92999
41
9
72034
9 79043
0.20957
9.92991
Q
19
• 4
8.0
42
9
72055
9 • 79 072
29
0.20 928
9.92983
18
.5
10.0
43
9
72075
9.79 100
. 20 900
9.92976
8
8
8
Q
17
.6
12.0
_4£_
4S
9
72096
20
9.79 128
28
0.20 872
9.92 968
16
15
•7
.8
14.0
16.0
18.0
9
72 116
9 79156
9 79 185
. 20 844
9.92960
46
9
72 137
29
28
0.20 815
9.92952
14
•9
47
9
72 157
9.79213
0.20 787
9.92944
D
13
1
48
9
72 177
9 79 241
0.20759
9.92936
12
1
49
50
9
72 198
20
9 79 269
28
28
20 731
9.92929
7
8
8
8
8
8
8
10
.1
8
0.8
7
07
9
.72218
9.79297
0.20 703
9.92-921
51
9
.72238
9 79 326
29
. 20 674
9 92913
9
2
I.b
14
•ia
9
72 259
9 79 354
. 20 646
9.92905
8
3
2.4
2. I
S3
9
.72 279
9.79382
28
0.20618
9.92897
7
4
32
2 8
54
SS
9
72 299
21
9.79410
28
28
. 20 590
9 92 889
6
5
6
4.0
3 b
4 2
4 9
9 72 320
9 -79 438
0.20 562
9.92881
,S6
9 72 340
9.79466
28
0.20534
9 92874
7
4
I
S7
9.72360
9 79 495
29
. 20 505
9.92866
3
Sb
9 72 381
9 79523
28
0.20477
9.92858
2
y 1
/ ■^ *' Ji
59
00
9 72 401
20
9 79551
28
28
. 20 449
9 92 850
8
I
9.72421
9 79 579
0.20421
9 . 92 842
L. Cos.
d.
L. Cotg.
c.d.
L. Tang.
L. Sin.
d.
Prop. Pts.
58^
1
58
TABLE II
32° 1
—
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
60
Prop. Pte.
9.72421
9-79 579
28
0.20 421
9.92842
8
8
8
8
7
8
9.72441
9.79607
0.20393
9.92834
59
2
9 72 461
9 79635
28
0.20 365
9.92 826
58
29 38
2.9 2.8
58 5-6
8.7 8.4
3
4
s
9 72482
9 72 502
20
20
9.79663
9.79691
28
28
28
20337
20 309
9.92 818
9 92 810
u
55
2
3
9.72522
9.79719
0.20 281
9 92 803
6
9.72542
9-79 747
0.20 253
9 92 795
8
54
4
II .6 II .2
7
9 72 562
9 79 776
28
0.20 224
9 92787
53
14. c
I4.0
8
9 72582
9.79804
28
0.20 196
9 92 779
52
6
17 4 16 8
9
10
9 . 72 602
20
9.79832
28
0.20 168
9.92771
8
8
8
8
8
8
8
8
8
51
50
•7
8
20.3 19.6
23 2 22.4
9 72 622
9 . 79 860
0.20 140
9 92763
II
9.72643
9.79888
28
0.20 112
9 92 755
49
9
26.]
I252
12
9 72 663
9.79916
28
0.20084
9 92 747
48
1
M
9 72683
9-79 944
28
0.20 056
9 92 739
47
1
14
9 72 703
20
9.79972
28
28
. 20 028
9.92 731
46
45
I
37
2.7
9 72 723
9.80000
0.20000
9.92 723
i6
9 72 743
9.80028
28
0.19972
9.92 715
44
.2
5 4
17
9.72 763
9 . 80 056
19944
9.92 707
43
3
8.1
i8
9 72 783
9.80084
28
28
28
0.19 916
9.92699
42
4 10.
'9
20
9.72803
20
9.80 112
0.19888
9.92691
8
8
8
8
8
8
8
8
8
8
8
8
8
41
40
I lit
9.72823
9.80 140
0.19 860
9.92683
21
9.72843
9.80 168
0.19 832
9 92675
39
22
9 72863
20
9.80195
0. 19 805
9,92667
38
9 24.3
2^
9 72883
9.80223
28
19777
9 92659
37
25
9.72902
20
9.80251
28
,0
0.19749
9 92651
36
35
9.72922
9.80279
0.19 721
9.92643
21 1 90 1
26
9.72942
9.80307
_Q
19 693
9 92635
34
J
I 20
27
9.72962
9-80335
28
28
28
0,19 665
9.92627
33
4
6.
8.
10.
2 40
3 6.0
4 8.0
5 10.0
28
29
30
9.72982
9.73002
20
20
9.80363
9.80391
0.19637
0.19 609
9 92 619
9.92 611
32
31
30
3
4
9.73022
9.80419
0.19 581
9 92 603
^i
9.73041
9.80447
o- 19 553
9 92 595
29-
5
12.
6 12.0
32
9.73061
9.80474
27
28
0.19526
9.92587
28
.7
14
7 14
33
9.73081
9.80 502
0.19498
9 92579
8
8
8
27
.8
16.
8 16.0
34
35
9 73 loi
20
9.80530
28
28
0.19470
9.92571
26
25
9
18.
9 18.0
9 73 121
9.80558
0.19442
9 92563
36
9 73 140
9.80586
28
28
27
28
28
„Q
0.19 414
9 92555
24
1
37
9.73 160
9.80 614
0.19386
9.92546
9
8
8
8
8
8
8
8
8
23
19
1 9
3«
39
40
41
9.73180
9.73200
20
20
9 . 80 642
9.80669
o. 19 358
0.19 331
9 92538
9 92530
22
21
20
19
.1
.2
3
.4
3
5-
7-
9 0.9
8 1.8
7 2.7
6 3.6
9.73219
9 -73 239
9.80697
9-80725
0.19303
0.19275
9.92522
9.92514
42
9 73259
9-80753
9.80 781
28
0.19247
9.92506
18
.5
9
5 4 5
43
9.73278
0.19 219
9.92498
17
.6
II
4 5 4
44
4S
9.73298
20
9.80808
28
28
28
0.19 192
9.92490
16
15
■I
13
15
3 6.3
2 7.2
9 73318
9.80836
0.19 164
9 92 482
46
9 73 337
^9
9.80864
0.19 136
9 92473
9
8
8
8
8
8
8
14
9
17
I 8.1
47
9 73 357
9.80892
0.19 108
9.92465
13
1
48
9-73 377
9.80919
27
28
28
28
0.19 081
9 92457
12
1
49
50
9 73396
^9
20
9.80947
0.19053
9.92449
II
10
I
8
0.
7
8 0.7
9 73416
9.80975
0.19025
9.92441
51
9 73 435
19
9.81 003
0.18997
9 92433
9
.2
I.
6 1.4
52
9 73 455
9.81 030
27
28
28
27
28
28
0. 18 970
9.92425
8
3
2.
4 21
S3
9 73 474
^9
9.81 058
9.81 086
0.18942
9.92416
9
8
8
8
8
7
-4
3
2 28
54
9 73 494
19
0.18914
9 . 92 408
6
5
i
4
4
3 5
8 42
6 49
\ is
9 73513
9.81 113
0.18887
9.92400
56
9 73 533
9 81 141
0.18859
9.92392
4
■'J
.8
6:
^S^
9 73552
*9
9.81 169
0.18 831
9 92384
3
9-73 572
9.81 196
27
28
28
0.18804
9 92376
2
y
y
59
60_
9 73591
19
20
9 81 224
0.18 776
9.92367
9
8
I
_0^
9.73 611
9.81 252
0.18 748
9 92 359
L. Cos.
d.
L. Cotg. Ic. d.
L. Tang.
L. Sin.
d.
Prop. Pte.
57° 1
LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS
59
33°
1
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
"60"
Prop. Pt8.
9-73611
19
9 81 252
27
0.18 748
9 92359
g
I
9 73630
20
9 81 279
28
0.18 721
9 92351
8
59
2
9-73 650
IQ
9 81 307
28
0.18 693
9 92343
58
38
87
2.7
5 4
3
9.73669
ao
981335
27
28
28
0.18665
9 92335
57
T
2.8
5.6
8.4
4
9.73689
19
19
9 81 362
0.18638
9 92326
8
8
8
56
55
.2
.3
9 73 708
9.81390
18 610
9 92318
6
9 73727
20
9 81 418
0.18 582
9,92310
54
4
II. 2
10.8
7
9 73 747
19
9.81445
28
0.18555
9,92302
9
8
53
14.0
13 5
8
9 73 766
9 81473
0.18527
9.92 293
52
.6
16.8
16.2
9
9 73785
20
9 81 500
28
0.18 500
9 92285
8
51
•7
19.6
18 9
10
9 73805
10
9.81 528
28
0.18472
9.92277
8
50
.8
22.4
21.6
II
9 73 824
9 81 556
27
28
0.18444
9.92269
9
8
49
9
25.2
243
12
9 73843
20
9 81 583
0.18 417
9.92 260
48
1
n
9 73863
19
19
9 81 611
0.18389
9.92252
47
15
9 73 882
9.81 638
28
0.18362
9 92244
9
8
46
45
1
30
2.0
9 73901
9.81 666
0.18334
9 92235
I
.16
9.73921
19
9 81 693
28
0.18 307
9 92227
8
44
8.0
17
9 73940
9 81 721
0.18 279
9.92219
8
43
•3
18
9 73 959
19
9.81 748
27
28
27
,0
0.18 252
9 92 211
42
•4
19
20
9 73978
^9
19
20
9.81 776
0. 18 224
9 92 202
9
8
g
41
40
'\
10.0
12
14
16
9 73 997
9 81 803
0.18 197
9.92 194
21
9 74017
19
19
19
19
9 81 831
0.18 169
9.92 186
39
22
9.74056
9 81 858
28
0.18 142
9.92 177
8
38
n
18
23
9 74055
9.81 886
0.18 114
9.92 169
37
24
25
9 74 074
9 81 913
28
0.18087
9.92 161
9
8
_36_
35
1
9 74093
9 81 941
0. 18 059
9.92152
19
26
9 74 "3
19
19
19
19
9 81 968
28
0. 18 032
9.92 144
8
34
¥
3:1
n
27
28
29
30
9 74 132
9 74 151
9.74170
9 81 996
9 82 023
9 82 051
27
28
27
28
0.18004
0.17977
0.17949
9.92 136
9.92 127
9.92 119
9
8
8
33
32
31
30
.2
•3
•4
• S
9 74 189
9.82078
0.17922
9.92 III
31
9.74208
19
9.82 106
0.17894
9.92 102
29
.6
I
r 1
32
9.74227
19
9 82133
28
0.17867
9.92094
8
28
.7
i^.\ 1
33
9.74246
^9
9.82 161
0.17839
9.92086
27
.8
) 2
34
35
9.74265
19
»9
9 82 188
27
27
28
0.17 812
9.92077
9
8
26
25
9
I
I
9,74284
9 82 215
0.17785
9.92069
1
36
9 74303
19
9,82243
0.17757
9.92060
8
24
1
37
9.74322
9 82 270
28
0.17730
9.92052
8
23
18
3«
9 74341
9.82 298
0.17 702
9.92044
22
1
I 8
39
40
9 74 360
19
9 82325
27
27
28
0,17675
9 92035
8
21
20
2
.3
36
5 4
9 74 379
9.82352
0.17648
9.92027
41
9 74398
9 82 380
0.17 620
9.92 018
19
.4
7.2
42
9 74417
^9
9 82 407
27
28
0.17593
9.92 010
18
9.0
43
9 74436
^9
9.82435
0.17565
0.17538
9 92 002
17
6
10.8
44
4S
9 74 455
»9
»9
9 . 82 462
27
27
28
9 91 993
9
8
lb
15
i
12.6
14.4
9 74 474
9.82489
0.175"
9.91 985
46
9 74 493
19
9.82517
0,17483
9,91976
9
8
14
•9
16.2
47
9 74512
»9
9.82 544
27
0.17456
9.91 968
13
1
48
9 74 53^
19
18
9 82571
27
28
27
0.17429
9 91 959
8
12
1
49
50
9 74 549
19
9.82599
0.17 401
9 91 951
9
II
10
. I
9
0.9
8
0.8
9 • 74 568
9.82 626
0.17374
9 91 942
51
9 74 587
19
9.82653
27
0.17347
9 9i 934
9
.2
1.8
1.6
52
9.74606
19
9.82681
0.17319
9.91 925
8
8
•3
2.7
2.4
S3
9 74 625
19
9.82 708
27
0.17 292
9.91 917
7
.4
36
32
54
55
9.74644
19
t8
9 82735
27
27
0.17265
9,91 908
9
8
6
5
i
I
9
4.5
5 4
6.3
7.2
8.1
n
72
9.74662
9.82 762
0.17 238
9.91 900
56
9 74681
19
9.82 790
0. 17 210
9.91 891
8
9
8
4
57
58
9 . 74 700
9 74 719
19
19
18
19
9 82817
9.82844
27
27
0.17 183
17 156
9,91883
9,91 874
3
2
59
60
9 74 737
9.82871
27
28
0.17 129
9,91 866
9
I
9 74756
9.82899
0.17 lOI
9 91 857
L. Cos.
T"
L. Cotg.
™d.
L. Tang*
L. Sin.
d.
/
Prop. Pts.
56°
1
60
TABLE II
34° 1
t
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
60
Prop. Pts.
"F
9-74 756
9.82 899
0.17 lOI
9 91 857
8
I
9
74 775
9
82 926
0.17074
9.91 849
5Q
2
3
9
9
74 794
74812
18
9
9
82953
82 980
27
28
0.17047
0.17 020
9.91 840
9.91 832
9
8
58
57
28
2.8
5.6
8.4
27
4
9
74831
19
18
9
83008
27
0.16 992
9.91 823
9
8
56
55
.2
•3
.4
2.7
9
74850
9
83035
0. 16 965
9.91 815
6
9
74868
9
83062
0.16938
9.91 806
9
8
54
II 2
10 8
7
9
74887
9
83089
28
0.16911
9.91 798
^^
14.0
8
9
74906
18
9
^Z 117
27
27
0.16883
9.91 789
9
8
9
52
t6 8
9
10
9
74924
19
9
83 144
0.16856
9.91 781
51
oO
•7
.8
19.6
22.4
189
21.6
9
74 943
9
83 171
0.16 829
9.91 772
II
9
74961
9-
83198
0.16 802
9.91 763
9
8
4Q
.9
25.2
24 3
12
9
74980
9
83225
0.16775
9 91 755
48
n
9
74 999
18
9
l^ ^p
28
0. 16 748
9 91 746
9
8
9
47
14
15
9
75017
19
t8
9
2,z 280
27
0.16 720
9 91 738
46
45
!
26
26
9
75036
9
83307
0.16 693
9 91 729
.1
16
9
75054
9
83334
0. 16 666
9.91 720
9
8
44
.2
M
17
9
75073
18
9
83 361
0.16 639
9.91 712
43
•3
18
9
75091
9
83 388
0. 16 612
9.91 703
9
8
9
42
.4
10.4
19
^0"
9
75 no
18
9
83415
27
28
0.16 585
9.91 695
41
40
i
13.0
;§2
9
75128
9
83442
0. 16 558
9.91 686
21
9
75 147
18
9
83470
27
27
27
27
0.16 530
9.91677
9
8
30
20 8
22
9
75 165
9
83497
0.16 503
9.91 669
38
23 4
23
9
75184
^9
9
83524
0.16 476
9.91 660
9
37
•y
24
2S
9
75202
19
9
83 551
16449
9,91 651
9
8
36
3S
1
9
75221
9
83 578
0. 16 422
9.91 643
X9
26
9
75239
9
83605
0.16395
0.16368
9.91 634
9
34
^
I - 1
27
9
75258
19
18
9
83632
9.91 625
9
8
33
\
28
9
75276
18
19
t8
9
83659
83686
0.16 341
9.91 617
32
3
•4
>.7
r6
) 5
29
30
9
75294
9
27
0.16 314
9.91 608
9
9
8
31
30
9
75313
9
83 713
0.16287
9-91 599
31
9
75331
9
83740
28
0. 16 260
9 91 591
29
(y
I
.4
32
9
75350
18
18
19
9
83 768
0.16232
9 91 582
9
28
7
I'
r'i
33
9
75368
9
83 795
0,16 205
9 91 573
9
27
8
1
52
7.1
34
35
9
75386
9
83822
27
0.16 178
9 .91 565
9
26
25
9
9
75405
9
83 849
0.16 151
9 91 556
1
Z^
9
75423
18
9
83876
0.16 124
9 91 547
9
24
1
37
9
75441
,Q
9
83 903
0.16 097
9 91 538
9
8
23
18
3«
9
75 459
9
83930
0.16 070
9 91 530
22
I 8
39
40
9
75478
^9
18
9
83957
27
0. 16 043
9.91 521
9
9
8
21
20
.2
.3
36
5 4
9
75496
9
83984
16 016
9.91 512
41
9
75514
9
84 on
0.15989
9.91 504
19
.4
72
42
9
75 533
9
84038
0.15 962
9 91 495
9
18
.«;
9.0
43
9
75551
18
18
18
9
84065
o- 15 935
9.91 486
9
17
.6
10.8
44
4.S
9
75569
9
84092
27
0.15908
9.91477
9
8
16
i
12 6
14 4
9
75587
9
84 119
0.15 881
9.91 469
4b
9
75605
9
84 146
27
0.15854
9.91 460
9
14
9
16.2
47
9
75624
19
18
9
84173
27
0.15827
9 91 451
9
13
1
48
9
75 ^f
9
.84200
27
0.15 800
9.91 442
9
12
1
49
50
51
9
75660
18
18
9
84227
27
27
26,
0.15 HZ
9 91 433
9
8
9
II
10
9
.2
9
8
08
I 6
9
9
75678
75696
9
9
'84280
0.15 746
0.15 720
9.91425
9 91 416
52
9
75 7M
9
.84307
27
9.91 407
9
8
•3
z 7
24
53
9
• 75 733
19
18
18
18
18
18
18
18
9
84334
27
0.15666
9.91 398
9
7
4
3 6
3 2
54
55
9
•75 751
9
.84361
27
27
0.15639
9.91 389
9
8
6
S
i
4 5
5 4
63
4
48
5 6
64
9
•75769
9
84388
0. 15 612
9.91 381
50
9
.75787
9
84415
27
0.15585
9.91 372
9
4
i
i^
9
.75805
9
.84442
27
0.15 558
9 91 363
9
3
l\
9
.75823
9
.84469
27
15 531
9 91 354
9
2
9
7^
59
G0_
9
.75841
9
.84496
27
27
0.15504
9 91 345
9
9
9
75859
9
■84523
0.15477
9 91 336
L. Cos.
d.
L. Cotg.
c.d.
L. Tang.
L. Sin.
d.
Prop. Pts.
55°
1
LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS
61
85^
2
3
I
7
8
_9_
10
II
12
13
iil
;i
17
18
i9.
20
21
22
23
24
25
26
27
28
_29
30
31
32
33
^\
36
37
38
_39
40
41
42
43
44
J^
47
48
49
50
51
52
53
54
55
56
57
58
ii.
60
L. Sin.
9 75 859
9 75877
9 75895
9 75913
9 75931
9 75 949
9 75 967
9 -75 985
9 . 76 003
9.76 021
9.76039
9.76057
9.76075
9.76093
9.76 III
9.76 129
9.76 146
9.76 164
9.76 182
9 76 200
9 76 218
9.76236
9 76253
9 76271
9 76 289
9 76 307
9 76 324
9.76342
9.76360
9.76378
9 76395
9 76413
9 76431
9.76448
9 . 76 466
9.76484
9.76501
9.76519
9 76537
9 76554
9.76572
9.76590
9 . 76 607
9 . 76 625
9 . 76 642
9 . 76 660
9.76677
9.76695
9.76 712
9 76 730
9.76747
9 76765
9 76 782
9 . 76 800
9.76 817
9 76835
9.76852
9 76 870
9.76887
9 76904
9 . 76 922
L. Cos. d.
L. Tang. c. d.
9-84 5£3
9.84550
9.84576
9,84 603
9.84630
9.84657
9 84 684
9.84 711
9 84738
9 84 764
9.84791
9.84818
o . 84 845
9.84872
9.84899
9.84925
9.84952
9.84979
9 . 85 006
9 85033
9 85059
9.85086
9 85 113
9.85 140
9.85 166
9 85 193
9 85 220
9.85247
9 85273
9 85 300
9 85327
9 85354
9 85 380
9 85407
9 85434
9 . 85 460
9.85487
9 85514
9 85540
9 85567
9 85 594
9.85 620
9 85647
9.85674
9.85 700
9,85 727
9 85 754
9.85 780
9 85 807
9 85834
9.85 860
9.85887
9 85 913
9 85 940
9 85 967
9 85 993
9 86 020
9 86 046
9 86 073
9 86 100
9.86 126
L. Cotg. c. d
27
26
27
27
27
27
27
27
26
27
27
27
27
27
26
27
27
27
27
26
27
27
27
26
27
27
27
26
27
27
27
26
27
27
26
27
27
26
27
27
26
27
27
26
27
27
26
27
27
26
27
26
27
27
26
27
26
27
27
36
L. Cotg.
0.15477
0.15450
C.15 424
c. 15 397
0.15370
15343
15 316
15289
15 262
15236
0.15 209
0.15 182
o 15 155
0.15 128
o. 15 lOI
o. 15 075
0.15 048
o. 15 021
0.14994
o. 14 967
0.14 941
0.14 914
0.14887
o . 14 860
0.14834
o. 14 807
0.14 780
0.14753
0.14 727
o. 14 700
0.14673
0.14 646
o. 14 620
o 14593
o. 14 566
o.
o.
o
o
o.
14540
14 513
14 486
14 460
14433
o. 14 406
0.14 380
o 14353
0.14326
0.14300
L. Cos.
0.14273
o. 14 246
o. 14 220
0.14 193
o. 14 166
0.14 140
0.14 113
o. 14 087
o. 14 060
0.14033
o. 14 007
o. 13 980
0.13954
0.13927
0.13 900
[3874
L. Tang.
54°
91 336
91 328
91 319
91 310
91 301
91 292
91 283
91 274
91 266
91 257
91 248
91 239
91 230
91 221
91 212
91 203
91 194
91 185
91 176
91 167
91 158
91 149
91 141
91 132
91 123
9 91 114
9 91 105
9 91 096
9.91 087
9.91 078
d.
91 069
91 060
91 051
91 042
91033
91 023
91 014
91 005
90 996
90987
90978
90 969
90 960
90951
90942
90933
90924
90915
90 906
90 896
90887
90878
90 869
90 860
90851
90 842
90 832
90 823
90 814
90805
9.90796
L. Sin.
60
59
58
55
54
53
52
51
50
49
48
47
_46_
45
44
43
42
41
40
39
38
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20"
19
18
17
16
15
14
13
12
II
To
9
8
7
6
Prop. Pts.
37
I
2.7
2
5 4
3
8. 1
4
10.8
. 5
13 5
.6
16.2
•7
18.9
.8
21.6
9
243
26
2.6
5-2
7.8
10.4
13.0
15 6
18 2
20,8
23 4
18
1.8
36
5 4
7.2
9.0
10 8
12 6
14.4
16.2
17
17
3 4
\\
85
10.2
11.9
136
15-3
10
i.o
20
30
40
60
7.0
8.0
90
9
.1
0.9
.2
1.8
•3
27
•4
36
i
4 5
5 4
I
63
7 2
9
8.1
8
08
1.6
24
3 2
7 2
Prop. Pts.
62
TABLE II
36^
9_
10
;i
i8
i2_
20
21
22
23
24
26
27
28
29
80
31
32
33
34
II
II
40
41
42
43
44
46
47
48
49
60
51
52
53
ii
55
50
II
GO
L. Sill.
76922
76939
76957
76974
76991
77009
77026
77043
77061
77078
77095
77 112
77130
77147
77164
77 181
77199
77216
77233
77250
77268
772S5
77302
77319
77336
77 353
77370
77 3^7
77405
77422
77 439
77456
77 473
77490
77507
77 5-4
77541
77558
77 575
77592
77G09
77626
77643
77 CGo
77677
77694
77 711
77728
77 744
77761
77 77^
77 795
77812
77829
77846
77862
77879
77896
77913
77930
77946
I L. Cos. I <1
17
18
J7
J7
18
»7
»7
18
17
17
17
18
17
17
17
x8
17
17
17
18
17
17
17
17
17
17
17
18
17
17
17
17
17
17
«7
17
17
17
17
17
17
17
17
17
16
17
Tangr.
c.d.
