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

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Wiley, 1913 - 453 pàgines
 

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Continguts

Fundamental relations
24
To express each of the functions in terms of a given one
25
Reduction of trigonometric expressions to their simplest form
27
Trigonometric identities
31
CHAPTER III
35
To find the angle less than oo corresponding to a given natural function
39
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
To find the logarithm of a given number
62
To find the number corresponding to a given logarithm
65
Directions for the use of logarithms
67
Application of logarithms
72
To compute a table of common logarithms
73
Relation between logN and logjN
75
Natural or hyperbolic logarithms
76
Tables of logarithmic trigonometric functions
78
To find the logarithmic trigonometric functions of an angle less than 90
79
To find the angle corresponding to a given logarithmic trigonometric function
81
Logarithmic functions of angles near o or oo
83
Use of 5 and T table
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 1800
118
The signs of the functions of an obtuse angle
119
Functions of supplementary angles
120
Functions of 900 + 6
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 13
131
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 15
150
i Auxiliary geometrical constructions 15 2
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 DC
177
Positive and negative angles
178
Sexagesimal measure of angles
179
Decimal division of degrees Io 87 Centesimal measure of angles
180
The circular or natural system of angular measures
181
Comparison of sexagesimal and circular measure
182
Relation between angle arc and radius
185
90a Area of circular sector J87 91 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
no Tangent of the sum and difference of two angles
215
in 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
The hyperbola
255
The tangent curve
257
The sinusoidal or simple harmonic curves
260
Angles as functions of time
262
Composition of sinusoidal curves
264
Theorem on composition of sinusoidal curves having equal wave lengths
267
Fouriers theorem
268
The logarithmic curve
269
The exponential curve
270
The general exponential curve 27
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 zn 1 o
292
To solve the equation zn + 1 o
293
The cube roots of unity
295
Solution of cubic equations
296
The irreducible case
298
To express sin n8 and cos n9 in powers of sin 6 and cos 8
301
CHAPTER XV
306
Convergent and nonconvergent series
307
Absolutely convergent series
308
Sum of an infinite series
310
Convergency test
311
Convergency of special seiies
313
The number e
316
The exponential series
318
The logarithmic series
320
Calculations of logarithms
321
Errors resulting from the use of logarithms
324
Limiting values of the ratios as x approaches zero
326
Limiting values of cos and sn A as n approaches infinity
328
The sine cosine and tangent series
329
Computation of natural functions table
330
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 ea cosfl + i sin 8
339
Exponential form of the sine and cosine 34
340
Hyperbolic functions defined
342
Duality of circular and hyperbolic functions
343
Table of formulas
345
Inverse hyperbolic functions
348
Geometrical representation of hyperbolic functions
349
Area of hyperbolic sector 35
353
Review
354
SPHERICAL TRIGONOMETRY CHAPTER I
1
Spherical trigonometry dependent on solid geometry 2
2
To express cos 9 and sin 6 in terms of sines and cosines of multiple angles 32
3
Colunar triangles 4
4
Polar triangles 6
6
The six cases of spherical triangles 7
7
The use of the polar triangle 8
8
Construction of spherical triangles 9
9
The general spherical triangle 11
11
CHAPTER II
14
Plane and spherical right triangle formulas compared 16
16
Napiers rules of circular parts 17
17
Proof of Napiers rules of circular parts 18
18
To determine the quadrant of the unknown parts 20
20
The ambiguous case of right spherical triangles 21
21
Solution of quadrantal triangles 25
25
Formulas for angles near o 90 180 26
26
Oblique spherical triangles solved by the method of right triangles 28
28
Properties of Oblique Spherical Triangles 24 The law of sines 33
33
The law of cosines 34
34
Relation between two angles and three sides 35
35
Analytical proof of the fundamental formulas 36
36
Fundamental relations for polar triangles 37
37
Formulas of half the angles in terms of the sides 39
39
Formulas of half the sides in terms of the angles 41
41
Delainbres or Gausss proportions 43
43
Art Page 33 Napiers proportions 44
44
Formulas for the area of a spherical triangle 45
45
Plane and spherical oblique triangle formulas compared 47
47
Derivation of plane triangle formulas from those of spherical triangles 48
48
CHAPTER IV
51
Case II Given the three angles S3 40 Case III Given two sides and the included angle 55
55
Case IV Given two angles and the included side 56
56
Case V Given two sides and the angle opposite one of them 58
58
Case VI Given two angles and the side opposite one of them 60
60
To find the area of a spherical triangle 62
62
Applications to geography and navigation 64
64
Applications to astronomy 66
66
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Passatges populars

Pàgina 55 - The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power.
Pàgina 131 - 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.
Pàgina 268 - ... 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.
Pàgina 219 - B) = cos A cos B - sin A sin B. cos (A - B) = cos A cos B + sin A sin B.
Pàgina 54 - The logarithm of the quotient of two numbers is equal to the logarithm of the dividend minus the logarithm of the divisor.
Pàgina 35 - 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...
Pàgina 126 - In any triangle, the sides are proportional to the sines of the opposite angles. That is, sin A = sin B...
Pàgina 49 - 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...
Pàgina 59 - FRACTION is a negative number, and is one more than the number of ciphers between the decimal point and the first significant Jigure.
Pàgina 18 - I. The sine of the middle part is equal to the product of the tangents of the adjacent parts.

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