Question

The number of solutions of $\cos 2\theta =\sin \theta$ in $(0,2\pi )$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$
• E. $0$

Answer 1

Answer: (C)

We have $\cos 2\theta =\sin \theta$
$\Rightarrow$ $\cos 2\theta =\cos \left(\frac{\pi }{2}-\theta \right)$
$\Rightarrow$ $2\theta =2n\pi \pm \left(\frac{\pi }{2}-\theta \right),n\in Z$
Taking + sign, we have $\theta =\frac{2n\pi }{3}+\frac{\pi }{6},n\in Z$
$\Rightarrow$ $\theta =\frac{\pi }{6}+\frac{5\pi }{6}\in (0,2\pi )$
Taking - sign, we have $\theta =2n\pi -\frac{\pi }{2},n\in Z$ $\theta =\frac{3\pi }{2}$
$\Rightarrow$ $\theta =\frac{3\pi }{2}\in (0,2\pi )$
Hence, there are three solutions.