Plane And Spherical Trigonometry Fourth Edition Thirteenth ImpressionPDF电子书下载

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