Question

The principal mod-amp form of $2-\sqrt{3}+i$ is

• A. $(\sqrt{6}-\sqrt{2})cis\frac{5\pi }{12}$
• B. $2\sqrt{3}cis\frac{\pi }{12}$
• C. $3\sqrt{2}cis\frac{5\pi }{12}$
• D. $2\sqrt{2}cis\frac{5\pi }{12}$

Answer 1

The correct option is A $(\sqrt{6}-\sqrt{2})cis\frac{5\pi }{12}$

$\theta =ta{n}^{-1}\left(\frac{b}{a}\right)$
$=ta{n}^{-1}\left(\frac{1}{2-\sqrt{3}}\right)$
$=\frac{5\pi }{12}$
Modulus $=\sqrt{(2-\sqrt{3}{)}^2+1}$
$=\sqrt{4+3-4\sqrt{3}+1}$
$=\sqrt{8-4\sqrt{3}}$
$=2\sqrt{2-\sqrt{3}}$

$=(\sqrt{6}-\sqrt{2})$