3. (a) By writing \(\frac { \pi } { 12 } = \frac { \pi } { 3 } - \frac { \pi } { 4 }\), show that
- \(\sin \left( \frac { \pi } { 12 } \right) = \frac { 1 } { 4 } ( \sqrt { 6 } - \sqrt { 2 } )\)
- \(\cos \left( \frac { \pi } { 12 } \right) = \frac { 1 } { 4 } ( \sqrt { 6 } + \sqrt { 2 } )\)
(b) Hence find the exact values of \(z\) for which
$$z ^ { 4 } = 4 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$$
Give your answers in the form \(z = a + i b\) where \(a , b \in \mathbb { R }\)