Mathematical Handbook

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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

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"

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).


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.


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.

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.


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


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


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.


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


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.


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".


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"+_


208. (a-6x)-'-I/'l+ ^ + ^+-^ + ...,\a\ a a2 a3 /

Convergent if Px2 " a2.

209. (l"x)-1=l =Fz+ x2=Far, + x4qFz5 +...


210. (l"x)-2= 1 =F2x+ 3x2qF4a^+ 5x4=F6a^+. . .


v 1 y I72.4 2.4.6 2.4.6.8


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


214. (1"i)^=1T^+- xt^"*** 1A71" x4+...* *

3.6 3.6.9 3.6.9.12


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.


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


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


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.


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


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.


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


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.


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.


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 A" 360c

364.

sin (A " 360") - sin A,cos (A " 360") - cos A,tan (A " 360") - tan A,etc.

TRIGONOMETRY 45

365.


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),


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.


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)


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=

ctnÂ -tanÂctn2 "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,

2tanÂ_

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.


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


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.


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 ""


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.


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 \


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.


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


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".



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.


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).

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