The Algebra of Coplanar Vectors and Trigonometry (Google eBook)

Front Cover
Macmillan and Company, 1892 - Numbers, Complex - 343 pages
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Contents

an interpretation in planar or twodimensional space?
11
CHAPTER I
12
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
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 Versor
48
Interpretation of 1
49
Vector expressed by a Complex Number or the sum of Project and Traject
50
General Conclusion
51
CHAPTER III
58
Introductory 5S 2 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 nil in powers of cos u sin tan u 9093
90
S Scries for cos nu and sin rewsin u in descending powers of cos u 9395
93
Series for cos nu and sin nu in ascending powers of sin u or cosm 9597
95
cos sinu in terms of cosines or sines of multiples of u 9799
97
To express cos sinw in terms of cosines or sines of multiples of u 99102
99
Formula connecting the sides and angles of a triangle
102
Examples on Chapter IV 104
104
CHAPTER V
106
If k is a positive scalar k i l where ij is a definite numerical constant
107
Hence 2 cos u i + 2i sin u I
108
Limit of sin uu when u vanishes
109
Circular MeasureRadian
110
Limit a lz when z vanishes Ill 7 Determination of e where e is such that limit 1z 1
111
n c2 4810475
114
Exponential Expressions for the Trigonometrical Functions
115
Particular Cases discussed 117119
117
General Theory of Logarithms
119
Logomoters to base n
120
Logometers to any numerical base
121
Logometers to a vector baseIllustrative Diagrams 124127
124
amh it 131132
131
Formulae for Excircular Functions 132134
132
Geometrical Interpretation of the Excircular Functions 134137
134
Roots of any order determined by roots of orders which are powers of primes 158
158
The sum of the mth roots 0 159
159
Sum of the mth powers of the nth roots 160
160
Sum of the products of every r of the th roots 161
161
1 Rationalising Factors 2 Cardans Solution of a Cubic 3 Gausss Theorem that n being a prime of the form 2a +1 162166
162
the circumference of a circle can be divided into n equal arcs with the usual conventions of Geometrical Construction
164
Examples 167168
167
INFINITE SERIESCONVERGENCY AND DIVERGENCY Geometric Series ART TACKi 1 Geometric Vector Series 1 + ri + rH + Character istic SSu...
169
Reduction of S and Sn to vector forms and the resulting Trigonometric Series 171174
171
Geometrical InterpretationTensor ratio r 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
Infinite Series in General 7 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 Series 197
197
Continuity of a Power Series
199
Case where the tensor r radius of Convergency k 201203
201
i 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 + zm 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 Series for sin 0 ros 0 sinh 0 oosh 230232
230
Logarithmic Series 233236
233
Series for Tan 1 x Tanh x 236238
236
Calculation of the value of ir 238240
238
Series for sin1 x sinh1 x cos1 r cosh1 z 240242
240
Summation by means of the foregoing series 242244
242
Sum of selected terms of a known series 245248
245
Summation of Trigonometrical Series by the Methoil of Differences 248252
248
Bernoullis Numbers 253
253
Expansion of xfte 1 254
254
Series for coth x cot x tanh x tan x coscch x cosec x 256258
256
Examples 259265
259
CHAPTER X
266
Resolution of z re into factors 271
271
Geometrical Interpretation including Cotess and Dc Moivres Properties of the Circle 273275
273
Resolution of zn cos u + ain into factors 275
275
Factor series for sine and cosine 276283
276
Geometrical Illustration 283285
283
Factor Series for sinh and cosh 285
285
The Factor Series for sine and cosine are periodic 286
286
Series for loge sin u c 287290
288
Walliss Theorem Deduction of approximate value of when ii 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 Convergenc of the series dependent on Bernoullis Numbers 294
294
Examples 296299
296
CHAPTER XI
300
Resolution of zn + 1 into the sum of n fractions 302
302

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