By expressing \(\sin \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\), show that
$$\sin ^ { 5 } \theta \equiv \frac { 1 } { 16 } ( \sin 5 \theta - 5 \sin 3 \theta + 10 \sin \theta ) .$$
Hence solve the equation
$$\sin 5 \theta + 4 \sin \theta = 5 \sin 3 \theta$$
for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
8 consists of the set of matrices of the form \(\left( \begin{array} { c c } a & - b b & a \end{array} \right)\), where \(a\) and \(b\) are real and \(a ^ { 2 } + b ^ { 2 } \neq 0\), combined under the operation of matrix multiplication.
Prove that \(G\) is a group. You may assume that matrix multiplication is associative.
Determine whether \(G\) is commutative.
Find the order of \(\left( \begin{array} { c c } 0 & - 1 1 & 0 \end{array} \right)\).