Question

If $\sin\theta + \sin 2\theta + \sin 3\theta = \sin\alpha$and

$\cos\theta + \cos 2\theta + \cos 3\theta = \cos\alpha$, then θ is equal to

• A. $\alpha/2$
• B. $\alpha$
• C. $2\alpha$
• D. $\alpha/6$

Answer 1

Answer: (A)

$\alpha/2$

Answer 2

$\sin\theta + \sin 3\theta + \sin 2\theta = \sin\alpha$

⇒ $2\sin 2\theta\cos\theta + \sin 2\theta = \sin\alpha$

⇒ $\sin 2\theta(2\cos\theta + 1) = \sin\alpha$ …..(i)

Now $\cos\theta + \cos 3\theta + \cos 2\theta = \cos\alpha$

$2\cos 2\theta\cos\theta + \cos 2\theta = \cos\alpha$

.$\cos 2\theta(2\cos\theta + 1) = \cos\alpha$ …..(ii)

From (i) and (ii), $\tan 2\theta = \tan\alpha$ ⇒ $2\theta = \alpha$ ⇒ $\theta = \alpha/2$.