Find the number of solutions of $\ln |\sin x|=-x^2+2x$ when $x\in [0,\pi]$ and when $x\in [-\pi/2, 3\pi/2]$

Question

The question can be easily solved by graphing the function. And it isnt particularly difficult to graph them either

However it is difficult to check whether the graphs will intersect or not, and at how many points, without a calculator at our disposal. Are there any methods to check this?