Question

Let $z$ be a complex number with modulus $2$ and argument $\frac{2 \pi}{3}$, then $z$ is equal to

• A. $-1+i \sqrt{3}$
• B. $1-i \sqrt{3}$
• C. $-\frac{1}{2}+\frac{i \sqrt{3}}{2}$
• D. None of these

Answer 1

Answer: (A)

Given: $|z|=2$ and $\arg (z)=\frac{2 \pi}{3}$
$\therefore $ If $z=r(\cos \theta+i \sin \theta)$,
then $r=2$ and $\theta=\frac{2 \pi}{3}$
$\therefore z =2\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)$
$=2\left(-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right) $
$=(-1+i \sqrt{3})$