Advanced Trigonometry

This volume will provide a welcome resource for teachers seeking an undergraduate text on advanced trigonometry, when few are readily available. Ideal for self-study, this text offers a clear, logical presentation of topics and an extensive selection of problems with answers. Contents include the properties of the triangle and the quadrilateral; equations, sub-multiple angles, and inverse functions; hyperbolic, logarithmic, and exponential functions; and expansions in power-series. Further topics encompass the special hyperbolic functions; projection and finite series; complex numbers; de Moivre's theorem and its applications; one- and many-valued functions of a complex variable; and roots of equations. 1930 ed. 79 figures.

Properties of the Triangle

1

(23)

Methods of solution

1

(3)

circumcentre, incentre, orthocentre, nine-point centre, polar circle

4

(6)

centroid

10

(5)

distance between special points

15

(5)

errors

20

(4)

Properties of the Quadrilateral

24

(9)

Cyclic quadrilateral

24

(3)

general quadrilateral

27

(1)

circumscribable quadrilateral

27

(6)

Equations, Sub-multiple Angles, Inverse Functions

33

(19)

General solutions

34

(7)

sub-multiple angles

41

(5)

inverse functions

46

(6)

A Hyperbolic Function and Logarithmic and Exponential Functions

52

(25)

Area-function for rect. hyperbola

52

(5)

differentiation

57

(3)

addition theorem

60

(3)

properties of log x and ex

63

(4)

useful inequalities

67

(2)

Euler's constant

69

(8)

Expansions in Power-Series

77

(27)

Convergence

77

(2)

expansions of sin x and cos x

79

(5)

expansion of log (1 + x)

84

(4)

expansion of tan-1x

88

(1)

evaluation of π

89

(1)

expansion of ex

90

(3)

lim n→∞

93

(11)

The Special Hyperbolic Functions

104

(14)

Definitions, sh x, ch x, th x

104

(1)

formulae

105

(2)

calculus applications

107

(3)

sh-1x, ch-1x, th-1x

110

(8)

Projection and Finite Series

118

(19)

Projections and general angles

118

(5)

cos(A + B), sin(A + B)

123

(2)

Σ cos[a + (r -- 1)β], etc

125

(5)

difference series

130

(7)

Complex Numbers

137

(25)

Definitions

138

(2)

notation and manipulation

140

(5)

modulus and amplitude

145

(3)

use of Argand Diagram

148

(2)

products and quotients

150

(5)

principal values

155

(7)

De Moivre's Theorem and Applications

162

(27)

De Moivre's theorem

162

(2)

principal values

164

(1)

values of zp/q

165

(1)

powers and roots in Argand Diagram

165

(4)

expansions of cos nθ, sin nθ

169

(3)

expansions of cos nθ, sin nθ, tan nθ

172

(2)

Σxr cos rθ, etc

174

(4)

cos nθ, sin nθ/sing θ as polynomials in cos θ, etc

178

(11)

One-Valued Functions of a Complex Variable

189

(15)

Absolute convergence

189

(1)

series of complex terms

190

(1)

exponential series and exponential function

191

(3)

modulus and amplitude of exp (z)

194

(3)

sin z, cos z, tan z

197

(1)

sh z, ch z, th z

198

(6)

Roots of Equations

204

(15)

Formation of equations

204

(2)

symmetric functions of the roots

206

(6)

essentially distinct roots

212

(7)

Factors

219

(22)

Algebraic functions

219

(3)

trigonometric functions, sin nθ, etc

222

(4)

x2n -- 2xn cos na + 1

226

(2)

comparison of series and products

228

(3)

partial fractions

231

(10)

Many-Valued Functions of a Complex Variable

241

(22)

Low w

241

(4)

expansion of log (1 + w)

245

(2)

circle of convergence

247

(5)

zw

252

(1)

binomial series

253

(1)

logarithms to any base

253

(3)

inverse functions

256

(7)

Miscellaneous Relations

263

(15)

(General identities

263

(2)

conditional identities

265

(3)

miscellaneous transformations

268

(2)

elimination

270

(4)

inequalities

274

(4)

Miscellaneous Examples on Chapters I-XIV

278

(7)

Answers

285

(48)

Index

333

(3)

Symbols

336

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Advanced Trigonometry

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Table of Contents

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