Question 1404c

Complex Numbers

Polar Form Arithmetic

Question

The complex number

{z}

is given by

z=\frac{\gamma}{\alpha^5 \beta^*},

where

\begin{align*} \alpha &= { \textstyle 2 \left( \cos \left( - \frac{1}{3} \pi \right) + \mathrm{i} \sin \left( - \frac{1}{3} \pi \right) \right)}, \\ \beta &= { \textstyle \cos \left( - \frac{3}{4} \pi \right) + \mathrm{i} \sin \left( - \frac{3}{4} \pi \right)}, \\ \gamma &= { \textstyle \cos \frac{1}{3} \pi + \mathrm{i} \sin \frac{1}{3} \pi}. \end{align*}

Find

{|z|}

and

{k,}

where

{\arg(z)=k\pi}

and

{-\pi < \arg(z) < \pi.}

Attempt

{|z| = }

{k = }

Answer

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