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

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

Complement and Supplement of an Angle
12
Examples
13
CHAPTER II
16
The Functions are always the Same for the Same Angle
18
Functions of Complemental Angles
20
Positive and Negative 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 Formulae
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
Formulae for transforming Sums into Products
55
Useful Formulae
56
Tangent of Sum and Difference of Two Angles
57
Formulae 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
ABT PAOe 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 22
69
If A + B + C 180 to And sin A + sin B + sin C etc
70
Inverse Trigonometric Functions
72
Table of Useful Formulas
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 to sin a m cos 6
128
To solve a sin p + b cos 0 c
129
To solve sin a + x m sin x
131
To solve tan + x m tan x
132
To solve tan a + x tan x m
133
To solve m sin 0 + x a to sin j + x b
134
To solve x cos a + y sin a to x sin a y cos a 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 I55 105 Distance between the Incentre and the Circumcentre
155
To find the Area of a Cyclic Quadrilateral
157
Examples
159
CHAPTER VII
165
ABT IAOK 109 Case L 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
CHAPTER VIII
204
Logarithmic Series
205
Computation of Logarithms
207
Sin 9 and tan fl are in Ascending Order of Magnitude
208
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 00
214
Sines of Angles Greater than 45
215
Formula? 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 n0 and sin n0 in Powers of sin 8 and cos O
233
To develop sin O and cos 8 in Series of Powers of O
234
Convergence of the Series
235
Expansion of sin0 in Terms of Cosines of Multiples of O
236
Expansion of sin 8 in Terms of Sines of Multiples of O
237
Exponential Values of Sine and Cosine
238
Gregorys Series
239
Eulers Series
240
Machins Series
241
Given sin0 x sin O + a expand O in Powers of x
242
Resolve x 1 into Factors
243
Resolve x + 1 into Factors
244
Resolve x2 2 cos O + 1 into Factors
245
De Moivres Property of the Circle
247
Cotes Properties of the Circle
248
Resolve cos fl into Factors
250
Sura the Series sin a + sin a+ 0 + etc
251
Sum the Series cos a + cosa + 0 + etc
252
Sum the Series sin sin + 3 + etc
254
ART WO 179 Sum the Series tan 9 + tan + tan + etc
255
Summation of Infinite Series
256
Examples
257
PART II
267
Fundamental Definitions and Properties
268
Formulae for Right Spherical Triangles
270
Napiers Rules
272
The Species of the Parts
273
Ambiguous Solution
274
Law of Sines
276
Law of Cosines
277
Relation between a Side and the Three Angles
278
To find the Value of cot a sin 6 etc
279
Useful Formulae
280
Formulae for the Half Angles
281
Formulae for the Half Sides
284
Napiers Analogies
286
Delambres or Gausss Analogies
287
Examples
288
CHAPTER XI
297
ART FAOE 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 i00 204 Case IV Given a Side and the Opposite Angle
301
Case V Given the Two Sides
302
20 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 Coordinates
339
Graphic Representation of the Spherical Coordinates
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 Formulae in Spherical Trigonometry
359
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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.

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