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If $\sin 2\theta = \cos \theta ,\,\,0 < \theta < \pi $, then the possible values of $\theta $ are
${90^o},{60^o},{30^o}$
${90^o},{150^o},{60^o}$
${90^o},{45^o},{150^o}$
${90^o},{30^o},{150^o}$
Solution
(d) $\sin 2\theta = \cos \theta $
$\Rightarrow \cos \theta = \cos \left( {\frac{\pi }{2} – 2\theta } \right)$
$ \Rightarrow $ $\theta = 2n\pi \pm \left( {\frac{\pi }{2} – 2\theta } \right) $
$\Rightarrow \theta \pm 2\theta = 2n\pi \pm \frac{\pi }{2}$
$i.e.$, $3\theta = 2n\pi + \frac{\pi }{2} $
$\Rightarrow \theta = \frac{1}{3}\left( {2n\pi + \frac{\pi }{2}} \right)$
and $ – \theta = 2n\pi – \frac{\pi }{2}$
$\Rightarrow \theta = – \left( {2n\pi – \frac{\pi }{2}} \right)$
Hence value of $\theta $ between $0$ and $\pi $ are $\frac{\pi }{6},\,\frac{\pi }{2},\,\frac{{5\pi }}{6}$
$i.e.$, ${30^o},\,{90^o},\,{150^o}$.
