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G"
MATHEMATICAL
HANDBOOK
CONTAINING
THE CHIEF FORMULAS OF ALGEBRA, TRIGONOMETRY,
CIRCULAR AND HYPERBOLIC FUNCTIONS,
DIFFERENTIAL AND INTEGRAL
CALCULUS, AND ANALYTI-CAL
GEOMETRY
TOGETHEB WITH
MATHEMATICAL TABLES
SELECTED AND ARRANGED
BY
EDWIN P. SEAVER, A.M., LL.B.
FORMERLY ASSISTANT PROFESSOR OF MATHEMATICS IN
HARVARD UNIVERSITY
NEW YORK
McGRAW PUBLISHING COMPANY
1907
-
\\v"nXv-^"\"\
,
01
/i
" \
r\sj X\S~ !X\* f \x,v% Cw
v
COPYBIGHT, 1907,
BY
EDWIN P. SEAVEB
WABAN, MASSACHUSETTS
J
Stanbope press , C
r. H. QILIOH COMPANY
BOSTON, U. S. A.
-
PREFACE.
The uses which this book may serve hardly need to be
pointed out. Some years ago the writer composed the part'
relating to Trigonometry and used it as a syllabus for in-struction
in his college classes. It served its purpose and
soon went out of print. But a stray copy of it found its
way to the table of a well-known civil engineer, to whom it
proved constantly useful, and by whom it was often referred
to as " his memory." This engineer has suggested a revision
and republication of the original book with important enlarge-ments.
Accordingly there have been added Sections on
Algebra, the Differential and Integral Calculus, and Analytic
Geometry. The subject of Hyperbolic Functions, which nowreceives much more attention than formerly, has been more
fully treated. Tables have been added, which include not
only those universally used, but also some " like those of the
Hyperbolic Functions, of the Natural Logarithms of Num-bers,
and that of the Velocity of Falling Bodies (v = 2\/gh) "that have been hitherto not readily accessible.
Of course no efforts have been spared to secure correctness
in the printing of the formulas and the tables ; but persons
experienced in such work need not be reminded of the im-probability
that the first edition of a book of this kind should
be absolutely free from error. The writer and the publishers
can only add, that notice of any errors that may be detected
will be thankfully received, and the necessary corrections
will be promptly made and published. Also, suggestions of
desirable additions to the book and of other improvements
are invited with a view to their use in possible future edi-tions.
E. P. S.
June, 1907.
iii
-
CONTENTS.
I. FORMULAS OF ALGEBRA.
PAGE
The general laws of ordinary algebra
The law of Association 1
The law of Commutation 1
The law of Distribution 2
Definitions and laws of the symbols, 0, 1, and oo. . .
.
3
Fractions and Ratios 3
Proportions 4
Powers 6
Products and Factors 6
The Binomial Theorem 8
Inequalities 9
Roots 10
Surds 11
The Imaginary Unit, i, and its powers 12
Complex Numbers 12
Logarithms 14
Permutations and Combinations 16
Determinants 17
Quadratic Equations 21
Equations of the nth degree 21
Cubic Equations: Cardan's Rule 23
Series:
Arithmetic 24
Geometric 24
Harmonic 25
Binomial 26
Exponential and Logarithmic.
27
Interest and Annuities 28
Probabilities 30
n. FORMULAS OF CIRCULAR FUNCTIONS AND OF
TRIGONOMETRY.
Definitions and fundamental relations of the functions
with reference to an acute angle 31
General definitions of angle, of arc, and of their functions 32
Cardinal values of angle and its functions 36
The fundamental relations of the functions generalized 37
v
-
vi CONTENTS.
PAOB
Inverse functions, or anti-functions 38
Values of functions for certain angles *39
Formulas expressing each function in terms of each of
the others 40
Positive and negative lines. Projections 41Positive and negative angles 42
Functions of the sum and of the difference of two angles 42
Functions of the sum of three angles 43
Functions of a negative angle 43
Functions of A-]" 90", 90" - A, A " 180", 180" - A,A " 270", 270" - A,A" 360", 360"- A 43
Solution of the equations sin A= a, cos A = a, tan A=*a. 45
Sums and products of functions 45
Functions of multiple angles 47
Functions of half an angle 48
Expressions equivalent to sin A, to cos A, to tan A, etc. 49
Functions of Periodic Values of the arc or Angle...
51
Equivalents of the inverse circular functions sin-1 x,cos-1 x, tan-1 x, etc 52
Relations of circular, exponential, and logarithmic func-tions
54
General Properties of plane triangles 56
Properties of a quadrilateral inscribed in a circle. .
59
Solutions of plane right triangles 60
Special formulas for plane right triangles in extreme
cases 61
Solutions of plane oblique triangles 62
General properties of spherical triangles 67
Solutions of spherical right triangles.
.
.* 72
Solutions of spherical oblique triangles 74
Special formulas for spherical right triangles in extreme
cases 82
Accurate computation of angles near 0" and near 90".
Uses of S and T 83
m. HYPERBOLIC FUNCTIONS.
Definitions 85
Relations of hyperbolic functions to one other....
87
Relations between hyperbolic and circular functions of
the same variable 88
Hyperbolic functions of a negative variable 88
Variations and Cardinal Values ' 89
Relations between hyperbolic and trigonometric formulas 89
The addition and subtraction formula and formulas de-duced
THEREFROM 89
-
CONTENTS Vii
PAGE
Hyperbolic functions of a complex variable 90
Periodicity of hyperbolic functions 91
Hyperbolic anti-functions expressed as logarithms...
91
The Gudermannian function and angle 92
IV. DIFFERENTIAL AND INTEGRAL CALCULUS.
Limits 93
Definitions and notation 93
Fundamental formulas 95
Differentials and integrals of the simpler functions of x 96
Additional integrals of simple form 102
Successive differentiation 104
Taylor's Theorem, Maclaurin's Theorem 105
Circular and hyperbolic functions expressed in series..
106
Bernoulli's and Euler's numbers 107
Evaluation of indeterminate forms.
109
Partial differential coeffcients 110
Change of independent variable 112
Maxima and minima 113
Integration of rational algebraic functions 114
Of rational proper fractions 115
Of irrational algebraic functions 119
Reduction formulas for the integration, of integral pow-ers
of the trigonometric functions 124
Miscellaneous integrals 126
Definite integrals ". 128
Approximate integration. Simpson's Rule 129
Differential equations of the first order 131
Homogeneous differential equations 132
Linear differential equations 132
Differential equations of the second order 133
Differential equations of the n** order with constant
coefficients 135
Vo ANALYTIC GEOMETRY.
The point and the straight line in a plane 137
Transformation of coordinates 142
The general equation of the second degree 145
Special formulas 148-160
for the Circle~
. ..
14"
for Conic Sections 150
for the Ellipse 151for the Hyperbola 152
for the Parabola 154
Diameters 158
-
VlH CONTENTS
General properties of plane curves paob
Tangents and Normals 160-165
Curvature 165
Evolutes 166
Areas 167
Lengths of arcs 168
Envelopes 168
Pedal curves 169
Trajectories 169The Cycloid 170
The Epicycloid and the Hypocycloid 171
The Epitrochoid and the Hypotrochoid 173
The Catenary and the Tractrix 174
The involute op a circle 175
Parabolic curves 176
The Spiral of Archimedes 177
Hyperbolic curves 177
The Hyperbolic Spiral 177
Logarithmic curves 178
The Logarithmic Spiral,
178
The Lemniscate, the Cissoid, Descartes' Folium, Quadri-
folium, the wltch of agnesi, the conchoid, the
llmacon, the folium, the logocyclic curve, the
Cubic Trisectrix, the Quadratrix, the Cartesian
Ovals,the Ovals of Cassini 179-181
Miscellaneous polar equations 181
Miscellaneous rectangular equations 181
The point, the straight line, and the plane in space. .
.
182
Transformation of coordinates 184
The general equation of a plane 185
The straight line in space 188
The general equation of the second degree in three
variables 191
Transformation of the general equation 192
Curved surfaces 195
Curves of double curvature 198
The Helix 201
-
TABLES.
PAGE
I. Squares, Cubes, Square Roots, Cube Roots, Cube
Roots of Squares, and Reciprocals of numbers
from 1 to 1000 205-224
II. Logarithms of numbers 226-243
III. Binomial coefficients and Factorials 244
IV. Natural Logarithms 245-248
V. Natural trigonometric functions, to three places 249
VI. Natural Sines and Cosines, to five places.
250-251
VII. Natural Tangents and Cotangents, to five places 252-253
VIII. Natural Secants and Cosecants, to five places 254-255
IX. Logarithms of Trigonometric Functions, to five
places 256-260
X. Arcs, Sines, Tangents, and Solid Angles 261
XI. Circumferences and Areas of Circles and Volumes
of Spheres 262-263
XII. Segments of a Circle 264-265
Xllla. Natural Values of the Hyperbolic Function
Sinhu= i(ea-e-") 266
XIII6. Common Logarithms of the same 267
XlVa. Natural Values of the Hyperbolic Function
Cosh u " i (e" + e~ ") 268XI Vb. Common Logarithms of the same 269
XVa. Natural Values of Tanh u 270
XV6. Common Logarithms of the same 270
Constants 271
Weights and Measures 272-276
Gravitation and the length of the Seconds Pen-dulum
277
Table of Velocities due to Gravity 279
lx
-
SECTION I.
ALGEBRA.
The General Laws of Common Algebra.
I. The Law of Association.
a+b+ c="a + (6+ c),
a+b" c = a + (b- c),
a " b + c = a" (b - c),
a"b" c = a" (6 + c),
abc= a (be) = (a") c,
ox!itc= cx (6-5-c),
Cv6-f-c= a.v (6 x c)
,
Av6xc= av (6-f-c),
wherein the concurrence of Me signs gives the direct sign
+ or x; and the concurrence of unlike signs gives the
indirect sign " or -*-. Thus,
+ ( + c) = + c, x ( x c) - x c,
-
(-
c) = + c, +( + c)-xc,
+ (-
c) = - c, X ( -s- c) - -*- c,
-
(-h c) =
- c, +(xc)-tc.
2* JVi6 Law o/ Commutation.
a + b = 6 + a,
a-6"=-6 + a,
a"= 6a,
ax6xc-axcx6,
ax"vc= o-fcx",
o-f6xc=
axc-f6.
1
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2 MATHEMATICAL HANDBOOK
3. The Law of Distribution.
For multiplication,
a(b+ c) = ab + ac,
("a"b)x("c"d) - +("a)x("c) + ("a)x("cf)+ ("b) X ("c) + ("6) x ("d) ="ac" ad"6c""d,
wherein the signs of each partial product are determined
by the following rule:If a partial product has factors with like signs, it must
have the sign + ; if factors with unlike signs, it must have
the sign -.
Thus,
+ ( + a) X ( + c) =" + ac, + ( + a) x ( - c) = - ac,+ ( - a) X ( - c) - + ac, + ( - a) X ( + c) - - ac.
For division,
("a"6) + ("c)-+("a) + ("c)+("6) + ("c),
with the following rule for signs:If the dividend and divisor of a partial quotient have like
signs, the partial quotient must have the sign + ; if theyhave unlike signs, it must have the sign -
.
Thus,
+ ( + a) + ( + c) = + (a -s- c), + ( + a) -s- ( - c) = - (a "*" c),
+ ( - a) + ( - c) = + (a + c), +(-fl) + ( + c) = -(flrc).
Otherwise expressed, this law is
"a"b ."a,"b"i "I-
,
"c " c "c
with the same rule for signs; that is,
"(77)-? +(^)-v
\" C / C I 4- /" / c
The divisor cannot be distributed.
-
ALGEBRA 3
Definitions and Laws of the Symbols
0, 1, and oo.
4. 0 = + a-a-*-a + a, l"Xa-*-a"-*-axa,
"6+0 =-"6-0, *axl = *a-*-l,
+0--0. xl-4-1.
Ox("6)=("6)xO = 0, 00 x("6) = ("6)xoo -"00,0 + ("6)-0, 00 +("6)-"oo,
+6-0 = + cc, j ("6)^("oo)=0.
