Question

If $2\sin x+1\ge 0$ and $xϵ[0,2\pi ]$, then the solution set for $x$ is

• A. $[0,\frac{7\pi }{6}]$
• B. $[0,\frac{7\pi }{6}]\cup [\frac{11\pi }{6},2\pi ]$
• C. $[\frac{11\pi }{6},2\pi ]$
• D. none of these

Answer 1

The correct option is B $[0,\frac{7\pi }{6}]\cup [\frac{11\pi }{6},2\pi ]$

$2\sin x+1\ge 0\Rightarrow \sin x\ge \frac{-1}{2}$

Also we know $-1\le \sin x\le 1$

considering both solution,
$\frac{-1}{2}\le \sin x\le 1$

Hence solution in the given interval is,
$x\in [0,\frac{7\pi }{6}]\cup [\frac{11\pi }{6},2\pi ]$