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 $\theta =\frac{2\pi }{3}$
$\therefore z=2\left(\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3}\right)$
$=2\left(-\frac{1}{2}+i\frac{\sqrt{3}}{2}\right)$
$=(-1+i\sqrt{3})$