If A + B + C = 180 &#x2218;<!-- ∘ --> </msup> , then prove that

Merati4tmjn

Merati4tmjn

Answered question

2022-04-12

If A + B + C = 180 , then prove that
sin 3 A + sin 3 B + sin 3 C = 3 cos A 2 cos B 2 cos C 2 + cos 3 A 2 cos 3 B 2 cos 3 C 2

Answer & Explanation

kazneni3tr2b

kazneni3tr2b

Beginner2022-04-13Added 17 answers

If A + B + C = ( 2 n + 1 ) π ,
sin A + sin B + sin C = 2 sin A + B 2 cos A B 2 + 2 sin C 2 cos C 2
Now sin A + B 2 = sin ( 2 n + 1 ) π C 2 = ( 1 ) n cos C 2
and cos A + B 2 = cos ( 2 n + 1 ) π C 2 = ( 1 ) n cos C 2
2 sin A + B 2 cos A B 2 + 2 sin C 2 cos C 2
= 2 ( 1 ) n cos C 2 cos A B 2 + 2 ( 1 ) n cos A + B 2 cos C 2
= 2 ( 1 ) n cos C 2 ( cos A B 2 + cos A + B 2 )
sin A + sin B + sin C = 4 ( 1 ) n cos A 2 cos B 2 cos C 2
if A + B + C = ( 2 n + 1 ) π
Here n=0,1

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