
Plane Transformation
Let $\displaystyle \theta$ be an angle, and let $\displaystyle m$ be the straight line through the origin with inclination $\displaystyle \theta$ to the positive $\displaystyle x$axis. Show that the plane transformation $\displaystyle Q = f(P)$ that maps the point $\displaystyle P(x,y)$ to the point $\displaystyle Q(x',y')$ where
$\displaystyle x' = x \cos 2 \theta + y \sin 2 \theta$
$\displaystyle y' = x \sin 2 \theta  y \cos 2 \theta$
is in fact the reflection $\displaystyle \rho_m$
Note that $\displaystyle \rho_m$ is the reflection through line $\displaystyle m$
Thanks guys

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 $\displaystyle [r, \alpha]$.
Now we rotate the entire graph clockwise by $\displaystyle \theta$ so that the line we are reflecting over becomes the x axis. $\displaystyle [r, \alpha] \to [r, \alpha\theta]$.
To reflect over the x axis, we simply multiply the angle by 1. $\displaystyle [r, \alpha  \theta] \to [r, \theta  \alpha]$.
Now rotate anticlockwise by $\displaystyle \theta$ to restore the original orientation and the transformation is complete. $\displaystyle [r, \theta  \alpha] \to [r, 2 \theta  \alpha]$
so $\displaystyle x' = r \cos (2\theta  \alpha)$
$\displaystyle
= r (\cos (2 \theta) \cos (\alpha) + \sin (2 \theta) \sin (\alpha))$
$\displaystyle
= r \cos (\alpha) \cos (2 \theta) + r \sin (\alpha) \sin (2 \theta)$
$\displaystyle
= x \cos (2 \theta) + y \sin (2 \theta)$
y' can be expanded in the same manner.