2022 RI P1 Q5

Do not use a calculator in answering this question.

The complex numbers \(z\) and \(w\) are given by

\(z=\sin \left( \frac{\pi }{6} \right)+\mathrm{i}\cos \left( \frac{\pi }{6} \right)\) and \(w=\sqrt{2}\left[ \sin \left( \frac{\pi }{6} \right)+\mathrm{i}\cos \left( \frac{\pi }{3} \right) \right]\).

1. Find \(\left| z \right|\) and \(\arg \left( z \right)\). Hence find the value of \({{z}^{3}}\).[3]
2. By considering some suitable form of \(w\) or otherwise, find \({{w}^{4}}\).[3]
3. Hence find the value of \({{z}^{2022}}-{{w}^{2020}}\).[2]