Question

prove that: acosA+bcosB+ccosC=2asinBsinC

Answer 1

a cos A + b cos B + c cos C, ... where ... a/sin A = ... = k 

= (k/2) [ 2 sin A cos A + 2 sin B cos B + 2 sin C cos C ] 

= (k/2) [ ( sin 2A + sin 2B ) + sin 2C ] 

= (k/2) [ 2 sin (A+B)· cos(A-B) + sin 2C ] 

= (k/2) [ 2 sin C. cos(A-B) + 2 sin C cos C ] 

= k sin C [ cos(A-B) + cos C ] ... here .. cos C = cos [ π - (A+B) ] = - cos (A+B) 

= k sin C [ cos(A-B) - cos(A+B) ] 

= k sin C [ 2 sin A sin B ] 

= 2 ( k sin A ) sin B sin C 

= 2 a sin B sin C