86
,86
126
o. '53
,86 179
86206
86232
,86259
,86285
86312
86338
.86365
.86392
1. 86418
1.86445
1.86 471
1.86498
1.86524
'•^5551
1.86577
1.86603
86 630
9.86656
9.86683
9.86 709
9.86736
9.86 762
9.86 789
9.86815
9.86842
9.86868
9.86894
9.86921
9.86947
9.86974
9.87000
9.87027
9-87053
9.87079
9.87 106
9.87 132
9-87 158
9.87 185
9.87 211
9.87238
9.87264
9.87290
9.87317
9-87343
9.87369
9.87396
9.87422
9.87448
9-87475
9.87501
9.87527
9-87554
9.87580
9.87606
9-87633
9.87659
9.87685
9.8771
L. Cotg,
27
26
27
26
27
26
27
26
»7
27
26
27
26
27
26
27
26
26
27
26
27
26
27
26
27
26
27
26
26
27
26
27
26
27
26
26
27
26
26
27
26
27
26
26
27
26
26
27
26
26
27
26
26
27
26
26
27
26
26
L« Cotg.
0.13874
0.13847
0.13 821
o. 13 794
0.13768
o. 13 741
0.13 715
0.13688
0.13 662
0-13635
0.13 608
0.13582
0.13555
0.13529
0.13502
0.13476
0.13449
0.13423
0.13397
0.13370
13344
^3317
13 291
13264
13 238
0.13211
0.13 185
0.13 158
0.13 132
0.13 106
0.13 079
0.13053
0.13026
0.13000
0.12973
0.12 947
0.12 921
0.12 894
0.12 868
0.12 842
0.12 815
0.12 789
0.12 762
0.12 736
0.12 710
c.d.
0.12683
0.12 657
0.12 631
0.12 604
0.12578
0.12 552
0.12 525
0.12 499
0.12473
0.12 446
0.12 420
0.12394
0.12367
0.12 341
0.12 315
[2 289
L. Tang.
53°
L, Cos.
90657
90648
90639
^ ,90 630
9.90 620
90 796
90787
90777
90 768
90759
90750
90741
90731
90 722
90713
90704
90 694
90685
90676
90667
90 611
90 602
90592
90583
90574
90565
90555
90546
90537
90527
90518
90509
90499
90490
90480
90471
90462
90452
90443
90 434
90424
90415
90405
90396
90386
90377
90368
90358
90349
90339
90330
90320
90 311
90301
90292
90 282
90273
90 263
90254
90 244
9 90 235
L. Sin.
GO
It
I
55
54
53
52
JL
60
49
48
46
45
44
43
42
41
40
39
38
36
35
34
33
32
31
Prop. Pis.
27
36
I
2.7
26
•2
-3
il
11
•4
10.8
10.4
. 1;
13-5
16.2
13.0
.6
IS. 6
• 7
18. q
.8
21.6
20.8
.9
243
23-4
18
1.8
36
5 4
7.2
9°
10.8
12.6
14.4
16.2
17
17
3-4
u
8.5
10.2
II. 9
136
15-3
x6
1.6
3 2
4.8
6.4
80
96
112
12 8
14.4
10
.1
I.O
.2
2.0
•3
•4
i
30
4.0
i
7.0
8.0
-9
9.0
9
09
I
2
3
4
I
Prop. Pts.
LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS
63
37° 1
L. Sin.
d.
L. Tang.
c.d.iL. Cotg. 1
L. Cos.
d.
60
Froi». V\s,
9 77946
9 87 711
0. 12 289
9 90 235
I
9 77963
9 87 738
26
0.12 262
9
90225
S9
2
9 77980
9 87 764
26
0.12 236
9
90 216
S8
^
9 77 997
9 87790
0.12 210
9
90 206
S7
27
,J_
9 78013
'7
9
87817
26
0. 12 183
9
90197
10
56
ss
.2
•3
.4
9 78030
9
87 843
0.12 157
9
90 187
6
9 78047
i5
9
87869
36
12 131
9
90178
S4
10 8
7
9 78 063
9
87895
12 105
9
90 168
S3
,5
'I 5
16 2
8
9 78080
9
87922
r>(\
0.12078
9
90 159
9
S2
,6
9
9 78097
[6
9
87948
26
26
0.12 052
9
90 149
10
51
50
:l
18.9
21.6
9 78 113
9
87974
0. 12 026
9
90139
1 1
9
78 130
9
88000
r%m
0.12000
9
90130
49
•9
24.3
12
9
78 147
[6
9
88027
on 973
9
90 120
48
1
n
9
78163
9
S°53
o.ii 947
9
90 III
47
1
•4
9
78 180
7
9
88079
26
26
o.ii 921
9
90 lOI
10
46
4S
.1
36
26
IS
9
78 197
9
11 '°5
O.II 895
9
90091
i6
9
78213
9
88 131
O.II 869
9
90 082
9
44
.2
^l
'7
9
78230
6
9
88158
27
26
26
26
26
on 842
9
90072
43
•3
78
i8
9 78 246
9
Z?> 184
O.II 816
9
90063
9
42
•4
10.4
19
20
9 78 263 ,
7
6
9
88210
on 790
9
90053
10
41
40
i
15 6
18 2
20 8
23 4
9 78280
9
88236
on 764
9
90 043
21
9 . 78 296
9
88262
on 738
9
90034
9
3Q
■ 7
.8
22
9
78313
9
88289
26
26
26
26
0. II 711
9
90024
38
2^
9
78329
9
88315
on 685
9
90014
37
•y
24
2S
9
78346
16
9
88341
II 659
9
90003
10
36
3S
1
9
78362
9
88367
on 633
9
8999s
17
1-7
26
9
78379
16
9
88393
on 607
9
8998s
34
.
27
9
78395
9
88420
27
26
O.II 580
9
89976
9
33
28
9
78412
^1
[6
'7
[6
9
88446
on 554
9
89966
32
34
29
BO
9
78428
9
88472
36
26
26
0. II 528
9
89956
9
31
80
•3
•4
10.2
9
78445
9
88498
on 502
9
89947
^i
9
78461
9
SS524
oil 476
9
89937
29
^2
9
78478
9
88 550
oil 450
9
89927
28
i
II. 9
136
IS.:?
1^
9
78494
,£.
9
88577
27
26
26
0,11423
9
89918
9
27
34
9
78510
17
t6
9
88603
II 397
9
89908
10
26
2S
.q
9
78527
9
88629
on 371
9
89898
1
^6
9
78543
9
88 6^5
88681
26
26
O.II 345
9
89888
24
1
V
9
78560
[6
[6
'7
[6
9
II 319
9
89879
•9
23
16
^«
9
78576
9
88707
II 293
9
89869
22
^
I 6
39
40
9
78592
9
88733
26
II 267
9
89859
10
21
20
.2
.3
U
9
78609
9
88759
88786
II 241
9
89849
41
9
78625
9
27
0. II 214
9
89840
9
19
.4
6.4
42
9
78642
16
t6
t7
[6
t6
[6
9
88812
on 188
9
89830
18
.1;
8.0
43
9
78658
9
88838
II 162
9
89820
17
.6
9.6
44
4S*
9
78674
9
88864
26
26
II 136
9
89810
9
16
IS
:l
n.2
12.8
9
78691
9
88890
0. II no
9
89801
46
9
78707
9
88916
on 084
9
89791
14
.9
14.4
47
9
78723
9
88942
26
O.II 058
9
89781
13
1
48
9
78739
9
88968
O.II 032
9
89771
12
1
49
50
_9
9
78756
78772
16
[6
9
88994
26
O.II 006
9
89761
9
II
10
.1
zo
I.O
9
0.9
9
89 020
. 10 980
9
89752
SI
9
78788
9
89046
26
0. 10 954
9
89 742
9
.2
2.0
1.8
S2
9
78805
I/'
[6
16
6
6
9
89073
27
0. 10 927
9
89732
8
3
30
2.7
S3
9
78821
9
89099
10 901
9
89722
7
4
4.0
36
54.
•>s
9
9
78837
9
89125
26
26
0.10875
9
89712
10
6
5
i
5
6
4.5
V.
78 85.3
9
89 151
0.10 849
9
89 702
S6
9 78 869
9
89 1 77
26
0.10823
9
89693
9
4-
i
8^o
S7
9 78886
16
6
[6
9
89 203
26
o.io 797
9
89 683
3
S«
9.78902
9
89 229
0. 10 771
9
89673
2
■9
9.0
59
60
9 78918
9
89255
26
0.10 745
9
89663
10
I
9 78934
9
89281
0. 10 719
9
89653
L. Cos. 4
i.
L. Cotg.
c."(i.
L. TaiifiT.
L. Sin.
d.
/
Prop. Fts.
52^ 1
64
TABLE II
38'
I
2
3
_4
I
7
8
_9_
10
II
12
13
;i
17
i8
i9_
20
21
22
23
24
26
27
28
29
31
32
33
34
36
37
38
39
40
41
42
43
44
45
46
47
48
49.
50
51
52
53
il
^^
GO
L. Sin,
78934
78 QW
78967
78983
78999
79015
79031
79047
79063
79079
79095
79 III
79 128
79 144
79 160
79176
79 192
79 208
79224
79240
79256
79272
79288
79304
79319
79 335
79351
79367
79383
79 399
79415
79431
79 447
79463
79478
79 494
79510
79526
79542
79558
79 573
79589
79605
79 621
79636
79652
79668
79684
79699
79715
79731
79 746
79 762
79778
79 793
79809
79825
79840
79856
79872
79887
L. Cos.
L. Taiig.
c. d.
89281
89307
89333
89359
89385
89 411
89437
89463
89489
89515
89541
89567
89593
89 619
89645
89671
89697
89723
89749
89775
89801
89827
89853
89879
89905
89931
89957
89983
90 009
90035
90061
90 086
90 112
90 138
90 164
90 190
90 216
90 242
90 268
90294
90 320
90346
90371
90397
90423
90449
90475
90 501
90527
90553
9.90578
9 . 90 604
90630
9.90
9 90
656
682
90 708
90734
90759
90785
90 811
9.90837
36
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
26
25
26
26
26
26
26
26
26
26
26
26
25
26
26
26
26
26
26
26
25
26
26
26
26
26
26
25
26
26
L. Cotg. Ic. (1
L. Cotg.
o.io 719
o.ic 693
o ic 667
o 10 64£
o. 10 615
O.IO 589
o. 10 563
0.10537
O.IO 511
o. 10 485
0.10459
0.10433
o. 10 407
O.IO 381
0.10355
0.10329
o . 10 303
O.IO 277
O.IO 251
o. 10 225
o. 10 199
O.IO 173
O.IO 147
O. 10 121
0.10095
O.IO 069
o. 10 043
O.IOOI7
0.09 991
o • 09 965
0.09939
0.09 914
0.09888
o . 09 862
0.09 836
0.09 810
0.09 784
0.09 758
0.09 732
0.09 706
o . 09 680
0.09 654
o . 09 629
o . 09 603
0.09577
0.09551
0.09 525
0.09499
0.09473
0.09447
o . 09 422
0.09 396
0.09 370
0.09344
0.09 318
0.09 292
0.09 266
o . 09 241
o 09 2 1 5
o 09 189
0.09 163
L. Tang.
5r
L. Cos.
89653
89643
89633
89 624
89 614
89 604
89594
89584
89574
89564
89554
89544
89534
89524
89514
89504
89495
89485
89475
89465
89455
89445
89435
89425
89415
89405
89395
89385
89375
89364
89354
89344
89334
89324
89314
89304
89294
89284
89274
89 264
89254
89244
89233
89 223
89213
89203
89193
8918:5
89173
89 Ib2
89152
89 142
89 132
89 122
89 112
89 lOI
89 091
89081
89071
89 060
9 . 89 050
L. Sin.
60
59
58
57
55
54
53
52
51
50
49
48
47
_46_
45
44
43
42
41
40
39
38
36
35
34
33
32
Jl
30
29
28
27
26
Prop. Vis.
26
I
26
2
S-2
3
7.8
4
10.4
5
13
6
15.6
7
18 2
8
20.8
9
23 4
17
2
1 ■
3
3
4
6
8.
.6
10.
.7
II
.8
13
9
15
25
25
50
7 5
10 o
12 5
15 o
17 5
20.0
22.5
16
I
1.6
2
32
3
4.8
4
S4
.1^
8 ol
.6
96
• 7
II. 2
.8
12.8
9
14-4
XI
I
I.
2
2
3
3
•4
4
I
7
I
7
.8
•9
8
9
zo
9
10
20
I
30
2
4.0
3
SO
4
6.0
S
7.0
6
8.0
I
90
15
30
6.0
7 5
9.0
10. 5
12.0
^3 5
Prop. Pis.
LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS
65
39° 1
t
L. Sin.
d.
L. Tang.
c.d.
L. Cotg.
L. Cos.
d.
60
Prop.
Pts.
9 79887
16
9.90837
26
0.09 163
9 89 050
■ 1
I
9
79903
TC
9
90«b3
26
0.09 137
9
89 040
S9
1
2
3
_4_
9
9
9
79918
79 934
79950
16
16
15
16
9
9
9
90889
90914
90940
2S
26
26
26
0.09 III
0,09086
. 09 060
9
9
9
89 030
89 020
89009
10
II
10
58
.1
.2
3
4
26
2.6
li
10.4
ill
208
9
79965
9
90966
0.09034
9
88999
6
9
79981
15
16
9
90992
oA
0.09008
9
88989
54
7
9
79996
9
91 018
0.08982
9
88978
S^
8
9
80012
15
16
16
9
91043
26
26
0.08957
9
88968
52
9
10
9
80027
9
91 069
0.08 931
9
88958
10
51
50
I
9
80043
9
91095
0.08905
9
88948
II
9
80058
9
91 121
26
0.08879
9
88937
49
9
23 4
12
9
80074
9
91 147
0.08853
9
88927
48
n
9
80089
16
9
91 172
25
26
26
26
26
0.08828
9
88917
47
1
14
15
9
80 105
IS
16
9
91 198
0.08802
9
88906
10
46
45
25
25
9
80 120
9
91 224
0.08 776
9
88896
16
9
80 136
9
91 250
0.08 750
9
88 886
44
.2
50
17
9
80 151
9
91 276
0.08 724
9
88875
43
•3
7-5
18
9
80166
16
9
91 301
25
26
26
26
0.08 699
9
88865
42
•4
10.0
19
20
9
80 182
15
16
9
91 327
0.08 673
9
88855
II
41
40
i
12.5
15
9
80197
9
91353
0.08 647
9
88844
21
9
80213
9
91 379
0.08 621
9
88834
39
i
17 5
22
9
80228
16
9
91404
25
26
26
26
0.08 596
9
88824
10
38
20.0
2S
9
80244
9
91430
0.08 570
9
88813
37
•9
22.5
24
2"^
^9
"9
80 259
15
16
9
91456
0.08 544
9
88803
10
36
35
1
80274
9
91482
0.08518
9
88793
16
26
9
80290
9
91 507
26
26
26
25
26
26
26
0.08493
9
88782
34
1.6
27
9
80305
9
91 533
0.08467
9
88772
33
28
9
80320
16
15
9
91 559
0.08 441
9
88761
32
8
29
30
9
8033^
9
91585
0.08 415
9
88751
10
10
31
80
■3
•4
9
80351
9
91 610
0.08 390
9
88741
31
9
80366
16
9
91 636
0.08 364
9
88730
11
9.6
112
32
9
80382
9
91 662
0.08338
9
88 720
10
28
i
33
9
80397
9
91688
08 312
9
88 709
II
27
12 8
34
3S
9
80412
16
9
91 713
25
26
26
26
0.08287
9
88699
10
II
26
25
.g
9 . .
lA A
9
80428
9
91 739
0.08 261
9
88 688
36
9
80443
9
91 765
0.08 235
9
88678
10
24
1
37
9
80458
9
91 791
08 209
9
88 668
10
23
X5
IS
30
38
9
80473
IS
16
15
9
91 816
25
0.08 184
9
88 657
II
22
39
40
9
80489
9
91842
26
0.08 158
9
88 647
10
11
21
20
.2
3
.4
9
80504
9
91 868
08 132
9
88636
41
9
80519
15
9
91 893
25
26
26
26
2S
26
0.08 107
9
88626
10
IQ
42
9
80534
15
9
91 919
0.08081
9
88615
II
18
7 5
43
9
80550
9
91 945
0.08 055
9
88605
17
5
90
44
4S
9
80565
IS
9
91 971
. 08 029
9
88594
10
16
15
i
10.5
12
9
80580
9
91 996
. 08 004
9
88584
46
9
80595
*s
9
92 022
0.07978
9
88 573
II
14
9
13 5
47
9
80610
IS
9
92 048
0.07952
9
88 S63
10
n
1
48
9
80625
IS
16
^5
9
92073
25
26
26
0.07927
9
88 552
II
12
1
49
60
9
80641
9
92099
0.07 901
9
88542
II
II
10
.1
II
I . I
10
1.0
9
80656
9
92 125
0.07875
9
88531
SI
9
80671
^s
9
92 150
2S
0.07850
9
88521
10
p
.2
2.2
2.0
52
9
80686
IS
9
92176
26
0.07 824
9
88510
II
8
3
3 3
30
S3
9
80 701
9
92 202
0.07 798
9
88499
I'
7
4
4 4
4.0
54
SS
9
80 716
15
15
9
92227
2S
26
0.07 773
9
88489
10
II
6
S
I
U
50
6.C
9
80731
9
92253
0.07 747
9
88478
S6
9 80 746
15
16
9
92279
0.07 721
9
88468
10
4
:i
ii
7.0
8.C
S7
9 80 762
9
92304
2S
0.07 696
Q
88457
IX
3
S8
9 80 777
15
9
92330
0.07 670
9
88447
10
2
•9
9 9
9.0
59
9 80792
15
15
9
9
92356
92381
2S
0.07 644
9 88 436
II
II
I
9 80 807
0.07 619
9.88425
L. Cos.
d.
L. Cotg.
c.d.
L. Tang.
L. Sin.
d.
f
Prop. Pis.
50^ 1
66
TABLE II
40° 1
t
L. Sin.
d.
L. Tang.
c. d.
L. Cotg.
L. Cos.
d.
w
Prop. Pts.
9.80807
9.92381
0.07 619
9.88425
I
9
80 822
9
92407
0.07593
9
88415
S9
2
9
80837
9
92433
0.07567
9
88404
S8
26
3
9
80852
9
92458
25
0.07 542
9
88 394
S7
T
2 6
4
9
80867
9
92484
26
0.07 516
9
88383
II
56
55
2
3
'7s
9
80882
9
92510
0.07490
9
88 372
6
9
80 897
9
92535
25
0.07465
9
88362
54
4
10 4
7
9
80 912
9
92561
26
0.07439
9-
88351
53
13
8
9
80927
9
92587
0.07413
9.
88340
52
(5
15 6
9
10
9
80942
9
92 612
25
26
0.07388
9
88330
II
51
50
7
8
182
20 8
9
80957
9
92 638
0.07362
9
88319
II
9
80972
9
92 663
25
26
26
0.07337
9
88308
49
9
23 4
12
9
80987
9
92689
0.07 311
9
88298
48
1
n
9
81 002
9
92 715
0.07 285
9-
88287
47
1
14
9 81 017
9
92 740
26
26
0.07 260
9
88276
10
46
45
25
25
9 81 032
9
92766
0.07234
9
88266
1
i6
9
81 047
9
92 792
0.07 208
9
88255
44
2
50
17
9
81 061
9
92817
25
26
0.07 183
9.
88244
43
3
7 5
i8
9
81 076
9
92843
0.07 157
9
88234
42
4
10
19
20"
9
81 091
9
92868
25
26
26
0.07132
9
88223
II
41
40
I
12.5
15
17 5
20
9
81 106
9
92894
0.07 106
9
88212
21
9
81 121
9
92 920
0.07 080
9
88201
39
22
9
81 136
9
92945
25
26
0.07055
9
88 191
°
38
n
22.5
2^
9
81 151
9
92971
0.07029
9
88180
37
24
2S
9
81 166
9
92996
25
26
26
0.07004
9
88 169
II
36
35
1
9
81 180
9
93022
0.06978
9
88158
IS
26
9
81 195
9
93048
0.06952
9
88 148
34
^
15
3
4 5
6
7 5
27
9
81 210
9
93073
25
26
0.06927
9
88137
33
2
28
9
81 225
9
93099
06 901
9
88 126
32
•3
•4
• S
29
30
_9
9
81 240
81 254
9
93 124
25
26
06876
9
88 115
10
31
30
9
93 150
0.06 850
9
%% 105
SI
9
81 269
9
93 175
25
0.06825
9
88094
29.
.6
90
32
9
81 284
9
93201
26
0.06 799
9
88083
28
.7
10 5
33
9
81 299
9
93227
0.06 773
06 748
9
88072
27
.8
12
34
3S
9
81 314
9
93252
25
26
9
88061
10
26
25
9
13 5
9
81 328
9
93278
0.06 722
9
88051
1
3^
9
81343
9
93303
25
06 697
9
88040
24
1
37
9
81358
9
93329
06 671
9
88029
23
14
3«
9
81372
9
93 354
25
06 646
9
88018
22
I
14
2 8
4 2
39
40
9
,81 387
9
93380
26
06 620
9
88007
11
21
20
.2
.3
9 81 402
9
93406
0.06594
9
87996
A\
9,81 417
9
93431
25
0.06 569
9
87985
19
• 4
56
42
9 81 431
9
93 457
26
. 06 543
9
87975
18
.5
7
43
9 81 446
9
93482
25
06 518
9
87964
17
.6
84
44
4S
9 81 461
9
93508
26
25
06 492
9
87953
II
16
^5
•7
.8
98
II 2
9^1 475
9
93 533
0.06467
9
87942
46
9 81 490
9
93 559
26
0.06 441
9
87931
14
9
12 6
47
9 81 505
9
93584
25
06 416
9 87 920
13
1
48
9 81 519
9
93 610
0.06390
9 87 909
12
1
49
50
9 «i 534
9
93636
26
25
06 364
9 87 898
II
II
10
I
XI
II
10
ID
9 81 549
9
93661
06339
9 87 887
.S»
9 8' 563
9 «• 578
9
93687
26
06313
9 87877
9
.2
22
2
S2
9
93712
25
0.06288
9
87866
8
3
3 3
3
S3
9 81 592
9
93 738
26
0.06 262
9
87855
7
4
4 4
4
54
9 81 607
9
93 763
25
26
0.06 237
9
87844
6
5
.^5
IS
5S
9 81 622
9
93 789
0.06 211
9
^l^ZZ
5
6
6 6
5^
9 81 636
9
93814
25
0.06 186
9
87822
4
.8
1^
^0
^s^
9 81 651
9
93 840
26
0.06 160
9 87 811
3
9 81 665
9
93865
25
0.06 135
9.87800
2
y
9 9
90
59
60
9 81 680
9
93891
26
25
0.06 109
9 87789
II
9 81 694
9
93916
06 084
9 87 778
L. Cos.
L. Cotg.
c. d.
L. Tang.
L. Sin.
f
Prop. Pt8.
49^ 1
LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS
67
41
9_
10
II
12
\i
i8
20
21
22
23
24
25
26
27
28
f9
30
31
32
33
Jl
36
37
38
39
40
41
42
43
44
46
47
48
49
50
51
52
53
_54
55
5^
00
L. Sin,
694
709
723
738
752
767
781
796
810
825
839
854
868
882
897
911
926
940
955
969
983
998
82 012
82 026
82 041
82055
82 069
82084
82098
82 112
82 126
82 141
82155
82 169
82 184
82 198
82 212
82 226
82 240
82255
82 269
82283
82 297
82 311
82_326_
82340
82354
82368
82382
82396
82 410
82 424
82439
82453
82467
82481
82495
82 509
82523
82537
82551
L. Cos.
d.
Tang.
93916
93942
93 967
93 993
94 018
94044
94 069
94095
94 120
94 146
94 171
94 197
94 222
94248
94273
94299
94324
94350
94 375
94401
94 426
94452
94 477
94503
94528
94 554
94 579
94 604
94630
94655
94 681
94 706
94 732
94 757
94783
94808
94834
94859
94884
94910
94 935
94961
94 986
95 012
95037
95 062
95088
95 113
95 139
95 164
95 190
95215
95 240
95 266
95291
95317
95 342
95368
95 393
95 418
c. d.
95 444
Cotff.
c. d.
