Plane Transformation

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 $[r, \alpha]$ .

Now we rotate the entire graph clockwise by $\theta$ so that the line we are reflecting over becomes the x axis. $[r, \alpha] \to [r, \alpha-\theta]$ .

To reflect over the x axis, we simply multiply the angle by -1. $[r, \alpha - \theta] \to [r, \theta - \alpha]$ .

Now rotate anticlockwise by $\theta$ to restore the original orientation and the transformation is complete. $[r, \theta - \alpha] \to [r, 2 \theta - \alpha]$

so $x' = r \cos (2\theta - \alpha)$
$<br /> = r (\cos (2 \theta) \cos (\alpha) + \sin (2 \theta) \sin (\alpha))$
$<br /> = r \cos (\alpha) \cos (2 \theta) + r \sin (\alpha) \sin (2 \theta)$
$<br /> = x \cos (2 \theta) + y \sin (2 \theta)$

y' can be expanded in the same manner.