Find the 4th roots of 16j, in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\) where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Illustrate the 4th roots on an Argand diagram.
Show that \(\left( 1 - 2 \mathrm { e } ^ { \mathrm { j } \theta } \right) \left( 1 - 2 \mathrm { e } ^ { - \mathrm { j } \theta } \right) = 5 - 4 \cos \theta\).
Series \(C\) and \(S\) are defined by
$$\begin{aligned}
& C = 2 \cos \theta + 4 \cos 2 \theta + 8 \cos 3 \theta + \ldots + 2 ^ { n } \cos n \theta
& S = 2 \sin \theta + 4 \sin 2 \theta + 8 \sin 3 \theta + \ldots + 2 ^ { n } \sin n \theta
\end{aligned}$$
Show that \(C = \frac { 2 \cos \theta - 4 - 2 ^ { n + 1 } \cos ( n + 1 ) \theta + 2 ^ { n + 2 } \cos n \theta } { 5 - 4 \cos \theta }\), and find a similar expression for \(S\).