Question

Domain of $f\left(x\right)=y=\sqrt{{\log }_3\cos \left(\sin x\right)}$ is

• A. $3\{\frac{2\pi }{2}:n\in Z\}$
• B. $\{2n\pi :n\in Z\}$
• C. $\{n\pi :n\in Z\}$
• D. None of thses

Answer 1

Answer: (C)

$\{n\pi :n\in Z\}$

Here $y$ would take real values if
${\log }_3\cos \left\{\right(\sin x\left)\right\}\ge 0\Rightarrow \cos \left(\sin x\right)\ge 1$
But $-1\le \cos \theta \le 1$

$\Rightarrow \cos \left(\sin x\right)=1$
$\Rightarrow \sin x=0$
$\Rightarrow x=n\pi ,n\in Z$
Hence, the domain consists of the multiples of $\pi$ i.e., domain $\left(y\right)=\{n\pi :n\in Z\}$