Changing coordinate system with non standard definitions

Question

The standard coordinate transformation to polar coordinates is $$ \begin{cases} x=r\cos(\varphi)\\ y=r\sin(\varphi) \end{cases} $$ with $r\in[0,\infty), \ \varphi\in[0,2\pi)$
The question is whether I can make the same transformations but with $r\in(-\infty,\infty), \ \varphi\in[-\frac{\pi}{2},\frac{\pi}{2})$.
Will this be valid, if not then why? Cause it looks like it can ease some calculations.