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) \(\frac{n \pi}{4}, n \pi \pm \frac{\pi}{6}\)
(C) \(\frac{n \pi}{4}, 2 n \pi \pm \frac{\pi}{3}\) (n is integer)
(D) \(\frac{\mathrm{n} \pi}{4}, 2 \mathrm{n} \pi \pm \frac{\pi}{6}\)

Answer 1

Ans: (A)
Hint \(: \sin 6 \theta+\sin 4 \theta+\sin 2 \theta=0 \Rightarrow 2 \sin 4 \theta \cdot \cos 2 \theta+\sin 4 \theta=0\)
\(\Rightarrow \sin 4 \theta=0\) or \(2 \cos 2 \theta+1=0\)
\(\Rightarrow 4 \theta=\mathrm{n} \pi \quad \Rightarrow \cos 2 \theta=-\frac{1}{2}\)
\(\Rightarrow \theta=\frac{n \pi}{4},(n \in \mathbb{Z}) \Rightarrow 2 \theta=2 n \pi \pm \frac{2 \pi}{3}\)
\(\Rightarrow \theta=\mathrm{n} \pi \pm \frac{\pi}{3}(\mathrm{n} \in \mathbb{Z})\)