2017 DHS P1 Q8
(c)
The complex number $z$ is given by $z=1+{{\text{e}}^{\mathbf{i}\,\alpha }}$.
(i) Show that $z$ can be expressed as $2\cos \left( \frac{1}{2}\alpha \right){{e}^{\mathbf{i}\,\left( \frac{1}{2}\alpha \right)}}$.
[2]
(ii) Given $\alpha =\frac{1}{3}\pi $ and $w=-1-\sqrt{3}\,\mathbf{i}$, find the exact modulus and argument of ${{\left( \frac{z}{{{w}^{3}}} \right)}^{*}}$.
[5]
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