5. Using A and B to represent any two algebraic ex-pressionsof quantity, '
If A xB = 0, either A - 0 or 5 = 0,
or both A - 0 and " - 0.
If A -^-" = 0 and B is not 0, then A - 0.
If A -^2? = 0 and A is not 0, then 5 " 00.
0 ooThe forms Oxoo, O-i-0 or-, 00-5-00 or "
,
and 00"000 00
require special investigations to determine their values in
the particular cases in which they arise. See pages 109, 110.
Fractions and Ratios.
6. Equivalent forms of notation,
a -s- 6 " ^ = a : 6 = a/6.6
7. Addition of fractions,
a jc ad+ be
b d~ bd '
8. Subtraction of fractions.
a_
c_
"d-~
be
6 d" 6d'
9. Multiplication of fractions,
a_c ac
6 d~ M"
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MATHEMATICAL HANDBOOK
io. Division of fractions,
bl
d ad
be
Proportions.
ii. If a : b = c : d, then ad = bc.
15. If
16. If
a : 6 = c : x, then x = "a
a : 6 " x : d, then ic " " "0
-
17- If
then
pA + qB + rC +. . .
1 8. If a : b = b : c, then b = Vac, one geometric meanbetween a and c.
19. If a : 6 = b : c = c : d, then 6 = i/a2d and c " ^ad?,two geometric means between a and d.
20. The reciprocalof a is " = a-1,a
of i - : - (r "of lisa-/^"1'
a \a/
21. If a : 6 = " : ",
then p and q are inversely or recivro-p q
cally proportional to a and 6; and the proportion may bewritten
a :b = q : p,
or a : 6 = p-1 : gr-1.
22. If x varies as y directly,
then
xi'
X2 " Vi*
IJ2
wherein xlf yx and x2, ?/2 denote simultaneous or correspond-ingvalues of the variables x and y.
23. If a: varies as y inversely, then
1 .1
2/1*2/2or xx : x2 - t/2 : yt.
For example, the force of gravitation, g, varies inversely
as the square of the distance, d2, that is
Qi" 02 - t: " tt " "*22: ^i2-1/1 *2
a\2 d32 2 x
"t/j " "*'2 ~"~ ~ """ " ~"
-
MATHEMATICAL HANDBOOK
Powers.
24. ( + a)n = + an. 3I- amxa-n-am + an.25. (-a)2n= + a2n. 32. am + a-n = amxan.26. (-a)2" + 1=--a2" + 1. 1
33. fl"-ffl-n = Cn =27. cwxcn = an
+n. a~n
28. am + an = am- n. 34. (am)n = amn = (an)"".
29. am -s- am " a0 = 1. 35. (a5)m " ambm.
30.a",a"-a-"-i. 36. (")"-".
\6/ 6-n an \a/
44. If a" 1, then a* =00, and a-00. " 0.
45. If a " 1, then a""
= 0, and a "co " 00.
46. log 0 = - 00.
47. log 1=0.
48. log 00 =00.
The forms 0", l00, 00 " require special investigation. See
page 109.
Products and Factors.
49. a2 - fc2= (a - b) (a + b).
50. a8 - b3 - (a - 6) (a2 + ab + 62).
51. a3 + "8 = (a + 6) (a2 - a" + 62).
52. an - bn =
(a -6) (a"-1 4- an"26 + an~zb2 +. . .
+ b"-1), always.
. 53. an -bn = (a + ")(an - * - an ~2b + an - W - . . . -6n _1),if n be even.
54. an + bn = (a + b)(an~ x - an~2b + a"-3^ -. .
.
+ b"-1),if n be odd.
55. (x + a) (x + b) - x2 + (a + 6) x + a".
-
ALGEBRA 7
56. (x + a)(x + b)(x 4- c) - a8 + (a 4- 6 4- c) a?+ (ab+bc + ca) x+ abc.
57. (x 4- a)(x 4- 6)(z 4- c)(x +d)=x4+(a+b + c + d)"+ (ab + ac+ ad+bc+ bd+ cd) 3?
4- (abc + abd + acd 4- bcd)x+ abcd.
58.* (a 4- 6)2 =- a2 4- 2a6 + b2 - a2 4- b2 4- 2a6.
59. (a-6)2-a2-2a6 4-62 = a24-62--2a6.
60. (a 4- 6)8 = a8 4- 3a26 + Sab2 4- 6s = a8 4- 68 + 3a6(a + 6).
61. (a - 6)8 - a8 - 3a26 + Sab2 - 63 - a3 - 63 - 3a6(a - 6).
For the general formula giving any power of a binomial,
see 78 to 82.
62. To square a polynomial. Square each term and add
to this square twice the product of that term by every term
that follows it. Thus,
(a 4- b + c 4- d + e)2 =
a2 + 2a(6 +c+d+e) + b2 + 26(c 4- d + e)4- c2 4- 2c(d + e) + (P+2de+ e2,
(a 4- 6 - c)2 = a2 4- 2a(6 - c) 4- 62 - 2bc + c2,
(a.- b - c)2 = a2 - 2a(6 4- c) 4- b2 4- 26c 4- c2.
63. a4 4- a262 4- 64 - (a2 4- a6 4- 62)(a2 - a" 4- 62).
64. a4 4- b4 = (a2 4- abV2 4- 62)(a2 - a6V2 4- 62).
65. f" + -Y - "2 + 4 + 2-\ a/ a2
66. fa 4- -Y - a3 4- -^ 4- 3 (a 4- -V\ a J ar \ a)
67. (a4-6 4-c)3 =a3 4- 63 4- c3 4- 3(62c 4- 6c2 4- c2** 4- ca2 4- a2b 4- aft2)+ 6a6c.
68. a2 4- 62 - c2 4- 2a6 = (a + 6)2 - c2,= (a+6+c)(c+6-c).
69. a2 - b2 - c2 4- 2bc = a2 - (6 - c)2,= (a 4- b " " c)(a - 6 4- c).
70. a8 4-68 4-c8 -3a6c - (a 4-6 4-c)(a24-62 4-c2 -6c -ca -ab).
-
8 MATHEMATICAL HANDBOOK
71. be2 + ft2c + ca2 + "a + aft2 + a2b + a3 + ft3+ c*
- (a+ft + c)(a2+ft2 + c2).
72. 6c2 + b2c + ca*+ "?a+ ab2 -f a2b + Sdbc
*= (a + b + c)(bc + ca + ab).
73. bc*+b2c + ca2+ "a + aft2 + a2b + 2abc
= (ft+ c)(c+a)(a + 6).
74. ftc2+ ft2*;+ ca2 + "?a + aft2 + a2ft - 2abc - a8 - ft3 - c8
= (6 + c " a) (c + a - 6)(a + b - c).
75. be2 - ft2c+ ca2 - "a + aft2 - a2ft = (6 - c)(c -a) (a - b).
76. 26V + 2c2a2 + 2a2ft2 - a4 - ft4- c4
" (a + ft+ c)(ft+ c - a)(c + a - ft)(a + ft - c).
77. a3 + 2a2ft + 2aft2 + ft" - (a + 6)(a2 + aft + ft2).
The Binomial Theorem.
78. (a + ft)"-
an+nan-ih + n(n^Van-2b,+ n(n~ l)(n- 2) "y1 1x2 1x2x3
wherein n may be positive or negative, integral or frac-tional.When n is a positive integer, the right hand mem-ber
has n + 1 terms; when n is negative or fractional, thenumber of terms is infinite.
79. The general expression for the (r + l)th term is
n (n - 1) (n - 2) (n - 3).
.,
(w - r + 1)an_r}f
1 X2x3x.
..
r
or
n (n - 1) (n - 2) 3x2x1aW_r5r
Ix2x3x...rx (n-r)(n-r-l) x. .
.2 xl'
or, using the factorial notation,
n!
r ! (n " r) !an~rbr;
and the formula may be written
(a + ft)-- V %/n'xf
a"-rftr. [N. B. 0! = 1r-or!(w-r)!
-
ALGEBRA 9
80. The coefficients of the several terms in the expan-sion
of the nth power of a binomial are conveniently desig-nated
by C0, Cl9 C2f etc. These are functions of n as follows:
C0"n"-1,
and in general
C,_
n(n - 1)(n - 2).
. .
(n -r 4- 1)1 X 2 X 3 X
.
. .
r
n!f
=
^ (n " 1) (n " 2)a=_
31x2x3
'
r!(w-r)!
c=
n (n - 1) (n - 2) (n - 3)4
1x2x3x4
Then
81. (a + 6)" = C0a" + 0,?-^ +C2an~262 + Csa"-S6s +. .
.
Also,
82. (a - 6)" - C0a* -CX"T-Xb + C2a"-2"2 -Cjf-*V +...
The numerical values of Clf C2, Cs,.
. .
for each power of
the binomial from the first (n =1) to the twentieth (n - 20)
power may be found in a table on page 244.
The numerical values of factorials from n = 0 to n = 20
may be found in a table on the same page.
Inequalities.
83. The value of the fraction
ai + a2 + fl3 +-
- "
+""
bt+ b2 + b3+.
. .
+ bn
is less than the greatest and greater than the least of the
fractions-*, -^ -*,
..
.
"
^ provided the denominators of the"1 "2 "3 "n
latter are all positive.
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10 MATHEMATICAL HANDBOOK
84. The arithmetical mean of two numbers is greaterthan the geometrical, and the geometrical is greater than
the harmonical. That is,
a+^"Vri" 2ab2 a+b
"
Also,
c a, + a, -f . . . 4- a.^
*/85. " ' 2 ""Vaxa2
.. .
a,.
n
The arithmetical mean of the powers is greater than the
power of the arithmetical mean, that is,
86.gm+"m
"
fc"*Y\2 \ 2 ) '
and, in general,
gqtm + q,m 4-
. ..
anm /al 4- a2 4-...
4- On\m
n \ n ) '
excepting when m is a positive proper fraction.
88. If a, b, c, be positive quantities,
89. If w " n " a,
//n-fjA" In 4- a\nb\m-a) \n-a)
1*"
Roots.
90. am=ya.
TO/ W" " " TOtty91. vax va = am xan = am" = vam+n.
TO/ TO/ " " 7/171/92. va+ va = an + am = a mn = Vaw_n.
93. ^aTO/ = am = Wo/ " Van.
( l\m (m, )m to."94. ^am/ = Wa; vam = a.
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ALGEBRA 11
95.(a")"
-
a"""-
y/^o=
\jVa-
mS/a.
mn
96. v amn = a n = am.
n
y to Ann A*
97. -V^ - *^ - V aF - aK = a".
m to111
98. "/afr= Vo x V6. 99. (a6)m = aro6m.1
101. (^*m
5m
Let A represent a positive number, and a the arithmeticalvalue of its indicated root. Then,
103,
/2n, 2"+l/ -V+A
= "a, v+A = +a,2"y 2n+l/ "-
v -A= "ia, v -A =-a,
.wherein i = V " 1.
Surds.
104. If a number partly rational and partly surd is
equal to another number also partly rational and partly
surd, the two rational parts are equal and the two surd
parts are equal. Thus if
tty" Hi
a+ vb = x+ vy"
wherein a and x are rational, and V6 and vy are surds,fchen
a - x,and b
= 2/.
105. If Va + vT r V* + \/2/,
then \/"- V" = V* - V2/-
106. y/a+y/b-y/a + b+2y/ab]
107. V/a-v/6=V/a+ 6 - 2Va6.108. (a "V")2 - a2 + b " 2aVb.
109. (a + Vb) (a - V") - a2 - 6.
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12 MATHEMATICAL HANDBOOK
The imaginary unit, ifand its powers,
no. By definition,
f-W-1, *--l, ^--V-l, "*- + !.%
in. Then, 112. Also,
'
- =i-x -i8
- " t;
i*"
t'U_
t4n + 3..
t'i_
_ i; I_
^_f_
^
i*-
** "^n ^4_ + L
i3
i"- + L
Complex Numbers.
