9. (a) Use mathematical induction to prove de Moivre's Theorem, namely that $$( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta$$ where \(n\) is a positive integer.
(b) (i) Use this result to show that $$\sin 5 \theta = a \sin ^ { 5 } \theta - b \sin ^ { 3 } \theta + c \sin \theta$$ where \(a , b\) and \(c\) are positive integers to be found.
(ii) Hence determine the value of \(\lim _ { \theta \rightarrow 0 } \frac { \sin 5 \theta } { \sin \theta }\)