2024 CJC P1 Q3

In an Argand diagram, the points \(A\) and \(B\) represent the complex numbers \(6{{\mathrm{e}}^{\mathbf{i}\,\alpha }}\) and \(6{{\mathrm{e}}^{\mathbf{i}\,\beta }}\) respectively, where \(0<\alpha <\frac{\pi }{2}\) and \(\frac{\pi }{2}<\beta <\pi \).

1. Mark the points \(A\) and \(B\) on an Argand diagram. You are expected to clearly label the relevant moduli and arguments of the complex numbers represented by the points \(A\) and \(B\), in terms of \(\alpha \) and \(\beta \).[2]
2. Show that \(\left| 6{{\mathrm{e}}^{\mathbf{i}\,\alpha }}-6{{\mathrm{e}}^{\mathbf{i}\,\beta }} \right|=p\sin \left( \frac{\beta -\alpha }{q} \right)\), where the integers \(p\) and \(q\) are to be determined. Hence write down, in terms of \(\alpha \) and \(\beta \), the perimeter of the triangle \(OAB\).[4]