Question

Find the modulus and principal argument of the complex number − √3 − i​

Answer 1

Question:

Find the modulus and Principle argument of complex number $- \sqrt{ 3} - i$

Solution:

$z = - \sqrt{3} - i \\ \\ z = ( - \sqrt{3} , - 1) \\ \\ |r| = \sqrt{ {x}^{2} + {y}^{2} }$

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

To Find Argument:

$\alpha = \tan^{ - 1} | \frac{y}{x} | \\ \\ \alpha = \tan^{ - 1} | \frac{-1}{ - \sqrt{3} } | \\ \\ \alpha = \tan^{ - 1} | \frac{1}{ \sqrt{3} } | \\ \\ \alpha = \frac{\pi}{6}$

z lies on the 3rd Quadrant$( θ= \alpha - \pi )$

$θ = \alpha - \pi \\ \\ \frac{\pi}{6} - \pi \\ \\ θ = \frac{ - 5\pi}{6} \\ \\ |r| = 2 \: \: \: \: ; θ = \frac{ - 5\pi}{6}$

Hope it helps ya!!

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