Question

If $f:[0,4\pi ]\to [0,\pi ]$ be defined by $f\left(x\right)=co{s}^{-1}\left(cosx\right)$Then, the number of points $xϵ[0,4\pi ]$ satisfying the equation $f\left(x\right)=\frac{10-x}{10}$, is ___

Answer 1

(i) Using definition of $f\left(x\right)=co{s}^{-1}\left(x\right)$, we trace the curve $f\left(x\right)=co{s}^{-1}\left(cosx\right)$.
(ii) The number of solutions of equations involving trigonometric and algebraic functions and involving both functions are found using graphs of the curves

We know that,$co{s}^{-1}\left(cosx\right)=\{\begin{array}{cc}x,& Ifxϵ[0,\pi ]\\ 2\pi -x,& Ifxϵ[\pi ,2\pi ]\\ -2\pi +x,& Ifxϵ[2\pi ,3\pi ]\\ 4\pi -x,& Ifxϵ[3\pi ,4\pi ]\end{array}$

For above graph, it is clear that $y=\frac{10-x}{10}$ and $y=co{s}^{-1}\left(cosx\right)$ intersect at three distinct points, so number of solutions is 3.