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The number of solutions of $\cos 2 \theta=\sin \theta$ in $(0,2 \pi)$ are
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Verified Answer
The correct answer is:
3
$$
\begin{aligned}
& \cos 2 \theta=\sin \theta \\
& \therefore 1-2 \sin ^2 \theta=\sin \theta \Rightarrow 2 \sin ^2 \theta+\sin \theta-1=0 \\
& \therefore(2 \sin \theta-1)(\sin \theta+1)=0 \Rightarrow \sin \theta=\frac{1}{2},-1
\end{aligned}
$$
We have $\theta \in(0,2 \pi)$
$\therefore$ Possible values of $\theta$ are $\frac{\pi}{6}, \frac{5 \pi}{6}, \frac{3 \pi}{6}$
\begin{aligned}
& \cos 2 \theta=\sin \theta \\
& \therefore 1-2 \sin ^2 \theta=\sin \theta \Rightarrow 2 \sin ^2 \theta+\sin \theta-1=0 \\
& \therefore(2 \sin \theta-1)(\sin \theta+1)=0 \Rightarrow \sin \theta=\frac{1}{2},-1
\end{aligned}
$$
We have $\theta \in(0,2 \pi)$
$\therefore$ Possible values of $\theta$ are $\frac{\pi}{6}, \frac{5 \pi}{6}, \frac{3 \pi}{6}$
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