The series \(C\) and \(S\) are defined as follows.
$$\begin{aligned}
& C = 1 + \binom { n } { 1 } \cos 2 \theta + \binom { n } { 2 } \cos 4 \theta + \ldots + \cos 2 n \theta
& S = \binom { n } { 1 } \sin 2 \theta + \binom { n } { 2 } \sin 4 \theta + \ldots + \sin 2 n \theta
\end{aligned}$$
By considering \(C + \mathrm { j } S\), show that
$$C = 2 ^ { n } \cos ^ { n } \theta \cos n \theta$$
and find a corresponding expression for \(S\).
Express \(\mathrm { e } ^ { \mathrm { j } 2 \pi / 3 }\) in the form \(x + \mathrm { j } y\), where the real numbers \(x\) and \(y\) should be given exactly.
An equilateral triangle in the Argand diagram has its centre at the origin. One vertex of the triangle is at the point representing \(2 + 4 \mathrm { j }\). Obtain the complex numbers representing the other two vertices, giving your answers in the form \(x + \mathrm { j } y\), where the real numbers \(x\) and \(y\) should be given exactly.
Show that the length of a side of the triangle is \(2 \sqrt { 15 }\).