What is the domain of $Z=\sin(\ln(x\,\arccos{y}))$?

Question

What is the domain of $Z=\sin(\ln(x\,\arccos{y}))$? I see that is should be $-\dfrac{\pi}{2}\leq \ln(x\,\arccos{y}) \leq \dfrac{\pi}{2}$ and then $e^{-\dfrac{\pi}{2}} \leq x*\arccos(y) \leq e^{\dfrac{\pi}{2}}$ and now I'm stuck.

Answer 1

Let's look at each part of the function and see what is required.

1. For $\arccos y$ to be defined requires $-1\leq y\leq 1$.
2. For $\ln(x\arccos y)$ to be defined requires $x\arccos y>0$. In other words, since $\arccos y$ is positive when $y<1$, we need $x>0$.
3. For $\sin(\ln(x\arccos y))$ to be defined requires no additional condition, given that $\ln(x\arccos y)$ is defined. This is because the sine function is defined for all possible inputs.

In summary, the domain is $\{(x,y)\,:\,x>0\text{ and }-1\leq y < 1\}$.