Question 6 - A-Level Maths

Find the three roots of $z ^ { 3 } = 1$, giving the non-real roots in the form $\mathrm { e } ^ { \mathrm { i } \theta }$, where $- \pi < \theta \leqslant \pi$.
Given that $\omega$ is one of the non-real roots of $z ^ { 3 } = 1$, show that $$1 + \omega + \omega ^ { 2 } = 0$$
By using the result in part (b), or otherwise, show that:
1. $\frac { \omega } { \omega + 1 } = - \frac { 1 } { \omega }$;
2. $\frac { \omega ^ { 2 } } { \omega ^ { 2 } + 1 } = - \omega$;
3. $\left( \frac { \omega } { \omega + 1 } \right) ^ { k } + \left( \frac { \omega ^ { 2 } } { \omega ^ { 2 } + 1 } \right) ^ { k } = ( - 1 ) ^ { k } 2 \cos \frac { 2 } { 3 } k \pi$, where $k$ is an integer.