Question

If $z=1+\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}$, then

• A. $Re\left(z^5\right)=\frac{\sqrt{3}}{2}$
• B. $Re\left(z^5\right)=\frac{1}{2}$
• C. $Im\left(z^5\right)=\frac{1}{2}$
• D. $Im\left(z^5\right)=\frac{\sqrt{3}}{2}$

Answer 1

Answer: (B)

$Re\left(z^5\right)=\frac{1}{2}$

$z=1+\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}=2{\cos }^2\frac{\pi }{3}+2i\sin \frac{\pi }{3}\cos \frac{\pi }{3}$
$\Rightarrow z=2\cos \frac{\pi }{3}(\cos \frac{\pi }{3}+i\sin \frac{\pi }{3})=\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}$
$\Rightarrow z^5={(\cos \frac{\pi }{3}+i\sin \frac{\pi }{3})}^5=\cos \frac{5\pi }{3}+i\sin \frac{5\pi }{3}$ ...{De Moivre's Theorem}

$\Rightarrow z^5=\frac{1-i\sqrt{3}}{2}$
$\therefore Re\left(z^5\right)=\frac{1}{2}\&Im\left(z^5\right)=\frac{-\sqrt{3}}{2}$
Hence, option B is correct.