L. Cotg.
o . 06 084
0.06 058
0.06033
0.06 007
0.05 982
0.05 956
0.05931
0.05 905
0.05 880
0.05 854
0.05 829
0.05 803
o 05 778
o 05 752
0.05 727
o 05 701
o 05 676
0.05 650
0.05 625
0.05 599
0.05 574
0.05 548
0.05 523
0.05 497
0.05 472
0.05 446
0.05 421
o 05 396
0.05 370
005 345
0.05 319
0.05 294
0.05 268
0.05 243
6.05 217
0.05 192
0.05 166
0.05 141
o 05 116
o . 05 090
0.05 065
o 05 039
o 05 014
o 04 988
o 04963
o 04 938
o 04 912
o 04 887
0.04 861
o . 04 836
0.04 5IO
0.04 785
o 04 760
o 04 734
0.04 709
o . 04 683
o . 04 658
o . 04 632
0.04 607
o . 04 582
0.04 556
L. Tang.
48°
L. Cos.
87778
87767
87756
87745
87734
87723
87712
87 701
87 69c
87679
87668
87657
87646
87635
87624
87613
87601
87590
87579
87 568
87557
87546
87535
87524
87513
87501
87490
87479
87468
87457
87446
87434
87423
87 412
87 401
87390
87378
87367
87356
9 87 345
9 87 334
9 87322
9 873"
9 87300
9.87288
87277
87266
87255
87243
87232
87221
87 209
87 198
9 87 187
9 87 175
9 87 164
9 87 153
9 87 141
9 87 130
9 87 119
9.87 107
L. Sin,
d.
«0
59
58
57
56
55
54
53
52
_5L
50
49
48
47
46
45
44
43
42
41
40
39
38
36
35
34
33
32
31
25
24
23
22
21
20
^9
18
17
16
15
14
13
12
II
9
8
7
6
Prop. Pts.
6
2
8
4
o
18.2
20.8
23 4
25
25
50
7 5
10.0
12.5
15 o
17 5
20.0
22.5
15
IS
30
6.0
7-5
9.0
10.5
12.0
13 5
X4
14
8
2
6
o
4
8
2
6
13
.1
12
.2
2.4
3
36
• 4
48
6.0
.6
7.2
• 7
8.4
.8
9.6
•9
0.8
Prop. Pts.
68
TABLE II
42^
2
3
A
5
6
7
8
9^
10
II
12
13
14
11
17
i8
i9_
20
21
22
23
24
26
27
28
29
30
31
32
33
34_
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
I
II
60
L. Sin.
82551
82565
82 579
82 593
82 607
82621
82 635
82 649
82663
82677
82 69_i
82 705
82 719
82 733
82 747
82 76_i
82775
82788
82802
82816
82830
82844
82858
82872
8288=:
82899
82913
82 927
82 941
82955
82968
82982
82 996
83 010
83023
83037
83051
83065
83078
83092
83 106
83 120
83133
83 147
83 161
83 174
83 188
83 202
83215
83 229
83242
83256
83 270
83283
83297
83310
83324
83338
83351
83365
983378
Cos.
d. L. Tang.
95 444
95 469
95 495
95 520
95 545
95 571
95596
95 622
95647
95 672
95698
95 723
95 748
95 774
95 799
95 825
95850
95875
95901
95 926
95952
95 977
96 002
96 028
96053
96 078
96 104
96 129
96155
96 180
96 205
96231
96 256
96281
96307
96332
96357
96383
96 408
96433
96459
96 484
96 510
96535
96 560
96586
96 611
96636
96 662
96687
96 712
96738
96 763
96788
96 814
96839
96864
96 890
96915
96 940
c.d,
96 966
d. I L. Cotg.
25
26
25
25
26
25
26
25
25
26
25
25
26
25
26
25
25
26
25
26
25
25
26
25
25
26
25
26
25
25
26
25
25
26
25
25
26
25
25
26
25
26
25
25
26
25
25
26
25
25
26
25
25
26
25
25
26
25
25
26
c.d.
L. Cotg.
0.04 556
o 04531
o . 04 505
o 04 480
o 04 455
o 04 429
o 04 404
0,04378
0.04353
0.04 328
o . 04 302
0.04 277
o 04 252
o . 04 226
0.04 201
0.04 175
0.04 150
0.04 125
o . 04 099
o . 04 074
o . 04 048
o . 04 023
0.03 998
0.03 972
0.03947
0.03 922
0.03 896
0.03871
0.03 845
0.03 820
0.03 795
0.03 769
0.03 744
0.03 719
0.03 693
0.03 668
0.03643
o 03617
0.03 592
0.03 567
0.03541
0.03 516
o . 03 490
o . 03 465
0.03440
0.03 414
0.03389
0.03364
0.03338
0.03313
0.03 288
0.03 262
0.03237
0.03 212
0.03 186
0.03 161
o 03 136
o 03 no
o 03 085
o , 03 060
0.03034
L. Tang.
470
I
. Cos.
d
•
60
9,87 107
9
87096
,
59
9
87085
58
9
87073
57
_9_
9
87062
2
56
55
87 050
9
87039
54
9
87028
5^
9
87016
52
9
87005
2
51
50
9
86993
9
86982
^
49
9
86970
48
9
86959
47
9
86947
I
46
45
9
86936
9
86924
44
9
86913
43
9
86902
42
9
86890
I
41
40
9
86879
9
86867
39
9
86 855
38
q
86844
37
9
86832
^
36_
35
9
86821
9
86809
M
9
86798
^
S3
9
86786
^2
9
86775
2
31
30
9
86763
9
86752
'
29-
9
86 740
28
9
86728
27
9
86717
2
26
25
9
86705
9
86694
'
24
9
86682
23
9
86670
22
9
86659
2
21
20
9
86647
9
86 63s
19
9
86624
18
9
86612
17
9
86600
I
16
15
9
86589
9
86577
14
9
86 565
13
9
86554
12
9
86542
2
II
10
9
86530
9
86518
9
9
86507
8
9
86495
7
9
86483
6
5
9
86472
9
86460
4
9
86448
3
9
86436
2
9 86 425
9 86413
I
.. Sin.
i
.
/
Prop. Pts,
36
2 6
25
25
o
5
5
7
10.0
12.5
14
14
2.8
13
.1
1 .
.2
2
•3
3
• 4
5
.5
6.
6
7
7
9
.8
10.
9
II .
12
.1
I 2
.2
3
•4
3I
4.8
6.0
9
II
I
2.2
3 3
4 4
II
7 7
8 8
9 9
Prop. Pts.
LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS
69
43^
L. Sin.
7
8
9_
10
II
12
13
lA
15
i6
17
i8
i9_
20
21
22
23
24
26
27
28
29
31
32
33
34
36
37
38
J9
40
41
42
43
_44_
46
47
48
49
50
51
52
53
id
55
56
57
58
i9_
60
983378
9 83392
9 83405
9.83419
9 83432
9.83446
9 83 459
9 83473
9 83 486
9 83 500
9 83513
9 83 527
9 83540
83554
83567
83581
83594
83608
83621
83,634^
83648
83661
83674
83688
83701
83715
83728
83741
83755
83768
83781
83795
83808
83821
83834
83848
83861
83874
83887
83901
83914
83927
83940
83954
83.967
83
83993
84 006
84 020
84033
9 . 84 046
9 84 059
9 84 072
9 84 085
9 84 098
84 112
84 125
84138
84 151
84 164
9 84177
L. Cos.
L. Tang.
9 96 966
9 96 991
9 97 016
9 97042
9.97067
97092
97 118
97 143
97168
97 193
97219
97244
97269
97295
97320
97 345
97371
97396
97421
97 447
97472
97 497
97523
97548
97 573
97598
97624
97649
97674
97 700
97725
97750
97776
97 801
97 826
97851
97877
97902
97927
97 953
97978
98 003
98 029
98054
98079
98 104
98 130
98 155
98 180
98 206
9 98 231
9 98 256
9 98 281
9 98307
9 98 332
9 98357
998383
9 . 98 408
9 98433
9.98458
9.98484
L. Cotg. c. (I
c. d.
L. Cotg.
0.03034
o 03 009
o . 02 984
0.02 958
0.02933
0.02 908
o 02 882
0.02 857
0.02 832
o 02 807
02 781
02 756
02 731
02 705
02 680
02 655
02 629
02 604
0.02 579
0.02 553
0.02 528
0.02 503
0.02 477
o . 02 452
0.02 427
o . 02 402
0.02 376
o 02 351
o . 02 326
o 02 300
o 02 275
o . 02 250
0.02 224
o 02 199
0.02 174
o 02 149
o 02 123
o . 02 098
o . 02 073
0.02 047
o . 02 022
o.oi 997
o.oi 971
o.oi 946
o.oi 921
O.OI 896
O.OI 870
0.01 845
O.OI 820
O.OI 794
O.OI 769
O.OI 744
O.OI 719
O.OI 693
O.OI 668
O.OI 643
o 01 617
o 01 592
O.OI 567
O.OI 542
O.OI 516
L. Tang.
46°
L
. Cos.
d.
9 86413
60"
9
86401
59
9
9
86389
86377
2
58
57
9
86366
2
2
56
55
S4
9
9
86354
86342
9
86330
■^^
9
86318
'^2
9
86306
51
50
9
86295
9
86283
49
9
86271
48
9
86259
47
9
86247
2
46
45
9
86235
9
8b 223
44
9
86 211
43
9
86200
42
9
86188
2
41
40
9
86176
9
86 164
i:^
39
9
86152
38
9
86 140
37
9
86128
[2
36
35
9
86 116
9
86 104
34
9
86092
33
9
86080
32
9
86068
[2
31
30
9
86056
9
86044
29
9
86032
28
9
86020
27
9
86008
12
26
25
9
85996
9
85984
24
9
85972
23
9
85 960
22
9
85948
C2
21
20
9
85936
9
85924
19
9
85 912
18
9
85900
17
9
85888
[2
16
IS
9
85876
9
85 864
14
9
85851
3
13
9
85839
12
9
85827
2
II
10
9
85815
9
85 803
9
9
85 791
8
9
85 779
7
■9
85 766
3
2
6
5
9
85754
9
85 742
2
4
9
85 730
3
9
85718
2
. 2
9
85706
2
I
9 85693
13
L. Sin.
d.
/
Prop. Pts.
26
I
2.
2
5
3
7-
4
10
13
6
7
18
8
20
9
23
25
I
25
2
50
3
7-5
4
10.0
5
12 s
6
15
7
175
8
20.0
9
22.5
14
,1
1-4
.2
2.8
•3
4.2
•4
5-6
•5
7.0
.6
.8.4
•7
98
.8
II. 2
•9
12.6
13
I
I .
2
2
3
3
4
5 .
6.
6
7-
7
9
8
10
9
II.
12
II
.1
1.2
I.
.2
2.4
2.
3
36
3
•4
4.8
4-
5
6.0
5-
6
7.2
6.
7
8.4
7-
8
9.6
8.
9
10.8
9-
Prop. Pts.
70
TABLE II
44° 1
/
L. Sin.
d.
L. Tang.
C.d.
L. Cotg.
L. Cos. d
I.
60
Prop. Pts.
9 84 177
9.98484
0.01 516
985693 ,
I
9 84 190
13
9 98509
25
0.01 491
9 85 681
SP
2
9
84203
84216
*3
9 98534
25
0.01 466
985669
58
9fi
S
9
9 98 560
0.01 440
9 85 657
S7
,
2 6
78
4
s
9
84229
»3
*3
998585
25
0.01 415
9.85645 ;
3
56
55
2
3
.4
9
84242
9 98 610
001 390
985632
6
9
84 2.S5
»3
9 98635
01 365
9 85 620
54
10 4
7
9
84269
9 98 661
01339
9.85608 '
5S
I
IT,
8
9
84 282
13
9 98686
01 314
985596 ;
52
15 6
9
10
9
84295
13
13
9 98 711
26
0.01 289
985583 ;
J
2
51
50
7
8
18 2
20 8
9 84 308
9 98737
01 263
985571
II
9 84 321
»3
9 98 762
01 238
985559 ,
49
9
23 4
12
9 84 334
9 98 787
01 213
01 188
985547 ,
48
1
IS
9 84 347
9 98812
26
25
985534 ■
47
1
14
IS
9 84 360
13
13
_9_
9
98838
9886^
98888
0.01 162
9.85522 ;
2
46
45
.1
25
2 5
9 84 373
0.01 137
985510
i6
9 84385
9.84398
9
0.01 112
985497
3
44
.2
5.0
17
13
9
98913
25
26
0.01 087
9.85485
43
•3
7 5
i8
9 84 411
13
9
98939
01 061
985473 ,
42
•4
10
19
20
9 84424
13
13
9
98964
25
25
26
01 036
9 85 460
3
3
41
40
■ i
.7
.8
•9
12 5
15
17 5
20.0
22 5
9 84 437
9
98989
o.oi on
985448 ,
21
9 84 450
13
9
99015
0.00 985
985436 ,
39
22
9.84463
9.84476
9
99040
. 00 960
985423 ,
3
38
23
9
99065
0.00935
9-85411
37
24
2S
9 84 489
13
9
99090
26
0.00910
9 85 399 ,
3
36
35
1
9 84 502
9
99 116
0.00 884
9.85386
14
26
984515
*3
9
99 141
25
0.00859
985374 ,
34
.1
27
9
84528
>3
9
99 166
25
0.00834
9.85361
3
33
42
56
7.0
28
9
84540
9
99 191
25
26
25
0.00809
985349
32
•3
•4
29
30
9
84553
13
13
9
99217
0.00 783
9 85 337
■3
31
30
9
84566
9
99242
0.00 758
9 85324
31
9
84579
*3
9
99267
25
26
0.00 733
9 85312
29
8 4
32
9
84592
9
99293
99318
0.00 707
9.85299
28
.7
9.8
33
9
84605
13
9
25
0.00 682
9.85287
27
8
II .2
34
9
84618
13
12
9
99 343
25
25
26
0.00657
9.85274
13
t2
26
25
.9
12.6
3S
9
84630
9
99368
0.00632
9 . 85 262
36
9
84643
84656
13
9
99 394
0.00606
9.85250
24
1
37
9
13
9
99419
25
0.00 581
9 85237
IJ
23
13
38
9
84669
13
9
99 444
25
0.00 556
9. 85 225
22
I
I 1
39
40
9
84682
13
12
9
99469
25
26
0.00531
9.85 212
13
t2
21
20
.2
.3
3 9
9
84694
9
99 495
. 00 505
9.85 200
41
9
84 707
»3
9
99520
25
. 00 480
9 85 187
13
19
.4
5.2
42
9
84 720
9
99 545
25
0.00455
9 85175
18
M
43
9
84733
13
9
99570
25
26
25
0.00430
9 85 162
13
17
.6
44
4S
9
84745
13
9
99596
. 00 404
9 85 150
'3
16
15
• 7
.8
91
10.4
9
84758
9
99621
0.00379
9 85 137
46
9
84771
»3
9
99646
25
0.00354
9.85 125
14
9
II. 7
:i
9
84784
13
9
99672
00 328
9.85 112
13
13
1
9
84796
9
99697
25
00 303
9.85 100
12
1
49
60
9 84 809
13
13
9
99722
25
25
0.00 278
0.00 253
9.85087
13
3
II
10
la
12
9 84822
9
99 747
9.85 074
SI
9
84835
13
9
99 773
26
0.00 227
9 . 85 062
9
.2
2.4
S2
9
84847
9
99 798
25
. 00 202
9,85049
3
8
•3
3^
S3
9
84860
13
9
99823
25
0.00177
9 85037
7
•4
4.8
54
SS
9
84873
13
12
9
99848
25
26
0.00 152
9 85 024
3
2
6
5
i
6.0
96
10.8
9
84885
84898
9
99874
0.00 126
9.85 012
S6
9
13
9
99899
25
0.00 lOI
984999 ;
3
4
■7
.8
■>7
9
84 911
13
9
99924
25
0.00076
984986 '
3
3
S8
9
84923
9
99 949
25
0.00 051
9.84974
2
•9
59
00
9
84936
13
13
9
99 975
26
25
0.00 025
9.84961 ]
3
2
_0_
9
84949
00 000
0.00000
9.84949
L. Cos.
d.
L. Cotg.
c.d.
L. Tanjr.
L. Sin. d
[.
Prop. Pts.
45°
TABLE III
NATURAL
TRIGONOMETRIC FUNCTIONS
FOR
EACH MINUTE
71
72
TABLE m
©°
1--
30
30 ■ —
4» 1
t
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
60
59
58
55
54
53
52
51
50
%
47
46
45
44
43
42
41
40
35
34
33
32
31
30
27
26
25
24
23
22
21
20
19
18
'\l
15
14
13
12
I
2
3
4
1
9
lO
II
12
~^
14
\l
19
20
21
22
23
26
27
28
29
30
.00000
.00029
.ocx)58
.00087
.00116
.00145
.00175
I. 00000
I .00000
I .00000
I. 00000
I .00000
I. 00000
I. 00000
■01745
.01774
.01803
.01832
.01862
.0189J
.01920
•99985
.99984
.99984
.99983
■99983
.99982
.99982
•03490
•03519
■03548
■03577
.03606
.03664
•99939
.99938
■99937
■99936
■99935
•99934
•99933
•05234
.05263
.05292
■05321
•05350
•05379
.05408
.99863
99861
.99860
.99858
■99857
■99855
.99854
.06976
■07005
■07034
.07063
.07092
.07121
.07150
■99756
■99754
■99752
■99750
■99748
■99746
■99744
.00204
.00233
.00262
.00291
.00320
.00349
I. 00000
I. 00000
I. 00000
I. 00000
.99999
.99999
.01949
.01978
.02007
.02036
.02065
.02094
.99981
.99980
.99980
.99979
99979
■99978
.03693
.03723
■03752
■03781
.03S10
•03839
.99932
99931
.99930
.99929
.99927
.99926
■05437
.05466
■05495
■05524
■05553
•05582
■99852
.99851
.99849
•99847
.99846
.99844
. .07208
■07237
.07266
.07295
•07324
.99742
.99740
■99738
.99736
■99734
•99731
.00378
.00407
.00436
.00465
.00495
.00524
.99999
.99999
.99999
.99999
•99999
.99999
.02123
.02152
.02181
.02211
.02240
.02269
.99977
•99977
.99976
.99976
•99975
■99974
.03868
.03897
.03926
•03955
.03984
.04013
.99925
.99924
.99923
.99922
.99921
.99919
.05611
.05640
.05669
.05698
.05727
■05756
.99842
.99841
.99839
.99838
.99836
.99834
•07353
.07382
.07411
.07440
.07469
.07498
.99729
.99727
•99725
•99723
.99721
.99719
•00553
.00582
.00611
.00640
.00669
.00698
•99998
.99998
.99998
•99998
.99998
.99998
.02298
.02327
.02356
.02385
.02414
.02443
.99974
■99973
.99972
.99972
.99971
.99970
.04042
.04071
.04100
.04129
.04159
.04188
.99918
.99917
.99916
■99915
■99913
.99912
■05785
.05814
.05844
■05873
.05902
■05931
99833
.99831
.99829
.99827
.99826
.99824
■07527
■07556
■07585
.07614
.07643
.07672
.99716
.99714
.99712
.99710
.99708
■99705
.00727
.00756
.00785
.00814
.00844
.00873
.99997
.99997
.99997
•99997
.99996
.99996
.02472
.02501
.02530
.02560
.02589
.02618
.99969
.99967
.99966
.99966
.99965
.99964
.99963
.99963
.99962
.99961
.04217
.04246
.04275
.04304
•04333
.04362
.99911
.99910
.99909
.99907
.99906
■99905
.05960
.05989
.06018
.06047
.06076
.06105
.06163
.06192
.06221
.06250
.06279
.99822
.99821
.99819
.99817
.99815
.99813
.07701
•07730
•07759
.07788
.07817
.07846
.99703
.99701
.99699
.99696
.99694
.99692
31
32
33
34
%
39
40
41
42
43
44
%
49
50
51
52
53
54
55
56
.00902
.00931
.00960
.00989
.01018
.01047
.99996
.99996
•99995
•99995
•99995
•99995
.02647
.02676
.02705
•02734
.02763
.02792
.04391
.04420
.04449
.04478
•04507
•04536
.99904
.99902
.99901
.99900
.99898
.99897
.99812
.99810
.99808
.99806
.99804
.99803
■07875
.07904
■07933
.07962
.07991
.08020
.99689
■99687
.99685
.99683
.99680
.99678
.01076
.01105
•01134
.01164
.01193
.01222
.99994
.99994
■99994
•99993
•99993
■99993
.02821
.02850
.02879
.02908
.02938
.02967
.99960
■99959
•99959
.99958
■99957
■99956
■04565
■04594
.04623
•04653
.04682
.04711
.99896
.99894
.99893
.99892
.99890
.99889
.06308
^06366
■06395
.06424
•06453
.99801
.99799
.99797
•99795
•99793
.99792
.08049
.08078
.08107
.08136
.08165
08194
.99676
■99673
.99671
.99668
.99666
.99664
.99661
•99659
■99657
.99654
.99652
.99649
.01251
.01280
.01309
•01338
.01367
.01396
.99992
.99992
.99991
.99991
.99991
.99990
.02996
.03025
03054
.03083
.03112
.03141
•99955
■99954
■99953
.99952
.99952
■99951
.04740
.04769
.04798
.04827
.04856
.04885
.99888
.99886
■99885
■99883
.99882
.99881
.06482
.06511
.06540
.06569
.06598
.06627
.06656
.06685
.06714
.06743
■06773
.06802
.06831
.06860
.06889
.06918
.06947
.06976
.99790
.99788
•99786
.99784
.99782
.99780
■99778
.99776
■99774
■99772
■99770
.99768
.08223
.08252
.08281
.08310
■08339
.08368
.01425
.01454
.01483
•01513
.01542
.01571
.99989
.99989
.99989
.03170
.03199
.03228
.03286
.03316
•99950
•99949
.99948
•99947
.99946
■99945
.04914
•04943
.04972
.05001
.05030
•05059
.99879
.99878
.99876
.99875
■99873
.99872
.08397
.08426
•08455
.08484
•08513
.08542
.99647
•99644
.99642
•99639
•99637
•99635
II
10
I
5
4
3
2
I
.01600
.01629
.01658
.C1687
.01716
•01745
.99987
.99987
.99986
.99986
99985
■99985
•Q3345
•03374
■03403
•03432
.03461
.03490
.99944
•99943
.99942
.99941
.99940
99939
.05088
.05117
.05146
■05175
.05205
■05234
.99870
.99869
.99867
.99866
.99864
■99863
.99766
.99764
.99762
.99760
•99758
99756
.08571
.08600
.08629
.08658
.08687
.08716
■99632
■99630
.99627
.99625
.99622
.99619
N. cos.
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
N. sine
89°
SS**
87°
86°
85°
TRIGONOMETRIC FUNCTIONS FOR EACH MINUTE
73
^^■^
5« 1
6° 1
7- 1
8» 1
»• 1
60
11
55
54
53
52
51
50
It
%
45
44
43
42
o
I
2
3
4
i
I
9
lO
i2
13
14
\l
\l
19
20
21
22
23
24
1
29
30
31
32
33
34
11
11
39
40
41
42
NT. sine ]
ST. COS.
N. sine ]
ST. COS.
NT. sine]
ST. COS.
^r. sine|]
ST. COS.
