Question

Find the number of solutions of equation $\sin ^{-1}\left(4 \sin ^2 \theta+\sin \theta\right)+\cos ^{-1}(-1+6 \sin \theta)=\frac{\pi}{2}$, in $\theta \in[0,5 \pi]$.

Answer 1

We must have, $4 \sin ^2 \theta+\sin \theta=-1+6 \sin \theta$
$\Rightarrow \sin \theta=1, \frac{1}{4} \Rightarrow 6$ solutions. $]$