The Algebra of Coplanar Vectors and Trigonometry (Google eBook)

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Macmillan and Company, 1892 - Exponential functions - 343 pages
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Contents

Equality of Vectors
13
Associative and Commutative Laws of Terms
14
Application to Geometry
16
Illustrative Examples 1728
17
Examples on Chap I 2830
30
CHAPTER II
36
ART PAGES 3 Vector Multiplication
38
Commutative Law of Multiplication t
39
Associative Law
40
Distributive Law
41
Reciprocal of a Vector
42
Vector Division
43
Multiplicity of Values of Scalar Powers
44
Illustrations
46
Vector as product of Tensor and Vcrsor 4S 13 Interpretation of V 1
49
Vector expressed by a Complex Number or the sum of Project and Traject
50
General Conclusion
51
CHAPTER III
58
Definitions of the Trigonometrical Ratios The fundamental equation i cos u + i sin u 5961
59
Fundamental Relations of the six ratios 62
62
The ratios in terms of versors
63
Ratios for the reversed angle supplement c 6769
67
Values of the ratios for certain acute angles 6972
69
Expressions for all angles having the same sine cosine or tangent 72
72
Inverse Functions 7376
73
Some Trigonometrical identities proved from the versor forms 7678
76
Examples on Chapter III
78
CHAPTER IV
79
Formulae for sine cosine tangent of u v 2m c 82
82
cosines as products of sines and cosines and the converse 84
84
Submultiple angle formula 8589
85
Functions of the sum of any number of angles 89
89
Functions of nu in powers of cos u sin tan u 9093
90
Series for cos nu and sin nusin it in descending powers of cos it 9395
94
cosit sinit in terms of cosines or sines of multiples of u 9799
97
To express cosmM sinit in terms of cosines or sines of multiples of it 99102
99
Formula connecting the sides and angles of a triangle
103
Examples on Chapter IV 104
104
CHAPTER V
106
If k is a positive scalar i lg where ij is a definite numerical constant
107
Hence 2 cos u + tj 2t sin u ij tj
108
Limit of sin nu when it vanishes
109
Circular MeasureRadian
110
Limit a lz when z vanishes Ill 7 Determination of c where e is such that limit clz 1
111
i c2 4810475
115
Particular Cases discussed 117119
117
General Theory of Logarithms
119
Logometers to base Tj
120
Logometers to any numerical base
121
Logometers to a vector baseIllustrative Diagrams 124127
124
by putting iu for u 128131
132
56 Geometrical Interpretation of the Excircular Functions 134137
134
Properties of the Excircle deduced from Excircidar Functions 137142
137
Graphic Construction of the Vectors sin u + vi cos u + vi 142144
142
cosh u + vi and sinh u + vi
145
cot u + vi
147
Miscellaneous Examples Chapters I VI 148151
148
CHAPTER VII
152
The Roots as powers of the Principal Root
155
Primary and Subordinate Roots 156158
156
Roots of any order determined by roots of orders which are powers of primes 158159
158
Geometrical InterpretationTensor ratio 1 the series fluctuating 174177
174
Tensor ratio 1 the series convergent Construction for the Characteristic 178182
178
Tensor Ratio 1 The series divergent
182
Definitions
184
Rate of Convergency 188190
189
Continuity or Discontinuity of an Infinite Series 190194
190
Fundamental Laws of Algebra in relation to Infinite Series 194197
194
Power Scries 197
197
Continuity of a Power Series
199
Case where the tensor r radius of Convergency k 201203
201
ExamplesApplication of the Tests to Particular Series 203206
203
Examples 207209
207
CHAPTER IX
210
Binomial Theorem 211213
211
Convergency or Divergency of the Binomial Series 213215
213
Illustrative Diagrams 215218
215
The Binomial Series equal to the prime value of 1 + 218220
218
Index a complex numberGeometrical Interpretation 220224
223
Trigonometrical Series derived from the Binomial Theorem 224
224
Exponential TheoremThree Proofs 225230
225
ART PAOES 9 Scries for sin 9 cos 0 sinh 6 cosh fl 230232
230
Logarithmic Scries 233236
233
Series for Tanl x Tanh1 x 236238
236
Calculation of the value of ir 238240
238
Series for sin1 x sinh1 x cos1 x cosh1 x 240242
240
Summation by moans of the foregoing series 242244
242
Sum of selected termsof a known series 245248
245
Summation of Trigonometrical Series by the Method of Differences 248252
248
Bernoullis Numbers 253
253
Expansion of xe 1 254255
254
Series for coth x cot x tanh x tan x cosech x cosec x 256258
256
Examples 259265
259
CHAPTER X
266
Resolution of s an into factors 271
271
Geometrical Interpretation including Cotess and De Moivres Properties of the Circle 273275
273
Resolution of z az cos u + a2 into factors 275
275
Factor series for sine and cosine 276283
276
Geometrical Illustration 283285
283
Factor Series for sinh and cosh 2S5 2S6 8 The Factor Series for sine and cosine are periodic 286
286
Scries for logs sin u c 287290
287
Walliss Theorem Deduction of approximate value of when n is large 290292
290
Deductions from the series for sines c
292
Relations between Bernoullis Numbers and Sums of inverse powers of the Natural Numbers
293
Limits of Convergency of the series dependent on Bernoullis Numbers 294
294
Examples 296299
296
CHAPTER XI
300
Resolution of 12 + 1 into the sum of n fractions 302
302
ART PAGES 3 Cot u and tan u expressed as the mean of n cotangents 303306
303
Geometrical Interpretation 306309
306
Cot u and tan u as the sum of an infinite series of fractions 309311
309
Series for cot u deduced from the Geometrical Interpretation 311
311
Cosec u and sec u as the sum of an infinite series of fractions
313
315317
315
RATIONAL AND INTEGRAL FUNCTIONS 1 Definition Notation c 318
318
23 Expression of fz in powers of z z0 319322
319
Continuity offz Every rational and integral equation has a root 322325
322
Expression of fz in vector factors 325
325
Tests for the number of roots within given limitsCauchys Theorem 326329
326
Conjugate Functions 329334
329
Discussion of a Cubic Function 334343
334
The Binomial Function z 1
343

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