Question

If $\sin 6\theta +\sin 4\theta +\sin 2\theta =0$, then general value of $\theta$ is

• A. $\frac{n\pi }{4},n\pi \pm \frac{\pi }{3}$
• B. $\frac{n\pi }{4},n\pi \pm \frac{\pi }{4}$
• C. $\frac{n\pi }{4},n\pi \pm \frac{\pi }{6}$
• D. $\frac{n\pi }{4},2n\pi \pm \frac{\pi }{6}$

Answer 1

Answer: (A)

$\sin 6\theta +\sin 4\theta +\sin 2\theta =0$
$\Rightarrow (\sin 6\theta +\sin 2\theta )+\sin 4\theta =0$
$\Rightarrow 2\sin 4\theta \cos 2\theta +\sin 4\theta =0$
$\Rightarrow \sin 4\theta (2\cos 2\theta +1)=0$
$\Rightarrow \sin 4\theta =0$ or $\cos 2\theta =-\frac{1}{2}=\cos \frac{2\pi }{3}$
$\Rightarrow 4\theta =n\pi$ or $2\theta =2n\pi \pm \frac{2}{3}$
$\Rightarrow \theta =\frac{n\pi }{4},\theta =n\pi \pm \frac{\pi }{3}\theta$