A Treatise on Plane and Advanced Trigonometry

by Hobson, E. W.

• ISBN13: 9780486441771
• ISBN10: 0486441776
• Edition: 7th
• Format: Hardcover
• Copyright: 2004-12-17
• Publisher: Dover Publications

Summary

This account of the theory of the circular functions and their applications to plane trigonometry will provide invaluable assistance to students of mathematics who intend to proceed further in the study of analysis. Other students, too, will benefit from the depth with which the more elementary parts of the subject are treated.

Table of Contents

CHAPTER I. THE MEASUREMENT OF ANGULAR MAGNITUDE.
1. Introduction	1	(1)
2-3. The generation of an angle of any magnitude	1	(2)
4. The numerical measurement of angles	3	(1)
5-10. The circular measurement of angles	4	(3)
11. The length of a circular arc	7	(3)
12. The area of a sector of a circle	10	(1)
Examples on Chapter I	10	(2)
CHAPTER II THE MEASUREMENT OF LINES. PROJECTIONS.
13-16. The measurement of lines	12	(1)
17. Projections	13	(2)
CHAPTER III. THE CIRCULAR FUNCTIONS.
18-21. Definitions of the circular functions	15	(4)
22-24. Relations between the circular functions	19	(1)
25. Range of values of the circular functions	20	(1)
26-29. Properties of the circular functions	21	(3)
30. Periodicity of the circular functions	24	(1)
31. Changes in the sign and magnitude of the circular functions	24	(2)
32. Graphical representation of the circular functions	26	(2)
33. Angles with one circular function the same	28	(1)
34. Determination of the circular functions of certain angles	29	(3)
35-38. The inverse circular functions	32	(1)
Examples on Chapter III	33	(3)
CHAPTER IV. THE CIRCULAR FUNCTIONS OF TWO OR MORE ANGLES.
39-43. The addition and subtraction formulae for the sine and cosine	36	(5)
44-45. Formulae for the addition or subtraction of two sines or two cosines	41	(3)
46. Addition and subtraction formulae for the tangent and cotangent	44	(1)
47. Various formulae	45	(2)
48. Addition formulae for three angles	47	(1)
49. Addition formulae for any number of angles	48	(2)
50. Expression for a product of sines or of cosines as the sum of sines or cosines	50	(2)
51. Formulae for the circular functions of multiple angles	52	(1)
52. Expressions for the powers of a sine or cosine as sines or cosines of multiple angles	53	(1)
53. Relations between inverse functions	54	(1)
54. Geometrical proofs of formulae	55	(3)
Examples on Chapter IV	58	(5)
CHAPTER V. THE CIRCULAR FUNCTIONS OF SUBMULTIPLE ANGLES.
55-63. Dimidiary formulae	63	(7)
64. The circular functions of one-third of a given angle	70	(2)
65-66 Determination of the circular functions of certain angles	72	(3)
Examples on Chapter V	75	(3)
CHAPTER VI. VARIOUS THEOREMS.
67. Introduction	78	(1)
68. Identities and transformations	78	(4)
69. The solution of equations	82	(2)
70. Eliminations	84	(1)
71. Relations between roots of equations	85	(2)
72. Maxima and minima. Inequalities	87	(2)
73. Porismatic systems of equations	89	(1)
74-77. The summation of series	90	(4)
Examples on Chapter VI	94	(10)
CHAPTER VII. EXPANSION OF FUNCTIONS OF MULTIPLE ANGLES.
78-79. Series in descending powers of the sine or cosine	104	(2)
80-83. Series in ascending powers of the sine or cosine	106	(3)
84. The circular functions of submultiple angles	109	(1)
85. Symmetrical functions of the roots of equations	110	(4)
86-91. Factorization	114	(6)
Examples on Chapter VII	120	(4)
CHAPTER VIII. RELATIONS BETWEEN THE CIRCULAR FUNCTIONS AND TIDE CIRCULAR MEASURE OF AN ANGLE.
92-95. Theorems	124	(3)
96. Miler's product	127	(3)
97-98. The limits of certain expressions	130	(1)
99. Series for the sine and cosine of an angle in powers of its circular measure	131	(4)
100. The relation between trigonometrical and algebraical identities	135	(1)
Examples on Chapter VIII	135	(4)
CHAPTER IX TRIGONOMETRICAL TABLES.
101. Introduction	139	(1)
102-105. Calculation of tables of natural circular functions	139	(4)
106. The verification of numerical values	143	(1)
707. Tables of tangents and secants	143	(1)
108. Calculation by series	144	(1)
109. Logarithmic tables	145	(1)
110-111. Description and use of trigonometrical tables	145	(2)
112-114. The principle of proportional parts	147	(5)
115-117 Adaptation of formulae to logarithmic calculation	152	(3)
CHAPTER X. RELATIONS BETWEEN TUUE SIDES AND ANGLES OF A TRIANGLE.
118-124. Theorems	155	(4)
125. The area of a triangle	159	(1)
126. Variations in the sides and angles of a triangle	160	(1)
