EXAMPLES. 1. If a, b, c be the sides of a spherical triangle, a', b', c' the sides of its polar triangle, prove sin a sin b: sin c = sin a' : sin b': sin c'. 2. If the bisector AD of the angle A of a spherical triangle divide the side BC into the segments CD = b', BD = c', prove sin b: sin c = sin b': sin c'. 3. If D be any point of the side BC, prove that cot AB sin DAC + cot AC sin DAB = cot AD sin BAC. cot ABC sin DC + cot ACB sin BD = cot ADB sin BC. 4. If a, ẞ, y be the perpendiculars of a triangle, prove that sin a sin α= sin b sin ẞ sin c sin = γ. 5. In Ex. 4 prove that sin a cos a = √cos2 b + cos2 c 2 cos a cos b cos c. 194. Useful Formulæ. Several other groups of useful formulæ are easily obtained from those of Art. 191; the following are left as exercises for the student: Applying these six formulæ to the polar triangle, we obtain the following six: sin A cos b = cos B sin C + sin B cos C cos a sin A cos c = sin B cos C+ cos B sin C cos a sin B cos a = cos A sin C + sin A cos C cos b. sin C cos a = cos A sin B + sin A cos B cos c sin C cos b = sin A cos B + cos A sin B cos c (7) To express the 195. Formulæ for the Half Angles. sine, cosine, and tangent of half an angle of a spherical triangle in terms of the sides. I. By (4) of Art. 191 we have .*. = cos a cos b cos c sin b sin c sin2 = sin § (a+b—c) sin ‡ (a − b+c) 2 sin b sin c (Art. 45) Let 2s = a+b+c; so that s is half the sum of the sides of the triangle; then a+b-c=2(sc), and ab+c=2(s—b). Sch. The positive sign must be given to the radicals in A, B, C are each less each case in this article, because where n2 sins sin(s - a) sin (s — b) sin (s — c). = EXAMPLES. 1-cos2a-cos2b-cos2c+2 cosa cos b cosc = 4 n2 sin'b sin c sin2b sin2 c' where 4n2 = 1- cos2 a cos2b-cos2c+2 cos a cos b cos c. C 2 2. Prove cos c = cos (a + b) sin2 + cos (a sin (s-a) sin (s-b) sin (s-e) sin a sin b sinc sin (a+b). sinc sin (a - b) sin c = 0. sin (ab) sinc |