113. A complex number is a collection of units partlyreal and partly imaginary. In its simplest form it is written
a "bi or a " ife,wherein a denotes the number of real units
and b the number of imaginary units in the collection.
Both a and 6 are real coefficients,the first of 1 the second
of i.
114. If two complex numbers are equal their real parts
are equal and their imaginary parts are equal.
Thus if A + iB = a + ib, then A " a and B = b.
115. The two complex numbers a+ib and a " ib are
conjugates the one of the other, and
(a + ib)(a - ib) - a2 + b2.
Product and quotient of two complex numbers.
1 16. (a + ib){c + id) = ac - bd + i(bc + ad).
a+ib ac+bd.
(frc-ocQXI7# c+w*" "? + "P c*+cP
'
-
ALGEBRA 13
1 1 8. Every complex number can be brought into theform
a " ib " r(cos 0 " % sin 0)
wherein r - y/ a2 + fe2 =* the modulus,
0" tan-1 " = cos-1- =sin_1" - the argument,
a r r
119. The product of two complex numbers is found by
multiplying their moduli and adding their arguments. Thus,
7^ (cos Bx" i sin 6t) x r2(cos 02 " i sin 62) =
Vjcosft + 02)" t sin (6,+ 02)]
120. The quotient is found by dividing one modulus bythe other and subtracting the argument of the divisor from
that of the dividend. Thus,
r(cos^"isin^).r,^, i-e2)"isin(0t-*2)].
r2(cos03"isin02) r2
12 1. Powers of a complex number.
[r(cos 0 + i sin 0)]m = rm(coa mO + i sin m0).
122. Roots of a complex number.
T 1" !"r infl+2for,
. .
mfl+2for1r(cos 0 + isin 0) I n - r" cos ~ + lBm ~ "
Relations of conjugates.
123. (a + ib)(a - ib) - rV
(a + #" = r(cos 0+i sin 6) =" re*,(a
-
ib= r(cos 0 - i sin 0) = re~i0.
125. (cos 0 + i sin 0) (cos 0 - i sin 0) = 1.
Roots of 1 and of - 1.
126. VI*- cos " isin. "n n
"/ " r (2fc4- l)ir,
.
.
(2fc+I"V
-
1 " cos " " i sin-* L-127.n n
-
14 MATHEMATICAL HANDBOOK
Logarithms.
The relation between a number, x, and its logarithm, u, is
expressed by the equations
128. x = au, u=*\ogaX
wherein a is the base of the system of logarithms intended.The relations between logarithms of the same number in
systems having different bases are thus shown.
If x = att,then u = loga x;
and if x = bv, then v = log" x.
Whence 1 = au-^ bv,
0 = u " v loga b,0
= u log6 a " v,
129. log, x = logb x x log, 6,
130. logo x = loga x x log6a,
131. lOga 6 X lOgftO-l.
The two systems of logarithms most in use are the Nat-ural
System, founded upon the Exponential Base, e, andthe Common System, founded upon the base 10. Loga-rithms
of the former system are often called hyperboliclogarithms or Naperian logarithms, and those of the latter
Briggs' logarithms or denary logarithms
Writing 10 for a and e for 6, the foregoing equationsbecome
"
132. log10x = log, xx log106,
133- l""geX = log10 X X l0ge 10,
134- l"gio e x lo" 10 = 1.
The Exponential Base, e, is the limit of (1 + " ]approaches oo.
135. *-1 + T+2! + ^T + 7T + "-
-
2.718 281 828 459. . "
m
as m
-
ALGEBRA 15
M" log10e is known as the modulus of the Common Sys-temof logarithms. Any logarithm in the Natural System
multiplied by the modulus gives the corresponding loga-rithmin the Common System; and, conversely, any loga-rithmin the Common System divided by the modulus gives
the corresponding logarithm in the Natural System. Thus,
I36. log10 X = M X loge X, loge X = M~X X log10X.
137. M - log10 e = 0.434 294 481 903.. .
138. M-1 - loge 10 = 2.302 585 092 994.. .
A positive number has an unlimited number of logarithms;but only one of these is real, namely the one obtained by
giving to the arbitrary integer k the value 0 in the following
general equations:
140. log,,( + ") - loge X " 2fclTl.
141. loge ( + 1) - 0" 2kiri.
Negative numbers have no real logarithms,
142. log. ( " x) =" log, x" (2k + l)iri.
143. loge ( - 1) = 0" (2* + l)irt.
Complex numbers have complex logarithms,
144, loge (a + ib) - -loge (a2 + b2)+ i (tan - l - " farY
-
16 MATHEMATICAL HANDBOOK
Imaginary numbers have imaginary logarithms,
145- log.*'- i*i, 146. i* - er** - 0.20788.
..
Rules for the practical use of logarithms are based onthe following principles:
147-
148.
149.
150.
151-
152.
If
then,
If
og (an/)- log x + log y.
og r - J - log x - log y.
og (#") - n log x.
og vx " " log x.n
og base - 1, log 1 = 0, log 0 - - 00 .
1 " x" +00,
0 " log x " + 00.
0"x" +1,
then, -00 " loga:"0.
That is,if a; is positiveand greater than 1, its
logarithm is positive ; if
positive and less than 1,[itslogarithm is negative.
Permutations and Combinations.
153. The number of permutations (sometimes called
arrangements) of n things taken all at a time is
n (n - 1) (n - 2) ...2x1, or n!
154. The number of permutations of n things taken r at
a time may be denoted by the symbol P(n, r).
P (n, r) - n (n - 1) (n - 2)...
to r factors,
n!
(n-r)\
155. The number of combinations of n things taken r at
a time may be denoted by the symbol C (n, r) .
n(n-l)(n-2). . .
(n - r+ 1)^
P(n, r)CAn,f,'S3
lx2x3...r r!
n!
r\(n- r)\=
C (n,n- r).
-
ALGEBRA 17
Comparing 155 with 79 it may be seen that
156. C (n, r) " the binomial coefficient of the (r + l)thterm of the development of (a + 6)n.
Numerical values of C(n r) up to n - 20 are found in thetable on page 244.
Determinants.
If there be n2 quantities whose symbols are arrayed in theform of a square of n rows and n columns, this array is
the symbol of a determinant. The n2 quantities forming the
array are the elements of the determinant. The deter-minant
itself is the algebraic sum of all the products that
can be formed of n elements taken one from each column
and each row in all possible ways, one half of these productsbeing written with the positive sign, the other half withthe negative.
157. An array of four elements, the symbol of a deter-minant
of the second order, gives 2!( "-2) terms, thus:
ai "i
a2 b2= axb2 - a2bv
158. An array of nine elements, the symbol of a deter-minant
of the third order, gives 3! ( " 6) terms, thus:
388 o,J"2c3- axbzc2+ a2bzcx - a2btc3+ ajbtc2 - aj)2cv
Note."
If the determinant array of the third, order
be written with the first two rows repeated as shown in
the margin, then the positive terms of its development
can be found by reading the three diagonal rows fromthe left downwards, and the negative terms by readingthe three diagonal rows from the left upwards.
-
18 MATHEMATICAL HANDBOOK
159, An array of sixteen elements, the symbol of adeterminant of the fourth order, gives 4!( =24) terms, thus:.
(h "i ^ "*i
4, bf'c d^2
a4 \ c4 dA
- alb2csd4- axb2c4d3+ a3btc2d4- a3bxc4d2+ axb3c4d2- atb3c2d4+ a3b2c4dx- a3b2cxd4+ axb4c2d3- axb4c3d2+ a3b4cxd2- a3b4c2dx- a2bxc3d4+ a2bxc4d3- a4bxc2d3+ a4bxc3d2- a2b3c4dx+ a2b3cxd4" a4b2c3dx-f a4b2cxd3- a2b4cxd3+ a2b4c3dx- a4bzcxd2+ a4bzc2dx
An array of twenty-five elements, the symbol of a de-terminantof the fifth order, gives 5!(=120) terms. In
general, a determinant of the nth order consists of n ! terms.
160. If the row and the column in which a given ele-mentstands be stricken out, the determinant formed of
the remaining elements is the minor determinant relative tothe given element.
If each element in one column of the major determinantbe multiplied by its relative minor determinant and the
positive sign be given to each element taken from an oddnumbered row and the negative sign to each element takenfrom an even numbered row, the algebraic sum of theresults is equal to the major determinant. Thus,
161.
= ai("2C3-"3C2)-a2("iC3-"3Cl) +a,(fr1C3-ftjCj.
l62.
ax bx cx dx
a2 b2 "2 d2
^3 "s C3 ^3
a4 b4 c4 d4
Thus can a determinant of any order, the nth, be made
to depend on n determinants of the (n - l)th order, and
-
ALGEBRA 19
each of these again on n - 1 determinants of the (n " 2)thorder, and so on, the ultimate result being that the originaldeterminant depends on a series of determinants of the
third or of the second order, which last are easily computeddirectly. This method of reduction makes easy the com-putation
of the value of any determinant with numerical
elements.
163. In any determinant the columns can be made rowsand the rows columns without changing its value. Thus,
164. If, in any determinant, two columns or two rows
change places with each other, the new determinant so,formed is equal to the first one with the opposite sign.
Thus,
=etc.
165. If the elements in two columns or in two rows are
equal or proportional each to each, the value of the deter-minant
is 0.
=n0=0.
166. A determinant is multiplied or divided by a number
by multiplying or dividing all the elements in one column
or in one row by that number.
pax pbx pcx
a2 ^2 c2
a3 b3 c8
-
20 MATHEMATICAL HANDBOOK
167. A determinant can be splitinto two or more deter-minants.
Thus,
ai + Pi + Qi "i ci
"*"2+ P3 + "a b2 c2
az + Ps + ?s "s cs
168. A determinant is not changed in value when the
elements of one column or row are each increased or dimin-ished
by n times the corresponding elements of a parallelcolumn or row. Thus,
ax " nbl bx cx
a2 " n^2 "2 C2
as " ^s "s C3
The Solution of Equations of the First Degree by Determi-nants.
169. The solution of
170. The solution of
axx+bxy + cxz= kx'
o"tP +fc"y + C2Z - K\" by puttinga3x+b3y+ cf-kti
ax bx cx
a2 b2 c2
a3 63 cs
-A
is x=" ^-Z); * = -z".
The same method applies to a set of n equations with nunknown quantities. Observe that the denominator of thevalue of each unknown quantity is the determinant formed
of the coefficients of all the unknown quantities, while the
numerator is the same determinant with the column of the
coefficients of that unknown quantity replaced by the
column of the absolute terms on the right-hand side of the
equations.
-
I7i. If
then
ALGEBRA
axx+ bxy + ctz~ 0)
a2x + b2y + c^** 0.
x : y : z =
21
= (6^3 - ftjcj: (c^ - c2at): (a^ - a2bt).
Quadratic Equations.
172. Solution of At? + Bx+ C = 0,
x
2A
173. Solution of x2 + px + q = 0,z ip" i^p2 - 4g.
174. The roots are
real when B2 " 4AC, or p2 " 4g,
imaginary when B2 " 4AC, or p2 " 4q,
real and equal when B2 " 4AC, or p2 = 4g.
175. In all cases, the sum of the roots = -p,
and the product of the roots - q.
176. If xx and x2 denote the roots, then the equation
x2" (xt+ x2) x + xxx2 = 0,
which may also be written
is identical with
(x - xx) (x - x2) - 0,
x2 4- px + q= 0.
177. To find two numbers whose sum and product are
known, form a quadratic equation, putting the negativeof the given sum for p and the given product for q, and
solve,
178. Any rational integral equation of the nth degreein x, may be written in the form
n " 3Xn + p^-1 + p2Xn~2 + p3"n-3 +. .
.
+pn - 0.
-
22 MATHEMATICAL HANDBOOK
179. If xv x2, xs,. . .
xn be the n roots of this equation,the first member is divisible by each of the factors x " xlf
x - x2, etc., and is the product of them all,thus,
(x " xt) (x - x2) (x - xs). .
.
(x - xn) = 0.
