Write the complex number

Question:

Write the complex number

$z=\frac{1-i}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}$ in polar from.

Solution:

According to the question,

We have,

$z=\frac{1-\mathrm{i}}{\cos \frac{\pi}{3}+\mathrm{i} \sin \frac{\pi}{3}}$

$=\frac{\sqrt{2}\left[\frac{1}{\sqrt{2}}-i \frac{1}{\sqrt{2}}\right]}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}$

$=\frac{\sqrt{2}\left[\cos \frac{\pi}{4}-i \sin \frac{\pi}{4}\right]}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}$

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

$=\sqrt{2}\left[\cos \left(-\frac{7 \pi}{12}\right)+i \sin \left(-\frac{7 \pi}{12}\right)\right]$

$=\sqrt{2}\left[\cos \frac{5 \pi}{12}+i \sin \frac{5 \pi}{12}\right]$

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