Prove uniqueness of polar-coordinates $(R>0, \theta)$ up to angle $\theta+ 2 \pi$. 
Prove uniqueness of polar-coordinates $(R>0, \theta)$ up to angle $\theta+ 2 \pi$.

Suppose we have $(x,y) \in \mathbb R^2$. Then we can transform this point to polar-cordinates $(R>0, \theta)$ where the polar-coordinates are stated to be unique up to $\theta+ 2 \pi$ .
How do I show that no other point $(a,b) \in \mathbb R^2$ different from $(x,y) \in \mathbb R^2$ doesn't have the same polar-coordinates ? I'm looking for an algebraic argument, not a "draw-and-see" argument.
I know I can prove uniquness of coordinate in $\mathbb R^n$ using linear algebra on the set $\{e_i\} $which form a basis.
More generally how are uniqueness of spherical coordinates proved ?

Uniquess of $R$ is easy, because $R=\sqrt{a^2+b^2}$. For the angle you have to do some trigonometric work (or quote that $e^{i\theta}=1$ iff $\theta$ is a multiple of $2\pi$)
