Question 1404c

Complex Numbers
Polar Form Arithmetic

Question

The complex number z{z} is given by
z=γα5β,z=\frac{\gamma}{\alpha^5 \beta^*},
where
α=2(cos23π+isin23π),β=3(cos13π+isin13π),γ=2(cos(23π)+isin(23π)).\begin{align*} \alpha &= { \textstyle 2 \left( \cos \frac{2}{3} \pi + \mathrm{i} \sin \frac{2}{3} \pi \right)}, \\ \beta &= { \textstyle 3 \left( \cos \frac{1}{3} \pi + \mathrm{i} \sin \frac{1}{3} \pi \right)}, \\ \gamma &= { \textstyle 2 \left( \cos \left( - \frac{2}{3} \pi \right) + \mathrm{i} \sin \left( - \frac{2}{3} \pi \right) \right)}. \end{align*}
Find z{|z|} and k,{k,} where arg(z)=kπ{\arg(z)=k\pi} and π<arg(z)<π.{-\pi < \arg(z) < \pi.}

Attempt

z={|z| = }
k={k = }

Answer

Question Progress

  • Start
  • Mastery