Express \(- 4 + 4 \sqrt { 3 } \mathrm { i }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
Solve the equation \(z ^ { 3 } = - 4 + 4 \sqrt { 3 } \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
The roots of the equation \(z ^ { 3 } = - 4 + 4 \sqrt { 3 } \mathrm { i }\) are represented by the points \(P , Q\) and \(R\) on an Argand diagram.
Find the area of the triangle \(P Q R\), giving your answer in the form \(k \sqrt { 3 }\), where \(k\) is an integer.
By considering the roots of the equation \(z ^ { 3 } = - 4 + 4 \sqrt { 3 } \mathrm { i }\), show that
$$\cos \frac { 2 \pi } { 9 } + \cos \frac { 4 \pi } { 9 } + \cos \frac { 8 \pi } { 9 } = 0$$