Question

Convert the following complex numbers into polar form.
$\frac{2+6\sqrt{3i}}{5+\sqrt{3i}}$

• A. $2e^{i\frac{\pi }{6}}$
• B. $e^{i\frac{\pi }{4}}$
• C. $2e^{i\frac{\pi }{3}}$
• D. $e^{i\frac{\pi }{6}}$

Answer 1

The correct option is C $2e^{i\frac{\pi }{3}}$

$\frac{2+6\sqrt{3}i}{5+\sqrt{3}i}$

$\frac{2+6\sqrt{3}i}{5+\sqrt{3}i}\times \frac{5-\sqrt{3}i}{5-\sqrt{3}i}$

$=\frac{(10+18)+(30\sqrt{3}+2\sqrt{3})i}{28}$

$\frac{28(1+i)}{28}=1+i$

$\sqrt{2}(\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}})$

$=\sqrt{2}(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4})$

$=\sqrt{2}{e}^{\frac{i\pi }{4}}$.