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113. When a Side and the Hypotenuse are nearly Equal
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166
167
168
169
172
114. Four Cases of Oblique Triangles ...
115. Case I.- Given a Side and Two Angles
116. Case II.—Given Two Sides and the Angle opposite One of them, 173
117. Case III. — Given Two Sides and the Included Angle....... 176
118. Case IV. - Given the Three Sides
177
124. Object observed from Two Points in Same Vertical Line.... 185
132. Sin and tan @ are in Ascending Order of Magnitude .
208
sin 0
133. The Limit of is Unity ..
209
134. Limiting Values of sin ✪ and cos 0.
135. To calculate the Sine and Cosine of 10" and of 1'.
211
136. To construct a Table of Natural Sines and Cosines..
137. Another Method..
142. Tables of Logarithmic Trigonometric Functions..
143. The Principle of Proportional Parts...
217
218
ART.
144. To prove the Rule for the Table of Common Logarithms... 218
219
220
221
222
223
145. To prove the Rule for the Table of Natural Sines...
146. To prove the Rule for a Table of Natural Cosines.
147. To prove the Rule for a Table of Natural Tangents.
148. To prove the Rule for a Table of Logarithmic Sines.
149. To prove the Rule for a Table of Logarithmic Cosines.
150. To prove the Rule for a Table of Logarithmic Tangents.
151. Cases of Inapplicability of Rule of Proportional Parts
152. Three Methods to replace the Rule of Proportional Parts... 224
Examples...
226
CHAPTER IX.
DE MOIVRE'S THEOREM. - APPLICATIONS.
153. De Moivre's Theorem.
and cos 0..
154. To find all the Values of (cos +V-1 sin 0)
155. To develop cos no and sin ne in Powers of sin
156. To develop sin and cos in Series of Powers of 0.
157. Convergence of the Series ......
158. Expansion of cos"
159. Expansion of sin"
in Terms of Cosines of Multiples of 0...
in Terms of Cosines of Multiples of 0..
160. Expansion of sin" 0 in Terms of Sines of Multiples of
161. Exponential Values of Sine and Cosine.....
162. Gregory's Series ..
163. Euler's Series.
164. Machin's Series.
165. Given sin = x sin (0+ a); expand
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Р
231
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234
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236
237
238
239
240
241
166. Given tan x = n tane; expand x in Powers of n.
168. Resolve x2 + 1 into Factors.
169. Resolve x2n 2n cos 0+1 into Factors.
170. De Moivre's Property of the Circle .
171. Cote's Properties of the Circle
172. Resolve sin @ into Factors
173. Resolve cos e into Factors
174. Sum the Series sin a + sin(α+ B) + etc.
175. Sum the Series cos a + cos(α + B) + etc...
176. Sum the Series sinm a + sinm (a + B) + etc..
177. Sum the Series sin α
178. Sum the Series cosec
sin(x+8) + etc.
+ cosec 20+ cosec 40+ etc..
ᎪᎡᎢ .
179. Sum the Series tan + tan- + tan + etc..
2 4
180. Sum the Series sin a + x sin (α+ B) + etc.
181. Summation of Infinite Series
Examples....
PART II.
SPHERICAL TRIGONOMETRY.
CHAPTER X.
255
256
257
192. Relation between a Side and the Three Angles
193. To find the Value of cot a sin b, etc.
.. 278
279
ᎪᎡᎢ.
201. Case I. Given the Hypotenuse and an Angle..
202. Case II. - Given the Hypotenuse and a Side
203. Case III. - Given a Side and the Adjacent Angle.
204. Case IV. Given a Side and the Opposite Angle..
205. Case V. - Given the Two Sides..
206. Case VI. Given the Two Angles.
-
207. Quadrantal and Isosceles Triangles
208. Solution of Oblique Spherical Triangles.
209. Case I.- Given Two Sides and the Included Angle.
Given Two Angles and the Included Side..
210. Case II.
Given Two Sides and One Opposite Angle...
Given Two Angles and One Opposite Side.
Given the Three Sides...
212. Case IV.
213. Case V.
214. Case VI.
Given the Three Angles..
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301
302
303
304
305
307
309
312
313
314
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CHAPTER XII.
THE IN-CIRCLES AND EX-CIRCLES.-AREAS.
215. The Inscribed Circle...
216. The Escribed Circles.
217. The Circumscribed Circle.
218. Circumcircles of Colunar Triangles...
219. Areas of Triangles. — Given the Three Angles..
324
325
326
328
329
330
220. Areas of Triangles. Given the Three Sides...
221. Areas of Triangles.—Given Two Sides and the Included Angle, 331
230. Small Variations in Parts of a Spherical Triangle
231. Inclination of Adjacent Faces of Polyedrons...
232. Volume of Parallelopiped.
353
356
357