Question

The value of $x$ in the interval $[0,2\pi ]$ for which $4{\sin }^{2}x-8\sin x+3\le 0$ is

• A. $[\frac{\pi }{6},\frac{\pi }{2}]$
• B. $[\frac{\pi }{3},\frac{\pi }{2}]$
• C. $[\frac{\pi }{6},\frac{3\pi }{2}]$
• D. $[\frac{\pi }{6},\frac{5\pi }{6}]$

Answer 1

The correct option is D $[\frac{\pi }{6},\frac{5\pi }{6}]$

$4{\sin }^{2}x-8\sin x+3\le 0\Rightarrow (2\sin x-1)(2\sin x-3)\le 0$
As $2\sin x<3$, so for the inequality to be true,
$\sin x\ge \frac{1}{2}$
So, from the graph


$\therefore x\in [\frac{\pi }{6},\frac{5\pi }{6}]$