Question

If ${\cos }^{-1}\left(\cos \frac{x}{5}\right)=\frac{x-10\pi }{5}$ holds good for some $x\in R,$ then the number of integral values of $x$ satisfying it is

Answer 1

let $t=\frac{x}{5}$
$\therefore {\cos }^{-1}\left(\cos t\right)=t-2\pi$
From the above graph clearly
${\cos }^{-1}\left(\cos x\right)=x-2\pi$ when $x\in [2\pi ,3\pi ]$
So, here $t\in [2\pi ,3\pi ]$
$\Rightarrow \frac{x}{5}\in [2\pi ,3\pi ]$
$\Rightarrow x\in [10\pi ,15\pi ]$
$\Rightarrow x\in [31.4,47.1]$
$\therefore$ Number of integral values of $x=16$