N. sine
N. COS.
.08716
.08745
.08774
.08803
.08831
.08860
.08889
.99619
.99617
.99614
.99612
.99609
.99607
.99604
. 10482
.10511
■ 10540
■ 10569
•10597
. 10626
•99452
•99449
•99446
•99443
•99440
99437
•99434
.12187
.12216
.12245
.12274
.12302
•12331
.12360
•99255
.99251
.99248
.99244
.99240
.99237
■99233
•I39I7
.13946
•13975
.14004
• 14033
.14061
. 14090
.99027
.99023
.99019
.99015
.99011
.99006
.99002
•15643
.15672
.15701
•15730
.15758
.15787
.15816
.98769
.98764
.98760
•98755
.98751
.98746
.98741
.08918
.08947
.08976
.09005
.09034
.09063
.99602
•99599
.99596
•99594
.99591
.99588
.10713
.10742
.10771
. 10800
•99431
.99428
.99424
.99421
.99418
■99415
.12389
.12418
.12447
.12476
.12504
•12533
.99230
.99226
.99222
.99219
.99215
.99211
.99208
.99204
.99200
.99197
•99193
.99189
.14119
.14148
.14177
• 14205
.14234
.14263
.98998
.98994
.98990
.98986
.98982
.98978
.15845
•15873
.15902
•15931
•98737
•98732
.98728
.98723
.98718
.98714
.09092
.09121
.09150
.09179
.09208
.09237
.99586
•99583
.99580
•99578
•99575
•99572
. 10829
'10887
.10916
.10945
.10973
.99412
•99409
.99406
.99402
•99399
•99396
.12562
.12591
.12620
.12649
.12678
.12706
. 14292
.14320
■14349
•14378
.14407
.14436
•98973
.98969
.98965
.98961
•98957
•98953
.16017
.16046
.16074
.16103
.16132
.16160
.98709
.98704
.98700
.98695
.09266
.09295
.09324
•09353
.09382
.09411
•99570
•99567
•99564
.99562
•99559
•99556
.11002
.11031
.11060
. 1 1089
.IIII8
.11147
•99393
•99390
.99386
•99383
.99380
•99377
■12735
.12764
.12793
.12822
.12851
.12880
.99186
.99182
.99178
•99175
.99171
.99167
.14464
•14493
.14522
.14551
.14580
. 14608
.98948
.98944
.98940
.98936
.98931
.98927
.16189
.16218
.16246
.16275
.16304
•16333
.98681
.98676
.98671
.98667
.98662
•98657
41
40
P
.09440
.09469
.09498
.09527
•09556
■09585
■99553
•99551
.99548
•99545
.99542
.99540
. II 1 76
.11205
.11234
.11263
.11291
.11320
•99374
•99370
■99367
.99364
•99360
•99357
.12908
.12937
.12966
.12995
.13024
•13053
.99163
.99160
.99156
.99152
.99148
.99144
.14666
.14695
•14723
•14752
.14781
■98923
.98919
.98914
.98910
.98906
.98902
.16361
.16390
.16419
.16447
.16476
•16505
.98652
.98648
•98643
.98638
•98633
.98629
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
\l
15
14
13
12
.09614
.09642
.09671
.09700
.09729
.09758
■99537
•99534
•99531
.99528
.99526
•99523
•I 1349
.11378
.11407
.11436
.11465
.11494
•99354
•99351
•99347
•99344
•99341
•99337
.13081
.13110
•13197
.13226
.99141
•99137
•99133
.99.129
.99125
.99122
.14810
• 14838
.14867
.14896
.14925
• 14954
.98897
.98884
.98880
.98876
•16533
.16562
.16591
.16620
.16648
.16677
.98624
.98619
.98614
.98609
.98604
.98600
.09787
.09816
.09845
.09874
.09903
.09932
.99520
•99517
•99514
•9951 1
.99508
.99506
•11523
•II552
.11580
.11609
.11638
.11667
•99334
•99331
.99327
•99324
.99320
■99317
•13254
•13283
•13312
•13341
•13370
•13399
.99118
.99114
.99110
.99106
.99102
.99098
.14982
.15011
.15040
.15069
•15097
.15126
.98871
.98867
.98863
.98858
.98854
.98849
.16706
•16734
.16763
.16792
.16820
.16849
•98595
.98590
.98585
.98580
•98575
.98570
43
44
11
.09961
.09990
.10019
.10048
.10077
.10106
•99503
•99500
.99497
•99494
.99491
.99488
.11696
.11725
■11754
.11783
.11812
.11840
■99314
.99310
•99307
•99303
.99300
.99297
•13427
•13456
•13485
•13514
•13543
•13572
•99094
.99091
.99087
•99083
•99079
•99075
•I5I55
.15184
.15212
.15241
.15270
•15299
.98845
.98841
.98836
.98832
.98827
.98823
.16878
.16906
•16935
.16964
.16992
.17021
.98565
.98561
.98556
.98551
.98546
.98541
49
50
51
52
53
54
55
56
11
i.
.10135
.10164
.10192
.10221
.10250
.10279
•99485
.99482
.99479
.99476
•99473
■99470
.11869
.11898
.11927
.11956
.11985
.12014
•99293
.99290
99286
.99283
.99279
.99276
.13600
.13629
.13716
•13744
.99071
.99067
.99063
.99059
•99055
.99051
•15327
•15356
•15385
•I54I4
.15442
•I547I
.98818
.98814
.98809
.98805
.98800
•98796
.17050
.17078
.17107
.17136
.17164
•17193
•98536
•98531
.98526
.98521
.98516
.98511
10
I
5
4
3
2
.10308
•10395
. 10424
•10453
■99467
.99464
.99461
.99458
■99455
■99452
.12043
.12071
.12100
.12129
.12158
.12187
.99272
.99269
•99265
.99262
.99258
99255
.13889
•13917
.99047
.99043
.99039
•99035
.99031
.99027
•15500
•15529
•'5557
.15586
•15615
•15643
.98791
•98787
.98782
.98778
•98773
.98769
. 1 7222
.17250
.17279
.17308
•17336
•17365
.98506
.98501
.98496
.98481
N. COS.
N. sine
N. COS.
M. sine
N. COS.
N.,sine
N. COS.
N. sine
N. COS.
N. sine
/
84'* 1
§3° 1
82° 1
81° 1
80°
74
TABLE III
t
o
I
2
3
4
I
I
9
lO
II
12
14
\l
17
i8
10° j
If
19' 1
1»»
.4^
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
!^. sine
N. COS.
N. sine
N. COS.
60
59
58
57
56
55
54
53
52
51
50
It
47
46
45
44
43
42
17365
17393
.17422
17451
■17479
.17508
•17537
.98481
.98476
.98471
.98466
.98461
98455
•98450
.19081
.19109
.19138
.19167
•19195
. 19224
.19252
.98163
.98157
.98152
.98146
.98140
98.35
.98129
.98124
.98118
.98112
.98107
.98101
.98096
20791
. 20820
.20848
.20877
.20905
.20933
.20962
•97815
.97809
.97803
.97797
.97791
•97784
•97778
•22495
•22523
•22552
.22580
.22608
.22637
.22665
•97437
•97430
.97424
•97417
.97411
•97404
•97398
.24192
.24220
.24249
•24277
•24305
•24333
.24362
.97030
•97023
•97015
.97008
.97001
■96994
.96987
■17565
•17594
.17623
.17651
.17680
.17708
.98445
.98440
•98435
.98430
.98425
.98420
.19281
.19309
•19338
.19366
•19395
.19423
.20990
.21019
.21047
.21076
.21104
.21132
.97772
.97766
.97760
•97754
.97748
.97742
.22693
.22722
.22750
•22778
.22807
.22835
•97391
•97384
•97378
•97371
•97365
•97358
•24390
.24418
.24446
.24474
•24503
•24531
.96980
•96973
.96966
•96959
.96952
.96945
.17766
•17794
.17823
.17852
.17880
.98414
.98409
.98404
•98399
tut
.19452
.19481
.19509
.19566
•19595
.98090
.98084
.98079
.98073
.98067
.98061
.21161
.21189
.21218
.21246
.21275
.21303
■97735
•97729
•97723
.97717
.97711
•97705
.22863
.22892
.22920
.22948
.22977
•23005
•97351
•97345
•97338
•97331
•97325
•97318
•24559
.24587
•24615
.24644
.24672
.24700
•96937
.96930
.96923
.96916
.96909
.96902
19
20
21
22
23
_^ -
25
26
27
28
29
30 .
31
32
33
34
It
.17909
•17937
.17966
•17995
.18023
.18052
•98383
■98373
:98368
.98362
•98357
.19623
.19680
.19709
.19766
.98056
.98050
.98044
•98039
•98033
.98027
•21331
.21360
.21388
.21417
.21445
.21474
.97698
.97692
.97686
.97680
.97667
•23033
.23062
.23090
.23118
.23146
•23175
973"
•97304
.97298
.97291
.97284
•97278
.24728
.24756
.24784
.24813
.24841
.24869
! 9688 7
.96880
•96873
.96866
.96858
41
40
It
11
.18081
.18109
.18138
.18166
.18195
.18224
•98352
•98347
.98341
•98336
•98331
•98325
.19794
.19823
.19908
•19937
.98021
.98016
.98010
.98004
.97998
.97992
.21502
•21530
•21559
.21587
.21616
.21644
.97661
.97648
.97642
.97636
•97630
.23203
.23260
.23288
.23316
•23345
.97271
.97264
•97257
•97251
.97244
•97237
.24897
.24925
.24954
.24982
.25010
.25038
.96851
.96844
•96837
.96829
.96822
.96815
35
34
33
32
31
30_
29
28
27
26
25
24
.18252
.18281
.18367
•18395
.98320
•98315
.98310
•98304
.98299
.98294
.19965
.19994
.20022
.20051
.20079
.20108
.97987
.97981
•97975
.97969
•97963
•97958
.21672
.21701
.21729
.21758
.21786
.21814
•97623
.97617
.97611
.97604
•97598
•97592
23373
.23401
•23429
•23458
•23486
•23514
.97230
•97223
.97217
.97210
•97203
.97196
.25066
.25094
.25122
•25151
•25179
•25207
.96807
.96800
till
•96778
.96771
11
39
40
41
42
.18424
^18481
•IP
.18567
.98288
•98283
.98277
.98272
.98267
.98261
.20136
.20165
•20193
.20222
.20250
.20279
•97952
•97946
.97940
•97934
.97928
.97922
•21843
.21871
.21899
.21928
.21956
.21985
•97585
•97579
•97573
•97566
.97560
•97553
•23542
•23571
•23599
.23627
.97189
.97182
.97176
.97169
.97162
•97155
•25235
.25263
.25291
•25320
•25348
•25376
.96764
.967.56
•96749
.96742
•96734
.96727
23
22
21
20
19
18
17
16
15
H
13
12
II
10
I
5
4
3
2
I
43
44
45
1
49
50
51
52
53
54
11
.18652
.18681
18710
.18738
.98256
.98250
•98245
.98240
.98234
.98229
.20307
.20336
20364
•20393
.20421
.20450
.97916
.97910
.97899
•97893
•97887
.22013
.22041
.22070
.22098
.22126
•22155
•97547
•97541
•97534
•97528
•97521
•97515
.23712
.23740
.23769
:2^^^5^
•23853
.97148
.97141
■97134
.97127
.97120
•97"3
•25404
•25432
.25460
.25488
•25516
•25545
.96719
.96712
•96705
.96697
.96690
.96682
.18767
' 18795
. 18824
.18852
.18881
.18910
.98223
.98218
.98212
.98207
.98201
.98196
.20478
.20507
•20535
.20563
.20592
.20620
.97881
•97875
.97869
•97863
•97857
■97851
.22183
.22212
.22240
.22268
.22297
•22325
.97508
.97502
.97496
.97489
•97483
.97476
.23882
.23910
.23966
•23995
.24023
.97106
.97100
.97093
.97086
•97079
.97072
•25573
.25601
.25629
■.25685
•25713
.96675
.96667
.96660
.96645
.96638
.18938
.18967
.18995
.19024
.19052
.19081
.98190
.98185
.98179
tit
.98163
.20649
.20677
.20706
.20734
.20763
.20791
•97845
•97839
•97833
•97827
.97821
97815
•22353
.22382
.22410
.22438
.22467
.22495 1
.97470
97463
•97457
.97450
•97444
97437
.24051
.24079
.24108
.24136
.24164
.24192
.97065
•97058
•97051
■97044
•97037
.97030
•25741
.25769
'25826
•25854
.25882
.96630
.96623
&
96600
■96593
N. COS.
N. sine
M. COS.
N. sine
N. cos.|N. sine
N. COS.
N. sine
N. CCS.
N. sine
/
1
79- 1
78^
yyo
76°
75°
TRIGONOMETRIC FUNCTIONS FOR EACH MINUTE
75
o
I
2
3
4
16° j
16" 1
17°
1§"
19°
60
It
11
55
54
53
52
51
50
tt
%
45
44
43
42
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
.25S82
25910
.25966
•25994
.26022
.26050
96593
96585
96578
.96570
.96562
96555
•96547
.27564
.27592
.27620
.27648
.27676
27704
27731
.96126
.96118
.96110
.96102
.96078
29237
.29265
.29293
.29321
.29348
.29376
.29404
•95630
.95622
•95613
•95605
•95596
•95588
•95579
.30902
.30929
•30957
•30985
.31012
.31040
.31068
.95106
•95079
.95070
.95061
.95052
•32557
•32584
.32612
.32694
.32722
•94552
•94542
•94533
•94523
•94514
•94504
•94495
I
9
lO
II
12
.26079
.26107
.26135
.26163
.26191
.26219
96540
•96532
.96524
•96517
.96509
.96502
■27759
.27787
•27815
27843
.27871
.27899
.96070
.96062
.96054
.96046
.96037
.96029
.29432
.29460
•29487
•29515
•29543
■29571
•95571
•95562
•95554
■95545
■95536
•95528
•31095
•31123
•31151
.31178
.31206
•31233
•95043
•95033
.95024
•95015
.95006
.94997
•32749
.32777
.32804
•32832
.94485
.94476
.94466
•94457
•94447
•94438
13
14
\i
18
19
20
21
22
23
24
.26247
.26275
.26303
•26331
•26359
.26387
•96494
.96486
.96479
.96471
.96463
•96456
.27927
•27955
.27983
.28011
.28039
.28067
.96021
.96013
.96005
•95997
.95989
.95981
.29599
.29626
.29710
■29737
•95519
•95511
•95502
•95493
•95485
•95476
.31261
.31289
•31316
•31344
•31372
•31399
.94988
•94979
.94970
.94961
.94952
•94943
.32914
.32942
•32969
•32997
•33024
•33051
.94428
.94418
.94409
•94399
.94390
.94380
•26415
•26443
.26471
.26500
.26528
.26556
.96448
.Q6440
•96433
.96425
.96417
.96410
.28095
.28123
.28150
.28178
.28206
•28234
■95972
.95964
•95956
.95948
•95940
■95931
.29765
•29793
.29821
.29849
.29876
.29904
•95467
•95459
•95450
•95441
■95433
•95424
•31427
•31454
.31482
•31510
•31537
•31565
•94933
•94924
•94915
.94906
'.lists
•33079
.33106
•33134
•33161
•33189
.33216
•94370
.94361
•94351
.94342
•94332
.94322
41
40
39
1
25
26
li
29
30
3J
32
33
34
35
36
11
39
40
41
43
44
45
46
47
48
.26584
.26612
^26696
.26724
.26752
.26780
.26808
.26836
.26864
.26892
.96402
•96394
.96386
96379
•96371
96363
•96355
•96347
.96340
.96332
•96324
.96316
.28262
.28290
.28318
.28346
•28374
.28402
■95923
•95915
•95907
.95898
.95890
.95882
■29932
.29960
•29987
■30015
■30043
.30071
.30098
.30126
■30154
.30182
.30209
.30237
•95415
•95389
•95380
•95372
•31593
.31620
.31648
•31675
•31703
•31730
.94860
.94851
.94842
.94832
•33244
•33271
•33298
•33326
•33353
•33381
•94313
•94303
•94293
.94284
•94274
.94264
•94254
•94245
•94235
.94225
.94215
.94206
35
34
33
32
31
30
27
26
25
24
23
22
21
20
\l
15
14
13
12
II
10
\
5
4
3
2
I
.28429
•28457
.28485
.28513
.28569
•95874
•95865
•95857
•95849
.95841
■95832
•95363
■95354
■95345
95337
■95328
95319
•31758
.31786
.31813
.31841
.31868
•31896
•94823
.94814
•94805
•94795
.94786
■94777
•33408
•33436
•33463
•33490
•33518
•33545
.26920
.26948
.26976
.27004
.27032
.27060
.96308
•96301
.96293
.96285
.96277
.96269
.28597
.28625
.28652
.28680
.28708
•28736
•95816
•95807
•95799
•95791
.95782
.30265
.30292
■30320
•30348
.30376
■30403
95310
•95301
•95293
.95284
•95275
.95266
•31923
•31951
•31979
.32006
•32034
.32061
.94768
•94758
.94749
.94740
•94730
.94721
•33573
.33600
.33627
•33710
.94196
.94186
.94176
.94167
•94157
.94147
.27088
.27116
.27144
.27172
.27200
.27228
.96261
•96253
.96246
.96238
.96230
.96222
.28764
.28792
28820
.28847
.28875
.28903
.95766
•95757
•95749
•95740
■95732
■30431
■30459
.30486
■30514
•30542
•30570
■95257
.95248
.95240
■95231
■95222
95213
.32089
.32116
.32144
.32171
.32199
.32227
.94712
.94702
•94693
.94684
.94674
•94665
•33737
•33764
•33792
•33819
•33846
•33874
•94137
.94127
.94118
.94108
49
50
51
11
54-
11
11
.27256
.27284
.27312
.27340
.27368
.27396
.96214
.96206
.96198
.96190
.96182
.96174
•28931
.29015
.29042
.29070
•95724
•95715
•95707
95698
.95690
.95681
•30597
.30625
.30708
•30736
■95204
•^5195
.95186
.95168
•95159
•32254
.32282
.32309
•32337
•32364
.32392
•94656
.94646
•94637
.94627
.94618
.94609
•33901
•33929
•33956
•33983
.54011
■34038
.94078
.94068
•94058
•94049
•94039
.94029
.27424
.27452
.27480
.27508
•27536
.27564
.96166
.96158
.96150
.96142
.96126
.29098
.29126
.29154
29182
.29209
.29237
•95664
•95656
•95647
95639
•95630
.30763
.30791
.30819
.30846
.30874
.30902
■95150
■95142
■95133
■95124
95"5
.95106
.32419
•32447
•32474
.32502
•32529
•32557
•94599
•94590
.94580
•94571
.94561
•94552
-34065
•34093
.34120
•34147
•34175
.34202
•94019
.94009
•93999
•93989
•93979
.93969
N. COS.
N. sine
N. COS.
N sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
N. sine
/
74" 1 73° 1 72"
71" 1 70"
76
TABLE III
f
o
I
2
3
4
"I
9
lO
II
12
20° 1
«»
22° 1
23° 1
24°
60
59
58
11
55
il_
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
11
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
\t
\l
15
14
13
12
10
I
5
4
3
2
I
N. sine
^T. COS.
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
.34202
.34229
.34284
•343"
•34339
•34366
.93969
•93959
93949
■93939
.93929
•93919
•93909
•35891
■35918
•35945
•35973
.36000
•93358
•93348
•93337
•93327
•93316
•93306
•93295
•37461
.37488
•37515
•37542
•37569
•37595
.37622
.92718
.92707
.92697
.92686
.92664
•92653
•39073
.39100
•39127
•39153
.39180
.39207
■39234
.92050
.92039
92028
.92016
.92005
.91994
.91982
.40674
.40700
.40727
•40753
.40780
.40806
•40833
.40860
.40886
.40913
.40992
•91355
91343
91331
•91319
•91307
.9 [295
.91283
•34393
.34421
.34448
•34475
•34503
•34530
•93899
.93889
•93879
.93869
•93859
•93849
.36027
.36108
■93285
•93274
•93264
•93253
•93243
.93232
•37649
.37676
■37703
•37730
•37757
•37784
.92642
.92631
.92620
.92609
.92598
•92587
.39260
.39287
•39314
•39341
•39367
■39394
.91971
•91959
.91948
.91936
.91925
.91914
.91272
.91260
.91248
.91236
.91224
.91212
13
14
•34557
•345^4
.34612
•34639
.34666
.34694
•93839
•93829
.93819
•93809
•93799
■93789
.36190
.36217
•36244
.36271
.36298
•36325
.93222
.93211
•93201
.93180
.93169
.37811
•37838
•37865
.37892
•37919
•37946
.92576
.92565
•92554
•92543
•92532
.92521
.39421
.39448
•39474
•39501
•39528
•39555
.91902
.91891
.91879
.91868
.91856
.91845
.41019
.41045
.41072
.41098
.41125
.41151
.91200
.91188
.91176
.91164
.91152
.91140
19
20
21
22
23
24
•34721
•34748
•34775
•34803
•34830
•34857
•93779
•93769
•93759
•93748
•93738
.93728
•36352
•36379
.36406
.36461
.36488
•93159
.93148
•93137
•93127
.93116
.93106
•37973
.38107
.92510
•92499
.92488
.92477
.92466
•92455
•39581
.39608
.39688
•39715
•91833
.91822
.91810
.91799
.91787
•91775
.41178
.41204
.41231
•41257
.41284
.41310
.91128
.91116
.91104
.91092
.91080
.91068
29
30
31
32
33
34
P
39
40
41
42
43
44
45
46
ti
49
50
51
52
53
54
55
56
,34884
.34912
•34939
.34966
•34993
•35021
•93718
.93708
.93698
.93688
.93677
.93667
•36515
•36542
.36569
.36596
•^^?^
.36650
•93095
.93084
•93074
.93063
•93052
.93042
.38188
.38215
.38241
.38268
•92444
.92432
.92421
.92410
•39741
•39768
•39795
.39822
•39848
•39875
.91764
•91752
.91741
.91729
.91718
.91706
•41337
•41363
.41390
.41416
•41443
.41469
.91056
.91044
.91032
.91020
.91008
.90996
•3^048
•35075
•35102
•35130
•35157
•35184
•93657
•93647
•93637
.93626
.93616
.93606
•36677
•36704
•36731
•36758
•93031
.93020
.93010
.92978
•38295
.38322
•38349
•38376
•38403
•38430
■$t
.38510
•38537
.38564
•38591
■92377
.92366
•92355
•92343
.92332
.92321
.92310
.92299
.92287
.92276
.92265
•92254
•39902
•39928
•39955
.39982
.40008
•40035
.91694
.91683
.91671
.91660
.91648
.91636
.41496
.41522
•41549
•41575
.41602
.41628
.90984
.90972
.90960
.90948
.90936
.90924
•35211
•35239
.35266
•35293
•35320
•35347
•93596
•93585
•93575
•93565
•93555
•93544
•36894
.36921
.36948
•36975
.92967
.92956
•92945
•92935
.92924
.92913
.40062
.40088
.40115
.40141
.40168
.40195
.91625
.91613
.91601
.91590
.91566
tell
.41707
•41734
.41760
.41787
.90911
.90899
.90887
•90875
.90863
.90851
•35375
•35402
•35429
•35456
•35484
•355"
