8. Given that \(z = e ^ { \mathrm { i } \theta }\)
- show that \(z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta\)
where \(n\) is a positive integer. - Show that
$$\cos ^ { 6 } \theta = \frac { 1 } { 32 } ( \cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta + 10 )$$
- Hence solve the equation
$$\cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta = 0 \quad 0 \leqslant \theta \leqslant \pi$$
Give your answers to 3 significant figures.
- Use calculus to determine the exact value of
$$\int _ { 0 } ^ { \frac { \pi } { 3 } } \left( 32 \cos ^ { 6 } \theta - 4 \cos ^ { 2 } \theta \right) d \theta$$
Solutions relying entirely on calculator technology are not acceptable.