Question

Find the modulus and the principal amplitude of the complex number $(-1-\sqrt{3}i)$

• A. $-2$ $and$ $\frac{2\pi }{3}$
• B. $2$ $and$ $\frac{-2\pi }{3}$
• C. $2$ $and$ $\frac{\pi }{3}$
• D. $-2$ $and$ $\frac{\pi }{3}$

Answer 1

The correct option is B $2$ $and$ $\frac{-2\pi }{3}$

$2=-1-\sqrt{3}$

$121=\sqrt{(-1{)}^2+(-\sqrt{3}{)}^2}=\sqrt{4}=2$

Arg $\left(2\right)=-\pi +{\tan }^{-1}\left(\sqrt{3}\right)$

$=-\pi +\frac{\pi }{3}$

$=\frac{-2\pi }{3}$