Question

Find the value of x for which satisfy equation: ${\cos }^{-1}\left(\cos x\right)=x$

• A. $[0,\frac{\pi }{4}]$
• B. $[0,\frac{\pi }{2}]$
• C. $[0,\pi ]$
• D. $[0,2\pi ]$

Answer 1

Answer: (C)

$[0,\pi ]$

$f\left(y\right)={\cos }^{-1}\left(y\right)$

Hence $yϵ[-1,1]$ and $f\left(y\right)ϵ[0,\pi ]$,
since $\cos \left(x\right)$ is a one one function in $[0,\pi ]$.
$\therefore {\cos }^{-1}\left(x\right)$ exists in $[0,\pi ]$

Therefore the above equation

$co{s}^{-1}\left(cos\right(x\left)\right)=x$ is satisfied only if $xϵ[0,\pi ]$