Plane and Sperical Trigonometry (with Five-place Tables): A Text-book for Technical Schools and Colleges

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J. Wiley, 1913 - 520 páginas
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Contenido

Fundamental relations 13 To express each of the functions in terms of a given
25
Reduction of trigonometric expressions to their simplest form
27
Trigonometric identities
31
CHAPTER III
35
To find the angle less than 90 corresponding to a given natural function
41
Accuracy of results
43
Solution of right triangles by natural functions
45
Triangles having a small angle
49
Historical note
50
Review
51
CHAPTER IV
53
Fundamental laws governing logarithms
54
Logarithms of special values
55
ART PAGE 27 The common system of logarithms
57
Rule for the characteristic
58
Table of common logarithms
61
4
62
9
63
To find the number corresponding to a given logarithm
65
Directions for the use of logarithms
67
17
71
19
72
To compute a table of common logarithms
73
Relation between logaN and logN
75
Natural or hyperbolic logarithms
76
Tables of logarithmic trigonometric functions
78
21
79
27
80
To find the angle corresponding to a given logarithmic trigonometric function
81
Logarithmic functions of angles near o or 90
84
Historical note
87
CHAPTER V
89
Number of significant figures
93
Applied problems involving right triangles
94
Heights and distances
95
Problems for engineers
96
Applications from physics
98
Problems in navigation
100
Geographical and astronomical problems
103
Geometrical applications
106
Oblique triangles solved by right triangles
110
CHAPTER VI
117
Definitions of the trigonometric functions of any angle less than 180
118
The signs of the functions of an obtuse angle
119
Functions of supplementary angles
120
Functions of 90 + 0
121
Angles corresponding to a given function
122
Review
123
CHAPTER VII
125
The projection theorem
126
The law of cosines
127
Arithmetic solution of triangles
128
The law of tangents
130
Formulas for the area of a triangle
132
Functions of half the angles in terms of the sides
134
CHAPTER VIII
138
Case II Given two sides and the angle opposite one of them
141
Case III Given two sides and the included angle
144
Case IV Given three sides
147
Practical applications
150
b Auxiliary geometrical constructions
152
c System of simultaneous equations
155
Miscellaneous heights and distances
158
Applications from physics
161
Applications from surveying and engineering
164
Applications from navigation
171
Problems from astronomy and meteorology
172
Geometrical applications
174
CHAPTER IX
177
Positive and negative angles
178
Sexagesimal measure of angles
179
Decimal division of degrees
180
The circular or natural system of angular measures
181
Comparison of sexagesimal and circular measure
182
Relation between angle arc and radius
185
Area of circular sector
187
Review
189
CHAPTER X
191
Signs of the functions in each of the quadrants
192
Changes in the value of the functions
195
Fundamental relations
196
Reduction of the functions to the first quadrant
199
Reductions from the fourth quadrant
201
Functions of negative angles
203
Table of principal reduction formulas and general rules
204
Generalization of the preceding reduction formulas
206
CHAPTER XI
209
Generalization of the addition theorems
210
Addition theorems Second proof
211
Subtraction theorems for the sine and cosine
212
Tangent of the sum and difference of two angles
215
Functions of double an angle
216
Sums and differences of sines or cosines transformed into products
219
CHAPTER XII
225
Formula for angles having a given sine
226
Formulas for angles having a given tangent
227
Trigonometric equations involving multiple angles
233
Trigonometric equations involving two or more variables
236
Solutions adapted to logarithmic computation
240
Inverse functions
245
Review
250
Fouriers theorem
268
The logarithmic curve
269
The exponential curve
270
The compound interest law
272
The catenary
274
The curve of damped vibrations
275
CHAPTER XIV
278
Geometric representation of complex numbers
280
Trigonometric representation of complex numbers
281
Geometric addition and subtraction of complex numbers
282
Physical applications of complex numbers
283
Historical note
285
Multiplication and division of complex numbers
287
Powers of complex numbers
288
Roots of complex numbers
289
To solve the equation z I O
292
To solve the equation z + 1 0
293
The cube roots of unity
295
Solution of cubic equations
296
The irreducible case
298
To express sin no and cos no in powers of sin and cos 0
301
To express cos and sin in terms of sines and cosines of multiple angles
302
CHAPTER XV
306
Convergent and nonconvergent series
