Question

The domain of $f\left(x\right)={\cos }^{-1}\left(\frac{2}{2+\sin x}\right)$ contained in $[0,2\pi ]$ is

• A. $[0,\frac{\pi }{2}]$
• B. $\left[\frac{\pi }{2},\pi \right]$
• C. $[0,\pi ]$
• D. $\left[\frac{-\pi }{2},\frac{\pi }{2}\right]$

Answer 1

Answer: (C)

$[0,\pi ]$

Domain of the function is value of $x$ in $f\left(x\right)={\cos }^{-1}\left(\frac{2}{2+\sin x}\right)$

Since value of $\cos x$ is between $-1$ to $1$ thus

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

$\Rightarrow \sin x\ge 0$

$\Rightarrow x\in [0,\pi ]$

Hence, option 'C' is correct.