ACJC Complex Numbers Tutorial Q13

On an Argand diagram, the points $P$ and $Q$ represent the complex numbers $p$ and $q$ respectively where $p=\cos \theta +\mathbf{i}\sin \theta $, $0<\theta <\frac{\pi }{4}$ and $q=2{{\text{e}}^{\mathbf{i}\,\,\phi }}$, $\frac{\pi }{4}<\,\phi <\frac{\pi }{2}$ and $\theta +\,\phi >\frac{\pi }{2}$. Find the moduli and arguments of the complex numbers $a$, $b$,$c$, $d$ and $\text{e}$, where $a=\mathbf{i}\,p$, $b={{q}^{2}}$, $c=\frac{1}{q}$ and $d=p+{{p}^{*}}$ and $e=pq$.