2022 ACJC P1 Q5

Do not use a calculator in answering this question.

Two complex numbers are \({{z}_{1}}=2\left( \cos \frac{\pi }{18}-\mathbf{i}\sin \frac{\pi }{18} \right)\) and \({{z}_{2}}=2\mathbf{i}\).

1. Show that \(\frac{{{z}_{1}}^{2}}{{{z}_{1}}^{*}}+{{z}_{2}}\) is \(\sqrt{3}+\mathbf{i}\).[3]
2. A third complex number, \({{z}_{3}}\), is such that \(\left( \frac{{{z}_{1}}^{2}}{{{z}_{1}}^{*}}+{{z}_{2}} \right){{z}_{3}}\) is real and \(\left| \left( \frac{{{z}_{1}}^{2}}{{{z}_{1}}^{*}}+{{z}_{2}} \right){{z}_{3}} \right|=\frac{2}{3}\). Find the possible values of \({{z}_{3}}\) in the form of \(r(\cos \theta +\mathbf{i}\sin \theta )\), where \(r>0\) and \(-\pi <\theta \le \pi \).[4]