Q77 / 155 · WBJEE 2018 · Mathematics

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

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

Solution

We have, $\sin 6 \theta+\sin 4 \theta+\sin 2 \theta=0$ $\Rightarrow \quad \sin 6 \theta+\sin 2 \theta+\sin 4 \theta=0$ $\Rightarrow \quad 2 \sin 4 \theta \cdot \cos 2 \theta+\sin 4 \theta=0$ $\Rightarrow \quad \sin 4 \theta(2 \cos 2 \theta+1)=0$ $\Rightarrow \quad \sin 4 \theta=0$ or $2 \cos 2 \theta+1=0$ $\Rightarrow \quad 4 \theta=n \pi$ or $\cos 2 \theta=\frac{-1}{2}=\cos \frac{2 \pi}{3}$ $\Rightarrow \quad \theta=\frac{n \pi}{4}$ or $2 \theta=2 n \pi \pm \frac{2 \pi}{3}$ $\Rightarrow \quad \theta=\frac{n \pi}{4}$ or $\theta=n \pi \pm \frac{\pi}{3},$ where $n \in z$

Hence, the correct answer is option: 'A'.

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