I really can't remember how I used to do these questions, but here is the way I would do it now. It is likely that I have forgotten a much simpler method.

Before the transformation, the polar coordinates of a point are .

Now we rotate the entire graph clockwise by so that the line we are reflecting over becomes the x axis. .

To reflect over the x axis, we simply multiply the angle by -1. .

Now rotate anticlockwise by to restore the original orientation and the transformation is complete.

so

y' can be expanded in the same manner.