Question

State true or false:

If $z={(\frac{\sqrt{3}}{2}+\frac{i}{2})}^5+{(\frac{\sqrt{3}}{2}-\frac{i}{2})}^5$, then $Im\left(z\right)=0$.

Answer 1

Put $\sqrt{3}/2=r\cos \theta ,\frac{1}{2}=r\sin \theta .$
$\therefore r=1$ and $\theta =\pi /6.$
$\therefore z={(\cos \theta +i\sin \theta )}^5+{(\cos \theta -i\sin \theta )}^5$
$=\cos 5\theta +i\sin 5\theta +\cos 5\theta -i\sin 5\theta ,$
$=2\cos 5\theta .$
$HenceIm\left(z\right)=0.$