Question

The number of solutions for the equation ${\cos }^{-1}\left(\cos x\right)=-\frac{1}{2}$ is

• A. 1
• B. infinite
• C. 0.0
• D. 2

Answer 1

The correct option is C 0.0

Here, it is easily solved if we use the graph of the function in hand. The function on the LHS is,
$f\left(x\right)={\cos }^{-1}\left(\cos x\right)\{\begin{array}{c}-2n\pi +x,xϵ[2n\pi ,(2n+1\left)\pi \right]\\ 2n\pi -x,xϵ\left[\right(2n-1)\pi ,2n\pi ,nϵI].\end{array}$
The solutions for the equation will be intersection of the following functions.
$y={\cos }^{-1}\left(\cos x\right)\&y=-\frac{1}{2}$.

We can see that the function $y={\cos }^{-1}\left(\cos x\right)$ gives only non-negative values and hence do not intersect with the line $y=-\frac{1}{2}$.
$\therefore$ Number of solutions for the given equation is zero.