180. Any equation the first member of which can be
separated into factors in the form just given is solved by-putting each factor equal to zero. For example, the equationa? + 3x* - 4x - 12 = 0, by separating the first member into
factors, becomes
(x + 3) (x + 2) (x - 2) - 0,
the roots of which are " 3,-2, and 4- 2.
Again, x8 + 4s2 + 4a; + 3 = 0, being reducible to the form
(z + 3) (Z2 + x + 1) = 0,
is solved by putting x 4- 3 = 0 and a? + x + 1 = 0.
Many cubic equations and equations of higher degrees
are easily solved by this method, if the first member is sep-arableinto factors of a degree not higher than the second.
181. If the factors of the first member of 179 be multi-plied
together, the result is
xn- (xt + x2 +
.#. .+ xn) xn~x + (xxx2+ x2x3 + . . .) xn -2
" \X-*X2Xq ~t~ "t'7 3 4 * * * / "' " " " "1 '*^i'*^2S ... Xfi ~"~ \J *
an equation which is identical with
xn 4- pxxn-1 + p2xn~2 4-. . .
+ pn = 0.
Therefore,
182. '
" py = the sum of the roots,
p2 = the sum of all the products of theroots taken two by two,
- p3 = the sum of all the products of theroots taken three by three,
.
.......
s
( - l)kpk = the sum of all the products of theroots taken k by k,
*( - l)nPn - the product of all the roots.
-
ALGEBRA 23
Cubic Equations.
183. To solve the general cubic equation
x3 + ax2 + bx + c = 0,
remove the second term by substituting for x an assumed
unknown, y - \a. The reduced equation takes the form
y* + w + q - 0.
*
184. The three roots of this last equation by Cardan'sRule are
2/1-^
- Iq+Vuqy+dp)* +"/-k-^(k)2+(i/")S
,.
-
l + tV3 -l-iV3wherein
^=
" w3 =
When the quantity dq)2 + ($p)8 is negative, the solutionmay be effected by means of circular or of hyperbolic func-tions
in the following way:
(1) When y* + py"q= 0, p and q being positive,computethe value of
"p from
Sinh^ i"_.
Then the roots are
fi/1=s=F2\/j^Sinh^,185. \y2 - " V"g Sinh \"p+ i Vp Cosh Jp,
[y3- " V^p Sinh ^ - i Vp Cosh Jp.
(it) When y* " py"q=0, p and q being positive and(ip)s " (bq)2"compute the value of
"p from
Cosh"p
"__
ipVip
-
24 MATHEMATICALHANDBOOK
Then the roots are
186.
0i -T 2v5pCoshi^,2/2 - " V^" Cosh Jp + i Vp Sinh "p,
2/s - " Vjp Cosh J^ - i Vp Sinh ^.
(Hi) When S/3- pr/ " g = 0, p and g being positiveand
(ip)8 " (M)2" compute the value ofthe angle "p from
cos"p " "
*2"
Then the roots are
it/i- =F 2VJ2 cos J?,2/3 = T2VJ2 cos (}p + 120"),yt - =F 2Vjp cos ftp + 240").
("0 When t/3- ?w/ " g = 0, p and g being positive,
and (ip)" = (fe)",
188. the roots are \Vl = T 2 x/*?)'_" 2/2 - 2/3 = " ^iP"
""
Series.
Arithmetic Series.
189. The n*1* term of the series
a, a + d, a + 2d, a + 3d, . . .
is a + (n - 1) d;
and the sum of n terms is
190.S = ^[2a+ (n-l)d].
Geometric Series.
191. The nth termof the series a, ar, ar2,ar9, ... is ar"-1
and the sum of n terms is
192.S = i^"
-
ALGEBRA 25
If the number of terms, n, be infinite and the ratio, r, be
a proper fraction, the series is convergent, and
193-.
S1-r
Harmonic Series.
194. The terms a, 6, c, d, etc., form a harmonic series if
their reciprocals" " "" " " " "etc, form an arithmetic series,abed
that is,when the relation subsisting between any three con-secutiveterms is
a a " b"^ as " " "
"
c b- c
195. The nth term in a harmonic series is
ab
(w-l)a- (n-2)b
196. The arithmetic mean between a and b = " ".
197. The geometric mean between a and b = Va6.
198. The harmonic mean between a and 6 = "a + b
199. A series partly arithmetic and partly geometric is
represented by
a, (a + d) r, (a 4- 2d) r2, (a + 3d) r8,etc.
The sum of n terms of this series,
o
_
a - [a + (n - 1) d\ r* rd(l - r^-1)1-r (1-r)2 "'
200. 1 + 2+3+4+5+. .
.+ (n-l) + n = n(n + *""
201. p + (p + 1) + (p + 2) + ...+ (?- 1) + q
_(q+V)(q-V+l)2
202. 2 + 4 + 6 + 8 +.
. .
+ (2n - 2) + 2n = n (n + 1).
203. 1 + 3 + 5 + 7 +. . .
+ (2n - 3) + (2n - 1) - ri".
-
26 MATHEMATICAL HANDBOOK
204. l" + 2" + 3"+4" +...
+n*-n"w+ 1H2n+ X".J..A.CJ
205. Is + 28 + 3s + 48 +. .
.
+ na=[n (n + 1)18'
Binomial Series.
206.
/1 1 w i 1 .w(n " 1)
o.
n(n " l)(n - 2).
,(1 " x)n " 1 " nx + " * '- x2 " "" ^ '- x* +...
2! 3!
Convergent if x2 " 1.
207.
(l"x)-n,iTwa;+n(!^Li)^Tn(n+l)(n+ 2)j"+_
Convergent if x2 " 1.
208. (a-6x)-'-I/'l+ ^ + ^+-^ + ...,\a\ a a2 a3 /
Convergent if Px2 " a2.
209. (l"x)-1=l =Fz+ x2=Far, + x4qFz5 +...
Convergent if x2 " 1.
210. (l"x)-2= 1 =F2x+ 3x2qF4a^+ 5x4=F6a^+. . .
Convergent if x2 " 1.
v 1 y I72.4 2.4.6 2.4.6.8
Convergent if x2 " 1.
2x2. (l"x)-i=l^,+||x^l|-^+||||x^...Convergent if x2 " 1.
213. (l"x)i=l"lx-^x2"^x*- 1'2m6m8a*"...0 ^ } ^ *3.6 3.6.9 3.6.9.12
Convergent if x2 " 1.
214. (1"i)^=1T^+- xt^"*** 1A71" x4+...* *
3.6 3.6.9 3.6.9.12
Convergent if x2 " 1.
-
ALGEBRA 27
Exponential and Logarithmic Series"
2l5'e-1+\+h+h+h+h + --
-
Limit of (l+ "Y for m= oo./*" /y*2 syrnt fwA
216. e* = l + -+-"-+" + " " +...
1 2! 3! 4!
[- 00 " X" + 00.
^_
"i
c2^2 cV c4x*217. al-1 + ca; +
^r+
-3r+-4j-+---
[-00 " x " +00.wherein c = log, a.
218. a*=i + ?ggLgx+ "lo"tg),a"+(h"'Oy +...
[-00 " x" + oo.
219. log, (1 " x) - " x - ia?" ix* - \x4" Jx5 - ..
.
[a?"l.
220. " loge-l" =^ * + Jx3 + Jx5 + jx7 +.
..
[x2" 1.1 " X
221. " log,?il = x-1 + JX-8 + iX"5 + |X"7 +.
..
[X2 " 1.X " 1
222. log. X
-2[^i+i(^)'+*(^M^)'+-l[0 " x " + 00
.
223. log, (a + x)
-logea+2[-^-+i(^-J+i(^-)5+...],[0 " a " + 00
,
- a " x " + 00.
224. log ( )= log (n + 1) - log n
_ 2 }1
4.i
I1
1
2n+l 3(2n+l)3 5(2n + l)5""
[0 "n" +00.
-
28 MATHEMATICAL HANDBOOK
225. log (x + Vl + X2)
* 2^ 2.4.5 2.4.6.7+
'" la?"1*
See Formulas 754 and 1036.
226. log, x - (x - 1) - \{x - l)s+\(x-\y-...
[0 " x " 2.
227. lo":c""^i+i(^iJ+i(^lJ+...
Interest and Annuities.
Let r be the rate, that is,
r " interest on one dollar for one year,
n = the number of years,
P=
the principal,A
=the amount in n years.
Then,
228. At simple interest, A = P (1 + nr).
229. At compound interest, A = P (1 + r)n.
230. If interest be compounded g times a year,
If A be an amount of money payable n years hence, and
P the present worth of A, then
;4
a=
p(i+
lY.
231. At simple interest, P "1 + nr
4232. At compound interest, P" -
(l + r)"
233. Discount " il - P.*
* This is Inte discount, so-called to distinguish it from commercial
discount, which, for commercial convenience, is based 01a simpler rule.
-
ALGEBRA 29
234. The amount of an annuity of
one dollar in n years at simple
"interest
235. Present value of such an an-)_
nuity ]
236. Amount at compound interest.
} "
237. Present value j "
_ ,
n(n--lL
n + $n(n" l)rm1 + nr
(l+r)"-l
(1 + r) - l'
1-
(1 + r)-".
(l+r)-l
238. Amount when the payments of
interest are made q times a
year
1
1
239. Present value
240. Amount when payments of the'
annuity are made m times a
year
241. Present value.
K)'- '(1 + r)"
-
1
m[(l+r)"-l]
1- (1 + r)
242. Amount when the interest is paid
q times and the annuity mtimes a year
243.
Present value
[(l+r)*-l]
m[(I+?M
-[KM
-
30 MATHEMATICAL HANDBOOK
Probabilities.
If there are a ways in which an event can happen, aad b
ways in which it must fail to happen, the chances (or odds)in favor of the event are said to be as a to b, and the
chances (or odds) against it as b to a.The probabilityof an event is the ratio of the number of
favorable chances to the total number of chances, both
favorable and unfavorable. In the case above stated,
" "" the probabilitythat the event may happen.
a + b
b the probability that the event may fail to244.
a + b happen.
The sum of these two probabilities is 1; and since the
event is certain either to happen or fail to happen,
245. Certainty = 1.
If p be the probability of an event, the probability that
that the event may fail is 1 " p.
If Ex and E2 are two possible and independent events, and
px and p2 are their respective probabilities,then
246. pxp2 " the probability that both Ex and E2 mayhappen.
247. 1 - pxp2 " the probability that not both Ex and E2
may happen.
248. (1 - Pi)p2 = the probability that Ex may fail and E2happen.
249* Pi(l ~~ V2) = the probability that Ex may happen andE2 fail.
250. px + p2 " 2pxp2 = the probability that one event
may happen, and the other fail.
251. (1 - px) (1 - p2) = the probability that both events
may fail.
The value of p may be determined, approximately at least,by observation of a large number of cases. Thus the expe-rience
of life assurance companies shows that out of 69,517
persons living in their fifty-first year 55,973 were living intheir sixty-firstyear. Therefore the probability that anassured person at the age of fifty may live ten years isthe ratio of these numbers, 0.805.
-
SECTION II.
CIRCULAR FUNCTIONS AND TRIGONOMETRY.
Definitions and Fundamental Relations with reference to an
Acute Angle.
Denoting the legs of a right angled triangle by a and 6,
the angles opposite them respectively by A and B, and
the hypotenuse by h, the functions of either acute angle
aredenned and expressed as follows:
/sin A
=--
= cos B,h
cos A =--
=
sin B,h
301
tan A=
"
=ctn B,
0
ctn A=
"
=tan B,
a
sec A ="
= esc B,b
,
csc A ="
= sec B.
a
The abbreviations are sin for sine, cos for cosine, tan foi
tangent, ctn for cotangent, sec for secant, and csc for cosecant
From these definitions follow at once the relations,
302
304.
306.
cosA1
303
305.
307
309
csc A
1
secA
1
ctn A
=
sin A,
= cos -4,
=tan A,
, ,icos
Actn A
=
-:
sin A
31
-
32 MATHEMATICAL HANDBOOK
And from the definitions together with the equation
h2-
a2 + V
follow the further relations,
313
'
sin (90"- it) - cos A,
cos (90" - ii) - sin A,tan (90" - A) = ctn A,ctn (90" - A) - tan A,sec (90" - it) - esc A,
"
esc (90" - A) = sec A.
