Polar form of $$$32 + 4 \sqrt{17} i$$$
Your Input
Find the polar form of $$$32 + 4 \sqrt{17} i$$$.
Solution
The standard form of the complex number is $$$32 + 4 \sqrt{17} i$$$.
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 $$$a = 32$$$ and $$$b = 4 \sqrt{17}$$$.
Thus, $$$r = \sqrt{32^{2} + \left(4 \sqrt{17}\right)^{2}} = 36$$$.
Also, $$$\theta = \operatorname{atan}{\left(\frac{4 \sqrt{17}}{32} \right)} = \operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}.$$$
Therefore, $$$32 + 4 \sqrt{17} i = 36 \left(\cos{\left(\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)} \right)} + i \sin{\left(\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)} \right)}\right).$$$
Answer
$$$32 + 4 \sqrt{17} i = 36 \left(\cos{\left(\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)} \right)} + i \sin{\left(\operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)} \right)}\right) = 36 \left(\cos{\left(\left(\frac{180 \operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{\pi}\right)^{\circ} \right)} + i \sin{\left(\left(\frac{180 \operatorname{atan}{\left(\frac{\sqrt{17}}{8} \right)}}{\pi}\right)^{\circ} \right)}\right)$$$A