Question

Find the domain of definition of the following function:
$y=arc\cos \frac{2}{2+\sin x}$

Answer 1

$y=\text{arc}\cos \frac{2}{2+\sin x},$

$-1\le \frac{2}{2+\sin x}\le 1$

$-2-\sin x\le 2$

$⟹\sin x=-4,$

This verifies for all $x$ and

$2\le 2+\sin x$

$⟹\sin x\ge 0,$

$\therefore$ $x\in [2n\pi ,(2n+1\left)\pi \right].$