307
Absolutely convergent series
309
Sum of an infinite series
310
The limit or r as n approaches infinity 166 The geometric infinite series 167 Convergency test
311
Convergency of special series
313
The number
316
333
317
The exponential series
319
The logarithmic series
320
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321
Calculations of logarithms
323
Errors resulting from the use of logarithms sin x tan x
324
Limiting values of the ratios
326
Limiting values of cos and as x approaches zero x x sin xn
327
The sine cosine and tangent series
329
Computation of natural functions table 178 Approximate equality of sine tangent and radian measure of small angles
333
CHAPTER XVI
336
Definition of the trigonometric functions of complex numbers
337
Eulers theorem ei cos 0+ i sin 0
339
Exponential form of the sine and cosine
340
Hyperbolic functions defined
342
Duality of circular and hyperbolic functions 186 Table of formulas PAGE 306
345
45
347
Inverse hyperbolic functions
348
Geometrical representation of hyperbolic functions
349
Area of hyperbolic sector 190 Use of hyperbolic functions
353
Review
354
INTRODUCTION ART 1 Definition of spherical trigonometry 2 The uses of spherical trigonometry
1
Spherical trigonometry dependent on solid geometry
2
Classification of spherical triangles
3
Polar triangles 8 The six cases of spherical triangles 9 Solution of spherical triangles
7
The use of the polar triangle
8
Construction of spherical triangles
9
The general spherical triangle
11
CHAPTER II
14
Generalization of the right triangle formulas
16
Napiers rules of circular parts
17
Proof of Napiers rules of circular parts
18
To determine the quadrant of the unknown parts
20
The ambiguous case of right spherical triangles 20 Solution of right spherical triangles
21
51
24
Solution of quadrantal triangles
25
Formulas for angles near o 90 180
26
Oblique spherical triangles solved by the method of right triangles
28
CHAPTER III
33
The law of cosines
34
Relation between two angles and three sides
35
Analytical proof of the fundamental formulas
36
Fundamental relations for polar triangles 29 Arithmetic solution of spherical triangles
37
Formulas of half the angles in terms of the sides
39
Formulas of half the sides in terms of the angles
41
Delambres or Gausss proportions
43
I 2
45
4
48
6
49
II
50
CHAPTER IV
51
FOOD ONE MO 14
53
21
54
Case III
55
25
56
26
57
Case V
58
Case VI
60
To find the area of a spherical triangle 45 Applications to geometry 46 Applications to geography and navigation 47 Applications to astronomy
62
28
63
35
65
39
66
308
2
310
4
316
7
349
8
354
9
Plane and spherical oblique triangle formulas compared
35
Derivation of plane triangle formulas from those of spherical triangles 48
48
Given two sides and the angle opposite one of them Given two angles and the side opposite one of them 450 44 47
57
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89
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Página 59 - The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power.
Página 135 - In the same way it may be proved that a : b : : sin. A : sin. B, and these two proportions may be written a : 6 : c : : sin. A : sin. B : sin. C. THEOREM III. t8. In any plane triangle, the sum of any two sides is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their difference. By Theorem II. we have a : b : : sin. A : sin. B.
Página 272 - ... an indispensable instrument in the treatment of nearly every recondite question in modern physics. To mention only sonorous vibrations, the propagation of electric signals along a telegraph wire, and the conduction of heat by the earth's crust, as subjects in their generality intractable without it, is to give but a feeble idea of its importance.
Página 223 - B) = cos A cos B - sin A sin B. cos (A - B) = cos A cos B + sin A sin B.
Página 58 - The logarithm of the quotient of two numbers is equal to the logarithm of the dividend minus the logarithm of the divisor.
Página 39 - Spherical Triangle the cosine of any side is equal to the product of the cosines of the other two sides, plus the product of the sines of those sides into the cosine of their included angle ; that is, (1) cos a = cos b...
Página 130 - In any triangle, the sides are proportional to the sines of the opposite angles. That is, sin A = sin B...
Página 53 - 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...
Página 63 - FRACTION is a negative number, and is one more than the number of ciphers between the decimal point and the first significant Jigure.
Página 22 - I. The sine of the middle part is equal to the product of the tangents of the adjacent parts.

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