Express in 5 +i (1- 5 ) polar form. - Vidyadip

Question

Express in $\sin\frac{\pi}{5}+\text{i}\Big(1-\cos\frac{\pi}{5}\Big)$ polar form.

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Answer

$\sin\frac{\pi}{5}+\text{i}\Big(1-\cos\frac{\pi}{5}\Big)$
$=2\sin\frac{\pi}{10}\cos\frac{\pi}{10}+\text{i}2\sin^2\frac{\pi}{10}$ $\big[\text{using }\sin2\theta=2\sin\theta\cos\theta\ { \&}\ 1-\cos2\theta=\sin^2\theta\big]$
$=2\sin\frac{\pi}{10}\Big(\cos\frac{\pi}{10}+\text{i}\sin\frac{\pi}{10}\Big)$

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