Polar form of $$$- \frac{5228171817}{100000000} - i$$$

Find the polar form of $$$- \frac{5228171817}{100000000} - i$$$.

The standard form of the complex number is $$$- \frac{5228171817}{100000000} - i$$$.

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 $$$a = - \frac{5228171817}{100000000}$$$ and $$$b = -1$$$.

Thus, $$$r = \sqrt{\left(- \frac{5228171817}{100000000}\right)^{2} + \left(-1\right)^{2}} = \frac{\sqrt{27343780548073081489}}{100000000}.$$$

Also, $$$\theta = \operatorname{atan}{\left(\frac{-1}{- \frac{5228171817}{100000000}} \right)} - \pi = - \pi + \operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}.$$$

Therefore, $$$- \frac{5228171817}{100000000} - i = \frac{\sqrt{27343780548073081489}}{100000000} \left(\cos{\left(- \pi + \operatorname{atan}{\left(\frac{100000000}{5228171817} \right)} \right)} + i \sin{\left(- \pi + \operatorname{atan}{\left(\frac{100000000}{5228171817} \right)} \right)}\right).$$$

$$$- \frac{5228171817}{100000000} - i = \frac{\sqrt{27343780548073081489}}{100000000} \left(\cos{\left(- \pi + \operatorname{atan}{\left(\frac{100000000}{5228171817} \right)} \right)} + i \sin{\left(- \pi + \operatorname{atan}{\left(\frac{100000000}{5228171817} \right)} \right)}\right) = \frac{\sqrt{27343780548073081489}}{100000000} \left(\cos{\left(\left(\frac{- 180 \pi + 180 \operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}}{\pi}\right)^{\circ} \right)} + i \sin{\left(\left(\frac{- 180 \pi + 180 \operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}}{\pi}\right)^{\circ} \right)}\right)$$$A