Therefore, if the values of all the functions of each anglefrom 0" to 45" are given (as in the table on page 249), thevalues of the functions of all angles from 90" to 45" are
given also.The functions of acute angles as above defined, when com-puted
and tabulated, are sufficient for the solution of right
triangles in all cases. They are also sufficient for the solu-tion
of an oblique triangle,if the latter be concerted into
the sum or the difference of two right trianglesby drawinga perpendicular from a vertex to the opposite side or tothe opposite side extended. For methods of solution, see
pages 60-66,
General Definitions of Angle, its Measures, and itsFunctions.
314. An angle is any amount of turning in a fixed plane
by which a straight line may be changed from one direc-tion
to any other direction in that plane.If the turning amount to less than a quarter of a revolu-tion
the angle is a geometric acute angle; if to more than a
-
TRIGONOMETRY 33
quarter and less than a half of a revolution, it is a geometricobtuse angle; if to more than a half and less than a whole
revolution, it is a so-called convex angle.The turning may amount to more than one whole revo-lution
or to more than any number of whole revolutions
however great. Moreover, the turning may be one way,positive, or the other way, negative. Therefore the generalvalue of an angle is expressed by
315. "A"k 360", or " a " 2kw,
wherein k is any integer or 0.
Angles are measured in degrees, minutes, and seconds; orin units of arc-measure, called radians. The arc-measure
is the ratio of the arc to the radius, the arc being the whole
arc described by any point of the turning line, and theradius the distance of that point from the centre of revo-lution.
The arc-measure of one whole revolution is the
circumference of a circle divided by its radius, or 2"r. Theinfinite range of value which an angle, A, or its arc-measure,
a,takes may be thus expressed,
in degrees, - oo" " A " + 00 ".
in radians, - 00 " a " + 00.
The two measures of an angle are thus related,
6
- (1 Radian-
57" 17' 44" .806310
( 180" - ir = 3.14159265 radians.
A table for converting either kind of measure into the
other is given on the next page.As a matter of notation in the following pages, capital
italic letters will, in general, indicate that the angles are tobe expressed in degrees, minutes and seconds, while Greek
letters or small italics will indicate that they are to be
expressed in arc-measure or radians. It is, however, in
many formulas a matter of indifference which notation is
used.
-
34 MATHEMATICAL HANDBOOK
3i7" TABLE
(a) For finding the Length of the Arc measuring any given Angle in
a Circle of which the Radius is i.
(6) For finding the Angle measured by any given Length of Arc ina Circle of which the Radius is 1.
-
TRIGONOMETRY 35
Functions of the General Angle Defined.
Drawing rectangular axes, xx' horizontal and y y' vertical,intersectingin o, and the line op in any required direction,let ox be the initial side and op the terminal side of any
angle whatever (denned as in 314).It is evident that op may fall in any one of the four
quadrants, the first xoy, the second yox', the third x'oy',or the fourth y'ox.
r
FlGUBB 1.
Let the coordinates of the point p in any situation be
x " the abscissa, or distance of p to the right or left
of the vertical axis,
y " the ordinate, or distance of p above or below the
horizontal axis.
Let r " the distance of p from o.
Then are the six functions of A (any angle whatever)defined as follows:
.,
ordinate ysin A =" " " *-,
318. cos A
tan A "
distance rabscissa
__
x
distance r'
ordinate yas
2-
abscissa x
ctn.A
sec A
esc A
abscissam
x
ordinate ydistance
_
r_
abscissa xdistance
__
r
ordinate y
-
36 MATHEMATICAL HANDBOOK
An angle is said to be an angle of the first,second, third,or fourth quadrant according as its terminal side falls inthe first,second, third, or fourth quadrant.
The functions of angles of different quadrants have posi-tiveor negative values dependent on the values of the
coordinates used in the definitions 318. The abscissa is
positive or negative according as p is to the right or leftof the vertical axis; the ordinate is positive or negativeaccording as p is above or below the horizontal axis ; thedistance op is positive in all situations.
Hence the values of the functions of angles of the several
quadrants are positive or negative as indicated below.
310
As an angle increases from 0" to 90", 180", 270", 360",
etc., its functions vary, some increasing, some decreasing,but all reaching maximum or minimum numerical values
for the cardinal values of the angle above mentioned.
When the function passes through the value 0 or 00 it
changes its sign, as is indicated in the following table.
320. Cardinal Values.
-
TRIGONOMETRY 37
Fundamental Relations Generalized.
From the definitions 318 follow at once the relations
321
tan A "
ctn A =
sin A
cosA!
cos A.
sin A'
and from the equation x1 + y2 " r2 follow the relations
322
sin2 A + cos2 A = 1,
1 + tan2 A = sec2 A,
1 + ctn2 A = esc2 A,
which are identical with 302-312, as they should be; but
these are applicable to angles (or arcs) of all magnitudespositive or negative, while those relate only to positive acute
angles.From 322 result six radical forms,
323
"Vl
-
cos2 A,
"Vl- sin2 A,
" Vsec2 A - 1,
"Vcsc2 A-l,
sec A - " Vl + tan2 A,
esc A - " Vl + ctn2 A.
sin A
cos A
tan A
ctn A
The interpretation of the double signs of these radicals
is found in the fact that to a given value of any one func-tion
belong two angles between 0" and 360"; and the other
five functions of these two angles are numerically equal each
to each; but four of them have opposite algebraic signs.These four are the ones which are given by the quadraticsolutions. The fifth is the reciprocal of the given one, and,like that, has the same value for the two angles.
-
38 MATHEMATICAL HANDBOOK
Anti-Functions.
If sin A= x, or
sin a = x, then A is the angle the sine of
which isx, or a
is the arc the sine of which is x, a relation
usually expressed by the notation
A=
sin - lx, or a =
sin-1x,
and read " A (or a) is the anti-sine of x."
Some writers use the notation arc-sin x instead of sin- * x.
Similarly, if tan B= y, B = tan-1 1/
sec C = z, C = sec-1 2.
The value of sin-1 x is not only A, as given above, but
any one of the infinite series of angles included in the gen-eral
expression A " k 360", or a" 2kir.
Hence, k being any integer, including 0,
rsin-1 x = A " k 360" - a " 2kir,
324.\
tan"1 y = B " k 360" = p " 2k*,Isec-1
z=C" A; 360"= y "
2far.
Also,
rcos-1x = (90"
-
A)"k 360"=
(""r-
a) " 2kir,
325.J
ctn-1 y - (90" - B) " * 360" = (J*- - 0) " 2far,Use"1
z-
(90"-
C) " * 360" = ("tt-
y) " 2far.
Whence
rsin-1 x 4- cos"1 x = 90" " k 360" - ^" 2for.
326. J tan-1 2/ 4- ctn-1 y = 90" " k 360" - ^r " 2kir.Isec-1
2 + esc-1 z = 90" " A; 360"-
^r" 2for.
-
TRIGONOMETRY 39
\/2=i.4i42i36
J\/2"=o. 7071068
\/a-y/2 = o. 7653669V2 + v/2 = 1.8477587
v/3 " 1.7320508
Jv/3=0-57735""3
K/3gai'i547""5Va" v/3-0.5176381V2+\/3=i-93i85i6
-
40 MATHEMATICAL HANDBOOK
Formulas expressing each junction in terms of each of the
others.
328. sin A - "Vl - cos2 A^4
"Vl + tan2 A
1 "Vsec2 A -I1
"Vl + ctn2 A secA esc
A
329. cos A -"Vl - sin2 A "
"Vl + tan2 A
ctn A_
^_
jVcsc2 A - 1"Vl + ctn2 A sec A esc A
330. Ua4^"
-
"Vl-
"*A- l
"\/l " sin2 A cos A ctn A'
-"Vsec2 A - 1 - iVcsc2 A - 1
.
,
"\/l-sin2A cos A331. ctnA = " :
"
-
sin A "Vl - cos2 Atan A
1
, */
"
T~a 1= "Vcsc2 A- Y
"V sec2 A - 1
-1"
332. sec A-
t
,.
=" --
"Vl + tan2 A,"vl - sin2 A cos A
"Vl + ctn2 A escA
ctn A "Vcsc2 A - 1
,
11 "\/l + tan2 A
333. cscA=- "-
=
,
/i " 2 A"" " 7
sin A "v 1 - cos2A tan A
/ t" ;sec A
- " v 1 + ctn2 A = / " ,"
"VW A - 1
-
TRIGONOMETRY 41 "
Positive and Negative Lines.
If the distance from a point a to any other point b ona straight line be reckoned as positive, then the distance
from b to a must be reckoned as negative; so that it isw
always true that
334- ab + ba = 0.
Let three points a, b, and c be arranged in any order on a
straight line. Then the algebraic sum of the distances aband bc is always ac, that is,
335- ab + bc - AC,
which by adding ca to each member becomes
336. ab + bc + ca " 0.
The same principle applies to any number of points,
arranged in any order whatever on a straight line,and their
distances, that is
337. AB + BC + CD -f.
.
.
+ MN + NA " 0.
Projections.The projections of a line ab upon the axes of x and y are*
-
( ab cos A = projection on the axis of x,Iab sin A " projection on the axis of y,
wherein A denotes the angle between the positive directionof the axis of x and the positive direction of the projectedline.
The sum of the projections of the sides of any closedpolygon, taken in order around the polygon, upon anychosen line is equal to 0.
In the case of a triangle abc placed anyhow in the planeof the axes ox and oy, if a', b', c' be the projections of thepoints a, b, c on the axis of x and a", b", c", the projec-tions
of the same points on the axis of y, then, whatever
the order in which the projections fall on either axis,
(a'b' + bV + c'a' = 0.339#
/ aV + bV + c"a" - 0.
-
42 MATHEMATICAL HANDBOOK
If the axes be rectangular and abc a right-angled tri-angle,
these equations give the formulas for the sine and
the cosine of the sum and of the difference of two angles.
Positive and Negative Angles.
If the angle aob be reckoned as positive, then the angle
boa must be reckoned as negative ; so that it is alwaystrue that
340. aob + boa = 0, or = " k 360", or = " 2kv.
Also, whatever the order of the lines radiating from o,
341. aob + boc = aoc, or = aoc " k 360",
or = aoc " 2kir.
342. aob + boc + coa = 0, or = " k 360", or = 2kv.
343. AOB + BOC + COD +. . .
+ MON + NOA " 0,
or = " k 360", or = " 2for.
Functions of the Sum and of the Difference ofTwo Angles.
344. sin (A + B) = sin A cos B + cos A sin B.
345. sin (A - B) - sin A cos B - cos A sin B.
346. cos (A + B) " cos A cos B " sin A sin B.
347. cos (A " B) = cos A cos B + sin A sin B.
tan A + tan B348. tan (A + B) =
349. tan (A - B) =
1"
tan A tan B
tan A"
tan B
1 + tan A tan B
4. / a , d\ctn B ctn A " 1
350. ctn (A + B) = "
351. ctn (A - 5) =
ctn i? + ctn A
ctn 1? ctn A + 1
ctn B-
ctn A
-
TRIGONOMETRY 43
Functions of the Sum of Three Angles.* #
352/ sin (A + B+ C)= -
sin A sin B sin C 4- sin A cos J5 cos C
4- cos A sin B cos C + cos A cos i? sin C.
353- cos (A + J5 + C)" cos A cos 5 cos C - cos A sin 5 sin C
"
sin A cos B sin C " sin A sin 5 cos C.
354
Functions of a Negative Angle.
355-
356.
Functions of A + 90".
-90".
357
Functions of 90" " A.
sin (90" - A) = cos A}cos (90" - A) = sin Attan (90" - A) = ctn A,etc. '
358. -
Functions of A + 180".
(sin (A + 180") = - sin A,cos (A + 180") = - cos Aytan (A + 180") = tan A,etc.
-
44 MATHEMATICAL HANDBOOK
359
^sin (A
cos (Atan (Aetc.
