I don't think your answer takes into account fully the ambiguities inherent in polar coordinates. You have negative radii possibilities, as well as changing the angle by certain well-defined amounts. How does this change your picture?
Sure, but that's cartesian coordinates. As soon as you start talking about radii and angles (i.e., polar coordinates), the game is different. Example:
In fact, in polar coordinates, the point is the same point as
So from this, you can see that setting
is simply not going to imply that and
Even if you throw in the other equation (the y equation), you're not going to resolve all the ambiguities.