How come the polar graph of or only has a period of a pi?
To find the period I would use , so why wouldnt the period of both these polar circles be
Consider , which is a circle of radius a/2, centered at a/2. From 0 to pi/2 , is positive, so r is positive. From pi/2 to pi, is negative, which makes r negative as well. On a graph, a negative r implies that the line starts from the origin and proceeds in the direction opposite to that which the corresponding angle points to.
So at theta=0, r = a, and theta=pi, r = -a, which are the same point.