Find the modulus and the arguments of each of the complex numbers. (2) z=

Find the modulus and the arguments of each of the complex numbers.

Q: 2 $z=-\sqrt{3}+i$

Answers (1)

Given the problem is
$z=-\sqrt{3}+i$
Now, let
$r\cos \theta = - \sqrt3 \ \ \ and \ \ \ r\sin \theta = 1$
Square and add both the sides
$r^2(\cos^2\theta +\sin^2\theta)= (-\sqrt3)^2+(1)^2$ $(\because \cos^2\theta +\sin^2\theta = 1)$
$r^2= 1+3$
$r^2 =4$
$r= 2$ $(\because r > 0)$
Therefore, the modulus is 2
Now,
$2\cos \theta = -\sqrt3 \ \ \ and \ \ \ 2\sin \theta = 1$
$\cos \theta = -\frac{\sqrt3}{2} \ \ \ and \ \ \ \sin \theta =\frac{1}{2}$
Since values of $\cos \theta$ is negative and value $\sin \theta$ is positive and we know that this is the case in II quadrant
Therefore,
Argument = $\left ( \pi - \frac{\pi}{6} \right )= \frac{5\pi}{6}$
Therefore, the argument is