If z = cos frac{π }{6} + isin frac{π }{6} then

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4-1.Complex numbers

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If $z = \cos \frac{\pi }{6} + i\sin \frac{\pi }{6}$ then

A

$|z|\, = 1,\,\,\,\,arg\,z = \frac{\pi }{4}$

B

$|z|\, = 1,arg\,z = \frac{\pi }{6}$

C

$|z|\, = \frac{{\sqrt 3 }}{2},\,arg\,z = \frac{{5\pi }}{{24}}$

D

$|z|\, = \frac{{\sqrt 3 }}{2},\,\,arg\,z = {\tan ^{ - 1}}\frac{1}{{\sqrt 2 }}$

Solution

(b)$z = \cos \frac{\pi }{6} + i\sin \frac{\pi }{6} = \frac{{\sqrt 3 }}{2} + \frac{i}{2}$
$\therefore \,\,|z|\, = \sqrt {\frac{3}{4} + \frac{1}{4}} = 1$
and $arg\,(z) = {\tan ^{ – 1}}\,\left( {\frac{y}{x}} \right) = {\tan ^{ – 1}}\left( {\frac{{1/2}}{{\sqrt 3 /2}}} \right) = {\tan ^{ – 1}}\left( {\frac{1}{{\sqrt 3 }}} \right)$
$ \Rightarrow \,\,arg(z)\, = {\tan ^{ – 1}}\left( {\tan \frac{\pi }{6}} \right) = \frac{\pi }{6}$.

Standard 11

Mathematics

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