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