If $A + B + C = 0^{o}$ then $sin 2 A + sin 2 B + sin 2 C = ?$

sin A cos B cos C

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2 sin A cos B sin C

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4 sin A sin B sin C

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- 4 sin A sin B sin C

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Solution

The correct option is D $- 4 sin A sin B sin C$

$Solve:- Given,$

$A + B + C = 0$
$\Rightarrow A + B = - C - (i)$

$\Rightarrow sin 2 A + sin 2 B + sin 2 C =$

$[2 sin (\frac{2 A + 2 B}{2}) \cdot cos (\frac{2 A - 2 B}{2})] + sin 2 C$

$= [2 sin (A + B) \cdot cos (A - B)] + 2 sin C cos C$

$sin c + sin D =$
$2 sin (\frac{c + D}{2}) \cdot cos (\frac{c - D}{2})$
$and sin 2 x = 2 sin x cos x$

by using equation (i) we get

$= [2 sin (- c) \cdot cos (A - B)] + 2 sin c cos c$

$= - 2 sin c [cos (A - B) - cos (c)]$

because $cos C = cos (- (A + B)) = cos (A + B)$

$S o, cos (A - B) - cos (A + B) = 2 sin A sin B$

$\Rightarrow sin 2 A + sin 2 B + sin 2 c = - 4 sin A sin B sin C$

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