Prove de Moivre's theorem

4. (a) Given that $$z = r ( \cos \theta + \mathrm { i } \sin \theta ) , \quad r \in \mathbb { R }$$ prove, by induction, that \(z ^ { n } = r ^ { n } ( \cos n \theta + \mathrm { i } \sin n \theta ) , \quad n \in \mathbb { Z } ^ { + }\) $$w = 3 \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right)$$ (b) Find the exact value of \(w ^ { 5 }\), giving your answer in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).

11 Prove de Moivre's theorem for a positive integral exponent: $$\text { for all positive integers } n , \quad ( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta \text {. }$$ Use de Moivre's theorem to show that $$\cos 7 \theta = 64 \cos ^ { 7 } \theta - 112 \cos ^ { 5 } \theta + 56 \cos ^ { 3 } \theta - 7 \cos \theta$$ Hence obtain the roots of the equation $$128 x ^ { 7 } - 224 x ^ { 5 } + 112 x ^ { 3 } - 14 x + 1 = 0$$ in the form \(\cos q \pi\), where \(q\) is a rational number.

9 Prove by mathematical induction that, for every positive integer \(n\), $$( \cos \theta + i \sin \theta ) ^ { n } = \cos n \theta + i \sin n \theta$$ Express \(\sin ^ { 5 } \theta\) in the form \(p \sin 5 \theta + q \sin 3 \theta + r \sin \theta\), where \(p , q\) and \(r\) are rational numbers to be determined.

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 \(\lim _ { \theta \rightarrow 0 } \frac { \sin 5 \theta } { \sin \theta }\)