A Treatise on Plane and Spherical Trigonometry: And Its Applications to Astronomy and Geodesy

Edward Albert Bowser

D. C. Heath & Company, 1892 - Trigonometry - 368 pages

Contents

• PLANE TRIGONOMETRY: 1
• CHAPTER II: 16
• Representation of the Functions by Straight Lines: 23
• Changes in Sine as the Angle increases from 0 to 360: 25
• Changes in Cosine as the Angle increases from 0 to 360: 26
• Changes in Tangent as the Angle increases from 0 to 360: 27
• Table giving Changes of Functions in Four Quadrants: 28
• Relations between the Functions of the Same Angle: 29
• Use of the Preceding Formulæ: 30
• To find the Trigonometric Functions of 45: 31
• To find the Trigonometric Functions of 60 and 30: 33
• Functions of Complemental Angles: 34
• To prove sin 90 + A cos A etc: 35
• To prove sin A sin A etc: 36
• Table giving the Reduced Functions of Any Angle: 37
• Periodicity of the Trigonometric Functions: 38
• Angles corresponding to Given Functions: 39
• General Expression for All Angles with a Given Sine: 40
• An Expression for All Angles with a Given Cosine: 41
• Trigonometric Identities: 43
• Examples: 44
• CHAPTER III: 50
• To find the Values of sin x y and cos x y: 52
• Formulæ for transforming Sums into Products: 55
• Useful Formulæ: 56
• Tangent of Sum and Difference of Two Angles: 57
• Formulæ for the Sum of Three or More Angles: 58
• Functions of Double Angles: 60
• Functions of 3 x in Terms of the Functions of x: 61
• Functions of Half an Angle: 63
• ART PAGE 53 Quadruple Values of Sine and Cosine of Half an Angle: 65
• Double Value of Tangent of Half an Angle: 66
• Triple Value of Sine of Onethird an Angle: 67
• Find the Values of the Functions of 221: 69
• Find the Sine and Cosine of 36º: 70
• Inverse Trigonometric Functions: 72
• Table of Useful Formulæ: 75
• Examples: 77
• CHAPTER IV: 87
• Common System of Logarithms: 91
• Comparison of Two Systems of Logarithms: 93
• Tables of Logarithms: 95
• Use of Tables of Logarithms of Numbers: 98
• To find the Logarithm of a Given Number: 99
• To find the Number corresponding to a Given Logarithm: 102
• 69a Arithmetic Complement: 103
• Use of Trigonometric Tables: 105
• Use of Tables of Natural Trigonometric Functions: 106
• To find the Angle whose Sine is Given: 108
• Use of Tables of Logarithmic Trigonometric Functions: 110
• To find the Logarithmic Sine of a Given Angle: 112
• To find the Angle whose Logarithmic Sine is Given: 114
• To find the Angle whose Logarithmic Cosine is Given: 115
• Angles near the Limits of the Quadrant: 116
• Examples: 117
• CHAPTER V: 126
• To solve m sin Ø a m cos º b: 128
• To solve a sin + b cos Q c: 129
• To solve sin a + x m sin x: 131
• To solve tan a + x m tan x: 132
• To solve tan a + x tan x m: 133
• To solve m sin 0 + x a m sin + x b: 134
• To solve x cos + y sin a m a sin Y COS L n: 135
• Adaptation to Logarithmic Computation: 137
• Trigonometric Elimination: 138
• Examples: 140
• CHAPTER VI: 146
• Oblique Triangles Law of Sines: 147
• Law of Cosines: 148
• Law of Tangents: 149
• Functions of Half an Angle in Terms of the Sides: 150
• To express the Sine of an Angle in Terms of the Sides: 152
• Expressions for the Area of a Triangle: 153
• Inscribed Circle: 154
• Escribed Circle: 155
• To find the Area of a Cyclic Quadrilateral: 157
• Examples: 159
• CHAPTER VII: 165
• ART PAGE 109 Case I Given a Side and the Hypotenuse: 166
• Case II Given an Acute Angle and the Hypotenuse: 167
• Case III Given a Side and an Acute Angle: 168
• Case IV Given the Two Sides: 169
• Four Cases of Oblique Triangles: 172
• Case II Given Two Sides and the Angle opposite One of them: 173
• Case III Given Two Sides and the Included Angle: 176
• Case IV Given the Three Sides: 177
• Sum the Series tan 8 + tan +tan + etc 2 4: 179
• Area of a Triangle: 180
• Heights and Distances Definitions: 181
• Heights of an Accessible Object: 182
• Geometric Principles: 183
• An Inaccessible Object above a Horizontal Plane: 184
• Object observed from Two Points in Same Vertical Line: 185
• Distance between Two Inaccessible Objects: 186
• The Species of the Parts: 187
• Problem of Pothenot or of Snellius: 188
• Examples: 189
• Law of Sines: 190
• Law of Cosines: 191
• Relation between a Side and the Three Angles: 192
• To find the Value of cot a sin b: 193
• Useful Formula: 194
• Formulæ for the Half Angles: 195
• Formulæ for the Half Sides: 196
• Napiers Analogies: 197
• Delambres or Gausss Analogies: 198
• CHAPTER VIII: 204
• Logarithmic Series: 205
• Computation of Logarithms: 207
• Sin 0 and tan 6 are in Ascending Order of Magnitude: 208
• sin e 133 The Limit of is Unity: 209
• To calculate the Sine and Cosine of 10 and of 1: 211
• To construct a Table of Natural Sines and Cosines: 213
• The Sines and Cosines from 30 to 600: 214
• Sines of Angles Greater than 45º: 215
• Formulæ of Verification: 216
• Tables of Logarithmic Trigonometric Functions: 217
• The Principle of Proportional Parts: 218
• To prove the Rule for the Table of Natural Sines: 219
• To prove the Rule for a Table of Natural Tangents: 220
• To prove the Rule for a Table of Logarithmic Sines: 221
• To prove the Rule for a Table of Logarithmic Cosines: 222
• Cases of Inapplicability of Rule of Proportional Parts: 223
• Three Methods to replace the Rule of Proportional Parts: 224
• Examples: 226
• CHAPTER IX: 229
• To develop cos no and sin no in Powers of sin 0 and cos 0: 233
• To develop sin 0 and cos 0 in Series of Powers of 0: 234
• Convergence of the Series: 235
• Expansion of sinn o in Terms of Cosines of Multiples of 0: 236
• Expansion of sinn e in Terms of Sines of Multiples of: 237
• Exponential Values of Sine and Cosine: 238
• Gregorys Series: 239
• Eulers Series: 240
• Machins Series: 241
• Given sin 0 x sin 0 + a expand a Powers of x: 242
• Resolve xn 1 into Factors: 243
• Resolve an + 1 into Factors: 244
• Resolve 2n 2 sin cos 0 + 1 into Factors: 245
• De Moivres Property of the Circle: 247
• Cotes Properties of the Circle: 248
• Resolve cos e into Factors: 250
• Sum the Series sin a + sina+B + etc: 251
• Sum the Series cos O + cosQ +B + etc: 252
• Sum the Series sin de sinQ + B + etc: 254
• 267: 267
• 274: 274
• Examples: 288
• CHAPTER XI: 297
• ART PAGE 201 Case I Given the Hypotenuse and an Angle: 298
• Case II Given the Hypotenuse and a Side: 299
• Case III Given a Side and the Adjacent Angle: 300
• Case IV Given a Side and the Opposite Angle: 301
• Case V Given the Two Sides: 302
• Quadrantal and Isosceles Triangles: 303
• Solution of Oblique Spherical Triangles: 304
• Case III Given Two Sides and One Opposite Angle: 309
• Case IV Given Two Angles and One Opposite Side: 312
• Case V Given the Three Sides: 313
• Case VI Given the Three Angles: 314
• Examples: 316
• CHAPTER XII: 324
• The Escribed Circles: 325
• The Circumscribed Circle: 326
• Circumcircles of Colunar Triangles: 328
• Areas of Triangles Given the Three Angles: 329
• Areas of Triangles Given the Three Sides: 330
• Areas of Triangles Given Two Sides and the Included Angle: 331
• Examples: 332
• CHAPTER XIII: 338
• Spherical Coördinates: 339
• Graphic Representation of the Spherical Coördinates: 341
• Problems: 342
• The Chordal Triangle: 346
• Legendres Theorem: 348
• Roys Rule: 350
• Reduction of an Angle to the Horizon: 352
• ART PAGE: 353
• Table of Formulæ in Spherical Trigonometry: 359

