Polar form of $$$-1 + \sqrt{3} i$$$
Your Input
Find the polar form of $$$-1 + \sqrt{3} i$$$.
Solution
The standard form of the complex number is $$$-1 + \sqrt{3} i$$$.
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 $$$a = -1$$$ and $$$b = \sqrt{3}$$$.
Thus, $$$r = \sqrt{\left(-1\right)^{2} + \left(\sqrt{3}\right)^{2}} = 2$$$.
Also, $$$\theta = \operatorname{atan}{\left(\frac{\sqrt{3}}{-1} \right)} + \pi = \frac{2 \pi}{3}$$$.
Therefore, $$$-1 + \sqrt{3} i = 2 \left(\cos{\left(\frac{2 \pi}{3} \right)} + i \sin{\left(\frac{2 \pi}{3} \right)}\right)$$$.
Answer
$$$-1 + \sqrt{3} i = 2 \left(\cos{\left(\frac{2 \pi}{3} \right)} + i \sin{\left(\frac{2 \pi}{3} \right)}\right) = 2 \left(\cos{\left(120^{\circ} \right)} + i \sin{\left(120^{\circ} \right)}\right)$$$A