Exercise8
We saw earlier that the rotation in the plane through an angle \(\theta\) is given by the matrix:
\begin{equation*} \left[\begin{array}{rr} \cos\theta \amp \sin\theta \\ \sin\theta \amp \cos\theta \\ \end{array}\right] \text{.} \end{equation*}We would like to find a similar expression for the matrix that represents the reflection in \(L_\theta\text{,}\) the line passing through the origin and making an angle of \(\theta\) with the positive \(x\)axis, as shown in Figure 22.

To do this, notice that this reflection can be obtained by composing three separate transformations as shown in Figure 23. Beginning with the vector \(\xvec\text{,}\) we apply the transformation \(R\) to rotate by \(\theta\) and obtain \(R(\xvec)\text{.}\) Next, we apply \(S\text{,}\) a reflection in the horizontal axis, followed by \(T\text{,}\) a rotation by \(\theta\text{.}\) We see that \(T(S(R(\xvec)))\) is the same as the reflection of \(\xvec\) in the original line \(L_\theta\text{.}\)
Using this decomposition, show that the reflection in the line \(L_\theta\) is described by the matrix
\begin{equation*} \left[\begin{array}{rr} \cos(2\theta) \amp \sin(2\theta) \\ \sin(2\theta) \amp \cos(2\theta) \\ \end{array}\right] \text{.} \end{equation*}You will need to remember the trigonometric identities:
\begin{equation*} \begin{aligned} \cos(2\theta) \amp {}={} \cos^2\theta  \sin^2\theta \\ \sin(2\theta) \amp {}={} 2\sin\theta\cos\theta \\ \end{aligned} \text{.} \end{equation*} Now that we have a matrix that describes the reflection in the line \(L_\theta\text{,}\) show that the composition of the reflection in the horizontal axis followed by the reflection in \(L_\theta\) is a counterclockwise rotation by an angle \(2\theta\text{.}\) We saw some examples of this earlier in Exercise 2.