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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Your Input

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

Answer

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