Discussion Forum

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

Answer:

Explanation:

Consider the equations

  $\cos 2 \theta=\sin \theta$

$\Rightarrow$     $1-2\sin^{2} \theta=\sin \theta$

$\Rightarrow$   $2 \sin^{2} \theta+\sin \theta-1=0$

$\Rightarrow$   $2 \sin \theta( \sin \theta+1)-1(\sin \theta+1)=0$

$\Rightarrow$   $(\sin \theta+1)(2 \sin \theta-1)=0$

$\Rightarrow$                $\sin \theta=-1$  or $\sin \theta=\frac{1}{2}$

$\Rightarrow$            $\theta=\frac{3 \pi}{2}$  or  $\theta= \frac{\pi}{6},\frac{5 \pi}{6}$

                                                $[\because \theta \in (0,2\pi)]$

Thus, number of solutions of the given equation is 3