Question 8 - A-Level Maths

1. Show that $\omega = \mathrm { e } ^ { \frac { 2 \pi \mathrm { i } } { 7 } }$ is a root of the equation $z ^ { 7 } = 1$.
2. Write down the five other non-real roots in terms of $\omega$.
Show that $$1 + \omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } + \omega ^ { 5 } + \omega ^ { 6 } = 0$$
Show that:
1. $\quad \omega ^ { 2 } + \omega ^ { 5 } = 2 \cos \frac { 4 \pi } { 7 }$;
2. $\cos \frac { 2 \pi } { 7 } + \cos \frac { 4 \pi } { 7 } + \cos \frac { 6 \pi } { 7 } = - \frac { 1 } { 2 }$.