An elementary treatise on plane and spherical trigonometry; with their applications to navigation, surveying, heights and distances, and spherical ... of Bowditch's navigator and the nautical

by Benjamin Peirce

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1861 ... = £(a--J + c); we have A + B = a + c, A--B = b; and (47) becomes sin. s--sin. (s--b) tang. b sin. s--sin. (s--b) tang. J (a-f-c)' This equation, substituted in (355), gives cos. (A--C) tang. J b cos. J (4--C)--tang. J (a--c)' which is the same as (354). 81. Scholium. In using (350) and (354), the signs of the terms must be attended to, by reference to PI. Trig. §§ 62 and 64; and it will be seen that, by (354), o-f-c is greater or less than 180, according as A--C is greater or less than 180. (355) 82. Examples. 1. Given in a spherical triangle two angles = 158 and--98, and the interjacent side = 144; to find the other two sides. Solution. By (350) and (354), J (-£+£) = 128 cosec. 10.10347 sec. 10.21066. i(A--C)= 30 sin. 9.69897 cos. 9.93753 £ b =72 tang. 10.48822 tang. 10.48822 i(a--c) = 62 53' 2" tang. 10.29066 J(a+c) =103 0'24" tang. 10.63641 Ans. a = 165 53' 26", c= 40 V 22". 2. Given in a spherical triangle two angles = 126 12' and = 109 16', and the interjacent side = 175 27'; to find the other two sides. Ans. 167 38' 19" and 14 30' 11"., 83. Problem. To solve a spherical triangle, when its three angles are given. B., p. 441. Solution. When the angles of the triangle ABC are given, the sides of its polar triangle A'B'C are readily found. The desired solution may, then, be obtained by applying to A'B'C' any of the methods of §§ 70-74. 84. Corollary. Applying (331) to A'B'C, we have, by (98) and (99), cos. C + cos. A cos. B cos. c =.-..--5; (357) sin. A sin. B which may also be derived from (317), and which may be used to find either side, when the angles are given. 85. Corollary. If we put S=i(A + B+C), (358) we have, in the polar triangle A'B'C, s' = 270--S, 89. Example. Given in a s...

64 pages, Paperback

First published September 27, 2010

About the author

American mathematician