•93534
•93524
•93514
•93503
•93493
•93483
.37002
.37029
•37056
•37083
.37110
•37137
.92902
.92892
.92881
.92870
•92859
.92849
.38617
.38644
.38671
.38698
•38725
•38752
.92243
.92231
.92220
.92209
.92198
.92186
.40221
.40248
.40275
.40301
.40328
•40355
•91555
•91543
91531
91519
.91508
.91496
.41813
.41840
.41866
.41892
.41919
•41945
•90839
.90826
.90814
.90802
.90790
■90778
•35538
•35565
•35592
•35619
•35647
•35674
•93472
•93462
•93452
•93441
•93431
.93420
•37164
•37191
.37218
•37245
.37272
•37299
.92838
.92827
.92816
•92805
.92794
.92784
.38912
•92175
.92164
.92152
.92141
.92130
.92119
.40381
.40408
.40434
.40461
.40488
.40514
.91484
.91472
.91461
.91449
•91437
.91425
.41972
.41998
.42024
•42051
•42077
.42104
.90766
•90753
.90741
9^729
.90717
.90704
.90692
.90680
.90668
90655
90643
.90631
•35701
•35728
•35755
•35782
•35810
•35837
.93410
•93400
•93389
•93379
•93368
•93358
•37326
•37353
•37380
•37407
•37434
•37461
.92773
.92762
.92751
.92740
.92729
.92718
•38939
.38966
•38993
.39020
.39046
•39073
.92107
.92096
.92085
.92073
.92062
.92050
.40541
.40567
.40594
.40621
.40647
.40674
.91414
.91402
.91390
.91366
•91355
•42130
.42156
.42183
.42209
•42235
.42262
N. COS.
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
N. sine
/
69° 1
68°
67° 1
66° 1
65°
TRIGONOMETRIC FUNCTIONS FOR EACH MINUTE
77
9
35° 1
26° 1
sr 1
28° 1
29° 1
60
59
58
57
56
55
54
53
52
51
50
:i
47
46
45
44
43
42
41
40
35
34
33
32
31
30
29
28
V,
25
24
23
22
21
20
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
o
I
2
3
4
c
l_
9
lO
12
13
14
:i
17
18
.42262
.42288
•42315
•42341
•42367
.42394
.42420
.90631
.90618
.90606
.90594
.90582
.90569
•90557
•43863
■43889
.43916
■43942
.43968
•43994
.89879
.89867
.89854
.89841
.89828
.89816
.89803
45399
•45425
•45451
■45477
•45503
•45529
■45554
.89101
.89087
.89074
.89061
.89048
■89035
.89021
.46947
•46973
•46999
.47024
.47050
.47076
.47101
.88295
.88281
.88267
.88254
.88240
.88226
.88213
.48481
.48506
■48532
■48557
■48583
.48608
.48634
■48659
.48684
.48710
■48735
.48761
.48786
.87462
.87448
■87434
.87420
.87406
■87391
■87377
•87363
•87349
•87335
.87321
•87306
.87292
.42446
•42473
•42499
•42525
•42552
.42578
•90545
•90532
.90520
.90507
.90495
.90483
.44020
.44046
.44072
.44098
.44124
•44151
•44177
■44203
.44229
■44255
.44281
•44307
.89790
•89777
.89764
•89752
■89739
.89726
.45580
.45606
•45632
■45658
■45684
•45710
.89008
.88995
.88981
.88968
.88955
.88942
.47127
•47153
.47178
.47204
.47229
•47255
.88199
.88185
.88172
.88158
.88144
.88130
.42604
•42631
.42709
•42736
.90470
.90458
.90446
•90433
.90421
.90408
■89713
.89700
.89687
.89674
.89662
.89649
•45736
•45762
•45787
•45813
•45839
•45865
.88928
•88915
: 88888
.88875
.88862
.47281
.47306
•47332
•47358
■47383
■47409
.88ii7
.88103
.88089
.88075
.88062
.88048
.48811
•48837
.48862
.48888
•48913
•48938
.87278
.87264
.87250
•87235
.87221
.87207
19
20
21
22
23
24
.42762
.42788
•42815
.42841
.42867
•42894
•90396
•90383
•90371
•90358
.90346
•90334
•44333
•44359
•44385
.44411
•44437
•44464
.89636
.89623
.89610
•89597
.89584
.89571
.45891
•45917
•45942
.45968
•45994
.46020
.88848
•88835
.88822
.88808
:i5i
■47434
.47460
■47486
■475"
■47537
■47562
•47588
.47614
•47639
.47665
.47690
.47716
.88034
.88020
.88006
■87993
.87979
.87965
.48964
.48989
.49014
.49040
•49065
.49090
•87193
.87178
.87164
.87150
.87136
.87121
25
26
27
28
29
3c
31
32
33
34
%
39
40
41
42
43
44
47
48
49
50
51
52
53
54
.42920
.42946
.42972
.42999
•43025
•43051
.90321
.90309
.90296
.90284
.90271
•90259
■44490
.44516
■44542
•44568
•44594
.44620
.89558
■89545
■89532
.89519
.89506
.89493
.46046
.46072
.46097
.46123
.46149
•46175
.88768
•88755
.88741
.88728
.88715
.88701
•87951
•87937
li
".87882
.49116
.49141
.49166
.49192
.49217
.49242
.87107
.87093
.87079
.87064
.87050
•87036
•43077
.43104
•43130
•43156
.43182
.43209
.90245
.90233
.90221
.90208
.90196
.90183
.44646
•44672
.44698
.44724
•44750
.44776
.89480
.89467
■89454
.89441
.89428
.89415
.46201
.46226
.46252
.46278
•46304
•46330
.88688
.88674
.88661
.88647
.88634
.88620
•47741
•47767
.47844
•47869
•47895
.47920
.47946
•47971
•47997
.48022
.87868
•87854
.87840
.87826
.87812
•87798
•87784
.87770
.87756
■87743
.87729
.87715
.49268
.49293
.49318
•49344
•49369
•49394
.87021
.87007
•86993
.86978
.86964
.86949
•43235
.43261
.43287
•43313
■43340
•43366
.90171
.90158
.90146
•90133
.90120
.90108
.44802
.44828
.44854
.44880
.44906
.44932
■44958
.44984
45010
•45036
■45062
.45088
.89402
.89389
■89376
•89363
•89350
•89337
.89324
.89311
.89298
.89285
.89272
.89259
•46355
.46381
.46407
•46433
.46458
.46484
.88607
.88566
.88539
.49419
•49445
•49470
•49495
.49521
■49546
•86935
.86921
.86878
.86863
•43392
.43418
•43445
•43471
•43497
•43523
.90095
.90082
.90070
.90057
.90045
.90032
.46510
•46536
•46561
.46587
.46613
■46639
.88526
.88512
.88499
.88485
.88472
.88458
.48048
•48073
.48099
.48124
.48150
•48175
.87701
.87687
■87673
■87659
•^645
.87631
■49571
■49596
.49622
.49647
.49672
.49697
■49723
■49748
•49773
.49798
.49824
•49849
.86849
.86834
.8682c
.86805
.86791
.86777
15
14
13
12
11
10
7
6
•43549
•43575
.43602
•43628
.90019
.90007
:&
.89968
.89956
•89943
•89930
.89918
■89905
.89892
•89879
■45114
.45140
.45166
.45192
.45218
■45243
■89245
■89232
.89219
.89206
.89193
.89180
.46664
.46690
.46716
.46742
.46767
■46793
.88445
.88431
.88417
.88404
.88390
•88377
.48201
.48226
.48252
.48277
■48303
.48328
.87617
.87603
.87589
•87575
.87561
.87546
.86762
.86748
■86733
.86719
.86704
.86690
•43706
43733
43759
.43785
.43811
•43837
•45269
•45295
•45321
•45347
•45373
•4539S
.89167
■89153
.89140
.89127
.89114
.89101
.46819
.46844
.4687c
.46896
.46921
■46947
.88363
.88349
■88336
.88322
.88308
.88295
•48354
•48379
.48405
.4843c
■48456
.48481
•87532
.87518
.87504
.87490
•87476
.87462
.49874
.49899
.49924
.49950
•49975
.5000c
.86675
.86661
.86646
.86632
.86617
.86603
5
4
3
2
I
N. COS.
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
N. sine
/
fij^
«;i°
62°
61°
60°
^^__
78
TABLE III
SO'' 1
31° 1
92^ 1
33°
34°
o
I
2
3
4
I
I
9
lO
II
12
13
14
i6
N. sine ]
ST. COS.
N. sine ]
N. COS.
N. sine
N. COS.
N. sine
NT. COS.
N. sine
N. COS.
60
^^
55
54
53
52
51
50
4I
47
46
45
44
43
42
50000
•5^5025
.50050
.50076
50101
50126
50151
.86603
.86588
8^573
86559
86544
86530
.86515
•51504
•51529
•51554
•5'579
•5'653
•85717
.85702
.85687
.85672
85657
•85642
.85627
.85612
•85597
.85582
.85567
.85551
•85536
52992
•530*7
•53091
53115
•53140
.84805
•84789
.84774
.84759
•84743
.84728
.84712
•54513
•54537
.54561
.54586
.54610
.83867
•83851
•83835
.83819
■83772
•55919
:^
•55992
.56016
.56040
.56064
.82871
.82855
.82839
.82822
.82806
.50176
.50201
.50227
.50252
.50277
.50302
.86501
.86486
.86471
.86457
.86442
.86427
.51678
•51703
.51728
5'753
.51778
.51803
•53164
53189
.84697
.84681
.84666
.84650
•84635
.84619
■54635
.54708
54732
■54756
83756
.83740
83724
.83708
.83692
■83676
.56088
.56112
itt
.56184
.56208
.82790
82773
•82757
.82741
mil
50327
50352
•50377
•50403
.50428
•50453
86413
.86398
•86384
.86369
•86354
.86340
.51828
.51852
•51877
.51902
,51927
•51952
.85521
.85506
.85491
.85476
.85461
.85446
•53312
•53337
•53361
•53386
•534"
•53435
.84604
.84588
•84573
•84557
.84542
.84526
.54781
.54805
•54829
•54854
■54878
■54902
.83660
•83645
.83629
•83613
•83597
.83581
tit
•56305
•56329
•56353
.82692
.82675
.82659
.82643
.82626
.82610
19
20
21
22
23
24
^1
27
28
29
30
.50478
•50503
.50528
50553
50578
.50603
.86325
.86310
.86295
.86281
.86266
.86251
51977
.52002
.52026
52051
.52076
52101
•85431
.85416
•85401
■85385
•85370
•85355
•53460
•53484
•53509
•53534
•53558
•53583.
•53607
•53705
•53730
.84511
•84495
.84480
.84464
.84448
•84433
■54927
•54951
•54975
■54999
■55024
■55048
•83565
•83549
•83533
•83517
•83501
.83485
•56377
.56401
.56425
•56449
•56473
•56497
•82593
.82577
.82561
•82544
.82528
.82511
41
40
'^
1?
35
34
33
32
31
30
29
28
V,
25
24
23
22
21
20
18
17
16
15
14
13
12
II
10
i
I
5
4
3
2
I
.50628
.50654
50679
•50704
.50729
•50754
86237
.86222
.86207
.86192
.86178
.86163
.52126
52151
•52175
.52200
.52225
•52250
•85340
•85325
.85310
.85294
.85279
.85264
.84417
.84402
.84386
•84370
•84355
•84339
■55072
■55097
.55121
•55145
•55169
•55194
•83469
•83453
•83437
.83421
•83405
•83389
•56521
.56641
•82495
.82478
.82462
.82446
.82429
.82413
3
32
33
34
11
•50779
.50804
. 50829
.50854
.50879
•50904
.86148
•86133
.86119
.86104
.86089
.86074
•52275
•52299
•52324
52349
•52374
•52399
•85249
.85234
.85218
.85203
.85188
•85173
•53754
•53828
•53853
•53877
.84292
•84277
.84261
.84245
•55218
.55266
•55291
55315
•55339
•83373
•83356
•83340
•83324
.83308
.83292
.56665
.56689
•56713
.56760
.56784
.82363
.82347
.82330
•82314
11
39
40
41
42
.50929
■50954
•50979
.51004
.51029
•51054
.86059
.86045
.86030
86015
.86000
.85985
•52448
•52473
•52498
•52522
•52547
.85157
.85142
.85127
.85112
.85096
.85081
•53902
•53926
•53951
•53975
.54000
•54024
.84230
.84214
.84198
.84182
.84167
.84151
•55412
•55436
•55460
•55484
•83276
.83260
•83244
.83228
.83212
•83195
.56808
•56832
■IS
.82297
.82281
.82264
.82248
.82231
.82214
43
44
tl
47
48
49
50
51
52
53
54
55
56
•51079
.51104
•51129
•51154
•51179
.51204
.85970
.85956
■85941
.85926
.85911
.85896
•52572
•52597
52621
.52646
.52671
.52696
.85066
.85051
•85035
.85020
•85005
.84989
•54049
•54073
•54097
•54122
.54146
•54171
•84135
.84120
.84104
.84088
.84072
•84057
•55509
•55533
•55557
.55581
•55605
•55630
•83179
•83163
•83147
•83J31
; 83098
.57000
.57024
•57047
•57071
.82198
.82181
.82165
.82148
.82132
•82115
.51229
•51254
•51279
•51304
•51329
•51354
•51379
.51404
.51429
•5H54
•5H79
•51504
.85881
.85866
.85851
•85836
.85821
.85806
.85792
•85777
.85762
•85747
•85732
.85717
.52720
•52745
.52770
•52794
.52819
.52844
•84974
.84959
•84943
.84928
.84913
.84897
•54195
.54220
•54244
.54269
•54293
•54317
.84041
.84025
.84009
•83994
•83978
.83962
•55654
•55678
•55702
•55726
•55750
•55775
.83082
.83066
•83050
•83034
•83017
.83001
•57095
•57119
•57143
•57167
•57I9I
•57215
.82098
.82082
.82065
.82048
.82032
.82015
.52869
•52893
.52918
•52943
.52967
.52992
.84882
.84866
.84851
.84836
.84820
.84805
•54366
■54391
•54415
•54440
■54464
•83946
•83930
i%l
.83883
.83867
•55799
.55823
•55847
.55871
•55895
•55919
.82985
.82969
•82953
.82936
.82920
.82904
•57238
.57262
.57286
•57310
•57334
•57358
.81999
.81982
.81965
.81949
.81932
.81915
N COS.
N. sine
N. COS.
Kfrgirie
N. COS.
N. sine
N. COS.
M. sine
N. COS.
N. sine
p
59" 1
68** 1
.«» 1
56°
55'
TRIGONOMETRIC FUNCTIONS FOR EACH MINUTE
79
ss*' 1
36°
37° 1
38° 1
39°
9
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. cos.
O
I
2
3
4
i
7
8
9
lO
12
13
14
15
16
57358
•57381
■57405
57429
•57453
•57477
■57501
•81882
.81865
•81848
.81832
•81815
•58779
•58802
.58826
.58849
.■|896
.58920
.80867
.80850
.80833
.80816
.80799
60182
. 60205
.60228
.60251
•60274
.60298
.60321
•79864
•79846
.79829
•79811
•79793
•79776
•79758
.61566
.61612
61635
.61658
.61681
.61704
.78801
•78783
.78765
•78747
•78729
•78711
•78694
.62932
•62955
.62977
•63000
•63022
•63045
•63068
•77715
. 77606
60
59
.77678
.77660
.77641
•77623
•77605
58
55
54
53
52
51
50
49
48
57524
•57548
•57572
•57596
•57619
•57643
•81798
•81782
•81765
•81748
•81731
•81714
■58943
.58967
■58990
.59014
■59037
.59061
.80782
.80765
.80748
•80730
•80713
.80696
.60344
.60367
.60390
.60414
.60437
.60460
•79741
•79723
•79706
•79688
•79671
•79653
.61726
•61749
.61772
•61795
.61818
.61841
•78676
•78658
• 78640
.78622
.78604
•78586
•63090
•63113
•63135
•63158
.63180
.63203
•77586
•77568
•77550
•77531
•77513
• 77494
•57667
•57691
•57715
•57762
.57786
•81698
.81681
•81664
•81647
•81631
•81614
.59084
.59108
■59131
•59154
•59178
•59201
.80679
•80662
.80644
80627
.80610
•80593
•60529
•60576
•60599
•79635
.79618
•79600
•79583
•79565
•79547
.61864
.61887
•61909
•61932
•61955
.61978
• 78568
•78550
■78532
•78514
•78496
•78478
.63225
•63248
.63271
IP
•63338
•77476
•77458
•77439
•77421
.77402
•77384
45
44
43
42
41
40
%
35
34
33
32
31
30
19
20
21
22
23
24
^i
27
28
29
30
31
32
33
34
35
36
39
40
41
42
43
44
t
47
48
•57810
•57833
•57904
•57928
•81597
•81580
•81563
•81546
.81530
•81513
•59225
.59248
•59272
•59295
•59318
59342
.80576
.80558
.8054J
.80524
•80507
•80489
.60622
.60645
•60668
•60691
•60714
•60738.
•79530
•79512
•79494
•79477
•79459
•79441
.62001
.62024
.62046
•62069
•62092
•62115
•78460
•78442
.78424
.78405
•78387
.78369
.63361
•63383
.63406
.63428
•63451
•63473
.77366
•77347
.77329
.77.310
.77292
•77273
•57952
•57976
•57999
•58023
•58047
•58070
•81496
81479
.81462
• 81445
•81428
.81412
•593^5
59389
.59412
•59436
.80472
•80455
•80438
.80420
.80403
•80386
•60761
•60784
.60807
.60830
•60853
•60876
•79424
.79406
•79388
•79371
•79353
•79335
•79318
.79300
. 79282
.79264
.79247
.79229
.79211
•79193
.79176
•79158
.79140
.79122
•62138
•62160
.62183
•62206
.62229
.62251
•78351
•78333
•78315
.78297
.78279
.78261
•63496
•63518
•63540
•63563
•77255
•77236
.77218
•77199
.77181
.77162
.58141
.58165
.58189
.58212
•81395
.81378
.81361
.81344
.81327
.81310
•59506
•59529
•59552
•59576
•59599
•59622
•80368
•80351
.80264
.80247
•80230
.80212
•80195
.80178
•60899
.60922
•60945
.60968
.60991
.61015
•61038
•61061
.61084
.61107
.61130
•61153
.62274
•62297
•62320
.62342
^62388
.78243
.78225
.78206
.78188
.78170
.78152
.63630
•63653
^63698
.63720
•63742
•77144
•77125
.77107
.77088
.77070
•77051
29
25
24
23
22
21
20
19
.58236
.58260
•58283
•58307
•58330
•58354
.81293
.81276
.81259
.81242
•81225
•81208
•59646
•59669
•59693
•59716
•59739
•59763
.62411
•62433
•62456
•62479
•62502
•62524
•78134
.78116
.78098
•78079
.78061
•78043
•63765
•63787
.63810
•63832
•63854
•63877
•77033
.77014
.76996
.76977
•76959
.76940
•58378
.58401
•58425
.58449
.58472
.58496
•81191
.81174
.81157
.81140
.81123
.81106
•59786
•59809
59832
•59856
•59879
•59902
•80160
•80143
•80125
•80108
•80091
■80073
.61176
.61199
.61222
•61245
61268
.61291
•79105
.79087
.79069
•79051
•79033
•79016
•62547
•62570
•62592
•62615
•62638
•62660
.78025
.78007
•77988
•77970
•77952
•77934
•63899
.63922
.63944
.63966
.63989
.64011
.76921
.76903
.76884
. 76866
.76847
•76828
15
14
13
12
11
10
\
5
4
3
2
I
49
50
51
52
53
54
•58519
•58637
.81089
.81072
•81055
.81038
.81021
.81004
•59926
•59949
•59972
•59995
•60019
.60042
.80056
.80038
.80021
•80003
.79986
•79968
•79951
•79934
.79916
•79899
•79881
•79864
.61314
.61383
.61406
.61429
•61451
.61474
•61497
•61520
•61543
.61566
•78998
•78980
.78962
.78944
.78926
.78908
•62683
.62706
.62728
•62751
•62774
•62796
.77916
•77897
•77879
.77861
•77843
.77824
•64033
•64056
•64078
.64100
•64123
•64145
•76810
•76791
•76772
•76754
•76735
.76717
.76698
.76679
•76661
•76642
•76623
.76604
•58661
.58684
.58708
58731
•58755
58779
.80987
.80970
•80953
•80936
•80919
•80902
•60065
•60089
.60112
•60135
•60158
•60182
.78891
•78873
•78855
•78837
.78819
.78801
•62819
.62842
•62864
•62887
•62909
•62932
.77806
.77788
.77769
•77751
■nnz
•77715
•64167
.64190
.64212
.64234
.64256
.64279
N. COS.
N. sine
N. COS. ]
V. sine
N. COS.
N^. sine
N. COS. ]
V. sme
N^. COS. ]
