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{\bf Question}

An anti-clockwise rotation of the plane (centre the origin)
through an angle $\alpha$ corresponds to the matrix

$$ C(\alpha) = \left(\begin{array}{rr} {\cos (\alpha)} &{-\sin
(\alpha)} \\ {\sin (\alpha)} & {\cos (\alpha)}  \end{array}
\right),$$

and the reflection of the plane in the line through the origin
making an angle $\beta$ with the x-axis (where $\beta$ is measured
anti-clockwise) corresponds to the matrix

$$ S(\beta) = \left(\begin{array}{rr} {\cos (2\beta)} &{\sin
(2\beta)} \\ {\sin (2\beta)} & {-\cos (2\beta)}  \end{array}
\right).$$

Show that
\begin{description}
\item[(i)]
$C(\alpha)C(\gamma)=C(\alpha + \gamma)$ (a rotation followed by a
rotation gives a rotation);
\item[(ii)]
$C(\alpha)S(\beta)=S(\frac{2\beta +\alpha}{2})$ (a reflection
followed be a rotation gives a reflection);
\item[(iii)]
$S(\beta)C(\alpha)=S(\frac{2\beta - \alpha}{2})$ (a rotation
followed by a rotation gives a reflection)
\item[(iv)]
$S(\beta)S(\gamma)=C(2\beta - 2\gamma)$ (a reflection followed by
a reflection gives a rotation)
\end{description}
(Hint: You will need the following expansions: $$\sin(A \pm B) =
\sin(A)\cos(B) \pm \cos(A)\sin(B)$$  $$\cos(A \pm B) =
\cos(a)\cos(B) \mp \sin(A)\sin(B)\  )$$


{\bf Answer}

\begin{description}

\item[(i)]
$\left(\begin{array} {cc} {\cos \alpha} & {-\sin \alpha}\\ {\sin
\alpha} & {\cos \alpha}
\end{array} \right) \left(\begin{array} {cc} {\cos \gamma} & {-\sin \gamma}\\ {\sin
\gamma} & {\cos \gamma}
\end{array} \right)$
\begin{eqnarray*}& = & \left(\begin{array} {cc} {\cos \alpha \cos \gamma -
\sin \alpha \sin \gamma} & {-[\sin \alpha \cos \gamma + \cos
\alpha \sin \gamma]}\\ {\sin \alpha \cos \gamma + \cos \alpha \sin
\gamma} & {\cos \alpha \cos \gamma - \sin \alpha \sin \gamma}
\end{array} \right)\\ & = & \left(\begin{array} {cc} {\cos (\alpha + \beta)} & {-\sin (\alpha + \beta)}\\
{\sin (\alpha + \beta)} & {\cos (\alpha+ \beta)} \end{array}
\right) \\ & = & C(\alpha + \gamma) \end{eqnarray*}

\item[(ii)]
$\left(\begin{array} {cc} {\cos \alpha} & {-\sin \alpha}\\ {\sin
\alpha} & {\cos \alpha}
\end{array} \right) \left(\begin{array} {cc} {\cos 2\beta} & {\sin 2\beta}\\ {\sin
2\beta} & {-\cos 2\beta}
\end{array} \right)$
\begin{eqnarray*}& = & \left(\begin{array} {cc} {\cos 2\beta \cos \alpha -
\sin 2\beta \sin \alpha} & {\sin 2\beta \cos \alpha + \cos 2\beta
\sin \alpha}\\ {\sin 2\beta \cos \alpha + \cos 2\beta \sin \alpha}
& {-[\cos 2\beta \cos \alpha - \sin 2\beta \sin \alpha]}
\end{array} \right)\\ & = & \left(\begin{array} {cc} {\cos (2\beta + \alpha)} & {\sin (2\beta + \alpha)}\\
{\sin (2\beta + \alpha)} & {-\cos (2\beta + \alpha)} \end{array}
\right)
\\ & = & S\left(\frac{2\beta + \alpha}{2} \right) \end{eqnarray*}

\item[(iii)]
$\left(\begin{array} {cc} {\cos 2\beta} & {\sin 2\beta}\\ {\sin
2\beta} & {-\cos 2\beta}
\end{array} \right) \left(\begin{array} {cc} {\cos \alpha} & {-\sin \alpha}\\ {\sin
\alpha} & {\cos \alpha}
\end{array} \right)$
\begin{eqnarray*}& = & \left(\begin{array} {cc} {\cos 2\beta \cos \alpha +
\sin 2\beta \sin \alpha} & {\sin 2\beta \cos \alpha - \cos 2\beta
\sin \alpha}\\ {\sin 2\beta \cos \alpha - \cos 2\beta \sin \alpha}
& {-[\cos 2\beta \cos \alpha + \sin 2\beta \sin \alpha]}
\end{array} \right)\\ & = & \left(\begin{array} {cc} {\cos (2\beta - \alpha)} & {\sin (2\beta - \alpha)}\\
{\sin (2\beta - \alpha)} & {-\cos (2\beta - \alpha)} \end{array}
\right)
\\ & = & S\left(\frac{2\beta - \alpha}{2} \right) \end{eqnarray*}

\item[(iv)]
$\left(\begin{array} {cc} {\cos 2\beta} & {\sin 2\beta}\\ {\sin
2\beta} & {-\cos 2\beta}
\end{array} \right) \left(\begin{array} {cc} {\cos 2\gamma} & {\sin 2\gamma}\\ {\sin
2\gamma} & {-\cos 2\gamma}
\end{array} \right)$
\begin{eqnarray*}& = & \left(\begin{array} {cc} {\cos 2\beta \cos 2\gamma +
\sin 2\beta \sin 2\gamma} & {-[\sin 2\beta \cos 2\gamma - \cos
2\beta \sin 2\gamma]}\\ {\sin 2\beta \cos 2\gamma - \cos 2\beta
\sin 2\gamma} & {\cos 2\beta \cos 2\gamma + \sin 2\beta \sin
2\gamma}
\end{array} \right)\\ & = & \left(\begin{array} {cc} {\cos (2\beta - 2\gamma)} & {-\sin (2\beta - 2\gamma)}\\
{\sin (2\beta - 2\gamma)} & {\cos (2\beta - 2\gamma)} \end{array}
\right)
\\ & = & C(2\beta - 2\gamma) \end{eqnarray*}
\end{description}


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