Question

The mod - amp form of $-1-\sqrt{3}i$ is :

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

Answer 1

Answer: (B)

$2cis(-\frac{2\pi }{3})$

$z=-1-\sqrt{3}i$
Now
$|z|=\sqrt{1+{\sqrt{3}}^2}$
$=\sqrt{4}$
$=2$
Hence
$z=2(\frac{-1}{2}-\frac{\sqrt{3}}{2}i)$
$=2(\cos (-\frac{2\pi }{3})+i\sin \left(\frac{-2\pi }{3}\right))$
$=2e^{\left(i\frac{-2\pi }{3}\right)}$
Hence
$|z|=2$ and
$arg\left(z\right)=\frac{-2\pi }{3}$