First slide

Algebra of complex numbers

Question

Principal argument of $z = \frac{i - 1}{i (1 \cos \frac{2 π}{7}) + \sin \frac{2 π}{7}}$ is

Moderate

A

$\frac{π}{28}$
B

$\frac{3 π}{28}$
C

$\frac{17 π}{28}$
D

$\frac{19 π}{28}$

Solution

$i (1 - \cos \frac{2 π}{7}) + \sin \frac{2 π}{7}$
$\begin{aligned} = 2 i \sin^{2} \frac{π}{7} + 2 \sin \frac{π}{7} \cos \frac{π}{7} \\ = 2 \sin (\frac{π}{7}) [\cos \frac{π}{7} + i \sin \frac{π}{7}] \end{aligned}$
Also , $i - 1 = \sqrt{2} (- \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}})$
$= \sqrt{2} (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})$
$∴ z = \frac{\sqrt{2}}{2 \sin (\frac{π}{7})} [\cos (\frac{3 π}{4} - \frac{π}{7}) + i \sin (\frac{3 π}{4} - \frac{π}{7})]$
$= \frac{\sqrt{2}}{2 \sin (\frac{π}{7})} [\cos (\frac{17 π}{28}) + i \sin (\frac{17 π}{28})]$
Thus , $\arg (z) = \frac{17 π}{28}$ .

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