Functions of A- 180c
180") - - sin A,180") - - cos A,180") - tan A,
360.
Functions of 180" -A.
sin (180"cos (180"tan (180"etc.
A) =" sin A,A) = - cos A,A) = - tan A,
361.
Functions of A + 270(
sin (A + 270")cos (A + 270")tan (A + 270")etc.
" cos A,
sin A,
-
ctn A,
362.
Functions of A- 270".
(sin (Acos (Atan (Aetc.
270")270")270")
cos A,
-
sin A,
-
ctn A,
363.
Functions of 270" -A.
Functions of A" 360c
364.
sin (A " 360") - sin A,cos (A " 360") - cos A,tan (A " 360") - tan A,etc.
-
TRIGONOMETRY 45
365.
Functions of 360" -A.
sin (360" -A) -- sin A,cos (360"- A) = cos A,tan (360" -A) - - tan A,
^ etc.
Solution of equations sin A=* a, cos A = a, and tan A " a.
If A is to be found from a given value a of its sine, that is,
if the equation sin A = a is to be solved for A, all the values
of A are given by the formula
366. sin-1 a = k 180" + ( - 1)*A,
wherein A; is 0 or any integer.In the same way, all the values of A obtainable
from the
equation cos A = a are given by
367. cos-1 a - k 360" " A.
And all values of A obtainable from tan A = a are given by
368. tan-1 a - " 180"+ A.
Sums and Products of Functions.
sin (A + B) + " sin (A - B).
sin (A + B) - " sin (A - B).
cos (A - B) - " cos (A + B).
cos (A - B) + " cos (A + B).
(A - B) - sin2 A - sin2 B,"
cos2 2?-
cos2 A.
(A - B) - cos2 A - sin2 ",= cos2 B - sin2 A.
2 sin " (A + B) cos " (A - B).
2 cos " (A + B) sin " (A - B).
2 cos " (A + ") cos " (A - B).
-
2 sin " (A + S) sin " (A - B),
-
46 MATHEMATICAL HANDBOOK
sin A + sin B_
tan j (A + B)sin A
"sin B tan $ (A " B)'
380.cos
^~ cos **
--
tan i (A + B) tan * (A - ").cos A + cos #
381.sin AA"sinI*
-ta.nl (A"B).cos A + cos B
3g2" sin^Tsing__ctnH^"g)cos A " cos 5
gsin (A T B) sin j(ATB)
3 3*on 4 " sin "
~
sin } (4 " ")'
g .sin (A T g) cos I (A T B)
.
3 4'sin A T sin 5
"
cos * (A " B)
385. tanA"tang-8itt(,A"B:"-cos A cos B
386. ctng"ctnA-.8?n^"g"-sin A sin 5
387. ctn 1? " tan A -C08 "A * g"
"
cos A sin zj
a88sin (A"B)
=
tan A " tan Z?=
ctn B " ctn A#
sin (A =F -B) tan A ^ tan 1? ctn B =F ctn A
~
cos (A"B)_
1^ tan A tan B^
ctn 1? T tan At3 9"
cos (A q= B)~
1 " tan A tan B~
ctn B " tan A
cos (A =f 5) ctn B " tan Aqoo. " !" " " "oy
sin (A"B) ctn"tanA"l
391. cos2 A + cos2 2? - 2 cos A cos B cps (A -h 5)-sin2 (A + 5).
"^
tan A + tan B,
^
A .^
D302. " : = tan A tan B."*
ctn A + ctn B
393. sec2 A + esc2 A = sec2 A esc2 A.
-
TRIGONOMETRY 47
Functions of Multiple Angles.
394. sin kA - 2 sin (Aj- I)A cos A - sin (k - 2)A,=
2 cos (k - 1)A sin A + sin (" - 2) A.
395. cos "A " 2 cos (" - 1)A cos A - cos (A;- 2)A,= "
2 sin (" " 1)A sin A + cos (A;" 2)A.m
396. tanfcA-tan (t - 1)A + tan A
1"
tan (k " 1) A tan A
397. sin 2A = 2 sin A cos A.
sin 3A= 3 sin A
-
4 sin3 A.
sin 4A=
4 sin A cos A - 8 sin8 A cos A.
sin 5A-
5 sin A-
20 sin3 A + 16 sin5 A.
sin 6A" 6 sin A cos A " 32 sin3 A cos A
+ 32 sin5 A cos A.
398. cos 2A - 2 cos2 A - 1.
cos 3A " 4 cos3 A - 3 cos A.
cos 4A = 8 cos4 A - 8 cos2 A + 1.
cos 5A = 16 cos5 A - 20 cos3 A + 5 cos A.
cos 6A - 32 cos6 A - 48 cos4 A + 18 cos2 A - 1.
399. tan2A = -^55^-.1
-
tan2 A
4*w% +""q"i3 tan A -tan3 A
400. tan 6A = " "1
-
3 tan2 A
" ~ _"i. o a
ctn2 A"
1 1"
tan2 A ctn A " tan ^401. ctnzA = " = = "
2 ctn A 2 tan A 2
**"* 0^ o jsec2 ^ ctn A + tan A
402. sec l A " = "1
-
tan2 A ctn A -" tan A
403. esc 2A = \ sec A esc A = " (tan A + ctn A).
404. 1 + sin 2A = (sin A + cos A)2.
405. 1 - sin 2A = (sin A - cos A)2. ";
406. 1 + cos 2A = 2 cos2 A.
407. 1 - cos 2A = 2 sin2 A.
408. esc 2A + ctn 2A =" ctn A.
-
48 MATHEMATICAL HANDBOOK
Functions of Half an Angle.
409. 1 " sin3 "A + cos3 $A.
410. sin A " 2 sin \A cos \A.
411. cos A " cos2 "A - sin2 ^A.
412. 1 + sin A =" (sin \A + cos "A)2.
413. 1 - sin A = (sin "A - cos ^A)3.
414. 1 + cos A = 2 cos2 Jit.
415. 1 - cos A = 2 sin2 "A.
416. sin "A -Vj(l - cos A),=jVl "+ sin A - jVl - sin A.
417. cos "A -V-"(l + cos A),
-
iVl + sin A + iVl - sin A.
418. tan"A=i/" " cosA 1
" cos A sin A
419. ctn
1 + sin A " cos A
. . . .
2 secA420. sec
sec A + 1
. " _ .
2 sec A421. CSC
sec A - 1
422. 1 + sin A - 2 sin2 (45" + \A) - 2 cos2 (45" - JA).
423. 1 - sin A = 2 sin2 (45" - JA) - 2 cos2 (45" + "A),
424. tan (45" " A) - ctn (45" T A) =1"tan^ ,1 T tan A
V 1 zb sin 2 A ^ cos A " sin Al:Fsin2A cos A =F sin A
"J
-
TRIGONOMETRY 49
425. tan (45" " *A) - ctn (45" T * A) - y/l"J5|L4,_l"SinA^seCil"tanA
cos A
cos A
1 =F sin A
426. tan (A - 45") =*an AA" *
.
tan A + 1
427. sin (45" + A) - cos (45" - A) -sin A + cob A
428. cos (45" + A) - sin (45" - A) -cos A
"im A.
V2
429. tan (45" + A) + tan (45" - A) - 2 sec 2A.
430. tan (45" + A) - tan (45" - A) = 2 tan 2A.
431. tan (45" + A) tan (45" - A) - 1.
432. sin (30" + A) + sin (30" - A) = cos A.
433. sin (30" + A) - sin (30" - A) = V3 sin A.
434. cos (30" + A) + cos (30" - A) - \/3 cos A.
435. cos (30" + A) - cos (30" - A) - - sin A.
Expressions Equivalent to sin A.
436. sin A " Vl - cos2 A =V(1 + cos A) (1 - cos A),
-cos^tanAcos A i"nA X
ctn A sec A esc A'
tan A 1 g\/sec2A-lrVl + tan2 A Vl + ctn2 A sec A
-
V^(l- cos 2A) =2 sin "A cos ^A,
2 tan frA 1
1 + tan2 iA ctn %A - ctn A
1 2
tan "A + ctn A tan "A + ctn \A
-
2 sin2 (45" + \A) - 1,
~1
-
2sin2 (45"- iA) -1
-
tan2 (45 " " ^A)1 + tan2 (45" - JA)
-
150 MATHEMATICAL HANDBOOK
Expressions Equivalent to cos A.
437. cos A - Vl - sin2 A = V(l + sin A) (1 - sin A),a\" a "*" a
sin ^ ctn A 1"= sin A ctn A = = "
,
tan A esc A sec A
_
ctn A 1 Vcsc2 A- 1
.
Vl + ctn2 A "VH tan2 A"
esc A
-VKl + cos2^)-^t2 sin A
= cos2 %A - sin2 \Ay
- 1 - 2 sin2 \A = 2 cos2 JA - 1 -* " tan2 *A
,
1 + tan2 \A
^
ctn2 \A - 1=
ctn^A -tan^Actn2 "A + 1 ctn "A + tan JA
1 1
tan A ctn \ A - 1 1 + tan A tan \A'
__J 2"
tan (45" + IA) + ctn (45" + }A) '
- 2 cos (45" + iA) cos (45" - *A),=
cos4 "A " sin4 \A.
Expressions Equivalent to tan A.
438. tan A - 23-4- " ^ - ^sec2 A - 1,
cos A ctn A
-
*
_
sin^=
Vl- cos2 At
Vcsc2 A- 1 Vl - sin2 A cos A
-v/1 - cos 2A sin 2A 1 - cos 2A1 + cos 2A 1 + cos 2A
=
sin 2A
= esc 2A - ctn 2A = ctn A - 2 ctn 2A,
2tan^A_
2 ctn jA=
1 - tan2 \A ctn2 \A - l'
=
2
ctn \A " tan \A '
tan (45" + jA) - tan (45" - jA)2
_
tan (45" + A) - 1_
1 - tan (45" - A)=
tan (45" + A) + 1 1 + tan (45" - A)
-
TRIGONOMETRY 51
Expressions equivalent to ctn A, sec A, and esc A arethe reciprocals of those above given for tan A, cos A, and
sin A, respectively,
Functions of Periodic Values of the Arc or Angle.
In the following equations, k is any integer positive,negative or 0.
sm Kir = 0,
439. \cos kir - ( - 1)*,
tan kir = 0,
sin
cos
2
2k+lir=0,
x
2fc+ltan it = oo
.
440. sin ^" = " sin (2"w " p) = =F sin [(2k + 1) it " "p],
/4"+1,
\J
/4*-l,
\- =Fcosf" " Tri^j= "cosf" " jt" p J.
441. esc "p= " csc (2far" ^")= etc.
442. cos ^ " cos (2far" "p)= - cos [(2A;+ 1) w " "p],
- sin f4"+1
"" ?") = - sin (4*-l
*"
V
443. sec "p= sec (2far:t ^) " etc.
(2k+ 1 \
445. ctn "p = " ctn (fori y")= etc.
The formulas 440, 442, and 444 give the only solutions
of the equations.
sin"p=" "sin a,
sin"p = " cos a,
and of the equivalent equations.
tan"p "- " tan a,
tan^
= " ctn a,
csc^"
" " csc a,
csc p = "sec a,
ctn^
= " ctn a,
ctn^" "tan a.
-
62 MATHEMATICAL HANDBOOK
If any two of the six elementary functions (not beingreciprocals of each other) have equal values for "p and athe only solution is
"p = 2kir + a.
Inverse Circular Functions.
446. sin-1 x = cos-1 Vl - x2 = tan-1x
Vl-x2'
-1447. cos-1 x = sin
ctn-1 -Vl-
x2=
sec-1,
1=
esc-1-*
s Vl-x2 *
2 sin-VjCl-^l-x2)= i sin-1 (2x Vl-x2),
2 tan-11-1*13
. j tan-2xVTHZ
.
x
*
1-2X2
i^r - cos-1 x " "*" - sin-1 Vl - x2
"
sin-1 ( " x),
iir+i sin-1 (2x* - 1) - i cos-1 (1 - 2X2).
dn-1 Vl-x2-
tan-1 -Vl-x2x
ctn-1"
x
=sec-"1
""
Vl-x2 *
CSC"1 "J=,-
2 cos-1 V* (1 + x),Vl-x2
I cos-1 (2X2 - 1) - 2 tan-\ /Ll_?V 1+x
-VB2 ctn-14 ,A^Xx
^^tan-f^1-^,*
V 2X2 - 1 /
= \k " sin-1 x = *r - cos-1 ( " x),
- \tr - COS-1 Vl - X2.
