Question

The values of $\theta$ satisfying $\sin 7\theta =\sin 4\theta -\sin \theta$ and $0<\theta <\frac{\pi }{2}$ are

• A. $\frac{\pi }{9},\frac{\pi }{4}$
• B. $\frac{\pi }{3},\frac{\pi }{9}$
• C. $\frac{\pi }{6},\frac{\pi }{9}$
• D. $\frac{\pi }{3},\frac{\pi }{4}$

Answer 1

The correct option is A $\frac{\pi }{9},\frac{\pi }{4}$

$\sin 7\theta =\sin 4\theta -\sin \theta$
$\sin 7\theta +\sin \theta =\sin 4\theta$
$2\sin 4\theta \cos 3\theta =\sin 4\theta$
$\Rightarrow 2\sin 4\theta \cos 3\theta -\sin 4\theta =0$
$\sin 4\theta (2\cos 3\theta -1)=0$
$\Rightarrow \sin 4\theta =0or2\cos 3\theta -1=0$
If $\sin 4\theta =0$
$\Rightarrow 4\theta =\pi$
$\Rightarrow \theta =\frac{\pi }{4}$
And if $2\cos 3\theta =1$
$\Rightarrow \cos 3\theta =\frac{1}{2}$
$\Rightarrow 3\theta =\frac{\pi }{3}$
$\Rightarrow \theta =\frac{\pi }{9}$