Given that \(z = \cos \theta + \mathrm { j } \sin \theta\), express \(z ^ { n } + \frac { 1 } { z ^ { n } }\) and \(z ^ { n } - \frac { 1 } { z ^ { n } }\) in simplified trigonometric form.
Hence find the constants \(A , B , C\) in the identity
$$\sin ^ { 5 } \theta \equiv A \sin \theta + B \sin 3 \theta + C \sin 5 \theta$$
Find the 4th roots of - 9 j in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), where \(r > 0\) and \(0 < \theta < 2 \pi\). Illustrate the roots on an Argand diagram.
Let the points representing these roots, taken in order of increasing \(\theta\), be \(\mathrm { P } , \mathrm { Q } , \mathrm { R } , \mathrm { S }\). The mid-points of the sides of PQRS represent the 4th roots of a complex number \(w\). Find the modulus and argument of \(w\). Mark the point representing \(w\) on your Argand diagram.