Answer
$x\approx3.553$ and $x\approx5.872$
Work Step by Step
One of the solutions can be found by using the inverse function:
$5~sin~x+2=0$
$5~sin~x=-2$
$sin~x=\frac{-2}{5}$
$x=arcsin(-0.4)=-0.412~rad~~$ (it lies in the range $[-\frac{\pi}{2},\frac{\pi}{2}]$)
$x=-0.412~rad+2\pi=5.872~rad~~$ (now it lies in the interval $[0,2\pi)$)
