Let $f:]0,\infty[ \times \mathbb{R}^2\rightarrow \mathbb{R}^3$ defined by $f(r,\theta,z)=(r \cos(\theta), r \sin(\theta),z)$.
By the inverse function theorem, we know that for each point there is a neighbourhood, on which a local inverse is defined. I would like to know is what would this neighbourhood look like, in its maximum 'size/amplitude'.
If $f(r,\theta,z)=(x,y,z)$ then $\cos(\theta)=\frac{x}{\sqrt{x^2+y^2}}$ and $\sin(\theta)=\frac{y}{\sqrt{x^2+y^2}}$. So, I was thinking of solving these last two equalities, in order to $\theta$. But then I get in trouble since $\arccos, \arcsin$ have different range, and to keep track of all possibilities seems to be too difficult.
Any help would be appreciated.
