Inverse function theorem question - multivariable calculus

Question

This is an exercise in Inverse Function Theorem http://en.wikipedia.org/wiki/Inverse_function_theorem

we are given the function $f:\mathbb R^2 \to \mathbb R^2$, $f(x,y)=(e^x \cos y,e^x \sin y)$

1) Show that $f$ is injective around every point in $\mathbb R^2$ - I managed to solve this

2) Find environments around the points $(0,\pi)$ and $(-1,\frac{\pi}{2})$ such that $f$ is injective in those environments, and find the inverse function in those environments

I managed to do question number 1, but am stumped by question number 2.

The answer to question 1 is that the determinant of the Jacobi matrix is $e^{2x}$ for all $(x,y) \in \mathbb R^2$ and that is always positive and so the determinant in every point on the plane is different than zero, and so we can apply inverse function theorem.

Question number 2...Could use a hand

Answer 1

Suppose we write \begin{cases}u&=&e^x\cos y\\ v&=&e^x\sin y\end{cases} so that $f(x,y)=(u,v)$.

Can you solve that system of equations for x and y? If so, what are the conditions for the solution to be a valid inverse around the points you mentioned?