1. ## Plane Transformation

Let $\theta$ be an angle, and let $m$ be the straight line through the origin with inclination $\theta$ to the positive $x$-axis. Show that the plane transformation $Q = f(P)$ that maps the point $P(x,y)$ to the point $Q(x',y')$ where

$x' = x \cos 2 \theta + y \sin 2 \theta$
$y' = x \sin 2 \theta - y \cos 2 \theta$

is in fact the reflection $\rho_m$

Note that $\rho_m$ is the reflection through line $m$

Thanks guys

2. 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 $x' = r \cos (2\theta - \alpha)$
$
= r (\cos (2 \theta) \cos (\alpha) + \sin (2 \theta) \sin (\alpha))$

$
= r \cos (\alpha) \cos (2 \theta) + r \sin (\alpha) \sin (2 \theta)$

$
= x \cos (2 \theta) + y \sin (2 \theta)$

y' can be expanded in the same manner.