Question

Sketch the graph for $y={\cos }^{-1}\left(\cos x\right)$

Answer 1

As $y={\cos }^{-1}\left(\cos x\right)$ is periodic with period $2\pi$
$\therefore$ to draw this graph we should draw the graph for one interval of length $2\pi$ and repeat for entire values of $x$ of length $2\pi$
As we know, $y={\cos }^{-1}\left(\cos x\right)=\{\begin{array}{c}x;0\le x\le \pi \\ 2\pi -x;0\le 2\pi -x\le \pi \end{array}$
$=\{\begin{array}{c}x;0\le x\le \pi \\ 2\pi -x;\pi \le x\le 2\pi \end{array}$
Thus, it has been defined for $0<x<2\pi$ that has length $2\pi$.So,its graph could be plotted as: