Question

Find the set of all values of $x$ in the interval $[0,\pi ]$ for which
$2si{n}^{2}x-3sinx+1\ge 0$

Answer 1

$2{\sin }^{2}x-3\sin x+1\ge 0\Rightarrow 2{\sin }^{2}x-2\sin x-\sin x+1\ge 0\Rightarrow (2\sin x-1)(\sin x-1)\ge 0\Rightarrow 2\sin x-1\le 0\sin x\ge 1$
$\Rightarrow \sin x\le \frac{1}{2}$ or $\sin x=1$
$\Rightarrow xϵ[0,\frac{\pi }{6}]\cup \left\{\frac{\pi }{2}\right\}\cup [\frac{5\pi }{6},\pi ]$