How many solutions will the equation $\sin(\sin(\sin(\sin(\sin(x))))=\frac{x}{3}$ have? How many real solutions does the following equation have?
$$\sin(\sin(\sin(\sin(\sin(x))))=\frac{x}{3}$$
I calculated the derivative of the LHS, but there was nothing I can find.
Could you please give me some hints?
 A: $x=0$ is an obvious solution.  For small $x, \sin (x)\approx x$, so the left side will have slope of $1$ as it leaves the origin, while the right will have slope $\frac 13$.  We know the left has value $0$ at $x=\pi$, so there must be one root in $(0,\pi)$.  The left side then becomes negative and hits zero at $x=3\pi$, but by then $\frac x3 \gt 1$, so there will be only one positive root.  Because both functions are odd, there will also be a single negative root.  Finding it is numeric, but we have a bracket so it will be easy.
A: Plotting, you probaly noticed that the positive solution is somewhere between $\frac \pi 2$ and $\frac {2\pi} 3$.
If you enjoy composing Taylor series, you should have
$$\sin (\sin (\sin (\sin (\sin (x)))))-\frac{x}{3}=\left(\sin (\sin (\sin (\sin (1))))-\frac{\pi }{6}\right)-\frac{1}{3}
   \left(x-\frac{\pi }{2}\right)+O\left(\left(x-\frac{\pi }{2}\right)^2\right)$$ which gives as an approximate solution
$$x=3 \sin (\sin (\sin (\sin (1))))\approx 1.88272 $$ which not too bad.
