Find number of solutions to the equation $$\sin(6\sin x)=\frac{x}{6}.$$
By plotting it on any graphing software one is able to see instantly that it has $15$ solutions. However I am not able to work it out manually.
Let $f(x)=\sin(6\sin x)$
Period of $f(x)$ is obviously $2\pi$
$f'(x)=\cos(6\sin x)\cdot 6\cos x$
Putting $f'(x)=0$ to obtain points of maxima/minima is the tricky part.
$x=\frac{\pi}{2},\frac{3\pi}{2}$ are quite obvious but $6\sin x=\frac{\pi}{2},\frac{3\pi}{2}$ is difficult to get. After this it becomes even more confusing. Could there be a way out.