Find the six roots of the equation \(z ^ { 6 } = 1\), giving your answers in the form \(\mathrm { e } ^ { \mathrm { i } \phi }\), where \(- \pi < \phi \leqslant \pi\).
It is given that \(w = \mathrm { e } ^ { \mathrm { i } \theta }\), where \(\theta \neq n \pi\).
Show that \(\frac { w ^ { 2 } - 1 } { w } = 2 \mathrm { i } \sin \theta\).
Show that \(\frac { w } { w ^ { 2 } - 1 } = - \frac { \mathrm { i } } { 2 \sin \theta }\).
Show that \(\frac { 2 \mathrm { i } } { w ^ { 2 } - 1 } = \cot \theta - \mathrm { i }\).
Given that \(z = \cot \theta - \mathrm { i }\), show that \(z + 2 \mathrm { i } = z w ^ { 2 }\).
Explain why the equation
$$( z + 2 \mathrm { i } ) ^ { 6 } = z ^ { 6 }$$
has five roots.
Find the five roots of the equation
$$( z + 2 \mathrm { i } ) ^ { 6 } = z ^ { 6 }$$
giving your answers in the form \(a + \mathrm { i } b\).