Answer
The domain is given by the set $\{(x,y)|x\neq 0, -1\leq y/x \leq 1\}$.
The range is given by the set $[-\pi/2,\pi/2]$.
Work Step by Step
1) Domain
Our function is $f(x,y) = \arcsin(y/x).$
The domain of $\arcsin$ function is a segment of the real line $[-1,1]$. This means that any argument of $\arcsin$ must take values from that segment so we need that
$$-1\leq\frac{y}{x}\leq1$$
Further, whenever we have a fraction, its denominator must not be zero so we additionally have $x\neq 0$. So the domain is the set of all ordered pairs $(x,y)$ such that $-1\leq y/x\leq 1$ and $x\neq 0$ which we will denote by $\{(x,y)|x\neq 0, -1\leq y/x \leq 1\}$.
2) Range
The range of $\arcsin$ function are angles between $-\pi/2$ and $\pi/2$. Because the argument $y/x$ can take any value from $-1$ to $1$ (any number between $-1$ and $1$ can be written as a quotient of other two numbers) we have that the range of the function $f(x,y) =\arcsin(y/x)$ is a segment $[-\pi/2,\pi/2]$.
