Convert the complex number in the polar form: i - Vidyadip

Question

Convert the complex number in the polar form: i

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Answer

Here z = $i = r(\cos \theta + i\sin \theta )$
$ \Rightarrow r\cos \theta = 0$ and $r\;\sin \theta = 1$ . . . (i)
Squaring both sides of (i) and adding
$$${r^2}({\cos ^2}\theta + {\sin ^2}\theta ) \Rightarrow {r^2} = 1 \Rightarrow r = 1$
$\therefore \cos \theta = 0$ and $\sin \theta = 1$
Since $\sin \theta $ and $\cos \theta $ are both positive
$\therefore \theta $ lies in first quadrant
$\therefore \theta = \frac{\pi }{2}$
Hence polar form of z is $\left( {\cos \frac{\pi }{2} + i\sin \frac{\pi }{2}} \right)$

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