Question

find modulus and argument of complex number -root 3+i

Answer 1

Modulus is 2 while the argument (angle) is 5 pie by 6.

Answer 2

Question:

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

Solution:

$z = - \sqrt{3} + i \\ \\ z = ( - \sqrt{3} ,1) \\ \\ |r| = \sqrt{ {x}^{2} + {y}^{2} } \\ \\ |r| = ( \sqrt{ {3}^{2} + {1}^{2} } \\ \\ |r| = \sqrt{4} \\ \\ |r| = 2$

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 2nd Quadrant$( θ= \pi - \alpha )$

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

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