WMC 2018 Pursuit - Pack 1, C2

$\begin{aligned} \arcsin (\sin x) & = \frac x 2 \\ \sin x & = \sin \frac x2 \\ 2 \sin \frac x2 \cos \frac x2 & = \sin \frac x2 & \small \color{#3D99F6} \text{For }\sin \frac x2 \ne 0 \\ 2 \cos \frac x2 & = 1 \\ \implies \cos \frac x2 & = \frac 12 \\ \frac x2 & = \pm \frac \pi 3 & \small \color{#3D99F6} \text{Note that }-\frac \pi 2 \le \arcsin x \le \frac \pi 2 \\ x & = \pm \frac {2\pi}3 \end{aligned}$

The difference between the largest and smallest solution is $\dfrac {4\pi}3 \approx \boxed{4.189}$ .