Polar Form of a Complex Number Calculator

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

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

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