7 The series \(C\) and \(S\) are defined for \(0 < \theta < \pi\) by $$\begin{aligned} & C = 1 + \cos \theta + \cos 2 \theta + \cos 3 \theta + \cos 4 \theta + \cos 5 \theta
& S = \quad \sin \theta + \sin 2 \theta + \sin 3 \theta + \sin 4 \theta + \sin 5 \theta \end{aligned}$$

1. Show that \(C + \mathrm { i } S = \frac { \mathrm { e } ^ { 3 \mathrm { i } \theta } - \mathrm { e } ^ { - 3 \mathrm { i } \theta } } { \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { i } \theta } - \mathrm { e } ^ { - \frac { 1 } { 2 } \mathrm { i } \theta } } \mathrm { e } ^ { \frac { 5 } { 2 } \mathrm { i } \theta }\).
2. Deduce that \(C = \sin 3 \theta \cos \frac { 5 } { 2 } \theta \operatorname { cosec } \frac { 1 } { 2 } \theta\) and write down the corresponding expression for \(S\).
3. Hence find the values of \(\theta\), in the range \(0 < \theta < \pi\), for which \(C = S\).