Question

The complex number $z=-1+i\sqrt{3}$ can be represented in polar form as $(\text{where}cis\theta =\cos \theta +i\sin \theta )$

• A. $\sqrt{2}cis\left(\frac{-2\pi }{3}\right)$
• B. $2cis\left(\frac{2\pi }{3}\right)$
• C. $\sqrt{2}cis\left(\frac{\pi }{3}\right)$
• D. $2cis\left(\frac{\pi }{3}\right)$

Answer 1

The correct option is B $2cis\left(\frac{2\pi }{3}\right)$

$z=-1+i\sqrt{3}=r(\cos \theta +i\sin \theta )$
Here, $\tan \alpha =\frac{|\sqrt{3}|}{|-1|}$
$\alpha =\frac{\pi }{3}$
As we know, this complex number lies in the second quadrant.
So, $\text{amp}\left(z\right)=\theta =(\pi -\alpha )$
$\Rightarrow \text{amp}\left(z\right)=(\pi -\frac{\pi }{3})=\frac{2\pi }{3}$
$r=\sqrt{(-1{)}^2+(-\sqrt{3}{)}^2}=\sqrt{1+3}=2$
$\therefore z=-1+i\sqrt{3}=2(\cos \left(\frac{2\pi }{3}\right)+i\sin \left(\frac{2\pi }{3}\right))$
$\Rightarrow z=2cis\left(\frac{2\pi }{3}\right)$