# Polar Form of a Complex Number Calculator

The calculator will find the polar form of the given complex number, with steps shown.

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Find the polar form of $\sqrt{3} + i$.

## Solution

The standard form of the complex number is $\sqrt{3} + i$.

For a complex number $a + b i$, the polar form is given by $r \left(\cos{\left(\theta \right)} + i \sin{\left(\theta \right)}\right)$, where $r = \sqrt{a^{2} + b^{2}}$ and $\theta = \operatorname{atan}{\left(\frac{b}{a} \right)}$.

We have that $a = \sqrt{3}$ and $b = 1$.

Thus, $r = \sqrt{\left(\sqrt{3}\right)^{2} + 1^{2}} = 2$.

Also, $\theta = \operatorname{atan}{\left(\frac{1}{\sqrt{3}} \right)} = \frac{\pi}{6}$.

Therefore, $\sqrt{3} + i = 2 \left(\cos{\left(\frac{\pi}{6} \right)} + i \sin{\left(\frac{\pi}{6} \right)}\right)$.

$\sqrt{3} + i = 2 \left(\cos{\left(\frac{\pi}{6} \right)} + i \sin{\left(\frac{\pi}{6} \right)}\right) = 2 \left(\cos{\left(30^0 \right)} + i \sin{\left(30^0 \right)}\right)$A