MATnEMATICS. SPHERICAL TRIGONOMETRY. 



[SINES AND oosrsES. 



the triangle, join O A. O B. O C, and produce the planes 

 It, HOC, CO A in 1. linitcly ; at A draw a piano 

 AIM perpendicular to O A, cutting the planes A O B, 

 BOG, CO A in Ap, pq, q.\ respectively, then since }>A 

 ia on the plane A O B, and perpendicular to O A, and 

 oA is on the plane C O A, and perpendicular to O A, 

 A<j u the angle between the planes, and .'.is equal to 

 the angle A of the triangle; also the angle pOq ia the 

 angle subtended by B C : i.e., in the angle a. 



Hence (Plane Trig., Art. 37) 



Aj* + A</' 2 Ap. Aq cos. A = pq* = Oj>' + <V 

 2 Oj>. Oq cos. a. 



Ap sin. 

 row - \ = tan. e = 



OP 



UA 



see. e 



COS. ,' 



Aq , Bin. b 



'. = tan. b 



OA 



cos. b 



COS.C 



CM.S. - L- 



' - * 



OA = = cos. b 



'e . sin. ! 6 .sin. e sin. 6 . 

 . + ^m 2. ,cos.A 



cos. 6 

 2 cos. a 



C03. S C 



2 sin. c sin. 6 



sin. 1 6 



cos.' 6 oon.*6 



and 



cos-'c 



Similarl y 8oU& 



2 sin. c sin. b 



8in.*c 



CM.-..- C 



sin." 



COS.- C 



cos. a c 



M. A 



2 cos. a 



2 



.'. cus. A 



* cos. c cos. b ( cos. 5 



.*. sin. c sin. 6 cos. A = cos. a cos. c cos. 6 

 cos. a cos. 6 cos c 



(3) Q. E. D. 



Bin. b sin. c 



COR. 1. If o.6.c. represent the lengths of the sides of 

 the triangle, and r the radius, 



Then are the angles of formula (3) in 

 circular measure. 



Hence, remembering that cos. 01 

 6' 



1.2.3.4' 



and sin. 5 = 



we have 



Cos. A 



(a* \ ! o 2 c* \ 1 



i 27,) ( 1 ~27 2 ~27 5 )+ tern " 1 fawaWl v;4' 



a 2 + terms involving 



-j + terms involving j 3-+ ' ' " 



Now in the limit when r is infinite, the terms involving 



. _L. . . . will all disappear ; hence in the limit, 

 r 3 



a A . 

 S. ^V ^ 



2 be + terms involving jjp 



2 be 



or o 2 = 6 2 + c 2 2iecos. A, 

 as in the case of the plane triangle. (Page 624). 



COB. 2. In formula (3) substitute 180 a' for A. 

 IStf" A' for a. 180 B' for b, and 180" C' for C, 

 and we shall have 



, coa. A' + cos. B' cos. C 7 



sin. H' sin. C' 



This is true of the sides and angles of every polar 

 triangle. Now it appears from Prop. XI., p. 699, that 

 every triangle may be regarded as the polar triangle of 

 some other ; hence the above formula is perfectly 

 general, and is true of every triangle, and we have 



cos. A 4- cos. B cos. C 



Cos. o =r - . ' . ^ - (4) 

 sin. B sin. 



COB. 3. Formulas similar to (3) and (4) are, of course, 

 true of Cos. B, and Cos. C, and of Cos. o and Cos. c. 



(3). To exjtress the Formulas of the, leal Article in a Form 

 adapted for Logarithmic Calculation. 



cos. o cos. 6 cos. c 

 Since Cos.A= gin . /, gin . c 



. OOB. a cos. b cos. c 

 ..1 + C08.A-1+- gin 6gin c 



COS. O COB. 6 CQg. C + 8 ' n - & B ' p - 



And 1 cos. A 1 



sin. b sin. c 

 cos, a cos. 6 cos, o 



: in- /. sia. ' 



cos. 6 COB, c 4" sin, b sin, c cos, o 

 sin. b sin. c 



A cos. o (cos. b cos. c sin. 6 sin. c) 



* O na 2 = . -. : 



2 sin . b sin. c 



cos, a cos. (6 -\- c) 

 sin. 6 sin. c 



. 2 A_ cos. (6 c) cos, a 

 And 2 sin. ^ - gin b gin c 



2 A 2 sin. ^ (a + b + c) sin, fr (6 + c a) 

 ' 2cos - "2 = " sin. b sin. c 



A 2 sin. 4 (a 6 + c) sin. 4. (a + b c) 

 And 2 sin. 2 ^ = siUi b sin _ c 



Now ifo-r& + c=2s, then b + c a = 2( a) 

 2 (g 6), and a -j- 6 c = 2 (s c). 



(5) 



sin. (s b) sin, (s e) 



sin. 6 sin c 



A sin, (s b) sin, (s c) 

 2 ~ sin. l sin . (s a) 



And since, 



2 A sin, s sin. ( a) 

 cos - ~2 = sin. 6 sin. c 



. t 

 Bm ' 2 = 



A A 

 sin. A = 2 sin. - cos. -3 



(G) 



sin. 2 A=- . , . . , sin, . sin. ( a) sin. (s b) 

 sin. 2 b sin.' c 



sin. (s c) 



these formulas are analogous to the formulas on p. 024, 

 of Plane Trigonometry, which can be shown to bo the 

 limits of those in the same uiauiior as iu Cor. (I), Art. (1), 

 and Cor. (1), Art. (2). 



It is to be observed that, since any two sides of a tr 

 angle are greater than the- third, s o, s b, s c, are 

 positive ; and since all the sides of a triangle less 

 t>,:m four right angles, is less than two right angles, 

 and d fortiori, s a, s b, s c, are each loss than two 

 riht angles : so that, sin. ., sin. ( a), sin. (s b), 

 sin. ( c), are each positive, also b and c are each less 

 than 180 ; so that sin. 6 and sin. c are always positive. 



