CHAPTER Ⅰ INTRODUCTION 1
1．Introductory remarks 1
2．Angles，definitions 2
3．Quadrants 3
4．Graphical addition and subtraction of angles 3
5．Angle measurement 4
6．The radian 5
7．Relations between radian and degree 6
8．Relations between angle，arc，and radius 8
9．Area of circular sector 10
10．General angles 12
11．Directed lines and segments 13
12．Rectangular coordinates 14
13．Polar coordinates 15
CHAPTER Ⅱ TRIGONOMETRIC FUNCTIONS OF ONE ANGLE 17
14．Functions of an angle 17
15．Trigonometric ratios 17
16．Correspondence between angles and trigonometric ratios 18
17．Signs of the trigonometric functions 19
18．Calculation from measurements 20
19．Calculations from geometric relations 21
20．Trigonometric functions of 30° 21
21．Trigonometric functions of 45° 22
22．Trigonometric functions of 120° 22
23．Trigonometric functions of 0° 23
24．Trigonometric functions of 90° 23
25．Exponents of trigonometric functions 25
26．Given the function of an angle，to construct the angle 26
27．Trigonometric functions applied to right triangles 28
28．Relations between the functions of complementary angles 30
29．Given the function of an angle in any quadrant，to construct the angle 31
CHAPTER Ⅲ RELATIONS BETWEEN TRIGONOMETRIC FUNCTIONS 34
30．Fundamental relations between the functions of an angle 34
31．To express one function in terms of each of the other functions 36
32．To express all the functions of an angle in terms of one function of the angle，by means of a triangle 37
33．Transformation of trigonometric expressions 38
34．Identities 40
35．Inverse trigonometric functions 42
36．Trigonometric equations 43
CHAPTER Ⅳ RIGHT TRIANGLES 47
37．General statement 47
38．Solution of a triangle 47
39．The graphical solution 48
40．The solution of right triangles by computation 48
41．Steps in the solution 49
42．Remark on logarithms 54
43．Solution of right triangles by logarithmic functions 54
44．Definitions 56
CHAPTER Ⅴ FUNCTIONS OF LARGE ANGLES 62
46．Functions of 1/2π－θ in terms of functions of θ 62
47．Functions of 1/2π＋θ in terms of functions of θ 63
48．Functions of π－θ in terms of functions of θ 63
49．Functions of π＋θ in terms of functions of θ 64
50．Functions of 3/2π－θ in terms of functions of θ 65
51．Functions of 3/2π＋θ in terms of functions of θ 65
52．Functions of-θ or 2π-θ in terms of functions of θ 66
53．Functions of an angle greater than 2π 67
54．Summary of the reduction formulas 67
55．Solution of trigonometric equations 71
CHAPTER Ⅵ GRAPHICAL REPRESENTATION OF TRIGONOMETRIC FUNCTIONS 76
56．Line representation of the trigonometric functions 76
57．Changes in the value of the sine and cosine as the angle increases from 0 to 360° 78
58．Graph of y=sin θ 79
59．Periodic functions and periodic curves 80
60．Mechanical construction of graph of sin θ 82
61．Projection of point having uniform circular motion 83
62．Summary 85
63．Simple harmonic motion 86
64．Inverse functions 87
65．Graph of y=sin-1 x，or y=arc sin x 87
CHAPTER Ⅶ PRACTICAL APPLICATIONS AND RELATED PROBLEMS 90
66．Accuracy 90
67．Tests of accuracy 91
68．Orthogonal projection 92
69．Vectors 93
70．Distance and dip of the horizon 95
71．Areas of sector and segment 99
72．Widening of pavements on curves 97
73．Reflection of a ray of light 102
74．Refraction of a ray of light 102
75．Relation between sin θ，θ，and tan θ，for small angles 103
76．Side opposite small angle given 105
77．Lengths of long sides given 105
CHAPTER Ⅷ FUNCTIONS INVOLVING MORE THAN ONE ANGLE 108
78．Addition and subtraction formulas 108
79．Derivation of formulas for sine and cosine of the sum of two angles 108
80．Derivation of the formulas for sine and cosine of the difference of two angles 109
81．Proof of the addition formulas for other values of the angles 110
82．Proof of the subtraction formulas for other values of the angles 110
83．Formulas for the tangents of the sum and the difference of two angles 113
84．Functions of an angle in terms of functions of half the angle 114
85．Functions of an angle in terms of functions of twice the angle 117
86．Sum and difference of two like trigonometric functions as a product 119
87．To change the product of functions of angles to a sum 122
88．Important trigonometric series 123
CHAPTER Ⅸ OBLIQUE TRIANGLES 130
89．General statement 130
90．Law of sines 130
91．Law of cosines 132
92．Case Ⅰ．The solution of a triangle when one side and two angles are given 132
93．Case Ⅱ．The solution of a triangle when two sides and an angle opposite one of them are given 136
94．Case Ⅲ．The solution of a triangle when two sides and the included angle are given First method 140
95．Case Ⅲ．Second method 140
96．Case Ⅳ．The solution of a triangle when the three sides are given 143
97．Case Ⅳ．Formulas adapted to the use of logarithms 144
CHAPTER Ⅹ MISCELLANEOUS TRIGONOMETRIC EQUATIONS 158
