The Algebra of Coplanar Vectors and Trigonometry (Google eBook)

Front Cover
Macmillan and Company, 1892 - Exponential functions - 343 pages
0 Reviews
  

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

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 PAGE 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 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 vorsors
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 formulae 8589
85
Functions of the sum of any number of angles 89
89
Functions of nit in powers of cos it sin it tan it 9093
90
Series for cos nit and sin nuam u in descending powers of eosii 9395
93
Scries for cos nit and sin nu in ascending powers of sin it or cos it 9597
95
cosit sinii in terms of cosines or sines of multiples of it 9799
97
To express C08tt sinn 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 A i lgt where j is a definite numerical constant
107
Hence 2 cos it it + ij 2i sin it ij
108
Limit of sin itit when it vanishes
109
Circular MeasureRadian
110
i Limit a 1 when s vanishes Ill 7 Determination of c where c is such that limit c 1z 1
111
z2 4810475
114
Exponential Expressions for the Trigonometrical Functions
115
Particular Cases discussed 117119
117
General Theory of Logarithms
119
Logomcters to base 7
120
Logomcters to any numerical base
121
Logomcters to a vector baseIllustrative Diagrams 124127
124
amh u 131132
132
56 Geometrical Interpretation of the Excircular Functions 134137
134
Properties of the Excircle deduced from Excircular Functions 137142
137
Graphic Construction of the Vectors sin u + vi cosw + vi 142144
142
cosh u + vi and sinh u + vi
145
cot it + 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
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 Convorgency 188190
188
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 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 PAOER 9 Series for sin 9 cos 0 sinh 0 cosh 0 230232
230
Logarithmic Series 233236
233
Series for Tanl x Tanhl x 236238
236
Calculation of the value of it 238240
238
Series for sinl x sinhl x cosl x coshl x 240242
240
Summation by means of the foregoing series 242244
242
Sum of selected terms of a known scries 245248
245
Summation of Trigonometrical Series by the Method of Differences 248252
248
Bernoullis Numbers 253
253
Expansion of xe 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 a into factors 271
271
Geometrical Interpretation including Cotess and De Moivros Properties of the Circle 273275
273
Resolution of zn az cos u + aSn into factors 275
275
Factor series for sine and cosine 276283
276
Geometrical Illustration 283285
283
Factor Series for sinh and cosh 2S5
286
Series for loge 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 ls + 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
Excircular Functions coth tanh c as the sum of an infinite series of fractions
314
Examples 31a317
317
CHAPTER XII
318
23 Expression offz in powers of z Zo 319322
319
Continuity of 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 si 1
343

Common terms and phrases

Popular passages

Page 4 - Symbolical Algebra" it is thus enunciated: "Whatever algebraical forms are equivalent, when the symbols are general in form but specific in value, will be equivalent likewise when the symbols are general in value as well as in form.
Page xix - On the geometrical representation of the powers of quantities, whose indices involve the square roots of negative quantities.
Page xxi - O'Brien (Rev. M.) Treatise on Plane Coordinate Geometry ; or the Application of the Method of Coordinates to the Solution of Problems in Plane Geometry. 8vo. Plates, 9s.
Page xx - Syllabus of a Course of Lectures upon Trigonometry and the Application of Algebra to Geometry. 8vo. 7*. 6d. MECHANICS AND HYDROSTATICS. Elementary Hydrostatics. By WH BESANT, MA, Late Fellow of St John's College. [Preparing. Elementary Hydrostatics for Junior Students. By R. POTTER, MA late Fellow of Queens...
Page 55 - The rectangle contained by the diagonals of a quadrilateral inscribed in a circle is equal to the sum of the rectangles contained by the two pairs of opposite sides.
Page 109 - ... perim. p : perim. p'. q. ED Carol. 1. And one of the triangular portions ABo, of a polygon circumscribing a circle, is to the corresponding circular sector, as the side AB of the polygon, to the arc of the circle included between AO and BO. Cor. 2. Every circular arc is greater than its chord, and less than the sum of the two tangents drawn from its extremities and produced till they meet. The first part of this corollary is evident, because a right line is the shortest distance between two given...
Page xix - Consideration of the Objections raised against the geometrical Representation of the Square Roots of Negative Quantities.
Page xxi - The SPIRIT of MATHEMATICAL ANALYSIS, and its Relation to a Logical System.
Page 270 - SX, which may be made as small as we please by taking n large enough.
Page 192 - Also e may be made as small as we please by taking n sufficiently great.

Bibliographic information