The infinite series \(C\) and \(S\) are defined as follows.
$$\begin{aligned}
& C = 1 + a \cos \theta + a ^ { 2 } \cos 2 \theta + \ldots
& S = a \sin \theta + a ^ { 2 } \sin 2 \theta + a ^ { 3 } \sin 3 \theta + \ldots
\end{aligned}$$
where \(a\) is a real number and \(| a | < 1\).
By considering \(C + \mathrm { j } S\), show that \(C = \frac { 1 - a \cos \theta } { 1 + a ^ { 2 } - 2 a \cos \theta }\) and find a corresponding expression for \(S\).
Express the complex number \(z = - 1 + \mathrm { j } \sqrt { 3 }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\).
Find the 4th roots of \(z\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\).
Show \(z\) and its 4th roots in an Argand diagram.
Find the product of the 4th roots and mark this as a point on your Argand diagram.