- (a) Use de Moivre's theorem to show that
$$\sin 5 \theta \equiv 16 \sin ^ { 5 } \theta - 20 \sin ^ { 3 } \theta + 5 \sin \theta$$
(b) Hence determine the five distinct solutions of the equation
$$16 x ^ { 5 } - 20 x ^ { 3 } + 5 x + \frac { 1 } { 5 } = 0$$
giving your answers to 3 decimal places.
(c) Use the identity given in part (a) to show that
$$\int _ { 0 } ^ { \frac { \pi } { 4 } } \left( 4 \sin ^ { 5 } \theta - 5 \sin ^ { 3 } \theta - 6 \sin \theta \right) \mathrm { d } \theta = a \sqrt { 2 } + b$$
where \(a\) and \(b\) are rational numbers to be determined.