-
TRIGONOMETRY 63
448. tan-1 x = sin-1 "=
Vl + x2cos
"
1
Vl + x2,
x
" sec Vl + ^-csc-^Vl + z2
"Jtt" tan-ii__
x
tan-1 ( - x),
2xi tan-1 = hr- ctn_1x,
1-
x2
1 " 1^X
1_
1X -" X
* sm-1
"= * cos-1 "*
1+x2*
1 + x2
2 cos V1 + Vl + X22Vl
+ X2
2 sin-JH^V 2V1 + X3
X2
=2 tan-
449. sin-1 x + sin-1 y
_t/-! + Vl + x2
x)
-
sin-1 (xVl-yt + y Vl - x2),- ir - sin-1 (x Vl -i/2 + y Vl - x2).
"
1 -1450. sin-1 x - sin-1 y
-Sin-1 (xVl-^-yVl-x2).
-1451. COS_1X" COS-1!/
45l-xy
453. tan-xx+ tan-1 1/ = w- tan-1-^-^-^,xi/~l
"x2 + 2/2a^l
"cos-1 (xi/T V(l - x2) (1 - y2)). J
2. tan-1 x + tan-1 v = tan- * " " H-* [xy"l.
[^2/^1
454. tan-1 x - tan-1 y = tan-1 " "^
"
1 + xy
-
54 MATHEMATICAL HANDBOOK
Relations of Circular, Exponential, and Logarithmic Func-tions.
455. e** - cosx + tsinx.
456. e_"x - cos x - i sin x.
457. (cos x + i sin x) (cos x - 1 sin x) = 1.
^1 + itan*.1 " i tan "
.
459. cosx = i(efa-he_fa).
460. sin a: - - # (e* - e"*), or t sin x - \ (e* - "r*).
.6*"-
1461. tang'-t^ ^ or t tan a: =
"2tx-
1
e,~ + 1
462. cos-1 x i loge (x + iVl- x2).
463. sin-1 x = - i loge (ix-h Vl - x2).
464. tan-x=-inoge^-^logej^g= iiloge^x^ ^ 1
" IX \+ IX
465. cos ix = " (e*'+en*) - Cosh x.
466. sin ix = ii {? - e~x) = i Sinh x.
% " x
467. tan ix = ?!(e* e X)
=iTanhx.* '
e*+e-*
See 714.
468. 6x+tV = e* (cos-1/ + i sin t/).
469. ax+iv = ax [cos (2/loge a) + i sin (y loge a)].
From 2 cos u = eitt+ e'*",
2isinw= 6*a-e-iM,
are obtained,
470. 2"-1 cosTO u = cos mu + Ct cos (m - 2) w+ C2 cos (w - 4)t"+ C3 cos (m - 6) u +
.
. .
V-
l)^"-1 sinw u = cos mw - Cx cos (m- - 2) u+ C2 cos (w-4) W-C3COS (m-6)u+
. .
.
when m is even,
(\w"" 1-
l ) v~2m~1 sinmu = sin mw - Ct sin (m-2)u+C2 sin (m. - 4) w - C3 sin (m - 6) w +
. .
.
when m is odd,wherein CVC2,C3... are the Binomial Coefficients.
47i
-
TRIGONOMETRY 55
473. (cos x" i sin x)n " cos nx" i sin nx.
",;
" 1" . " x +2kir
.
.
.
x + 2for
474. vcos "" ^ sin x " cos " 1 sin "
n n
475. sin (x + ty) - i(e^ + erv) sin a; + %i (e* - e-v) cos x.
476. cos (x+ it/) " i(eV + e~"v) cos x "" (eV ~~ e~v) sm x-
477. loge (*"iy) - i log. (z2 + y*) + i (tan- "^ 2faA,
whena:
is positive.
478. loge (x" ty)-
iloge (a? + tf) +t7ten-1J"(2fc + 1) A
whenx
is negative.
479. log, (Z""\ - 2t (tan-1 " " 2far).\x
-
ty/ a;
480..
rloge ( + 1) = " 2kiri,
lo"i -"(2*+J)iri,
log.(-l)-"(2i + l)wi,
log,(-i)-"(2*+i)"t.
481.
e(2* + *)iri_
t-=
^
e(2A + l)iri 1-t*,482#
g*^
gtt+2Jbri
"* + "2*+iMf
=_
gtf+^Jfe+lM
_
J0* + "2fc + |)*".
Let^
beany variable, real or imaginary; and let r be its
modulus and 0 its argument. Then,
483. z - r (cos 0+i sin 0) = re* - re"tf+2*ir)i,
484. log, 2 - log, r + (0 + 2for) i.
-
66 MATHEMATICAL HANDBOOK
PLANE TRIANGLES.
General Properties.
Formulas expressing the general properties of triangles
usually occur in sets of three, of which only one needs to be
printed. The others are obtained from it by a cyclic changeof letters,that is,by changing a to 6, 6 to c, and c to a, alsoA to B, B to C, and C to A, always in this fixed order.
This process applied to the first equation of 501 gives the
second, applied to the second gives the third, and appliedto the third gives the first again. And so in all cases.
ca = b cos C + c cos B,
501. \b = c cos A + a cos C,[c
=" a cos B + b cos A.
All the relations between the six parts of a plane triangleare implicitly contained in, and can, by algebraic transfor-mations,
be derived from the three equations, 501.
502. A +B+C = 180".
a b c503
504.
sin A sin B sin C
a+b=
tan \ (A + B)m . .
@*a-b ta,ni(A-B)
K0"^"_6
=
cos iU-B). . .
".""
c cos i (A + B)
506. gLz" = sinHA-") . . . (3).c sin J (A + B)
507. a2 = b2 + c2 - 26c cos A. . .
(3).
508. a2 + ft2+ c2 - 26c cos A + 2ca cos 2? + 2a6 cos C.
* The symbol " indicates that there is a full set of three equationsof which only one is printed, the other two being obtainable from thisone by a cyclicchange of letters.
-
TRIGONOMETRY 67
If d denote the diagonal of a parallelogram drawn fromthe point where the sides a and b meet making an angle Cwith each other,
509. d? = a2 + 62 + 2ab cos C.
510. Let s=i(a+b + c),
whence, s-a=$(-a+b + c),8 - b = % (a - 6-f c),
s - c = " (a + 6 - c).
Six. sin }A - yAs~ W8 - c" " " """.
512. cos \A -i/s(s"a) " " -"-6c
513. tani-i- "/("-fc)("-c)" " .".V s (s - a)
.
Let r denote the radius of the inscribed circle.
514. r.y/("-")("-ft)("-"0,- s tan "A tan ^Z? tan \C,
=
a sin fr# sin \C. . .
(3).cos "A
5i5- tan|A ". . .
(3).s " a
Let T denote the area of the triangle.
516- T = sr = Vs (s - a) (s - b) (s - c),
- i V^c2 + 2c2a2 + 2a262 - a4 - b4 - c4,
" cos \A cos ^2? cos \Cya+ b + c
- 1 (a + 6 + c)2 tan "A tan $B tan "C,c2 sin A sin 5
2 sin (A + 5)= "a6 sin C
...
(3).
OT7 9"r
517. sin A - " - " " "-fa
" '
be be ""
-
68 MATHEMATICAL HANDBOOK
Let Pat pi, pe denote perpendiculars from the vertices
upon the sides a, b, and c respectively. Then
-
o t " n ' r"be sin A /"\
518. pa =* 0 sin C =* c sin B =" " " (J) "a
sin 5 sin C"
2T^
:" a= "
. .
.(").sin A a
Let ra, r6, and rc denote the radii of the three escribed
circles touching externally the sides a, b, and c respectively.
519. r"- " tan jA " ^ " ^ . " coe |B cos jC @s - a s " a cos 4 4
520. T" Wraryc.
521. rar6rc = a"c cos "^4 cos \B cos "C.
522. r - Vrarb + VV6rc + Vrcrfl
523. tan "4 = i/^5..
.".
M.
1.1.1 1 1.1.1524, " l 1" = " = 1 1
ra rb re r pa Pt Pc
Let R denote the radius of the circumscribed circle.
525. R = \a esc A
= iV(b + c)28ec2iA+(b-c)2c8c2$A..
.(f),
= Js sec \A sec %B sec "C,
-i (ra+ r6 + rc-r)--jy-
526. 2flra5c
a+ 6+ c
527. iZ + r " ^ (a ctn A + " ctn J5 + c ctn (7),= sum of the perpendiculars to the sides from
the centre of the circumscribing circle.
528, ^R? - 2Rr = distance between the centres of theinscribed and circumscribed circles.
-
TRIGONOMETRY 59
If o be the centre of the inscribed circle its distance from
the vertex a of the triangle is
520. oa=
cos "A. .
.
".
a+b+ c
a,
d,
n 1.
2a sin 2? sin C (o\530. cos A + cos B + cos C " 1 H : " " "W'
a + b + c
531* a cos A + b cos 2? + c cos C " 4R sin 4 sin 2? sin C,
=2a sin 2? sin C
.
.
.(3).
If A + " + C - 180", then follow 532-540,
532. sin A + sin B + sin C = 4 cos |A cos %B cos "C,533. sin A + sin 5 - sin C " 4 sin "A sin "J5 cos \C"
534. cos A. + cos B + cos C = 4 sin $A sin "2? sin JC + 1,
535. cos A + cos B - cos C = 4 cos %A cos "2? sin "C- 1,
536. tan A + tan 2? + tan C " tan A tan B tan C,
537. ctn \A + ctn "2? + ctn J C " ctn "A ctn "2? ctn "C,
538. sin 2A + sin 2B + sin 2C " 4 sin A sin 2? sin C,
539. cos 2A + cos 2B + cos 2C " - 4 cos A cos J? cos C" 1,
540. ctn A ctn 2? + ctn B ctn C+ ctn C ctn A = 1.
If A + J3+C-900,
541. tan A tan B + tan 2?'tan C + tan C tan A - 1.
The Quadrilateral Inscribed in a Circle.
542. Angles, A + C - 180", Sides, ab " a, cd " c,J5 + D - 180". bc - b, da - d.
. Da2 + fe2 - c2 - d2
543' C0Sjg"2(a6 + cd)
'
544. Diagonal, Tc2 - (ac+6f)(af+6c)-ab + cd
%
Put s = * (a+ 6 + c+ d), X
:/.u
.1
o
545- 0 -\/(8 - a) (s - 6) (s - c) (s - d)" area of the quadrilateral.
-
60 MATHEMATICAL HANDBOOK
20546. sin A - sin C - - " ^-r- "
bc+ da
20547. sin B = sin Z) -
^"
ao + ca
Radius of the circumscribed circle,
548. R - -=- V(a6 + erf)(ac + 6d) (ad + 6c).
Solution of Right Triangles.
549. Case I. Given an angle and the hypotenuse,A. and ft,to find B, a, and 6.
B=
90"-
A.
"
a = " sin A. log a = log ft + log sin A.
6=
ft cos A. log 6 = log ft + log cos A.
Test. The computed values of a and 6 should satisfy
2 log a - log (h+b) + log (ft- b),2 log 6 = log (ft+ a) + log (ft- a).
550. Case II. Given an angle and the leg opposite,A and o, to find B, ft,and 6.
B=
90"-
A.
ft= a esc -4. log ft = log a + log esc A.
b= a ctn A. log 6 = log a + log ctn A.
Test. The computed values of ft and b should satisfy
2 log a = log (ft+ 6) + log (ft- b),2 log b - log (ft+ a) + log (ft- a).