V. sine
54" 1
53° 1
52° 1
51° 1
50° 1
80
TABLE III
o
I
2
3
4
1
9
lO
II
12
33
\l
18
40- 1
41** 1
42° 1
«» 1
44°
60
It
%
55
54
53
52
51
50
It
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
N. sine
N. COS.
.64279
.64301
64323
64346
.64368
.64390
.64412
.76604
.76586
.76567
.76548
•76530
.76511
.76492
.65606
.65628
•65650
.65672
•65694
.65716
•65738
•75471
•75452
•75433
•75414
•75395
•75375
•75356
66913
•66935
.66956
.66978
.66999
.67021
•67043
.67064
.67086
.67107
.67129
.67151
.67172
•74314
•74295
•74276
•74256
•74237
•74217
.74198
.74178
•74159
•74139
.74120
.74100
.74080
.68200
.68221
.68242
.68264
.68285
.68306
•68327
•73135
.73116
.73096
■73076
•73056
•73036
.73016
.69466
•69487
.69508
•69529
•69549
.69570
.69591
•71934
.71914
.71894
■71873
•71853
•71833
.71813
•64435
•64457
•64479
.64501
.64524
.64546
•76473
■76455
.76436
.76417
.76398
.76380
•65759
•65803
.65825
.65847
.65869
■7S337
•75318
•75299
.75280
•75261
•75241
•68349
.68370
•68391
.68412
•68434
•68455
.68476"
•68497
.68518
•68539
.68561
.68582
.72996
.72976
•72957
.72937
.72917
.72897
.69612
.69633
.69654
•69675
.69696
.69717
.71792
.71772
•71752
.71732
.71711
.71691
.64568
.64590
.64612
■64635
.64657
.64679
•76361
•76342
•76323
.76304
.76286
.76267
.65891
•65913
•65956
.65978
.66000
.75222
•75203
•75184
•75165
•75146
.75126
.67194
.67215
•67237
.67258
.67280
.67301
.74061
.74041
.74022
.74002
•73983
•73963
.72877
.72857
•72837
.72817
•72797
■72777
•69737
•69758
.69779
.69800
.69821
.69842
.71671
.71650
.71630
.71610
•71590
•71569
%
45
44
43
42
19
20
21
22
23
24
11
11
29
3c
31
32
33
34
35
36
11
39
40
41
42
43
44
:i
47
48
49
50
51
52
53
54
55
56
11
.64701
.64723
.64746
.64768
.64790
.64812
.76248
.76229
.76210
.76192
.76173
•76154
•76135
.76116
.76097
.76078
•76059
.76041
.76022
.76003
•75984
•75965
•75946
•75927
.75908
.75889
•75870
.75851
•75832
•75813
.66022
.66044
.66066
.66088
.66109
.66131
.75069
•75050
•75030
.75011
•67323
67344
•67366
.67409
•67430
•73944
•73924
•73904
■73885
•73865
.73846
.68603
.68624
.68645
.68666
.68688
.68709
■72757
■72737
•72717
.72697
.72677
.72657
.69862
•69883
.69904
.69946
.69966
•71549
•71529
.71508
.71488
.71468
•71447
41
40
35
34
33
32
31
30
29
28
%
25
24
23
22
21
20
19
\l
15
14
13
12
II
10
i
7
6
.64834
64856
64878
.64901
64923
64945
64967
.64989
.65011
•65033
■65055
■65077
.65100
.65122
.65144
.65166
.65188
.65210
66175
.66197
.66218
.66240
.66262
•74992
•74973
•74953
•74934
•74896
.67452
■67473
■67538
•67559
.73826
•73806
•73787
•73767
•73747
•73728
•68730
.68751
.68772
68793
.68814
■68835,
.72637
.72617
•72597
•72577
•72557
•72537
•69987
.70008
.70029
.70049
.70070
.70091
.71427
.71407
.71386
.71366
■71345
■71325
.66284
.66306
•66327
•66349
•66393
.66414
.66436
.66458
.66480
.66501
•66523
.74876
•74857
■.Ifsfs
.67580
.67602
.67623
.67688
.73708
•73688
73669
•73649
.73629
.73610
.68857
.68878
.68899
.68920
.68941
.68962
•72517
.72497
•72477
•72457
•72437
.72417
.70112
.70132
•70153
.70174
•70195
•70215
•71305
.71284
.71264
•71243
.71223
.71203
.74760
•74741
.74722
•74703
■74683
.74664
.67709
.67730
.67752
•67773
•67795
.67816
•73590
•73570
•73551
•73531
•735"
•73491
.68983
.69004
.69025
.69046
.69067
.69088
•72397
■72377
■72357
■72337
.72317
.72297
.70236
•70257
.70277
.70298
.70319
•70339
.71182
.71162
.71141
.71121
.71100
.71080
.65232
•65254
•65276
.65298
•65320
•65342
•75794
•75775
•75756
•75738
•75719
■75700
"66566
66588
.66610
.66632
•66653
.74644
.74606
•74586
•74567
•74548
•67837
.67901
•67923
.67944
•73472
•73452
•73432
•73413
•73393
■73373
.69109
.69130
.69151
.69172
.69193
.69214
.72277
•72257
.72236
.72216
.72196
.72176
.70360
.70381
.70401
.70422
•70443
•70463
•71059
.71039
.71019
.70998
•70978
•70957
•65364
.65386
.65408
•65430
•65452
•65474
.75680
.75661
•75642
•75623
.75604
•75585
.66675
.66697
.66718
.66740
.66762
•66783
.74528
•74509
•74489
•74470
•74451
•74431
.67965
.68029
.68051
.68072
•73353
•73333
•73314
•73294
•73274
•73254
•69235
.69256
.69277
.69298
.69319
.69340
.72156
•72136
.72116
.72095
•72075
•72055
.70484
•70505
•70525
.70546
•70567
•70587
.70937
.70916
.70896
.70875
•70855
.70834
•65496
.65518
•65540
65562
.65584
.65606
•75566
•75547
•75528
•75509
•75490
•75471
.66805
.66827
.66848
.66870
.66891
•66913
.74412
•74392
•74373
•74353
•74334
•74314
.68093
.68115
.68136
.68157
.68179
.68200
•73234
•73215
•73195
•73175
•73155
•73135
•69361
.69382
.69403
.69424
•72035
•72015
•71995
•71974
•71954
71934
.70608
.70628
.70649
.70670
.70690
.70711
.70813
•70793
.70772
•70752
•70731
.70711
5
4
3
2
I
t
N. COS. ]
NT. sine
f^. COS. ]
N". sine
NT. COS. ]
N". sine
^J. CO.S. ^
ST. sine
N'. cos. ]
V. sine
49° 1
4§*' 1
47° 1
46° 1
45- 1
NATURAL TANGENTS AND COTANGENTS
81
/
0^^
1°
2^"
3°
40
/
Tang Cotg
Tang Cotg
Tang Cotg
Tang Cotg
Tang Cotg
I
2
3
4
0000 Infinite
0003 3437-75
0006 1718.87
0009 1145.92
0012 859.436
0175 57.2900
0177 56.3506
0180 55.4415
0183 54.5613
0186 53.7086
0349 28.6363
0352 28.3994
0355 28.1664
0358 27.9372
0361 27.7117
0524 19.0811
0527 18.9755
0530 18.8711
0533 18.7678
0536 18.6656
0699 14.3007
0702 14.2411
070^ 14.1821
0708 14.1235
0711 14.0655
60
5
6
7
8
9
0015 687.549
0017 572957
0020 491.106
0023 429.718
0026 381.971
0189 52.8821
0192 52.0807
0195 51.3032
0198 50.5485
0201 49.8157
0364 27.4899
0367 27.2715
0370 27.0566
0373 26.8450
0375 26.6367
0539 18.5645
0542 18.4645
0544 18.3655
0547 18.2677
0550 18.1708
0714 14.0079
0717 13-9507
0720 13.8940
0723 138378
0726 13.7821
55
54
53
52
51
10
II
12
13
14
0029 343-774
0032 312.521
0035 286.478
0038 264.441
0041 245.552
0204 49.1039
0207 48.4121
0209 47.7395
0212 47.0853
0215 46.4489
0378 26.4316
0381 26.2296
0384 26.0307
0387 25.8348
0390 25.6418
0553 18.0750
0556 17.9802
OS59 17.8863
0562 17.7934
^565 17-7015
0729 13.7267
0731 13.6719
0734 13-6174
0737 13-5634
0740 13.5098
50
49
48
47
46
17
i8
19
0044 229.152
0047 214.858
0049 202,219
0052 190.984
0055 180.932
0218 45.8294
0221 45.2261
0224 44.6386
0227 44.0661
0230 43.5081
0393 25.4517
0396 25.2644
0399 25.0798
0402 24.8978
0405 24.7185
0568 17.6106
0571 17.5205
0574 1 7.43 H
0577 17.3432
0580 17.2558
0743 13.4566
0746 13.4039
0749 13.3515
0752 13.2996
0755 13.2480
45
44
43
42
41
20
21
22
23
24
25
26
27
28
29
0058 171.885
0061 163.700
0064 156.259
0067 149.465
0070 143.237
0233 42.9641
0236 42.4335
0239 41-9158
0241 41.4106
0244 40.9174
0407 24.5418
0410 23.3675
0413 24.1957
0416 24.0263
0419 23.8593
0582 17.1693
0585 17.0837
0588 16.9990
0591 16.9150
0594 16.8319
0758 13.1969
0761 13.1461
0764 13.0958
0767 13.0458
0769 12.9962
40
39
38
31
36
0073 137-507
0076 132.219
0079 127.321
0081 122.774
0084 118.540
0247 40.4358
0250 39.9655
0253 39.5059
0256 39.0568
0259 38.6177
0422 23.6945
0425 23.5321
0428 23.3718
0431 23.2137
0434 23.0577
0597 16.7496
0600 16.6681
0603 16.5874
0606 16.5075
0609 16.4283
0772 12.9469
0775 12.8981
0778 12.8496
0781 12.8014
0784 12.7536
35
34
32,
32
31
30
31
32
34
0087 114-589
0090 110.892
0093 107.426
0096 104.171
0099 101.107
0262 38.1885
0265 37.7686
0268 37-3579
0271 36.9560
0274 36.5627
0437 22.9038
0440 22.7519
0442 22.6020
0445 22.4541
0448 22.3081
0612 16.3499
0615 16.2722
0617 16.1952
0620 16.1190
0623 16.0435
0787 12.7062
0790 12.6591
0793 12.6124
0796 12.5660
0799 12.5199
30
29
28
27
26
35
36
39
0102 98.2179
0105 95.4895
0108 92.9085
oiii 904633
01 13 88.1436
0276 36.1776
0279 35.8006
0282 35-4313
0285 35-0695
0288 34.7151
0451 22.1640
0454 22.0217
0457 21.8813
0460 21.7426
0463 21.6056
0626 15.9687
0629 15.8945
0632 15.8211
0635 15.7483
0638 15.6762
0802 12.4742
0805 12.4288
0808 12.3838
0810 12.3390
0813 12.2946
25
24
23
22
21
40
41
42
43
44
01 16 85.9398
01 19 83.8435
0122 81.8470
0125 79.9434
0128 78.1263
0291 34.3678
0294 34.0273
0297 336935
0300 33.3662
0303 33-0452
0466 21.4704
0469 21.3369
0472 21.2049
0475 21.0747
0477 20.9460
0641 15.6048
0644 15-5340
0647 15.4638
0650 15.3943
0653 15.3254
0816 12.2505
0819 12.2067
0822 12.1632
0825 12.1201
0828 12.0772
20
19
18
17
i6
45
46
47
48
49
0131 76.3900
0134 74.7292
0137 73.1390
0140 71.6151
0143 70.1533
0306 32.7303
0308 32.4213
0311 32.1181
0314 31.8205
0317 31.5284
0480 20.8188
0483 20.6932
0486 20.5691
0489 20.4465
0492 20.3253
0655 15.2571
0658 15.1893
0661 15.1222
0664 15.0557
0667 14.9898
0831 12.0346
0834 11.9923
0837 11.9504
0840 11.9087
0843 11.8673
15
14
13
12
11
50
51
52
53
54
59
0146 68.7501
0148 67.4019
0151 66.1055
0154 64.8580
0157 63.6567
0320 31.2416
0323 30.9599
0326 30.6833
0329 30.4116
0332 30.1446
0495 20.2056
0498 20.0872
0501 19.9702
0504 19.8546
0507 19.7403
0670 14.9244
0673 14-8596
0676 14.7954
0679 14-7317
0682 14.6685
0846 11.8262
0849 11.7853
0851 11.7448
0854 11.7045
0857 11.6645
10
I
7
6
5
4
3
2
I
0160 62.4992
0163 61,3829
0166 60.3058
0169 59.2659
0172 58.2612
0335 29.8823
0338 29.6245
0340 29.3711
0343 29.1220
0346 28.8771
0509 19.6273
OCil2 19.5156
0515 19.4051
0518 19.2959
0521 19.1879
0685 14.6059
0688 14.5438
0690 14.4823
0693 14.4212
0696 14.3607
0860 11.6248
0863 11.5853
0866 11.5461
0869 11.5072
0872 11.4685
60
0175 57.2900
0349 28.6363
0524 19.0811
0699 14.3007
0875 11.4301
(.otg Tang
Cotg Tang
Cotg Tang
Cotg Tang
Cotg Tang
/
8^0
8H0
H7^ i H(i^
85^
/
82
TABLE III
/
50
0°
7"
8'-^
90
/
Tang Cotg
Tang
totg
Tang Cotg
Tang Cotg
Tang
Cotg
60
0875 1 1. 4301
1051
9.5144
1228 8.1443
1405 7.1154
1584
6.3138
I
0878 11.3919
1054
9-4878
1231 8.1248
1408 7.1004
1587
6.3019
59
2
0881 11.3540
1057
9.4614
1234 8.1054
141 1 7.0855
1590
6.2901
58
^
0884 1 1. 3 1 63
1060
94352
1237 8.0860
1414 7.0706
1593
6.2783
57
4
0887 11.2789
1063
9.4090
1240 8.0667
1417 7-0558
1596
6.2666
56
s
0890 1 1.24 1 7
1066
9-3831
1243 8.0476
1420 7.0410
1599
6.2549
55
6
0892 11.2048
1069
9-3572
1246 8.0285
1423 7.0264
1602
6.2432
54
7
0895 1 1. 1 681
1072
9-3315
1249 8.0095
1426 7.0117
1605
6.2316
53
8
0898 11.1316
1075
9.3060
1 25 1 7.9906
1429 6.9972
1608
6.2200
52
9
0901 11.0954
1078
9.2806
1254 7.9718
1432 6.9827
1611
6.2085
51
10
0904 11.0594
1080
9-2553
1257 7.9530
1435 6.9682
1614
6.1970
50
II
0907 11.0237
1083
9.2302
1260 7.9344
1438 6.9538
1617
6.1856
49
12
9910 10.9882
1086
9.2052
1263 7.9158
1441 6.9395
1620
6.1742
48
1,3
0913 10.9529
1089
9.1803
1266 7.8973
1444 6.9252
1623
6.1628
47
14
0916 10.9178
1092
9-1555
1269 7.8789
1447 6.91 10
1626
6.1515
46
IS
0919 10.8829
1095
9.1309
1272 7.8606
1450 6.8969
1629
6.1402
45
i6
0922 10.8483
1098
9.1065
1275 7.8424
1453 6.8828
1632
6.1290
44
17
0925 10.8139
IIOI
9.0821
1278 7.8243
1456 6.8687
1635
6.1178
43
18
0928 10.7797
1104
9-0579
1281 7.8062
1459 6.8548
1638
6.1066
42
19
0931 10.7457
1 107
9-0338
1284 7.7883
1462 6.8408
1641
6-0955
41
20
0934 10.71 19
mo
9.0098
1287 7.7704
1465 6.8269
1644
6.0844
40
21
0936 10.6783
1113
8.9860
1290 7.7525
1468 6.8131
1647
6.0734
39
22
0939 10.6450
1116
8.9623
1293 7.7348
1471 6.7994
1650
6.0624
38
23
0942 10.61 18
1119
8.9387
1296 7.7171
1474 6.7856
1653
6.0514
37
24
0945 10.5789
1122
8.9152
1299 7.6996
1477 6.7720
1655
6.0405
36
2S
0948 10.5462
1125
8.8919
1302 7.6821
1480 6.7584
1658
6.0296
35
26
0951 10.5136
1128
8.8686
1305 7.6647
1483 6.7448
1661
6.0188
34
27
0954 10.4813
1131
8.8455
1308 7.6473
i486 6.7313
1664
6.0080
33
28
0957 10.4491
1 134
8.8225
1 31 1 7.6301
1489 6.7179
1667
5-9972
32
29
0960 10.4172
1 136
8.7996
1314 7.6129
1492 6.7045
1670
5.9865
31
30
0963 10.3854
1 139
8.7769
1317 7-5958
1495 6.0912
Tell
5-9758
30
31
0966 10.3538
1 142
8.7542
1319 7.5787
1497 6.6779
5-9651
29
32
0969 10.3224
1 145
8.7317
1322 7.5618
1500 6.6646
1 679
5.9545
28
33
0972 10,2913
1148
8.7093
1325 7.5449
1503 6.6514
1682
5.9439
27
34
0975 10.2602
1151
8.6870
1328 7.5281
1506 6.6383
1685
1688
5-9333
5.9228
26
25
3S
0978 10.2294
1154
8.6648
1331 7.5113
1509 6.6252
36
0981 10.1988
1157
8.6427
1334 7.4947
1512 6.6122
1691
5.9124
24
37
0983 10.1683
1 160
8.6208
1337 7.4781
1515 6.5992
1694
5-9019
23
38
0986 10.1381
1163
8.5989
1340 7.4615
15 18 6.5863
1697
5.8915
22
39
0989 10.1080
1166
8-5772
1343 7.4451
1521 6.5734
1700
5.8811
21
40
0992 10.0780
1 169
8.5555
1346 7.4287
1524 6.5606
1703
5.8708
20
41
0995 10.0483
1172
8.5340
1349 7.4124
1527 6.5478
1706
5.8605
19
42
0998 10.0187
1 175
8.5126
1352 7.3962
1530 6.5350
1709
5.8502
18
43
looi 9.9893
1178
8.4913
1355 7.3800
1533 6.5223
1712
5.8400
17
44
1004 9.9601
8.4701
1358 7.3639
1536 6.5097
1715
6.8298
16
15
45
1007 9.9310
1 184
8.4490
1361 7.3479
1539 6.4971
1718
58197
46
loio 9.9021
1187
8.4280
1364 7.3319
1542 6.4846
1721
5-8095
14
47
IOI3 9-8734
1 189
8.4071
1367 7.3160
1545 6.4721
1724
5-7994
13
48
1016 9.8448
1192
8.3863
1370 7.3002
1548 6.4596
1727
5-7894
12
49
IOI9 9.8164
"95
8.3656
1373 7.2844
1551 6.4472
1730
5-7794
II
50
1022 9.7882
1 198
8.3450
1376 7.2687
1554 6.4348
1733
5-7694
10
SI
1025 9.7601
1 201
8.3245
1379 7.2531
1557 6.4225
1736
5-7594
9
S2
1028 9-7322
1204
8.3041
1382 7.2375
1560 6.4103
1739
5-7495
8
S3
1030 9.7044
1207
8.2838
1385 7.2220
1563 6.3980
1742
5.7396
7
54
1033 9-6768
1210
8.2636
1388 7.2066
1566 6.3859
1745
5.7297
6
5S
1036 9-6499
1213
8.2434
1391 7.1912
1569 6.3737
1748
5.7199
5
5^
1039 9.6220
1216
8.2234
1394 7.1759
1572 6.3617
1751
5.7101
4
S7
1042 9-5949
1219
8.2035
1397 7.1607
1575 6.3496
1754
5.7004
3
5«
1045 9.5679
1222
8.1837
1399 7.1455
1578 6.3376
1757
5.6906
2
59
1048 9-541 1
1225
8.1640
1402 7.1304
1581 6.3257
1760
5.6809
I
00
1051 9-5144
1228
8.1443
1405 7.1 154
1584 6.3138
Cotg Tang
1763
Cotg
5-6713
Tang
Cotg Tang
Cotg
Tang
Cotg Tang
L.
84°
83°
82^
81"
80"
/
NATURAL TANGENTS AND COTANGENTS
83
/
10^ 1
11^
12°
13°
14°
/
Taug
Cotg
Tang
Cotg
Tang
Cotg
Tang
Cotg
Tang
Cotg
60
^I^Z
5-6713
1944
5.1446
2126
4.7046
2309
4-3315
2493
4.0108
I
1766
5.6617
1947
5.1366
2129
4-6979
2312
4-3257
2496
4.0058
5?
2
1769
5-6521
1950
5.1286
2132
4.6912
2315
4.3200
2499
4.0009
58
^
1772
5-6425
1953
5.1207
2135
4.6845
2318
4-3143
2503
3-9959
57
4
1775
5-6330
1956
5.1128
2138
4.6779
2321
4.3086
2506
3.9910
5^
S
1778
5-6234
1959
5-I049
2141
4.6712
2324
4-3029
2509
3-9861
55
6
I78I
5.6140
1962
5-0970
2144
4.6646
2327
4-2972
2512
3-9812
54
7
1784
5.6045
1965
5.0892
2147
4.6580
2330
4.2916
2515
3-9763
53
8
1787
5-5951
1968
5.0814
2150
4.6514
233.3
4-2859
2518
3-9714
52
9
10
1790
5-5857
1971
5-0736
2153
4.6448
2336
4.2803
2521
3.9665
51
1793
5-5764
1974
5.0658
2156
4.6382
2339
4.2747
2524
3-9617
50
II
1796
5-5671
1977
5-0581
2159
4-6317
2342
4.2691
2527
3-9568
49
12
1799
5-5578
1980
5-0504
2162
4.6252
2345
4-2635
2530
3.9520
48
i,S
1802
5-5485
1983
5.0427
2165
4.6187
2349
4.2580
2533
3-9471
47
14
1805
5-5393
1986
5-0350
2168
4.6122
2352
4.2524
2537
3-9423
46
IS
1808
5-5301
1989
5-0273
2171
4-6057
2355
4.2468
2540
3-9375
45
i6
I8II
5-5209
1992
50197
2174
4.5993
2358
4.2413
2543
3-9327
44
17
I8I4
5.5118
1995
5.0121
2177
4-5928
2361
4.2358
2546
3-9279
43
18
I8I7
5.5026
1998
5-0045
2180
4-5864
2364
4.2303
2549
3-9232
42
19
1820
5-4936
2001
4.9969
2183
4.5800
2367
4.2248
2552
3.9184
41
20
1823
5-4845
2004
4.9894
2186
4-5736
2370
4.2193
2555
3-9136
40
21
1826
5-4755
2007
4.9819
2189
4.5673
2373
4.2139
2558
3.9089
39
22
1829
5-4665
2010
4.9744
2193
4.5609
2376
4.2084
2561
3.9042
38
23
1832
5-4575
2013
4.9669
2196
4-5546
2379
4.2030
2564
3-8995
31
24
i«35
5-4486
2016
4-9594
2199
4.5483
2382
4.1976
2568
3-8947
36
2S
1838
5-4397
2019
4.9520
2202
4.5420
2385
4.1922
2571
3.8900
35
26
1841
5-4308
2022
4-9446
2205
4.5357
2388
4.1868
2574
3-8854
34
27
1844
5-4219
2025
4-9372
2208
4.5294
2392
4.1814
2577
3-8807
33
28
1847
5-4131
2028
4.9298
221 1
4.5232
2395
4.1760
2580
3-8760
32
29
1850
5-4043
2031
4.9225
2214
4.5169
2398
4.1706
2583
3-8714
31
30
i8S3
5-3955,
2035
4.9152
2217
4-5107
2401
4.1653
2586
38667
30
31
1856
5.3868
2038
4.9078
2220
4.5045
2404
4.1600
2589
3.8621
29
32
i«59
5-3781
2041
4.9006
2223
4.4983
2407
4-1547
2592
3-8575
28
33
1862
5-3694
2044
4-8933
2226
4.4922
2410
4.1493
2.595
3-8528
27
34
1865
5-3607
2047
4.8860
2229
4.4860
2413
4.1441
2599
3.8482
26
35
1868
5-3521
2050
4.8788
2232
4.4799
2416
4.1388
2602
3-8436
25
3^
1871
5-3435
2053
4.8716
2235
4.4737
2419
4.1335
2605
3-8391
24
37
1874
5-3349
2056
4.8644
2238
4.4676
2422
4.1282
2608
3-8345
23
38
1877
5-3263
2059
4.8573
2241
4-4615
2425
4.1230
261 1
3-8299
22
39
1880
5-3178
2062
4-8501
2244
4-4555
2428
4.1178
2614
3-8254
21
40
1883
5-3093
2065
4-8430
2247
4.4494
2432
4.1126
2617
3.8208
20
41
1887
5-3008
2068
4-8359
2251
4.4434
2435
4.1074
2620
3-8163
19
42
1890
5-2924
2071
4.8288
2254
4-4374
2438
4.1022
2623
3.81 18
18
43
1893
5-2839
2074
4.8218
2257
4.4313
2441
4.0970
2627
3-8073
17
44
45
1896
1899
5-2755
2077
4.8147
2260
4-4253
2444
4.0918
2630
3.8028
16
5.2672
2080
4-8077
2263
4.4194
2447
4.0867
2633
3-7983
15
4b
1902
5.2588
2083
4.8007
2266
4-4134
2450
4-0815
2636
3-7938
14
47
1905
5-2505
2086
4-7937
2269
4.4075
2453
4.0764
2639
3-7893
13
48
1908
5-2422
2089
4.7867
2272
4.4015
2456
4.0713
2642
3-7848
12
49
1911
5-2339
2092
4.7798
2275
4.3956
2459
4.0662
2645
3.7804
II
50
1914
5-2257
2095
4-7729
2278
4.3897
2462
4.061 1
2648
3.7760
10
51
1917
5-2174
2098
4-7659
2281
4.3838
2465
4.0560
2651
3-7715
9
52
1920
5.2092
2101
4-7591
2284
4.3779
2469
4.0509
2655
3-7671
8
53
1923
5.2C1 1
2104
4.7522
2287
4-3721
2472
4.0459
2658
3.7627
7
54
55
1926
5.1929
2107
4-7453
2290
4.3662
2475
4.0408
2661
3-7583
b
1929
5.1848
2110
4.7385
2293
4.3604
2478
4.0358
2664
3-7539
'5
5<^
1932
^•^lll
2113
4-7317
2296
4-3546
2481
4.0308
2667
3-7495
4
57
1935
5.1686
2116
4.7249
2299
4-3488
2484
4.0257
2670
3-7451
3
5«
i93«
5.1606
2119
4.7181
2303
4.3430
2487
4.0207
2673
3.7408
2
59
1 941
5-1526
2123
4.7114
2306
4.3372
2490
4.0158
2676
3.7364
I
00
1944
5.1446
2126
4.7046
2309
4.3315
2493
4.0108
2679
3-7321
Cotg
Tan^
Cotg
Tang
Cotg
Tang
Cotg
Tang
Cotg
