Question

Find the modulus and the arguments of each of the complex numbers is $z=-\sqrt{3}+i$

Answer 1

$z=-\sqrt{3}+i$

$\left|z\right|=\sqrt{{\left(-\sqrt{3}\right)}^2+1^2}=\sqrt{3+1}=2$

$\tan \varphi =\left|\frac{y}{x}\right|=\left|-\frac{1}{\sqrt{3}}\right|=\frac{1}{\sqrt{3}}$

$=\varphi =\frac{\pi }{6}$

Since $z$ lies in second quadrant so, argument of $z$=$z=\pi -\varphi$

$=\pi -\frac{\pi }{6}$

$=\frac{5\pi }{6}$