If A + B + C = π , then sin 2 A + sin 2B + sin 2C is equal to :

Question

If A + B + C = π , then sin 2 A + sin 2B + sin 2C is equal to : (A)  4 sin A sin B sin C  (B)  4 cos A cos B cos C  (C)  2 cos A cos B cos C  (D)  2 sin A sin B sin C

Tardigrade Admin · Accepted Answer

We have, A +B +C = π or B+ C = π -A ⇒ sin (B+C) = sin (π -A)= sin A ∴ sin 2 A + sin 2 B +sin 2 C = 2 sin A cos A + 2 sin (B+C)cos(B-C) = 2 sin A[cos A +cos(B-C)] <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/chemistry/f0aeddca37a6c9696156f03c8cadab75-.png" /> = 2 sin A [ cos(B-C) -cos(B+C)] = 2 sin A[2 sin B sin C] =4 sin A sin B sin C

Q. If $A + B + C = π$ , then $s in 2 A + s in 2 B + s in 2 C$ is equal to :

$4 s in A s in B s in C$

$4 cos A cos B cos C$

$2 cos A cos B cos C$

$2 s in A s in B s in C$

Q. If A+B+C=π , then sin2A+sin2B+sin2C is equal to :

4sinAsinBsinC

4cosAcosBcosC

2cosAcosBcosC

2sinAsinBsinC

We have, A+B+C=π or B+C=π−A ⇒sin(B+C)=sin(π−A)=sinA ∴sin2A+sin2B+sin2C =2sinAcosA+2sin(B+C)cos(B−C) =2sinA[cosA+cos(B−C)] =2sinA[cos(B−C)−cos(B+C)] =2sinA[2sinBsinC] =4sinAsinBsinC

Q. If $A + B + C = π$ , then $s in 2 A + s in 2 B + s in 2 C$ is equal to :

$4 s in A s in B s in C$

$4 cos A cos B cos C$

$2 cos A cos B cos C$

$2 s in A s in B s in C$