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 \(q=2{{\mathrm{e}}^{\mathbf{i}\,\,\phi }}\), \(\frac{\pi }{4}<\,\phi <\frac{\pi }{2}\) and \(\theta +\,\phi >\frac{\pi }{2}\). Find the moduli and arguments of the complex numbers \(a\), \(b\),\(c\), \(d\) and \(\mathrm{e}\), where \(a=\mathbf{i}\,p\), \(b={{q}^{2}}\), \(c=\frac{1}{q}\) and \(d=p+{{p}^{*}}\) and \(e=pq\).

On your Argand diagram, sketch and label the points \(P\), \(Q\), \(A\), \(B\), \(C\),\(D\) and \(E\) representing the complex numbers \(p\), \(q\), \(a\), \(b\), \(c\), \(d\) and \(e\) respectively.