Question

If $A+B+C=\pi$ then $\sin 2A+\sin 2B+\sin 2C=$

• A. $4\sin A\sin B\cos C$
• B. $4\sin A\sin B\sin C$
• C. $4\cos A\sin B\sin C$
• D. None of these

Answer 1

Answer: (B)

$4\sin A\sin B\sin C$

$\sin 2A+\sin 2B+\sin 2C$
$=2\sin \left(\frac{2A+2B}{2}\right)\cos \left(\frac{2A-2B}{2}\right)+\sin (2\pi -2(A+B))$
$=2\sin (A+B)\cos (A-B)-\sin 2(A+B)=2\sin (A+B)\cos (A-B)-2\sin (A+B)\cos (A+B)=2\sin (A+B)(\cos (A-B)-\cos (A+B))=2\sin (\pi -C)\left(2\sin A\sin B\right)=4\sin A\sin B\sin C$