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

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 V 1
49
Vector expressed by a Complex Number or the sum of Project and Trajcct
50
General Conclusion
51
CHAPTER III
58
Definitions of the Trigonometrical Ratios The fundamental equation i cos u + i sin u 59
59
Fundamental Relations of the six ratios 62 00
63
Ratios for the reversed angle supplement c 07
72
Inverse Functions 73
73
Some Trigonometrical identities proved from the versor forms 76
76
Examples on Chapter III
78
CHAPTER IV
79
Formulae for sine cosine tangent of u v 2m c 82
82
Submultiple angle formulae 8589
89
Functions of nu in powers of cos u sin u tan u 9093
90
Series for cos nu and sin nusin 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
cosn sinnu in terms of cosines or sines of multiples of u 9799
97
To express cosmm sinu in terms of cosines or sines of multiples of u 99102
99
connecting the sides and angles of a triangle
103
Examples on Chapter IV 104
104
CHAPTER V
106
If k is a positive scalar fc i gjjfc where n is a definite numerical constant
107
Hence 2 cos u r + V1 2i sin u i tj
108
Limit of sin uu when ft vanishes
109
Circular MeasureEadian
110
Limit a 1j when z vanishes Ill 7 Determination of c where c is such that limit C 1z 1
111
t c2 4810475
114
Exponential Expressions for the Trigonometrical Functions
115
Particular Cases discussed 117119
117
General Theory of Logarithms
119
Logometers to base n
120
Logometers to any numerical base
121
Logometers to a vector baseIllustrative Diagrams 124127
124
CHAPTER VI
128
EXCIRCULAR OR HYPERBOLIC TBIGONOMKTRY ART PAGES 1 2 The excircular Functions obtained from the circular Functions by putting iu ...
132
56 Geometrical Interpretation of the Excircular Functions 134137
134
Properties of the Excirele deduced from Excircular Functions 137142
137
Graphic Construction of the Vectors sin u + vi cosm + vi 142144
142
Sum of the mth powers of the ?sth roots 160161
160
Sum of the products of every r of the mth roots 161162
161
1 Rationalising Factors 2 Cardans Solution of a Cubic 3 Gausss Theorem that n being a prime of the form 2 +1 J 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
Geometrical InterpretationTensor ratio r 1 the series
174
Tensor Ratio 1 The series divergent
182
Rate of Convergency 188190
189
Power Series 197
197
CHAPTER IX
210
The Binomial Series equal to the prime value of 1 + 218220
218
Trigonometrical Series derived from the Binomial Theorem 224
224
ART PAOES 9 Series for sin 9 cos 0 sinh 8 cosh 9 230232
230
Logarithmic Series 233236
233
Series for Tani x Tanhl 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 means of the foregoing scries 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 254255
254
Scries for coth x cot r tanh x tan x eosech 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 Moivrcs Properties of the Circle 273275
273
Resolution of zn az cos m + a8 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 log 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
303
Introductory 30
318

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 160 - ... with the last, the second with the last but one, and so on.
Page xix - The Principles and Application of Imaginary Quantities, Book I. ; to which are added, some observations on Porisms ; being the first of a series of original tracts in various parts of the Mathematics ; by Benj.
Page xix - Consideration of the objections raised against the geometrical representation of the square roots of negative quantities.
Page 22 - The straight lines which join the vertices of a tetrahedron to the centroids* of the opposite faces meet in a point which is a point of quadrisection of each line.
Page 213 - But it is clear that there is a wide margin of cases in which more subtle tests will be needed. Examples LXVIII. 1. Apply Cauchy's and d'Alembert's tests (as specialised in 7 above) to the series Znkrn, where k is a positive integer.

Bibliographic information