Find the number of solutions of $\cos x=\frac{\lvert x \rvert}{80}$

Question

Find the number of solution of $\cos x=\frac{\lvert x \rvert}{80}$

Domain of $x$ is $[0,\pi]$ and the range of $x$ is $[-80,80]$. I am not able to proceed.

Answer 1

Calculate the number of full cosine "hills" from 0 to 80 (12). Each such hill gives two zeros. And you have one 'semi-hill' between 0 and $\pi/2$. So the total number of zeros is 12$\cdot$2+1=25 for $x>0$. Multiply this by 2 (because LHS and RHS a are symmetric with respect to $y$-axis) and you get 50 as the final answer.