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So, I understand that the general coordinates in the Polar Coordinate system takes the form $(r,\theta) $. From what I've read, $\theta$ is the angle made with positive x-axis and $r$ is the distance from the origin. Clearly, distance cannot be negative, so why can r be negative when dealing with polar coordinates?

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    You could see [this post](http://math.stackexchange.com/questions/1390581/how-can-r-be-negative-when-dealing-with-polar-coordinates?rq=1) which is very similar – 2017-01-23

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If r is negative you start on the other side (the negative x-axis). So $(-3, \frac{\pi}{3})$ is actually the same point as $(3, \frac{4\pi}{3}).$