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Prove a CosA + b CosB + c CosC = 2aSinBSinC

Asked by Ambili Thalekkare 18th August 2015, 8:13 PM
Answered by Expert
Answer:
W e space k n o w space t h e space sin e space r u l e colon fraction numerator a over denominator sin A end fraction equals fraction numerator b over denominator s i i n B end fraction equals fraction numerator c over denominator sin C end fraction equals R... left parenthesis 1 right parenthesis C o n s i d e r space L H S colon a cos A plus b cos B plus c cos C equals R sin A cos A plus R sin B cos B plus R sin C cos C.... left parenthesis f r o m space e q u a t i o n space left parenthesis 1 right parenthesis right parenthesis equals R open square brackets s i n A c o s A plus s i n B c o s B plus s i n C c o s C close square brackets equals R over 2 open square brackets 2 s i n A c o s A plus 2 s i n B c o s B plus 2 s i n C c o s C close square brackets equals R over 2 open square brackets sin 2 A plus sin 2 B plus sin 2 C close square brackets A l s o comma space w e space k n o w space t h a t comma space sin C plus sin D equals 2 sin fraction numerator C plus D over denominator 2 end fraction cos fraction numerator C minus D over denominator 2 end fraction T h u s comma space L H S equals R over 2 open square brackets 2 s i n open parentheses A plus B close parentheses cos open parentheses A minus B close parentheses plus s i n 2 C close square brackets space space space space space space space equals R over 2 open square brackets 2 s i n open parentheses straight pi minus straight C close parentheses c o s open parentheses A minus B close parentheses plus s i n 2 C close square brackets space space space space space space space equals R over 2 open square brackets 2 s i n C c o s open parentheses A minus B close parentheses plus 2 s i n C c o s C close square brackets space space space space space space space equals R sin C open square brackets c o s open parentheses A minus B close parentheses plus c o s C close square brackets space space space space space space space equals R s i n C open square brackets c o s open parentheses A minus B close parentheses plus c o s open parentheses straight pi minus open parentheses straight A plus straight B close parentheses close parentheses close square brackets space space space space space space space equals R s i n C open square brackets c o s open parentheses A minus B close parentheses minus c o s open parentheses straight A plus straight B close parentheses close square brackets A n d space w e space k n o w space t h a t comma space cos open parentheses A minus B close parentheses minus cos open parentheses A plus B close parentheses equals 2 sin A sin B... left parenthesis 2 right parenthesis A g a i n comma space L H S equals 2 R s i n C sin A sin B space space space space left square bracket u sin g space e q u a t i o n space left parenthesis 2 right parenthesis right square bracket space space space space space space space space space space space space space space space space space space equals 2 R s i n C s i n A s i n B space space space space space space space space space space space space space space space space space space equals 2 R s i n C cross times a over R cross times s i n B space space space space space open square brackets f r o m space e q u a t i o n space left parenthesis 1 right parenthesis comma space sin A equals a over R close square brackets space space space space space space space space space space space space space space space space space space equals 2 a s i n B s i n C H e n c e space p r o v e d.
Answered by Expert 19th August 2015, 8:56 AM
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