The infinite series \(C\) and \(S\) are defined as follows.
$$\begin{gathered}
C = a \cos \theta + a ^ { 2 } \cos 2 \theta + a ^ { 3 } \cos 3 \theta + \ldots
S = a \sin \theta + a ^ { 2 } \sin 2 \theta + a ^ { 3 } \sin 3 \theta + \ldots
\end{gathered}$$
where \(a\) is a real number and \(| a | < 1\).
By considering \(C + \mathrm { j } S\), show that
$$S = \frac { a \sin \theta } { 1 - 2 a \cos \theta + a ^ { 2 } }$$
Find a corresponding expression for \(C\).
P is one vertex of a regular hexagon in an Argand diagram. The centre of the hexagon is at the origin. P corresponds to the complex number \(\sqrt { 3 } + \mathrm { j }\).
Find, in the form \(x + \mathrm { j } y\), the complex numbers corresponding to the other vertices of the hexagon.
The six complex numbers corresponding to the vertices of the hexagon are squared to form the vertices of a new figure. Find, in the form \(x + \mathrm { j } y\), the vertices of the new figure. Find the area of the new figure.