10. In your answers to this question, the elements of each matrix should be expressed in exact form in surds where necessary.

The transformation $U$, represented by the $2 \times 2$ matrix $\mathbf { P }$, is a rotation through $45 ^ { \circ }$ anticlockwise about the origin.
\begin{enumerate}[label=(\alph*)]
\item Write down the matrix $\mathbf { P }$.

The transformation $V$, represented by the $2 \times 2$ matrix $\mathbf { Q }$, is a rotation through $60 ^ { \circ }$ anticlockwise about the origin.
\item Write down the matrix $\mathbf { Q }$.

The transformation $U$ followed by the transformation $V$ is the transformation $T$. The transformation $T$ is represented by the matrix $\mathbf { R }$.
\item Use your matrices from parts (a) and (b) to find the matrix $\mathbf { R }$.
\item Give a full geometric description of $T$ as a single transformation.
\item Deduce from your answers to parts (c) and (d) that $\sin 75 ^ { \circ } = \frac { 1 + \sqrt { 3 } } { 2 \sqrt { 2 } }$ and find the\\
exact value of $\cos 75 ^ { \circ }$, explaining your answers fully.
\end{enumerate}

\hfill \mbox{\textit{Edexcel F1 2017 Q10 [9]}}