551. Case III. Given an angle and the leg adjacent,A and 6, to find By ft,and a.
J5-
90"-
A. v"
ft=
b sec A. log ft = log b 4- log sec A. w ft
a = b tan A. log a = log 6 4- log tan A.
Test. The computed values of ft and a should satisfy
2 log a = log (ft4- b) 4- log (ft- "),2 log 6 - log (ft4- a) + log (ft- a).
-
TRIGONOMETRY 61
552. Case IV. Given the hypotenuse and a leg,A and a,to find A, B, and b.
"
sin A- cos B = ",
lo" sin A " loS cos B
h" log a+ co-log h" 10.
whence ii-sin-"",h
6"
\/{h + a) (A - a), log b - " [log (^ + a) 4- log (h - a)].
Test. The computed values of A and b should satisfy
log b 4- log tan A = log a.
553. Case V. Given the two legs, a and b7 to find A, B,
and A.
+ aa 1"S tan -A " log ctn B
tan Jx =s ".
iiii-k
0 " log a 4- co-log 6 " 10.
ctn B ="
.
0
4= tan - " "
.
b
B-ctn-1?-.
^= a esc .4, log h = log a 4- log esc A.
-VW"2.
Test. The computed value of A should satisfy
2 log a - log (h 4- 6) 4- log (A - 6),2 log 6 = log (h 4- a) 4- log (^ - a).
Special Formulas for Plane Right Triangles.
554. h - b = 2h sin2 "A,
gives h " b with great accuracy when A is small, or A with
great accuracy when h and 6 are nearly equal.
555. tan \A - y/|-fr_ft- 6A 4- 6 a \
-
62 MATHEMATICAL HANDBOOK
55"". tan (45" " A) - """oT a
557. BinCg-^^^+y-^cos^-^-^.
558. t"n (B - A) = {b + a) (b ~ a\2ab
a formula which gives B-A with great accuracy when aand 6 are given nearly equal, or b - a with great accuracywhen A and B are given nearly equal.
For the use of S and T7,and a table of the values of these
functions up to 2", see page 83.
Solution of Plane Oblique Triangles.
An oblique triangle can be solved in two ways, (1) byconverting it into the sum or the difference of two righttriangles, formed by drawing a perpendicular from anyvertex, and solving these right triangles by methods above
given (549-553), or (w) by substituting the given parts in
general formulas, and working out the required parts.Either method serves well to test the accuracy of the results
obtained by the other. The outlines of solutions by bothmethods are given in the following formulas.
To preserve a uniformity of notation let the perpendicu-larsfrom the vertices a, b, and c upon the sides a, b, and c
of the triangle be marked ap, bq, cr respectively; so that
always
559- a = bp 4- pc, b = cq + qa, c = ar + rb.
These equations hold as well when the perpendicularfalls without as when it falls within the triangle, if regardbe had to the principle stated in 335.
Case I. Given two angles and a side, A, B, and c, tofind a, b, and C.
C= 180"- (A + B).
-
TRIGONOMETRY 63
560. First Method. By drawing a perpendicular from
either end of the given side.
Perpendicular ap. Perpendicular bq.
ap " c sin B. bq = c sin A.
bp " c cos B. aq = c cos A.
pc " ap ctn C. qc = bq ctn C.
a = BP + PC. b = AQ + QC.
b= ap csc C. a " bq csc C
Either process affords a test of the results obtained bythe other.
561. Second Method. By the general formulas.
a " 2R sin A. Compute 2R (or log 2R) byb= 2R sin B. aid of the last of these equa-
c =" 2R sin C. tions, then a and b by aid of the
other two.
562. Otherwise by the formulas
a +b-cVHt^-W-S.
cos i (A + B)
6, rinl(A-g ,jsin J (A + B)
a="(/S+D).6- J(iS-i)).
which, being in different form,afford a good test of the re-sults
obtained by the formu-las
first used.
Case II. Given two sides and the included angle, a, 6,and C, to find A, B, and c.
563, First Method. By drawing a perpendicular to
the longer given side from the end of the shorter. Let
b " a.
Perpendicular bq.
bq - a sin C. tan A = ",
or A = tan-1 " "
QA QA
cq - a cos C. B= 180" - (C + A).
QA " b - CQ. C = BQ CSC A = QA Sec ii.
-
64 MATHEMATICAL HANDBOOK
564. Second Method. By the general formulas. "
i (A + B) - 90" - JC.
tan i (A - 5) - " ^ tan " (A + B).a + b
A= i(A + B) + l(A-B).
B- " (A + 5) - I (A - 5).
a sin C 6 sin C0 D " ^
c:
"" " " =
2R sin C.sin A sin B
Or, as a test, c may be computed by either of the equations
c=(a+b)coaiU + B)~
(q-6)sinH^+ fl).V
cosi(A-B)V
^sini(A-S)
Case III. Given two sides and the angle opposite one of
tnem a, b, and A, to find B, C, and c.1
565. First Method. By drawing a perpendicular fron
the point where the given sides meet.
Perpendicular cr.
cr " b sin A.
ar " b cos A.
rb = \/(a 4- cr) (a - cr).c = ar " rb, a double solution.
Cx = AR + RB )
:2 = AR " RB )C2
Angle acr = 90" - A.
Angle rcb - sin-1 " = cos-1" "a a
Cx = acr 4- RCB
C2 = ACR - RCB
Bx - 180" - (A + Cx) - A + C2"2 = 180" - (A + C2) = A 4- Cx
If cr = 6 sin A " a, there is no real solution.
-
TRIGONOMETRY 65
566. Second Method. By the general formulas.
sm b -b8mA
,
giving two values of B, B1 " 90" " B2.a
C1 = B2- A, C2 = BX- A.
a sin C, ' a sin C2c = * y c2 = * '
sin A sin A
If sin B comes out greater than 1, there is no real solution.
If C2 comes out negative the second solution is inad-missible.
Case IV. Given the three sides a, 6, and c, to find the
angles A, B, and C.
567. First Method. By drawing a perpendicular tothe longest side, or to the side that is not less than either
of the other two.
Let 0 b" a. Or, let c be not less than either b or a.
Perpendicular cr.
ar + rb = c.
(b + a) (b- a)ar - rb = ^ L-^
'--
c
cosA=^" or^=cos-1^-6 b
cos 5=52
" or B= cos-1 " "
a a
C= 180"- (A + B),
Test, by drawing another perpendicular, which may falloutside the triangle.
Perpendicular ap (which is supposed to fall outside the
triangle),bp - cp " a.
(c+b) (c- b)BP + CP = " L'
a
BP BP
cos B = " " or B = cos-1 " "
\- - w
A-
180"-(B + C).
cos C = " "orC = cos-1 (" - ) " 90".
-
66 MATHEMATICAL HANDBOOK
568. Second Method. By the general formulas.
J(b-
a) (s-
b) (s-
c),
tan*;!"
""
tani"--V" tan*C=
"
"
s-as-b 8-c
Test. A + B + C-
180".
The half-angles may also be computed by
8in \A-
\/(s " fr" (S ~ C)" "
-CD
or byooB}A-\A(8~a)'
"
*"'
" oc
Special Formulas for the case of two nearly equal sides or
angles,
569. a b -*"
cos * (A + B) sin l(A~B)" .
.".
sin A
570.8inH^-*)-9(a-^n/m-.-(3).
2ocos " (A + 2?)
For theuse
of S and T7 anda
table of the values of these
functionsup to 2", see page 83.
-
TRIGONOMETRY 67
SPHERICAL TRIANGLES.
General Properties.
Let a, b, c, denote the sides and A, B, C, the angles of a
spherical triangle; and let a', V ',c', denote the sides andA'
}B', C", the angles of its polar triangle. Then
601. a + Af - 180".
b + B' = 180".
c + C" = 180".
A + a' - 180".
5 + 6' - 180".
C + c' - 180".
The perpendicular great-circle arcs ap, bq, cr, drawn inthe spherical triangle coincide with the perpendicular great-circle arcs a'p',b'q', c'r', similarly drawn in its polar tri-angle.
The fundamental equations, from which are or can be
derived all other general equations relating to sphericaltriangles,are these three,
(cosa = cos b cos c + sin b sin c cos A,cos b " cos c cos a + sin c sin a cos B,cos c = cos a cos b + sin a sin 6 cos C
603.sin a sin 6 sine
sin A sin 5 sin C=
M=
jTfo Modulus.
cos A = - cos 5 cos C + sin 2? sin C cos a,
604. "" cos B = - cos C cos A + sin C sin A cos 6,
cos C = - cos A cos 5 + sin A sin 5 cos a
605.
f
ctn a sin b
ctn 6 sin c
ctn c sin a
ctn a sin c
ctn 6 sin a
ctn c sin b
cos 6 cos C+ sin C ctn A,
cos c cos A + sin A ctn 5
cos a cos B + sinB ctn C,
cos c cos i? + sin B ctn A,
cos a cos C +sin C ctn B,
cos 6 cos A + sin A ctn C.
-
68 MATHEMATICAL HANDBOOK
606.
607.
r
sin a cos B
sin b cos C
sin c cos A
sin a cos C
sin 6 cos A
y.sin c cos B
r
sin A cos b
sin 5 cos c
sin C cos a
sin A cos c
sin 5 cos a
sin C cos b
sin c cos 6 " cos c sin 6 cos A,
sin a cos c " cos a sin c cos J5,
sin 6 cos a " cos 6 sin a cos C,
sin b cos c " cos b sin c cos A,
sin c cos a " cos c sin a cos B,
sin a cos 6 " cos a sin 6 cos C.
sin C cos B + cos C sin B cos a,
sin A cos C 4- cos A sin C cos 6,
sin B cos A 4- cos B sin A cos c,
sin B cos C + cos B sin C cos a,
sin C cos A. + cos C sin A cos 6,
sin A cos B + cos A sin B cos c.
608. cos a = cos (6 + c) + 2 sin 6 sin c cos2 "A. .
" cos (b - c) - 2 sin b sin c sin2 "A.
.
"
(8),
.
(3).
609,
610.
cos A " - cos (B 4- C) - 2 sin " sin C sin2 "a.
- - cos (B - C) +2 sin 5 sin C cos2 "a.
s = " (a+ 6 + c).
,
,.
4
/sin (s - 6) sin (s - c) /?n611. sin \A - 1/ * " :" f" :" * '-- " -(A)-" sin o sin c
,, A .
/sins sin (s-a)~
612. cos "A = V/ " :" "7 -. . .".Y sin o sin c
613. tanU-v/Sin-(8"")S)n(8;C)-"
-"-
" sin s sin (s " a)
614. S-J(A + B + C).
Spherical Excess of a Spherical Triangle " 2E.
615. 2E - A + B + C - 180" = 2"S - 180".
616. cos "a - 1/. p "
X L-"
-CD,t sm 5 sin C
v/cosQS-j^cobQS-C)sin J5 sin C
. .
.(3).
-
TRIGONOMETRY 69
6iy. dn^i/*1?"? "*-*"...",t sin B sin C
-
4/-cos "S cos (aS- A)
~
V sin B sin C
9
*," "*" i 4/sin(ff-fl)sin(C-ff)
^6i8. ctn io = V.
n
'
, * n*" "
-""
v'
sin E sin (A - 2?)
cos (S - fl) cos (S - C)(3)
- cos S cos (" " A)
619. tan "15 - Vtan "s tan " (s - a) tan"(s-6) tan"(s " c).
Let r denote the polar radius of the circle inscribed in
the spherical triangle.
620. tan r -J^n (s - a) Bin (s-b) sin (s-c)V sin s
Let R denote the polar radius of the circle circumscribed
about the spherical triangle.
621. ctn WaMA-g"8in (B~E) 8ln iC-E)V sin 2?
622. tan"A=.
tfnr" . "
".
sin (s " a)
623. ctnfri- .C^fi",-" "($)"sin (A - E)
624. ctnfi-l/"7^"^cos (j?~ *" cos ^H.V
- cos "
625. ctn"a~ct*g
" "-".
cos (o - A)
626..
2 tan r sin s = sin A sin b sin c. . "
(3).