What does it mean to write a polar coordinate with an angle in the range $-2\pi \le \theta < 0$?
Say i have $(r,\theta)$ in the $0 \le \theta < 2\pi$, how i would i translate that polar coordinate into the range $-2\pi \le \theta < 0$?
It all seems to me, like its just a matter of reference point, why does the range matter if this is just a point and not a vector?
Here is the particular dilemmer i am dealing with!
The range of $\theta$ doesn't matter, as long as you cover a full circle. You have multiple choices
$$0\leq\theta<2\pi$$ $$-\pi\leq\theta<\pi$$ $$-2\pi\leq\theta<0$$
and so on. However, in all cases you can still use
$$\left(x,y\right)=\left(r\cos\theta,r\sin\theta\right)$$
EDIT: As for your edited question, you need to understand that if
$$\theta_{1}=\theta_{2}+2\pi$$
then both describe the same point. So find a $\theta$ that fits to this point, then add/subtract $2\pi$ to get into the desired domain.