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यदि $\sin 2\theta = \cos \theta ,\,\,0 < \theta < \pi $, तो $\theta $ के सम्भव मान हैं
${90^o},{60^o},{30^o}$
${90^o},{150^o},{60^o}$
${90^o},{45^o},{150^o}$
${90^o},{30^o},{150^o}$
Solution
$\sin 2\theta = \cos \theta \Rightarrow \cos \theta = \cos \left( {\frac{\pi }{2} – 2\theta } \right)$
$ \Rightarrow $ $\theta = 2n\pi \pm \left( {\frac{\pi }{2} – 2\theta } \right)$
$\Rightarrow \theta \pm 2\theta = 2n\pi \pm \frac{\pi }{2}$
अर्थात्, $3\theta = 2n\pi + \frac{\pi }{2} $
$\Rightarrow \theta = \frac{1}{3}\left( {2n\pi + \frac{\pi }{2}} \right)$
तथा $ – \theta = 2n\pi – \frac{\pi }{2} \Rightarrow \theta = – \left( {2n\pi – \frac{\pi }{2}} \right)$
अत: $\theta $ के मान $0$ और $\pi $ के बीच $\frac{\pi }{6},\,\frac{\pi }{2},\,\frac{{5\pi }}{6}$
अर्थात् ${30^o},\,{90^o},\,{150^o}$ हैं।