98．Types of equations 158
99．To solve r sin θ + s cos θ = t for θ when r， s， and t are known 160
100．Equations in the form p sin α cos β = a， p sin α sin β3 = b， p cos α =c， where p，α，and β are variables 161
101．Equations in the form sin （α + β） = c sin α， where β and c areknown 161
102．Equationsin the form tan （α + β） = c tan α， where β3 and c areknown 162
103．Equations of the form t = θ + φ sin t， where θ and φ are givenangles 162
CHAPTER Ⅺ COMPLEX NUMBERS， DEMOIVRE＇S THEOREM， SERIES 165
104．Imaginary numbers 165
105．Square root of a negative number 165
106．Operations with imaginary numbers 166
107．Complex numbers 166
108．Conjugate complex numbers 167
109．Graphical representation of complex numbers 167
110．Powers of i 169
111．Operations on complex numbers 169
112．Properties of complex numbers 171
113．Complex numbers and vectors 171
114．Polar form of complex numbers 172
115．Graphical representation of addition 174
116．Graphical representation of subtraction 175
117．Multiplication of complex numbers in polar form 176
118．Graphical representation of multiplication 176
119．Division of complex numbers in polar form 176
120．Graphical representation of division 177
121．Involution of complex numbers 177
122．DeMoivre＇s theorem for negative and fractional exponents 178
123．Evolution of complex numbers 179
124．Expansion of sin nθ and cos nθ 182
125．Computation of trigonometric functions 184
126．Exponential values of sin θ， cos θ， and tan θ 184
127．Series for sinn θ and cosn θ in terms of sines or cosines of multiples of θ 185
128．Hyperbolic functions 187
129．Relations between the hyperbolic functions 188
130．Relations between the trigonometric and the hyperbolic functions 188
131．Expression for sinh x and cosh x in a series．Computation 189
131．Forces and velocities represented as complex numbers 189
CHAPTER Ⅻ SPHERICAL TRIGONOMETRY 193
132．Great circle， small circle， axis 193
133．Spherical triangle 193
134．Polar triangles 194
135．Right spherical triangle 195
136．Derivation of formulas for right spherical triangles 196
137．Napier＇s rules of circular parts 197
138．Species 198
139．Solution of right spherical triangles 198
140．Isosceles spherical triangles 200
141．Quadrantal triangles 201
142．Sine theorem （law of sines） 202
143．Cosine theorem （law of cosines） 202
144．Theorem 204
145．Given the three sides to find the angles 204
146．Given the three angles to find the sides 205
147．Napier＇s analogies 206
148．Gauss＇s equations 208
149．Rules for species in oblique spherical triangles 209
150．Cases 210
151．Case Ⅰ．Given the three sides to find the three angles 211
152．Case Ⅱ．Given the three angles to find the three sides 212
153．Case Ⅲ．Given two sides and the included angle 212
154．Case Ⅳ．Given two angles and the included side 213
155．Case Ⅴ．Given two sides and the angle opposite one of them 213
156．Case Ⅵ．Given two angles and the side opposite one of them 215
157．Area of a spherical triangle 215
158．L＇Huilier＇s formula 216
159．Definitions and notations 217
160．The terrestrial triangle 217
161．Applications to astronomy 218
162．Fundamental points， circles of reference 219
Summary of formulas 222
Useful constants 225
INDEX 226
LOGARITHMS AND EXPLANATIONS OF TABLES 240
1．Use of Logarithms 240
2．Exponents 240
3．Definitions 241
4．Notation 241
5．Systems of Logarithms 242
6．Properties of Logarithms 242
7．Logarithms to the Base 10 243
8．Rules for Determining the Characteristic 245
9．The Mantissa 246
10．Tables 246
11．To Find the Mantissa of the Logarithm of a Number 247
12．Rules for Finding the Mantissa 248
13．Finding the Logarithm of a Number 249
14．To Find the Number Corresponding to a Logarithm 249
15．Rules for Finding the Number Corresponding to a Given Loga-rithm 251
16．To Multiply by Means of Logarithms 252
17．To Divide by Means of Logarithms 253
18．Cologarithms 253
19．To Find the Power of a Number by Means of Logarithms 254
20．To Find the Root of a Number by Means of Logarithms 254
21．Proportional Parts 255
22．Suggestions 255
23．Changing Systems of Logarithms 259
24．Use of Table Ⅱ 260
25．Table Ⅲ．Explanatory 262
26．To Find Logarithmic Function of an Acute Angle 262
27．To Find the Acute Angle Corresponding to a Given LogarithmicFunction 263
28．Angles near 0 and 90° 265
29．Functions by Means of S and T 265
30．Functions of Angles Greater Than 90° 266
31．Table Ⅳ．Explanatory 268
32．To Find the Natural Function of an Angle 268
33．To Find the Angle Corresponding to a Given Natural Function 269
34．Table Ⅴ．Explanatory 270
35．Errors of Interpolation 271
Table Ⅰ．Logarithms of Numbers 274
Table Ⅱ．Conversion of Logarithms 295
Table Ⅲ．Logarithms of Trigonometric Functions 296
Table Ⅳ．Natural Trigonometric Functions 348
Table Ⅴ．Radian Measure 372
Table Ⅵ．Constants and Their Logarithms 373