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.
- 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.
- 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 \(\mathbf { R }\).
- Use your matrices from parts (a) and (b) to find the matrix \(\mathbf { R }\).
- Give a full geometric description of \(T\) as a single transformation.
- 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.