Tang
/
7
0°
78°
77 '
7
6^
7
5°
f
84
TABLE III
/
15°
1(>°
17°
18^
193
/
ian^
Cot^
Tang
Cotg
Tang
Cotg
Tang
Cotg
Tang
Cotg
2679
3.7321
2867
3-4874
3057
3-2709
3249
3.0777
3443
2.9042
60
I
2683
3-7277
2871
34836
3060
3-2675
3252
3.0746
3447
2.9015
S9
2
2686
3-7234
2874
3-4798
3064
3.2641
3256
3.0716
3450
2.8987
S8
3
2689
3-7191
2877
3.4760
3067
3.2607
3259
3.0686
3453
2.8960
S7
4
2692
3-7148
2880
3.4722
3070
3.2573
3262
3.0655
3456
2-8933
'^6
5
2695
3-7105
2883
3-4684
.3073
32539
3265
3.0625
3460
2.8905
SS
6
2698
3.7062
2886
3-4646
3076
3.2506
3269
3.0595
3463
2.8878
S4
7
2701
3-7019
2890
3.4608
3080
3-2472
3272
3.0565
3466
2.8851
S3
8
2704
3.6976
2893
34570
.3083
3.2438
3275
3.0535
3469
2.8824
S2
9
2708
3-6933
2896
3-4533
3086
3-2405
3278
3-0505
3473
2.8797
51
50
10
2711
3.6891
2899
3-4495
3089
3-2371
3281
3.0475
.3476
2.8770
II
2714
3.6848
2902
3-4458
3092
3-2338
3285
3.0445
3479
2.8743
49
12
2717
3.6806
2905
3.4420
3096
3.2305
3288
3.0415
.3482
2.8716
48
1.3
2720
3.6764
2908
3-4383
3099
3.2272
3291
30385
.3486
2.8689
47
14
2723
3.6722
2912
3-4346
3102
3-2238
3294
3-0356
3489
2.8662
46
15
2726
3.6680
2915
3-4308
3105
3.2205
3298
3.0326
3492
2.8636
4S
lb
2729
3.6638
2918
3-4271
3108
3.2172
3301
3.0296
3495
2.8609
44
17
2733
3.6596
2921
3-4234
3111
3.2139
3304
3.0267
3499
2.8582
43
18
2736
3-6554
2924
3-4197
3"5
3.2106
3307
3.0237
3502
2.8556
42
19
2739
3.6512
2927
3.4160
3118
3.2073
3310
3.0208
3505
2.8529
41
20
2742
3.6470
2931
3.4124
3121
3.2041
3314
3.0178
3So8
2.8502
40
21
2745
3.6429
2934
3.4087
3124
3.2008
3317
3.0149
3512
2.8476
39
22
2748
3.6387
2937
3.4050
3127
3.1975
3320
3.0120
3515
2.8449
38
23
2751
3-6346
2940
3-4014
3131
3.1943
3323
3.0090
3518
2.8423
37
24
2754
3-6305
2943
3-3977
3134
3.1910
3327
3.0061
3522
2.8397
36
25
2758
3.6264
2946
3-3941
3137
3.1878
3330
3.0032
.3525
2.8370
3S
26
2761
3.6222
2949
3-3904
3140
3.1845
3333
3.0003
3528
2.8344
34
27
2764
3.6181
29S3
3-3868
3143
3-1813
3336
2.9974
3S3I
2.8318
33
28
2767
3.6140
2956
3.3832
3147
3.1780
3339
2-9945
3535
2.8291
32
29
2770
3.6100
2959
3-3796
3150
3-1748
3343
2.9916
3538
2.8265
31
30
2773
3-6059
2962
3.3759
3153
3.1716
3346
2.9887
3541
2.8239
30
31
2776
3.6018
2965
3.3723
3156
3.1684
3349
2.9858
3S44
2.8213
29
32
2780
3-5978
2968
3.3687
3I.S9
3-1652
3352
2.9829
3548
2.8187
28
^^
2783
3-5937
2972
3.3652
3163
3.1620
3.356
2.9800
3551
2.8161
27
34
2786
3.5897
2975
3-3616
3166
3-1588
3359
2.9772
3554
2.8135
26
35
2789
3-5856
2978
3-3580
3169
3-1556
3362
2.9743
3558
2.8109
25
3b
2792
3.5816
2981
3.3544
3172
3.1524
3365
2.9714
3561
2.8083
24
37
2795
3-5776
2984
3.3509
3175
3.1492
3369
2.9686
3564
2.8057
23
38
2798
3-5736
2987
3-3473
3179
3.1460
.3372
2.9657
3567
2.8032
22
39
40
2801
3.5696
2991
3.3438
3182
3.1429
3375
2.9629
3571
2.8006
21
2805
3.5656
2994
3.3402
318s
3.1397
3378
2.9600
3574
2.7980
20
41
2808
3-5616
2997
3.3367
3188
3.1366
3382
2.9572
3577
2.7955
19
42
2811
3-5576
3000
3-3332
3191
3.1334
3385
2.9544
3581
2.7929
18
43
2814
3-5536
3003
3.3297
3195
3.1303
3388
2.9515
3584
2.7903
17
44
2817
3-5497
3006
3.3261
3198
3.1271
3391
2.9487
3587
2.7878
16
4S
2820
3-5457
3010
3.3226
3201
3.1240
3395
2.9459
3590
2.7852
15
46
2823
3.5418
3013
3.3191
3204
3.1209
3398
2.9431
3594
2.7827
14
47
2827
3-5379
3016
3.3156
3207
3.1178
3401
2.9403
3597
2.7801
13
48
2830
3-5339
3019
3.3122
3211
3-1146
3404
2.9375
3600
2.7776
1 2
49
2833
3.5300
3022
3.3087
3214
3-i"5
3408
2.9347
3604
2.7751
II
50
2836
3.5261
3026
3-3052
3217
3.1084
3411
2.9319
3607
2.7725
10
SI
2839
3.5222
3029
3-3017
3220
3.1053
3414
2.9291
3610
2.7700
9
S2
2842
3.5183
3032
3-2983
3223
3.1022
3417
2.9263
3613
2-7675
8
S3
2845
3-5144
.S03S
3.2948
3227
3.0991
3421
2-9235
3617
2.7650
7
54
2849
3-5105
3038
3-2914
3230
3.0961
3424
2.9208
3620
2.7625
b
ss
2852
3-5067
3041
3.2880
3233
3-0930
3427
2.9180
3623
2.7600
5
S6
2855
3-5028
3045
3-2845
3236
3.0899
3430
2.9152
3627
2-7575
4
S7
28s8
3.4989
3048
3.281 1
3240
3.0868
3434
2.9125
3630
2.7550
3
S8
2861
3-4951
.SOS I
3-2777
3243
3-0838
3437
2.9097
3633
2.7525
2
59
2864
3.4912
3054
3-2743
3246
3.0807
3440
2.9070
3636
2.7500
I
GO
2867
3.4874
3057
3.2709
3249
3.0777
3443
2.9042
3640
2.7475
Cotg
Tang
Cotg
Tang
Cotg
Tang
Cotg
Tang
Cotg
Tang
/
74° 1
73^ 1
7
2°
71°
7
0° 1 / 1
NATURAL TANGENTS AND COTANGENTS
85
/
20°
21°
2
52°
23°
24°
/
Tang
Cotg
Tang
Cotg
Tang
Cotg
Tang
Cotg
Tang
Cotg
3640
2.7475
3839
2.6051
4040
2.4751
4245
2.3559
4452
2.2460
60
I
3643
2.7450
3842
2.6028
4044
2.4730
4248
2.3539
4456
2.2443
59
2
3646
2.7425
3845
2.6006
4047
2.4709
4252
2.3520
4459
2.2425
58
3
3650
2.7400
3849
2.5983
4050
2.4689
4255
2.3501
4463
2.240S
57
4
S
3653
2.7376
3852
2.5961
4054
2.4668
4258
2.3483
4466
2.2390
56
55
3656
2.7351
3855
2.5938
4057
2.4648
4262
2.3464
4470
2.2373
6
3659
2.7326
3859
2.5916
4061
2.4627
4265
2.3445
4473
2.2355
54
7
3663
2.7302
3862
2.5893
4064
2.4606
4269
2.3426
4477
2.2338
53
8
3666
2.7277
.3865
2.5871
4067
2.4586
4272
2.3407
4480
2.2320
52
9
3669
2.7253
3869
2.5848
4071
2.4566
4276
2:3388
4484
2.2303
51
10
3673
2.7228
.S872
2.5826
4074
2.4545
4279
2.3369
4487
2.2286
50
II
367b
2.7204
.S875
2.5804
4078
2.4525
4283
2.3351
4491
2.2268
49
12
3679
2.7179
3879
2.5782
4081
2.4504
4286
2.3332
4494
2.2251
48
1.3
3(>^3
2.7155
3882
2-5759
4084
2.4484
4289
2.3313
4498
2.2234
47
14
3b86
2.7130
3885
2.5737
4088
2.4464
4293
2.3294
4501
2.2216
46
IS
3689
2.7106
3889
2-5715
4091
2.4443
4296
2.3276
4505
2.2199
45
I6
3693
2.7082
3892
2.5693
4095
2.4423
4300
2.3257
4508
2.2182
44
17
3696
2.7058
3895
2.5671
4098
2.4403
4303
2.3238
4512
2.2165
43
18
3699
2.7034
3899
2.5649
4101
2.4383
4307
2.3220
4515
2.2148
42
19
3702
2.7009
3902
2.5627
4105
2.4362
4310
2.3201
4519
2,2130
41
20
3706
2.6985
3906
2.5605
4108
2.4342
4314
2.3183
4522
2.2113
40
21
3709
2.6961
3909
2.5533
4111
2.4322
4317
2.3164
4526
2.2096
39
22
3712
2.6937
3912
2.5561
4115
2.4302
4320
2.3146
4529
2.2079
38
23
37^^
2.6913
3916
2.5539
4118
2.4282
4324
2.3127
4533
2.2062
37
24
3719
2.6889
3919
2.5517
4122
2.4262
4327
2.3109
4536
2.2045
36
25
3722
2.6865
3922
2.5495
4125
2.4242
4331
2.3090
4540
2.2028
35
2b
3726
2.6841
3926
2.5473
4129
2.4222
4334
2.3072
4543
2.201 1
34
27
3729
2.6818
3929
2.5452
4132
2.4202
4338
2.3053
4547
2.1994
33
28
3732
2.6794
.3932
2.5430
413s
2.4182
4341
2.3035
4S50
2.1977
32
29
373^
2.6770
3936
2.5408
4139
2.4162
4345
2.3017
4554
2 i960
31
30
3739
2.6746
3939
2.5386
4142
2.4142
4.348
2.2998
4557
2.1943
30
31
3742
2.6723
3942
2.5365
4146
2.4122
4352
2.2980
4561
2.1926
29
32
3745
2.6699
3946
2.5343
4149
2.4102
4355
2.2962
4564
2.1909
28
33
3749
2.6675
3949
2.5322
4152
2.4083
4359
2.2944
4568
2.1892
27
34
3752
2.6652
3953
2.5300
4156
2.4063
4362
2.2925
4571
2.1876
26
35
3755
2.6628
3956
2.5279
4159
2.4043
4365
2.2907
4575
2.1859
25
3^
3759
2.6605
3959
2.5257
4163
2.4023
4369
2.2889
4578
2.1842
24
37
3762
2.6581
3963
2.5236
4166
2.4004
4372
2.2871
4582
2.1825
23
3«
3765
2.6558
3966
2.5214
4169
2.3984
4376
2.2853
4585
2.1808
22
39
3769
2.6534
3969
2.5193
4173
2.3964
4379
2.2835
4589
2.1792
21
40
3772
2.65 II
3973
2.5172
4176
2.3945
4383
2.2817
4592
2.1775
20
41
3775
2.6488
3976
2.5150
4180
2.3925
4386
2.2799
4596
2.1758
19
42
3779
2.6464
3979
2.5129
4183
2.3906
4390
2.2781
4599
2.1742
18
43
37^2
2.6441
3983
2.5108
4187
2.3886
4393
2.2763
4603
2.1725
17
44
37^5
2.6418
3986
2.5086
4190
2.3867
4397
2.2745
4607
2.1708
16
15
45
3789
2.6395
3990
2.5065
4193
2.3847
4400
2.2727
4610
2.1692
4b
3792
2.6371
3993
2.5044
4197
2.3828
4404
2.2709
4614
2.1675
14
47
3795
2.6348
3996
2.5023
4200
2.3808
4407
2.2691
4617
2.1659
13
48
3799
2.6325
4000
2.5002
4204
2.3789
441 1
2.2673
4621
2,1642
12
49
3802
2.6302
4003
2.4981
4207
2.3770
4414
2.2655
4624
2.1625
II
50
3805
2.6279
4006
2.4960
4210
2.3750
4417
2.2637
4628
2.1609
10
51
3809
2,6256
4010
2.4939
4214
2.3731
4421
2.2620
4631
2.1592
?
52
3812
2.6233
4013
2.4918
4217
2.3712
4424
2.2602
4635
2.1576
8
53
3«i5
2.6210
4017
2.4897
4221
2.3693
4428
2.2584
4638
2.1560
7
54
3«i9
2.6187
4020
2.4876
4224
2.3673
4431
2.2566
4642
2.1543
b
55
3822
2.6165
4023
2.4855
4228
2.3654
4435
2.2549
4645
2.1527
5
5t>
3«25
2.6142
4027
2.4834
4231
2.3635
4438
2.2531
4649
2.1510
4
57
3829
2.6119
4030
2.4813
4234
2.3616
4442
2.2513
4652
2.1494
3
5«
3«32
2.6096
4033
2.4792
4238
2.3597
4445
2.2496
4656
2.1478
2
59
3835
2.6074
4037
2.4772
4241
2.3578
4449
2.2478
4660
2.1461
I
60
3839
2.6051
4040
2.4751
4245
2.3559
4452
2.2460
4663
2.1445
Cotg
Tang
Cotg
Tang
Cotg
Tang
Cotg
Tang
Cotg
Tang
/
69°
68°
67°
66° 1
d
5°
/
86
TABLE TTT
/
25°
26°
270
28°
29°
/
2
3
4
Tang totg
Tang Cotg
Tang Cotg
Tang Cotg
Tang Cotg
4663 2.1445
4667 2.1429
4670 2. 14 1 3
4674 2.1396
4677 2.1380
487"7 2.0503
4881 2.0488
4885 2.0473
4888 2.0458
4892 2.0443
5095 1.9626
5099 1. 96 1 2
5103 1.9598
5106 1.9584
5110 1.9570
5317 1.8807
5321 1.8794
5325 1.8781
5328 1.8768
5332 1.8755
5543 1.8040
5547 1.8028
5551 1. 8016
5555 1-8003
5558 1.7991
60
59
58
57
56
7
8
9
4681 2.1364
4684 2. 1 348
4688 2.1332
4691 2.1315
4695 2.1299
4895 2.0428
4899 2.0413
4903 2.0398
4906 2.0383
4910 2.0368
51 14 1.9556
51 17 1.9542
5121 1.9528
5125 1.9514
5128 1.9500
5336 1.8741
5340 1.8728
5343 1-8715
5347 1.8702
5351 1.8689
5562 1.7979
5566 1.7966
5570 1.7954
5574 1.7942
5577 1-7930
55
54
53
52
51
10
II
12
13
14
4699 2.1283
4702 2.1267
4706 2. 1 25 1
4709 2.1235
4713 2.1219
4913 2.0353
4917 2.0338
4921 2.0323
4924 2.0308
4928 2.0293
5132 1.9486
5136 1.9472
5139 1.9458
5143 1.9444
5147 1.9430
5354 1.8676
5358 1.8663
5362 1.8650
5366 1.8637
5369 1.8624
5581 1.7917
5585 1.7905
5589 1-7893
5593 1.7881
5596 1.7868
50
49
48
11
17
19
4716 2.1203
4720 2.1 187
4723 2.1 171
4727 2.1155
4731 2.1 139
4931 2.0278
4935 2.0263
4939 2.0248
4942 2.0233
4946 2.0219
5150 1.9416
5154 1.9402
5158 1.9388
5161 1.9375
5165 1.9361
5373 1861 1
5377 1-8598
5381 1.8585
5384 1.8572
5388 1.8559
5600 1.7856
5604 1.7844
5608 1.7832
5612 1.7820
5616 1.7808
45
44
43
42
41
20
21
22
23
24
4734 2. 1 1 23
4738 2.1 107
4741 2.1092
4745 2.1076
4748 2.1060
4950 2.0204
4953 2.0189
4957 2.0174
4960 2.0160
4964 2.0145
5169 1.9347
5172 1.9333
5176 1.9319
5180 1.9306
5184 1.9292
5392 1.8546
5396 1.8533
5399 1.8520
5403 1.8507
5407 1.8495
5619 1.7796
5623 1.7783
5627 1.7771
5631 1.7759
5635 1.7747
40
38
37
36
27
28
29
4752 2.1044
4755 2.1028
4759 2.1013
4763 2.0997
4766 2.0981
4968 2.0130
4971 2.0115
4975 2.0101
4979 2.0086
4982 2.0072
5187 1.9278
5191 1.9265
5^95 1-9251
5198 1.9237
5202 1.9223
541 1 1.8482
5415 1.8469
5418 1.8456
5422 1.8443
5426 1.8430
5639 1.7735
5642 1.7723
5646 1.7711
5650 1.7699
5654 1.7687
35
34
33
32
31
30
31
32
33
34
4770 2.0965
4/ 73 2.0950
4777 2.0934
4780 2.0918
4784 2.0903
4986 2.0057
4989 2.0042
4993 2.0028
4997 2.0013
5000 1.9999
5206 1.9210
5209 1.9196
5213 1.9183
5217 1.9169
5220 1.9155
5430 1. 84 1 8
5433 1.8405
5437 1.8392
5441 1.8379
5445 1.8367
5658 1.7675
5662 1.7663
5665 1.7651
5669 1.7639
5673 1.7627
30
29
28
27
26
39
4788 2.0887
4791 2.0872
4795 2.0856
4798 2.0840
4802 2.0825
5004 1.9984
5008 1.9970
50" 1-9955
5015 1. 9941
5019 1.9926
5224 1.9142
5228 1.9128
5232 1.9115
5235 1.9101
5239 1.9088
5448 1.8354
5452 1.8341
5456 1.8329
5460 1. 83 1 6
5464 1.8303
5677 1.7615
5681 1.7603
5685 1.7591
5688 1.7579
5692 1.7567
25
24
23
22
21
40
41
42
43
44
4806 2.0809
4809 2.0794
4813 2.0778
4816 2.0763
4820 2.0748
5022 1. 9912
5026 1.9897
5029 1.9883
5033 1.9868
5037 1.9854
5243 1.9074
5426 1. 9061
5250 1.9047
5254 1.9034
5258 1.9020
5467 1. 8291
5471 1.8278
5475 1-8265
5479 1.8253
5482 1.8240
5696 1.7556
5700 1.7544
5704 1.7532
5708 1.7520
5712 1.7508
20
19
18
17
16
45
46
47
48
49
4823 2.0732
4827 2.0717
4831 2.0701
4834 2.0686
4838 2.0671
5040 1 .9840
5044 J. 9825
5048 1. 98 II
5051 1.9797
5055 1.9782
5261 1.9007
5265 1.8993
5269 1.8980
5272 1.8967
5276 1.8953
5486 1.8228
5490 1.8215
5494 1.8202
5498 1. 8 1 90
5501 1.8177
5715 1.7496
5719 1.7485
5723 1.7473
5727 1. 7461
5731 1.7449
15
14
13
12
II
50
51
52
53
54
4841 2.0655
4845 2.0640
4849 2.0625
4852 2.0609
4856 2.0594
5059 1.9768
5062 1.9754
5066 1.9740
5070 1.9725
5073 1.9711
5280 1.8940
5284 1.8927
5287 1.8913
5291 1.8900
5295 1.8887
5505 1-8165
5509 1.8152
5513 1.8140
5517 1.8127
5520 1.8115
5735 1-7437
5739 1.7426
5743 1-7414
5746 1.7402
5750 1.7391
10
9
8
7
6
59
4859 2.0579
4863 2.0564
4867 2.0549
4870 2.0533
4874 2.0518
5077 1.9697
5081 1.9683
5084 1.9669
5088 1.9654
5092 1.9640
5298 1.8873
5302 1.8860
5306 1.8847
5310 1.8834
5313 1.8820
5524 1.8103
5528 1.8090
5532 1.8078
5535 1.8065
5539 1.8053
5754 1-7379
5758 1.7367
5762 1.7355
5766 1.7344
5770 1.7332
5
4
3
2
I
60
4877 2.0503
5095 1.9626
5317 1.8807
5543 1.8040
5774 1-7321
Cotff Tang
Cotg Tang
Cotg Tang
Cotg Tang
Cotg Tang
/
64°
63°
620
61°
60° / 1
NATURAL TANGENTS AND COTANGENTS
87
/
30°
31°
32°
38°
34° / 1
Tang Cotg
Tang Cotg
Tang Cotg
Tang Cotg
Tang Cotg
I
2
3
4
5
6
7
8
9
5774 1.7321
5777 1-7309
5781 1.7297
5785 1.7286
5789 1.7274
6009 1.6643
6013 1.6632
6017 1.6621
6020 1. 66 10
6024 1.6599
6249 1.6003
6253 1.5993
6257 1.5983
6261 1.5972
6265 1.5962
6494 1.5399
6498 1.5389
6502 1.5379
6506 1.5369
6511 1-5359
6745 1.4826
6749 1.4816
6754 1.4807
6758 1.4798
6762 1.4788
60
59
58
57
56
5793 1-7262
5797 1-7251
5801 1.7239
5805 1.7228
5808 1.7216
6028 1.6588
6032 1.6577
6036 1.6566
6040 1.6555
6044 1.6545
6269 1.5952
6273 1.5941
6277 1.5931
6281 1.5921
6285 1. 59 1 1
6515 1-5350
6519 1.5340
6523 1-5330
6527 1-5320
6531 1-53"
6766 1.4779
6771 1.4770
6775 1.4761
6779 1.4751
6783 1.4742
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
10
II
12
13
5812 1.7205
5816 1.7193
5820 1.7182
5824 1.7170
5828 1.7159
6048 1.6534
6052 1.6523
6056 1.65 1 2
6060 1. 6501
6064 1.6490
6289 1.5900
6293 1.5890
6297 1.5880
6301 1.5869
6305 1.5859
6536 1.5301
6540 1.5291
6544 1.5282
6548 1.5272
6552 1.5262
6787 1.4733
6792 1.4724
6796 1.4715
6800 1.4705
6805 1.4696
15
i6
17
i8
19
5832 1. 7147
5836 1,7136
5840 1.7124
5844 1.7113
5847 1. 7 102
6068 1.6479
6072 1.6469
6076 1.6458
6080 1.6447
6084 1.6436
6310 1.5849
6314 1.5839
6318 1.5829
6322 • 1.5818
6326 1.5808
6556 1.5253
6560 1.5243
6565 1.5233
6569 1.5224
6573 1.5214
6809 1.4687
6813 1.4678
6817 1.4669
6822 1.4659
6826 1.4650
20
21
22
23
24
5851 1.7090
5S55 1-7079
5859 1.7067
5863 1.7056
5S67 1.7045
6088 1.6426
6092 1. 641 5
6096 1.6404
6100 1.6393
6104 1.6383
6330 1.5798
6334 1-5788
6338 1.5778
6342 1.5768
6346 1.5757
6577 1.5204
6581 1.5195
6585 1.5185
6590 1.5 1 75
6594 1.5 166
6830 1. 464 1
6834 1.4632
6839 1.4623
6843 1. 46 14
6847 1.4605
25
26
27
28
29
5871 1.7033
5875 1.7022
5879 1.7011
5883 1.6999
5887 1.6988
6108 1.6372
6112 1.6361
61 16 1.6351
6120 1.6340
6124 1.6329
6350 1.5747
6354 1.5737
6358 1.5727
6363 1-5717
6367 1-5707
6598 1.5156
6602 1.5 147
6606 1.5 137
6610 1. 5127
6615 1.5118
6851 1.4596
6856 1.4586
6860 1.4577
6864 1.4568
6869 1.4559
35
34
33
32
31
30
31
32
33
34
5890 1.6977
5894 1.6965
5898 1.6954
5902 1.6943
5906 1.6932
6128 1.6319
6132 1.6308
6136 1.6297
6140 1.6287
6144 1.6276
6371 1.5697
6375 1-5687
6379 1-5677
6383 1.5667
6387 1.5657
6619 1.5108,
6623 1.5099
6627 1.5089
6631 1.5080
6636 1.5070
6873 1.4550
6877 1.4541
6881 1.4532
6886 1.4523
6890 1.45 14
30
29
28
27
26
35
36
37
38
39
5910 1.6920
5914 1.6909
5918 1.6898
S922 1.6887
5926 1.6875
6148 1.6265
6152 1.6255
6156 1.62,1/1
6160 1.6234
6164 1.6223
6391 1.5647
6395 1-5637
6399 1-5627
6403 1. 56 1 7
6408 1.5607
6640 1. 506 1
6644 1.505 1
6648 1.5042
6652 1.5032
6657 1.5023
6894 1.4505
6899 1.4496
6903 1.4487
6907 1.4478
691 1 1.4469
25
24
23
22
21
40
41
42
43
44
5930 1.6864
5934 1.6853
5938 1.6842
5942 1.6831
5945 1.6820
6168 1.6212
6172 1.6202
6176 1.6191
6i8o 1.6181
6184 1. 6170
6412 1.5597
6416 1.5587
6420 1.5577
6424 1.5567
6428 1.5557
6661 1.5013
6665 1.5004
6669 1.4994
6673 1.4985
6678 1.4975
6916 1.4460
6920 1.445 1
6924 1.4442
6929 1.4433
6933 1-4424
20
19
18
17
16
45
46
47
48
49
50
51
52
53
54
5949 1.6808
5953 1-6797
5957 1-6786
5961 1.6775
5965 1.6764
6188 1.6160
6192 1.6149
6196 1.6139
6200 1. 6 1 28
6204 1. 61 18
6432 1.5547
6436 1.5537
6440 1.5527
6445 1-55 1 7
6449 1-5507
6682 1.4966
6686 1.4957
6690 1.4947
6694 1.4938
6699 1.4928
6937 1.4415
6942 1.4406
6946 1.4397
6950 1.4388
6954 1-4379
15
14
13
12
5969 1.6753
5973 1-6742
5977 1-6731
5981 1.6720
5985 1.6709
6208 1.6107
6212 1.6097
6216 1.6087
6220 1.6076
6224 1.6066
6453 1-5497
6457 1.5487
6461 1.5477
6465 1.5468
6469 1-5458
6703 1.4919
6707 1. 49 10
6711 1.4900
6716 1. 489 1
6720 1.4882
6959 1-4370
6963 1.4361
6967 1-4352
6972 1.4344
6976 1-4335
10
9
8
7
6
55
56
11
59
5989 1.6698
5993 1.6687
5997 1-6676
6001 1.6665
6005 1.6654
6228 1.6055
6233 1.6045
6237 1-6034
6241 1.6024
6245 1. 6014
6473 1-5448
6478 1.5438
6482 1.5428
6486 1.5418
6490 1.5408
6724 1.4872
6728 1.4863
6732 1.4854
6737 1.4844
6741 1.4835
6980 1.4326
6985 1.43 1 7
6989 1.4308
6993 1-4299
6998 1.4290
5
4
3
2
GO
6009 1.6643
6249 1.6003
6494 1-5399
6745 1.4826
7002 1. 4281
Cotg Tang
Cotg Tang
Cotg Tang
Cotg Tang
Cotg Tang
/
51)°
5S°
57°
56°
55°
/
88
TABLE III
/
35°
36°
37°
38°
31)°
/
Tang Cotg
Tang Cotg
Tang Cotg
Tang Cotg
Tang Cotg
I
2
3
4
7002 1. 428 1
7006 1.4273
70 II 1.4264
7015 1.4255
7019 1.4246
7265 1.3764
7270 1.3755
7274 1.3747
7279 1-3739
7283 1.3730
7536 1.3270
7540 1.3262
7545 1.3254
7549 1.3246
7554 1.3238
7813 1.2799
7818 1.2792
7822 1.2784
7827 1.2776
7832 1.2769
S098 1.2349
8103 1.2342
8107 1.2334
81 12 1.2327
8117 1.2320
60
59
58
I
7
8
9
7024 1.4237
7028 1.4229
7032 1.4220
7037 1.4211
7041 1.4202
7288 1.3722
7292 1.3713
7297 1.3705
7301 1.3697
7306 1.3688
7558 1.3230
7563 1.3222
7568 1.3214
7572 1.3206
7577 1.3198
7836 1.2761
7841 1.2753
7846 1.2746
7850 1.2738
7855 1-2731
8122 1. 2312
8127 1.2305
8132 1.2298
