Question

What will be the argument of the complex number $ z=\frac{i-1}{e^{i\pi/3}} $ ?

• A. $ \pi /4 $
• B. $ \pi /3 $
• C. $ 5 \pi /12 $
• D. $ 3\pi /8 $

Answer 1

Answer: (C)

We have, $z =\frac{i -1}{e^{i\pi/3}}$
$= \frac{i -1}{cos(\pi/3) + i\,sin(\pi/3)}$
$\Rightarrow z = 2\left[\frac{\left(i-1\right)}{1+i\sqrt{3}}\times \frac{1-i\sqrt{3}}{1-i\sqrt{3}}\right] $
$ = 2\left[ \frac{i -1 +\sqrt{3} +i\sqrt{3}}{1+3}\right] $
$= \frac{1}{2}\left[\left(\sqrt{3} -1\right)+i\left(\sqrt{3}+1\right)\right] $
$ \therefore arg\left(z\right) = tan^{-1}\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right)$
$= tan^{-1}\left(\frac{1+1/\sqrt{3}}{1-1/\sqrt{3}}\right) $
$ = tan^{-1}\left(1\right)+tan^{-1}\left(\frac{1}{\sqrt{3}}\right)$
$ = \frac{\pi}{4} +\frac{\pi}{6} = \frac{5\pi}{12}$