The complex number $z$ is given by $z=1+\cos \alpha +\mathrm{i}\sin \alpha $, where $0Show that $z$

Question

The complex number $z$ is given by $z=1+\cos \alpha +\mathrm{i}\sin \alpha $, where $0Show that $z$ can be expressed as $\left( 2\cos \frac{\alpha }{2} \right)\left( \cos \frac{\alpha }{2}+\mathrm{i}\sin \frac{\alpha }{2} \right)$. [2] Find $z{{z}^{*}}$, where ${{z}^{*}}$ denotes the conjugate of $z$. [2] Given that $\alpha =\frac{\pi }{3}$, without using a calculator, find the values of $\left| {{z}^{6}} \right|$ and $\arg \left( {{z}^{6}} \right)$. Deduce the value of ${{z}^{6}}$. [3] The complex numbers $u$ and $v$ are such that $\frac{\mathrm{i}\left( u+v \right)}{u-v}$ is real. Show that $\left| u \right|=\left| v \right|$. [3]

Accepted Answer

$z^6 = -27$

2016 RI Promo Q8

2016 RI Promo Q8

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