If I have a circle of radius $a$ center at $(a,0)$. If I want to change coordinate from Cartesian coordinate to polar coordinate. I have convinced myself that $r=2a\cos\theta$ and $\frac{-\pi}{2}\le\theta<\frac{\pi}{2}$.
But what if a circle of radius $a$ center at $(0,a)$? I know that $r=2a\sin\theta$ but why $0\le\theta<\pi?$ Looking at the graph it seems like $\theta$ can vary from $-\pi$ to $\pi.$
The second circle is in the upper-half plane. Also, the angle parametrizing the first circle ranged over $\pi$ radians. Why would the second be any different?
Actually, any interval of length $\pi$ seems to work!
The first circle can be parametrized as:
$$r=2a\cos\theta \text{ and }\phi_0\le\theta<\phi_0+\pi,$$ and the second as
$$r=2a\sin\theta \text{ and }\phi_0\le\theta<\phi_0+\pi,$$
for any $\phi_0.$