2022 JPJC MYE P2 Q6

Do not use a calculator in answering this question.

1. It is given that \(w=1+\mathrm{i}\) and \(z=-\sqrt{3}-\mathrm{i}\).
   1. Find \(\frac{{{w}^{3}}}{z}\) in the form \(r\left( \cos \theta +\mathrm{i}\sin \theta \right)\), where \(r>0\) and \(-\pi <\theta \le \pi \).[3]
   2. Find the smallest positive integer value of \(n\), such that \({{\left( \frac{{{w}^{3}}}{z} \right)}^{n}}\) is real.[3]
2. The complex number \(q\) is given by \(\frac{{{\mathrm{e}}^{\mathrm{i}\theta }}}{{{\mathrm{e}}^{-\mathrm{i}\theta }}-\mathrm{i}}\), where \(0 < \theta < \pi \).
   Show that the real part of \(q\) is \(\frac{1}{2}\left( 1-2\sin \theta  \right)\).[4]