127-128. Relations between the sides and angles of polygons	161	(1)
129. The area of a polygon	162	(2)
Examples on Chapter X	164	(3)
CHAPTER XI. TILE SOLUTION OF TRIANGLES.
130. Introduction	167	(1)
131-133. The solution of right-angled triangles	167	(2)
134-140. The solution of oblique-angled triangles	169	(7)
141-144. The solution of polygons	176	(2)
145-149. Heights and distances	178	(4)
Examples on Chapter XI	182	(8)
CHAPTER XII. PROPERTIES OF TRIANGLES AND QUADRILATERALS.
150. Introduction	190	(1)
151. The circumscribed circle of a triangle	190	(1)
152-154. The inscribed and escribed circles of a triangle	191	(4)
155. The medians	195	(1)
156. The bisectors of the angles	196	(1)
157. The pedal triangle	197	(1)
158. The distances between special points	198	(3)
159. Expressions for the area of a triangle	201	(1)
160-163. Various properties of triangles	201	(2)
164-167. Properties of quadrilaterals	203	(5)
168. Properties of regular polygons	208	(1)
169. Examples	209	(4)
Examples on Chapter XII	213	(11)
CHAPTER XIII. COMPLEX NUMBERS.
170. Introduction	224	(1)
171-174. The geometrical representation of a complex number	224	(3)
175-177. The addition of complex numbers	227	(2)
178. The multiplication of complex numbers	229	(2)
179. Division of one complex number by another	231	(1)
180-185. The powers of complex numbers	232	(5)
186-187. De Moivre's theorem	237	(2)
188. Factorization	239	(2)
189. Properties of the circle	241	(1)
190 Examples	241	(2)
Examples on Chapter XIII	243	(3)
CHAPTER XIV THE THEORY OF INFINITE SERIES.
191. Introduction	246	(1)
192-196 The convergence of real series	246	(5)
197. The convergence of complex series	251	(2)
198. Continuous functions	253	(1)
199-201. Uniform convergence	253	(4)
202. The geometrical series	257	(1)
203-208. Series of ascending integral powers	258	(7)
209. Convergence of the product of two series	265	(1)
210. The convergence of double series	266	(2)
211-212. The Binomial theorem	268	(4)
213-217. The circular functions of multiple angles	272	(7)
218-219. Expansion of the circular measure of an angle in powers of its sine	279	(1)
220-222. Expression of powers of sines and cosines in sines and cosines of multiple angles	280	(4)
CHAPTER XV. THE EXPONENTIAL FUNCTION. LOGARITHMS.
223-227. The exponential series	284	(4)
228. Expansions of the circular functions	288	(1)
229-230(¹). The exponential values of the circular functions	288	(3)
231-232. Periodicity of the exponential and circular functions	291	(1)
233-237. Analytical definition of the circular functions	291	(5)
238-239. Natural logarithms	296	(1)
240-244. The general exponential function	297	(3)
245. Logarithms to any base	300	(1)
246-248. Generalized logarithms	300	(2)
249-250. The logarithmic series	302	(2)
251. Gregory's series	304	(1)
251(¹)-251(³). The quadrature of the circle	305	(5)
252-254. The approximate quadrature of the circle	310	(1)
255. Trigonometrical identities	311	(1)
256-257. The summation of series	312	(3)
Examples on Chapter XV	315	(7)
CHAPTER XVI. THE HYPERBOLIC FUNCTIONS.
258. Introduction	322	(1)
259. Relations between the hyperbolic functions	322	(1)
260-261. The addition formulae	323	(1)
262. Formulae for multiples and submultiples	324	(1)
263-265. Series for hyperbolic functions	324	(2)
266. Periodicity of the hyperbolic functions	326	(1)
267-270. Area of a sector of a rectangular hyperbola	326	(5)
271. Expressions for the circular functions of complex arguments	331	(1)
272-274. The inverse circular functions of complex arguments	331	(2)
275-276. The inverse hyperbolic functions	333	(2)
277. The solution of cubic equations	335	(1)
278. Table of the Gudermannian function	336	(1)
Examples on Chapter XVI	337	(1)
CHAPTER XVII. INFINITE PRODUCTS.
279-281. The convergence of infinite products	338	(5)
282-292. Expressions for the sine and cosine as infinite products	343	(11)
292(¹). Representation of the exponential function by an infinite product	354	(1)
293-295. Series for the tangent, cotangent, secant, and cosecant	355	(5)
296-299. Expansion of the tangent, cotangent, secant, and cosecant in powers of the argument	360	(5)
300. Series for the logarithmic sine and cosine	365	(2)
301. Examples	367	(2)
Examples on Chapter XVII	369	(5)
CHAPTER XVIII. CONTINUED FRACTIONS.
302-303. Proof of the irrationality of a	374	(1)
304. Transformation of the quotient of two hypergeometric series	375	(1)
305. Euler's transformation	376	(1)
Examples on Chapter XVIII	376	(2)
Miscellaneous Examples	378