Question:
If $4 \sin ^{2} x=1$, then the values of $x$ are
(a) $2 n \pi \pm \frac{\pi}{3}, n \in Z$
(b) $n \pi \pm \frac{\pi}{3}, n \in Z$
(c) $n \pi \pm \frac{\pi}{6}, n \in Z$
(d) $2 n \pi \pm \frac{\pi}{6}, n \in Z$
Solution:
(c) $n \pi \pm \frac{\pi}{6}, n \in Z$
Given;
$4 \sin ^{2} x=1$
$\Rightarrow \sin ^{2} x=\frac{1}{4}$
$\Rightarrow \sin x=\frac{1}{2} \quad$ or $\quad \sin x=-\frac{1}{2}$
$\Rightarrow \sin x=\sin \frac{\pi}{6} \quad$ or $\sin x=\sin \left(-\frac{\pi}{6}\right)$
$\Rightarrow x=n \pi+(-1)^{n} \frac{\pi}{6}, n \in Z \quad$ or $x=n \pi+(-1)^{n}\left(-\frac{\pi}{6}\right), n \in Z$
$\Rightarrow x=n \pi \pm \frac{\pi}{6}, n \in Z$