The complex number $z$ is given by $z=1+{{\mathrm{e}}^{\mathbf{i}\,\alpha }}$. Show that $z$ can be

Question

The complex number $z$ is given by $z=1+{{\mathrm{e}}^{\mathbf{i}\,\alpha }}$. 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] 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]

Accepted Answer

$\left|\left(\frac{z}{w^3}\right)^*\right| = \frac{\sqrt{3}}{8}$, $\arg\left(\frac{z}{w^3}\right)^* = -\frac{\pi}{6}$

2017 DHS P1 Q8

2017 DHS P1 Q8

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