Simplify $\arcsin(\sin(x))$ when $\frac{\pi}{2} \leq x \leq \frac{3\pi}{2}$
I realize that $\arcsin(\theta)$ is restriced to $\frac{-\pi}{2} \leq \theta \leq \frac{\pi}{2}$ in order to be one to one and therefore invertible; however, I can't seem to connect the new domain given in the question to the provided result of $\pi - x$.
How does the shifted domain change the result to $\pi - x$ instead of just $x$? Isn't the function just the same in the new interval, i.e., it's still one to one?
I have seen a few explanations of this topic in textbooks; however, I can't seem to connect the logic. How do I go about solving these type of problems in general?