9 In this question you must show detailed reasoning.
You are given the complex number \(\omega = \cos \frac { 2 } { 5 } \pi + \mathrm { i } \sin \frac { 2 } { 5 } \pi\) and the equation \(z ^ { 5 } = 1\).
- Show that \(\omega\) is a root of the equation.
- Write down the other four roots of the equation.
- Show that \(\omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } = - 1\).
- Hence show that \(\left( \omega + \frac { 1 } { \omega } \right) ^ { 2 } + \left( \omega + \frac { 1 } { \omega } \right) - 1 = 0\).
- Hence determine the value of \(\cos \frac { 2 } { 5 } \pi\) in the form \(a + b \sqrt { c }\) where \(a , b\) and \(c\) are rational numbers to be found.