SINES, COSIXES, ETC.] M ATHEMAT ICS. SPHERIC AL TRIGONOMETRY. 



659 



Hence, (5), (6), (7), (8), are always positive, and .-. the 



A A A 



values of cos. -3- sin. 75 tan. -= and sin. A derived from 



a 21 31 



them are always real. 



COB If we consider the case of the polar triangle A 

 = 180 of :. ^ = 90 -> o= 180 A', 6 = 180 



& 



B', c=180 c 7 . 

 .'. = 270 - | (A' + B' + C') = 270 S' 



. ., . C , A sm. (s 6) , 



Similarly, tan. tan. = ---- . - and 

 2 sin. s 



A , B sin. (s c) 

 tan. - tan. _ = - ^ -- '- 

 2 2 sm. s. 



But tan. 



C tan '2 tan -o- 



22 A, B 



1 tan. Q 'an- a" 



sin. (s a) sin. (s 6) 



sm. s sin. s 



sin. (s c) 



