Find the modulus and the arguments of the complex number:z = - 3 … - Vidyadip

Question

Find the modulus and the arguments of the complex number:$z = - \sqrt 3 + i$

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Answer

Here $z = - \sqrt 3 + i$$ = r(\cos \theta + i\sin \theta )$
$ \Rightarrow r\cos \theta = - \sqrt 3 $ and $r\sin \theta = 1$
Squaring both sides of (i) and adding
${r^2}({\cos ^2}\theta + {\sin ^2}\theta ) = 3 + 1$$ \Rightarrow {r^2} = 4 \Rightarrow r = 2$
$\therefore 2\cos \theta = \frac{{ - \sqrt 3 }}{2}$ and $2\sin \theta = 1$
$ \Rightarrow \cos \theta = \frac{{ - \sqrt 3 }}{2}$ and $\sin \theta = \frac{1}{2}$
Since $\sin \theta $ is positive and $\cos \theta $ is negative
$\therefore \theta $ lies in second quadrant
$\therefore \theta = \left( {\pi - \frac{\pi }{6}} \right) = \frac{{5\pi }}{6}$
$\therefore |z| = 2$ and arg $(z) = \frac{{5\pi }}{6}$

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