Question

Convert the following complex number into polar form.
$\frac{1+3i}{1-2i}$

• A. $\sqrt{2}{e}^{i\frac{3\pi }{4}}$
• B. $\sqrt{3}{e}^{i\frac{3\pi }{4}}$
• C. $\sqrt{2}{e}^{i\frac{\pi }{4}}$
• D. $\sqrt{3}{e}^{i\frac{5\pi }{4}}$

Answer 1

Answer: (A)

$\sqrt{2}{e}^{i\frac{3\pi }{4}}$

$\frac{(1+3i)}{(1-2i)}$

$\frac{(1+3i)}{(1-2i)}\times \frac{(1+2i)}{(1+2i)}=\frac{1+2i+3i+6i^2}{5}=\frac{1+5i-6}{5}$

$=\frac{-5+5i}{5}$

$=-1+i$

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

$\sqrt{2}(\cos \left(\frac{3\pi }{4}\right)+i\sin \left(\frac{3\pi }{4}\right))$

$\sqrt{2}e\frac{3\pi i}{4}$.