G6 BOOK OF MATHEMATICAL PROBLEMS. 



307. Having given sin 6 + sin <j> = a, cos 6 + cos < = bj prove 



that 



6 d> 4a 



(1) tantan = . 



___, 



(a a +b*)*-4a a 



(3) cos cos d> = , , r- . 

 4 (a + b ) 



in 



w ^ 



(5) 



M/ T I/ 



(6) cos30 + cos3(f> = &(46 2 -3-3 '- ->. 



( a +6 J 



308. Having given 



e cos (/? + y) + cos (J3 y) = e cos (y + a) + cos (y a) 



= e cos (a + )8) + cos (a - /3), 



a, ft y being unequal and less than tr ; prove that each member of 



e'l 



these equations is equal to - - , and that 



J 



sin (/3 + y) + sin (y + a) + sin (a + ft) = 0. 



309. Having given 



A cos {ft - y) + B (cos ft + cos y) + C (sin /3 + sin y) + D = 0, 

 A cos (y - a) + B (cos y + cos a) + C (sin y + sin a) + D = 0, 

 4 cos (a-fi) + B (cos a + cos ft) + C (sin a 4- sin /?) + Z> = ; 

 prove that 



A* + B* + C* = 2 AD, ftnless sin ~^- sin y ~ a sin ?^~ = 0. 



222 



310. Having given the equation 



A cos (ft- y) + J5 cos (J3+y) + C sin (/? + y) 



+ -4 ' (cos + cos y) + B' (sin /? + sin y) + C' = 0, 