Popular passages

Page 148 - In any triangle the square of any side is equal to the sum of the squares of the other two sides minus twice the product of these two sides and the cosine of their included angle.

Page 147 - Law of Sines. — In any triangle the sides are proportional to the sines of the opposite angles.

Page 278 - AB'C, we have by (4) cos a' — cos b cos c' + sin b sin c' cos B'AC, or cos(тг— a) = cos b cos(тг— c) + sin b sin(тт — C)COS(тг —A). .-. cos a = cos b cos с + sin b sin с cos A.

Page 278 - ... cos a = cos b cos с + sin b sin с cos A ; (2) cos b = cos a cos с + sin a sin с cos в ; ^ A. (3) cos с = cos a cos b + sin a sin b cos C.

Page 278 - A cos 6 = cos a cos c + sin a sin c cos B cos c = cos a cos 6 + sin a sin 6 cos C Law of Cosines for Angles cos A = — cos B...

Page 6 - Radian is the angle subtended, at the centre of a circle, by an arc equal in length to the radius...

Page 17 - If the cosine of A be subtracted from unity, the remainder is called the versed sine of A. If the sine of A be...

Page 89 - The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power.

Page 149 - In every plane triangle, the sum of two sides is to their difference as the tangent of half the sum of the angles opposite those sides is to the tangent of half their difference.

Bibliographic information

• Title: A Treatise on Plane and Spherical Trigonometry: And Its Applications to Astronomy and Geodesy
• Author: Edward Albert Bowser
• Publisher: D. C. Heath & Company, 1892
• Original from: the University of California
• Digitized: Nov 29, 2007
• Length: 368 pages