Answer
$0.8959, 2.2457$
Work Step by Step
Step 1. Let $u=sin(x)$; we have $2u^2+u-2=0$ which gives $u=\frac{-1\pm\sqrt {17}}{2}$. That is $u=-1.2808, 0.7808$
Step 2. For $sin(x)=u=-1.2808$, we have no solutions in $[0,2\pi)$
Step 3. For $sin(x)=u=0.7808$, we have solutions in $[0,2\pi)$ as $x=sin^{-1}0.7808\approx0.8959$ and $x=\pi-0.8959\approx2.2457$
Step 4. Thus, we have all the solutions in $[0,2\pi)$ as $x=0.8959, 2.2457$
