Question

If $4{\sin }^2\mathrm{\theta }=1$, then the values of θ are
(a) $2n\pi \pm \frac{\pi }{3},n\in Z$

(b) $n\pi \pm \frac{\pi }{3},n\in Z$

(c) $n\pi \pm \frac{\pi }{6},n\in Z$

(d) $2n\pi \pm \frac{\pi }{6},n\in Z$

Answer 1

(c) $n\pi \pm \frac{\pi }{6},n\in Z$

Given:
$$\begin{gathered} 4{\sin }^2\theta =1 \\ \Rightarrow {\sin }^2\theta =\frac{1}{4} \\ \Rightarrow \sin \theta =\frac{1}{2}\mathrm{or}\sin \theta =-\frac{1}{2} \\ \Rightarrow \sin \theta =\sin \frac{\mathrm{\pi }}{6}\mathrm{or}\sin \theta =\sin \left(-\frac{\mathrm{\pi }}{6}\right) \\ \Rightarrow \theta =n\mathrm{\pi }+(-1{)}^n\frac{\mathrm{\pi }}{6},n\in Z\mathrm{or}\theta =n\mathrm{\pi }+(-1{)}^n\left(-\frac{\mathrm{\pi }}{6}\right),n\in Z \\ \Rightarrow \theta =n\mathrm{\pi }\pm \frac{\mathrm{\pi }}{6},n\in Z \end{gathered}$$