8136 1.2290
8141 1.2283
55
54
53
52
51
10
II
12
13
14
7046 1. 41 93
7050 1. 4 1 85
7054 1.4176
7059 1.4167
7063 1. 41 58
7310 1.3680
7314 1.3672
7319 1.3663
7323 1.3655
7328 1.3647
7581 1.3190
7586 1.3182
7590 1.3175
7595 1.3167
7600 1. 3 1 59
7860 1.2723
7865 1.2715
7869 1.2708
7874 1.2700
7879 1.2693
8146 1.2276
8151 1.2268
8156 1. 2261
8i6i 1.2254
8165 1.2247
60
49
48
47
46
19
7067 1.4150
7072 1.4141
7076 1.4132
7080 1. 4 1 24
7085 1.4115
7332 1.3638
7337 1.3630
7341 1.3622
7346 1.3613
7350 1.3605
7604 1.3151
7609 1.3143
7613 1.3135
7618 1.3127
7623 1.3119
7883 1.2685
7888 1.2677
7893 1.2670
7898 1.2662
7902 1.2655
8170 1.2239
8175 1.2232
8180 1.2225
8185 1.2218
8190 1. 2210
45
44
43
42
41
20
21
22
23
24
7089 1. 4 1 06
7094 1.4097
7098 1.4089
7102 1.4080
7107 1.4071
7355 1-3597
7359 1.3588
7364 1.3580
7368 1.3572
7373 1-3564
7627 1.3111
7632 1. 3103
7636 1.3095
7641 1.3087
7646 1.3079
7907 1.2647
7912 1.2640
7916 1.2632
7921 1.2624
7926 1.2617
8195 1.2203
8199 1. 2196
8204 1. 21 89
8209 1.2181
8214 1.2174
40
39
38
27
36
25
26
27
28
29
71 II 1.4063
71 15 1.4054
7120 1.4045
7124 1.4037
7129 1.4028
7377 1.3555
7382 1.3547
7386 1.3539
7391 1.3531
7395 1.3522
7650 1.3072
7655 1.3064
7659 1.3056
7664 1.3048
7669 1.3040
7931 1.2609
7935 1.2602
7940 1.2594
7945 1.2587
7950 1.2579
8219 1. 2167
8224 1.2 1 60
8229 1.2153
8234 1.2145
8238 1.2138
35
34
32
31
30
31
32
zz
34
7133 1.4019
7137 1.4011
7142 1.4002
7146 1.3994
7151 1.3985
7400 1.35 14
7404 1.3506
7409 1.3498
7413 1.3490
7418 1. 3481
7673 1.3032
7678 1.3024
7683 1.3017
7687 1.3009
7692 1.3001
7954 1.2572
7959 1.2564
7964 1.2557
7969 1.2549
7973 1.2542
8243 1.2131
8248 1. 21 24
8253 1.2117
8258 1.2109
8263 1.2102
30
29
28
27
26
37
38
39
7155 1-3976
7159 1.3968
7164 1.3959
7168 1.3951
7173 1.3942
7422 1.3473
7427 1.3465
7431 1.3457
7436 1.3449
7440 1.3440
7696 1.2993
7701 1.2985
7706 1.2977
7710 1.2970
7715 1.2962
7978 1.2534
7983 1.2527
7988 1.2519
7992 1.25 1 2
7997 1.2504
8268 1.2095
8273 1.2088
8278 1.2081
8283 1.2074
8287 1.2066
25
24
23
22
21
40
41
42
43
44
7177 1.3934
7181 1.3925
7186 1.3916
7190 1.3908
7195 1-3899
7445 1.3432
7449 1.3424
7454 1.3416
7458 1.3408
7463 1.3400
7720 1.2954
7724 1.2946
7729 1.2938
7734 1.2931
7738 1.2923
8002 1.2497
8007 1.2489
8012 1.2482
8016 1.2475
8021 1.2467
8292 1.2059
8297 1.2052
8302 1.2045
8307 1.2038
8312 1.203 1
20
»9
18
17
16
45
46
47
48
49
7199 1.3891
7203 1.3882
7208 1.3874
7212 1.3865
7217 1.3857
7467 1.3392
7472 1.3384
7476 1.3375
7481 1.3367
7485 1.3359
7743 1.2915
7747 1.2907
7752 1.2900
7757 1.2892
7761 1.2884
8026 1.2460
8031 1.2452
8035 1.2445
8040 1.2437
8045 1.2430
8317 1.2024
8322 1. 2017
8327 1.2009
8332 1.2002
8337 1.1995
15
14
13
12
II
50
51
52
53
54
7221 1.3848
7226 1.3840
7230 1.3831
7234 1.3823
7239 1.3814
7490 1.335 1
7495 1-3343
7499 1.3335
7504 1.3327
7508 1.3319
7766 1.2876
7771 1.2869
7775 1.2861
7780 1.2853
7785 1.2846
8050 1.2423
8055 1.2415
8059 1.2408
8064 1. 240 1
8069 1.2393
8342 1. 1 988
8346 1.1981
8351 1. 1974
8356 1. 1967
8361 1. 1960
10
9
8
7
6
55
56
59
7243 1.3806
7248 1.3798
7252 1.3789
7257 1.3781
7261 1.3772
7513 i-2>2,^i
7517 1.3303
7522 1.3295
7526 1.3287
7531 1.3278
7789 1.2838
7794 1.2830
7799 1.2822
7803 1.28 1 5
7808 1.2807
8074 1.2386
8079 1.2378
8083 1. 237 1
8088 1.2364
8093 1.2356
8366 1. 1953
8371 1. 1946
8376 1. 1939
8381 1. 1932
8386 1. 1925
5
4
3
2
60
7265 1.3764
7536 1.3270
7813 1.2799
8098 1.2349
8391 1.1918
Cotg Tang
Cotg Tang
Cotg Tang
Cotg Tang
Cotg Tang
_
/
54°
53°
52°
51°
50°
zl
NATURAL TANGENTS AND COTANGENTS
/
40°
41°
42o
43°
44°
/
Tang Cotg
Tang Cotg
Tang Cotg
Tang Cotg
Tang Cotg
60
59
58
11
I
2
3
4
8391 1.1918
8396 1.1910
8401 1. 1903
8406 1. 1 896
8411 1. 1889
8693 1.1504
8698 1.1497
8703 1.1490
8708 1.1483
8713 1.1477
9004 I.I 106
9009 I.I 100
9015 1.1093
9020 1.1087
9025 1.1080
9325 1.0724
9331 1.0717
9336 1.0711
9341 1.0705
9347 1.0699
9657 1-0355
9663 1.0349
9668 1.0343
9674 1-0337
9679 1.0331
1,
9
8416 1. 1882
8421 I. 1875
8426 1. 1 868
8431 1.1861
8436 1. 1 854
8718 1.1470
8724 1.1463
8729 1. 1456
8734 1.1450
8739 1. 1443
9030 1.1074
9036 1.1067
9041 1.1061
9046 1. 1 054
9052 1. 1048
9352 1.0692
9358 1.0686
9363 1.0680
9369 1.0674
9374 1.0668
9685 1.0325
9691 1.0319
9696 1. 03 1 3
9702 1.0307
9708 1. 030 1
55
54
53
52
51
50
49
48
47
46
10
II
12
13
14
8441 1. 1 847
8446 1. 1 840
8451 1.1833
8456 1. 1826
8461 1.1819
8744 1.1436
8749 1.1430
8754 1.1423
8759 1.1416
8765 1.1410
9057 1.1041
9062 1.1035
9067 1.1028
9073 1.1022
9078 1. 1016
9380 1.0661
9385 1.0655
9391 1.0649
9396 1.0643
9402 1.0637
9713 1.0295
9719 1.0289
9725 1.0283
9730 1.0277
9736 1.0271
15
16
17
18
19
8466 1.1812
8471 1. 1806
8476 I.I 799
8481 1. 1792
8486 1.17^5
8770 1.1403
8775 1-1396
8780 1.1389
8785 1.1383
8790 1.1376
9083 1.1009
9089 1. 1 003
9094 1.0996
9099 1.0990
9105 1.0983
9407 1.0630
9413 1.0624
9418 1.0618
9424 1. 06 1 2
9429 1.0606
9742 1.0265
9747 1-0259
9753 1-0253
9759 1.0247
9764 1.0241
45
44
43
42
41
20
21
22
23
24
8491 I. 1778
8496 1.1771
8501 1.1764
8506 1. 1757
8511 1. 1750
8796 1. 1 369
8801 1. 1363
8806 1.1356
8811 1.1349
8816 1.1343
9110 1.0977
9115 1.0971
91 21 1.0964
9126 1.0958
9131 1.0951
9435 1-0599
9440 1.0593
9446 1.0587
9451 1.0581
9457 1-0575
9770 1.0235
9776 1.0230
9781 1.0224
9787 1.0218
9793 1. 02 1 2
40
39
3^
11
25
26
27
28
29
8516 1. 1743
8521 1. 1736
8526 1. 1729
8531 1. 1722
8536 1.1715
8821 1. 1336
8827 1.1329
8832 1.1323
8837 1.1316
8842 1.1310
9137 1.0945
9142 1.0939
9147 1.0932
9153 1.0926
9158 1.0919
9462 1.0569
9468 1.0562
9473 1-0556
9479 1-0550
9484 1.0544
9798 1.0206
9804 1.0200
9810 1.0194
9816 1.0188
9821 1.0182
35
34
33
32
31
30
31
32
33
34
8541 I. 1708
8546 I.I 702
8551 1-1695
8556 1. 1688
8561 1.1681
8847 1.1303
8852 1.1296
8858 1.1290
8863 1.1283
8868 1.1276
9163 1.0913
9169 1.0907
9174 1.0900
9179 1.0894
9185 1.0888
9490 1.0538
9495 1-0532
9501 1.0526
9506 1.0519
9512 1.0513
9827 1.0176
9833 1.0170
9838 1. 01 64
9844 1. 01 58
9850 1.0152
30
29
28
27
26
37
38
39
8566 1.1674
8571 1.1667
8576 1.1660
8581 I. 1653
8586 1. 1 647
8873 1.1270
8878 1.1263
8884 1.1257
8889 1. 1250
8894 1.1243
9190 1.0881
9195 1.0875
9201 1.0869
9206 1.0862
9212 1.0856
9517 1.0507
9523 1.0501
9528 1.0495
9534 1.0489
9540 1.0483
9856 1.0147
9861 1.0141
9867 1.0135
9873 1.0129
9879 1.0123
25
24
23
22
21
40
41
42
43
44
i
49
8591 I. 1640
8596 I. 1633
8601 1. 1 626
8606 1. 1619
8611 1. 1612
8899 1-1237
8904 I.I 230
8910 1.1224
8915 1.1217
8920 1.1211
9217 1.0850
9222 1.0843
9228 1.0837
9233 1. 083 1
9239 1.0824
9545 1-0477
9551 1.0470
9556 1.0464
9562 1.0458
9567 1.0452
9884 1.0117
9890 l.OIIl
9896 1.0105
9902 1.0099
9907 1.0094
20
19
18
17
16
8617 1. 1606
8622 1. 1599
8627 1. 1592
8632 1.1585
8637 1.1578
8925 1.1204
8931 1.1197
8936 1.1191
8941 I.I 184
8946 1.1178
9244 1. 08 1 8
9249 1. 08 1 2
9255 1.0805
9260 1.0799
9266 1.0793
9573 1.0446
9578 1.0440
9584 1.0434
9590 1.0428
9595 1.0422
9913 1.0088
9919 1.0082
9925 1.0076
9930 1.0070
9936 1.0064
15
14
13
12
II
50
51
52
53
54
8642 1.1571
8647 1. 1 565
8652 1.1558
8657 1.1551
8662 1.1544
8952 1.1171
8957 1.1165
8962 1.1158
8967 1.1152
8972 1.1145
9271 1.0786
9276 1.0780
9282 1.0774
9287 1.0768
9293 1.0761
9601 1.0416
9606 1. 04 10
9612 1.0404
9618 1.0398
9623 1.0392
9942 1.0058
9948 1.0052
9954 1.0047
9959 I -004 1
9965 1.0035
10
i
7
6
57
58
59
8667 1. 1538
8672 1.1531
8678 1.1524
8683 1.1517
8688 1. 1510
8978 I.I 139
8983 1.1132
8988 1.1126
8994 1.1119
8999 I-III3
9298 1.0755
9303 1-0749
9309 1.0742
9314 1.0736
9320 1.0730
9629 1.0385
9634 1-0379
9640 1.0373
9646 1.0367
9651 1.0361
9971 1.0029
9977 1.0023
9983 1.0017
9988 1. 00 1 2
9994 1.OC06
5
4
3
2
I
(>0
8693 1.1504
9004 I.I 106
9325 1.0724
9657 1-0355
1000 1. 0000
Cotg Tang
Cotg Tang
Cotg Tang
Cotg Tang
Cotg Tang
/
490
48°
47°
46°
45°
/
TABLE IV
*
SQUARES OF NUMBERS
No.
Square.
No.
Square.
No.
Square.
No.
Square.
No.
Square.
I
O
20
21
400
40
41
1600
m
61
3600
80
81
64CX)
I
441
1681
3721
6561
2
4
22
484
42
1764
62
3844
82
6724
3
9
23
529
43
1849
63
3969
83
6889
4
i6
24
576
44
1936
64
4096
84
7056
5
25
25
625
45
2025
65
4225
85
7225
6
36
26
676
46
2116
66
4356
86
7396
7
49
27
729
47
2209
67
4489
87
7569
8
64
28
784
48
2304
68
4624
88
7744
9
10
1 1
81
29
30
31
841
49
50
51
2401
69
70
71
4761
89
90
91
7921
8100
100
900
2500
4900
121
961
2601
5041
8281
12
144
32
1024
52
2704
72
5184
92
8464
'3
169
33
1089
53
2809
73
5329
93
8649
14
196
34
1 1 56
54
2916
74
5476
94
8836
'5
22s
35
1225
55
3025
75
5625
95
9025
i6
256
36
1296
56
3136
76
5776
96
9216
17
289
37
1369
57'
3249
77
5929
97
9409
i8
324
38
1444
58'
3364
78
6084
98
9604
«9
•20
361
400
39
40
1521
59
GO
3481
79
80
6241
99
100
9801
1600
3600
6400
lOOOO
91
92
TABLE IV
00
!♦♦
244
$♦♦
4^#
&♦♦
6#^
T4^
§♦♦
9^^
u
00
Diff'
I
100
400
900
1600
2500
3600
4900
6400
8100
OI
02
03
102
104
106
404
408
412
906
912
918
1608
1616
1624
2510
2520
2530
3612
4914
4928
4942
6416
6432
6448
8118
8136
8154
01
04
09
3
5
7
04
108
no
112
416
420
424
924
930
936
1632
1640
1648
2540
2550
2560
3648
3660
3672
4956
6464
6480
6496
8172
8190
8208
16
25
36
9
II
13
11
09
114
116
118
428
432
436
04.2
948
954
1656
1664
1672
2570
2580
2590
3696
3708
4998
5012
5026
6512
6528
6544
8226
8244
8262
1'.
IS
17
19*
10
121
441
961
1681
2601
3721
5041
6561
8281
00
21
II
12
13
123
125
127
445
449
453
967
973
979
1689
1697
1705
2611
2621
2631
3733
3745
3757
5055
5069
5083
6577
8299
8317
8335
21
44
69
23
25
27
14
129
132
134
til
466
985
992
998
1713
1722
1730
2641
2652
2662
3769
3782
3794
5097
5112
5126
6625
6642
6658
8353
8372
8390
96
5^
29*
31
33
19
136
139
141
470
475
479
1004
ion
1017
1738
1747
1755
2672
2683
2693
3806
3819
3^3^
5140
5155
51^9
6691
6707
8408
8427
8445
89
6i
35*
37
39*
20
144
484
1024
1764
2704
3844
5184
6724
8464
00
4»
21
22
23
146
148
151
488
492
407
1030
1036
1043
1772
17S0
1789
2714
2724
2735
3881
5198
5212
5227
6756
^773
8482
8500
8519
41
84
29
43
45*
47
24
153
lit
501
506
510
1049
1056
1062
1797
1806
1814
2745
2766
3893
3906
3918
5241
5256
5270
6789
6806
6822
8537
8556
8574
76
25
76
49*
5»
53*
11
29
161
166
515
519
524
1069
1075
1082
1823
1831
1840
2777
2787
2798
3931
3943
3956
5285
5299
53H
6839
6855
6872
8630
29
84
41
55
51*
59*
30
169
529
1089
1849
2809
3969
5329
6889
8649
00
61
31
32
33
171
III
542
1095
1102
1 108
1857
1866
1874
2819
2830
2840
3981
3994
4006
5343
5358
5372
6905
6922
6938
8667
8686
8704
61
89
63*
65
67*
34
35
36
184
547
l^
"15
1122
1128
1883
1892
1900
2862
2872
4019
4032
4044
5387
5402
5416
6955
C972
6988
8723
8742
8760
56
11
69*
7»
73*
39
187
190
193
571
"35
1142
1 149
1909
1918
1927
2883
2894
2905
4057
4070
4083
5431
5446
5461
7005
7022
7039
8779
8798
8817
69
44
21
75*
77*
79*
40
196
576
1 156
1936
2916
4096
5476
7056
8836
00
81
41
42
43
198
201
204
580
5«5
590
1 162
1 1 76
1944
1953
1962
2926
2937
2948
4108
4121
4134
5490
5505
5520
7072
7089
7106
8854
8S73
8892
81
64
49
83*
85*
87^
44
45
46
207
210
213
595
605
1 183
1 190
1197
1971
1980
1989
2959
2970
2981
4147
4160
4173
5535
5550
5565
7123
7140
7157
891 1
8930
8949
36
2
89*
91"
93*
49
216
219
222
610
620
1204
I2n
1218
1998
2007
2016
2992
3003
3014
4186
4199
4212
5580
7174
7191
7208
8968
8987
9006
09
04
01
95*
97*
99*
50
225
625
1225
2025
3025
4225
S62S
7225
9025
00
SQUAKES OF NUMBERS
93
60
!♦♦
24^
3##
4##
5##
€♦♦
74^
«♦♦
94^
U
00
Diff.
z
225
625
1225
2025
3025
4225
5625
7225
9025
51
52
53
228
231
234
630
640
1232
1239
1246
2034
2043
2052
3036
3047
3058
4238
4251
5640
5655
5670
7242
7259
7276
9044
9063
9082
01
04
09
3
5
7
54
55
5^^
237
240
243
645
650
655
1253
12G0
1267
2061
2070
2079
3069
3080
3091
4277
4290
4303
5685
5700
5715
7293
7310
7327
9101
9120
9139
16
25
36
9
11
13'
11
59
246
249
252
660
665
670
1274
12S1
1288
20S8
2097
2106
3102
3113
3124
4316
4329
4342
5730
5760
7344
7361
7378
9158
9177
9196
49
Si
15
17
19*
00
256
676
1296
2116
3136
4356
5776
7396
9216
00
21
61
62
63
259
262
265
68i
686
691*
1303
1310
1317
2125
2134
2143
3147
3158
3169
4369
4382
4395
^^6
5821
7413
7430
7447
9235
9254
9273
21
69
23
25
27
^4
268
272
275
696
702
707
1324
1332
1339
2152
2162
2171
3180
3192
3203
4408
4422
4435
5836
^5^7
7464
7482
7499
9292
9312
9331
96
25
56
29*
31
33
69
278
282
285
712
718
723
1346
1354
1361
2180
2190
2199
3214
3226
3237
4448
4462
4475
5882
5898
5913
7516
7534
7551
9350
9370
9389
89
24
61
35*
37
39*
70
289
729
1369
2209
3249
4489
5929
7569
9409
00
41
71
72
73
292
295
299
734
739
745
1376
1383
1391
2218
2227
2237
3260
3271
3283
4502
4515
4529
5944
5959
5975
7586
7603
7621
9428
9447
9467
29
43
45"
47
74
302
306
309
750
It
1398
1406
1413
2246
2265
3294
3306
3317
4542
4556
4569
5990
6021
7638
7656
7673
9486
9506
9525
76
49*
51
S3*
77
78
79
113
316
320
767
772
778
1421
1428
1436
2275
2284
2294
3329
3340
3352
4583
4596
4610
6037
6052
6ob8
7691
7708
7726
9545
9564
9584
41
55
57*
59*
80
324
784
1444
2304
3364
4624
6084
7744
9604
00
61
81
82
83
327
331
334
789
1451
2313
2323
2332
3375
4637
4651
4664
6099
6115
6130
7761
,7779
7796
9623
9662
61
It
63*
67*
338
342
345
806
812
817
1482
1489
2342
2361
3410
3422
3433
4678
4692
4705
6146
6162
6177
7814
7832
7849
9682
9702
9721
56
25
96
69*
71
73*
89
349
353
357
823
829
835
1497
1505
1513
2371
2381
2391
3445
3457
3469
4719
4733
4747
6193
6209
6225
7867
7885
7903
9741
9761
9781
69
44
21
75*
77*
79*
90
361
841
1521
2401
3481
4761
6241
7921
9801
00
81
91
92
93
372
846
852
858
1528
1536
1544
2410
2420
2430
3492
4774
4788
6256
6272
6288
7938
7956
7974
9820
9840
9860
81
64
49
83*
85*
87*
94
384
864
870
876
1560
1568
2440
2450
2460
3528
3540
3552
4816
4830
4844
6304
6320
6336
7992
8010
8028
9880
9900
9920
36
89*
9x*
93*
99
388
392
396
882
888
894
1592
2470
2480
2490
3564
4858
6pi
6384
8046
8064
8082
9940
9960
9980
09
04
01
99*
100
400
900
1600
2500
3600
4900
6400
8100
lOOOO
00
MATHEMATICAL SERIES
While this series has been planned to meet the
needs of the student who is preparing for engineer-
ing work, it is hoped that it will serve equally well
the purposes of those schools where mathematics is
taken as an element in a liberal education. In order
that the applications introduced may be of such char-
acter as to interest the general student and to train
the prospective engineer in the kind of work which
he is most likely to meet, it has been the policy of the
editors to select as joint authors of each text, a
mathematician and a trained engineer or physicist.
The problems as well as the applications intro-
duced in the text are of such a character as to draw
upon the student's general information which will
be of use to him later in the application of mathe-
matics. Without sacrificing the value of mathe-
matical study as a discipline, it is the purpose of the
series so to correlate the mathematics with the phy-
sical applications as to stimulate the interest and
train the student to use his mathematics as a means
of investigation and stating the laws of physical
phenomena.
The following texts have appeared :
I. Calculus.
By E. J. TowNSEND, Professor of Mathematics in the
University of Illinois, and G. A. Goodenough, Professor of
Mechanical Engineering, University of Illinois. $2.50.
II. College Algebra.
By H. L. RiETZ, Assistant Professor of Mathematics in
the University of Illinois, and Dr. A. R. Crathorne, Asso-
ciate in Mathematics in the University of Illinois. $1.40.
III. Trigonometry.
By A. G. Hall, Professor of Mathematics in the Uni-
versity of Michigan, and F. G. Frink, Professor of Railway
Engineering in the University of Oregon. $1.25.
HENRY HOLT AND COMPANY
NEW YORK CHICAGO
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Hoskins's Hydraulics.
By L. M. HosKiNS, Professor in Leland Stanford Uni-
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A comprehensive text-book, intended for the
fundamental course in the subject usually offered
in schools of engineering, but somewhat more com-
pact in treatment than the ordinary treatise now
available.
Russell's Text-book on Hydraulics.
By George E. Russell, Assistant Professor of Civil En-
gineering, Massachusetts Institute of Technology. viii +
183 pp. 8vo. $2.50.
This book is designed primarily for classroom
use rather than for reference for practicing engi-
neers. It avoids discussion of specialized topics
which are taught separately with special books and
devotes itself to the consideration of the more com-
mon and important subjects. At the end of each
chapter are given problems to illustrate the appli-
cation of the principles just preceding.
Benjamin's Machine Design.
By Charles H. Benjamin, Professor in Purdue Uni-
versity. i2mo. 202 pp. $2.00.
Machinery : — We know of no v^^ork on Machine Design
which can be more heartily recommended to the average
student than this. . . . The work has the characteristics of
Professor Benjamin's writing in general ; that is, clearness
and simplicity. It is brought up to date, containing, for
example, a summary of the paper on the collapsing strength
of lap-welded steel tubes presented by Professor Stewart
before the spring meeting of the A. S. M. E. in 1906.
Leffler's The Elastic Arch.
With special reference to the Reinforced Concrete Arch.
By Burton R. Leffler, Engineer of Bridges on the Lake
Shore and Michigan Southern Railway. viii-}-57 pp. i2mo.
$1.00.
HENRY HOLT AND COMPANY
NEW YORK CHICAGO
14 DAY USE
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290ct59lii
n^C'D LP
PCTisigsg
8JuI'6UM
RECn 1 D
JUN 841961
FEB 7 1969 3 8
RECEIVED —
M^*^ 25 '69 -8 AM
LOAN DEPT.
MAR 6 1975 4
REC. CIR. MAR 6 '75
KTTt
1^
Wl
jtECCIR- J>Pli?9
31
LD 21A-50m-4,'59
(A1724sl0)476B
General Librsuy
University of California
Berkeley
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