Fixed Point Iteration is Not Converging to the Desired Root

This is the visual interpretation of @Robert Israel's answer.
Captured from my old exercise book ($\epsilon$ used is $0.0005$). Basically, the problem lies within how we truncate the sine-shaped graph when we draw arcsine graph $y = \frac{1}{\pi} sin^{-1}(-\pi/2)$, while in fact we are solving an equation involving sine, that is, $2sin(\pi x)+x=0 $. The $y_1$ and $y_2$ in the picture above is the "extended version" of the arcsine graph.

Note that if $1 \le x \le 2$, you don't want the principal branch of arcsin. You could try $x = 2 + \dfrac{\arcsin(-x/2)}{\pi}$ (for the solution in $[1.5,2]$, or $x = 1 + \dfrac{\arcsin(x/2)}{\pi}$ (for the solution in $[1,1.5]$).