Polarform von $$$- \frac{5228171817}{100000000} - i$$$

Bestimmen Sie die Polarform von $$$- \frac{5228171817}{100000000} - i$$$.

Die Standardform der komplexen Zahl ist $$$- \frac{5228171817}{100000000} - i$$$.

Für eine komplexe Zahl $$$a + b i$$$ ist die Polarform durch $$$r \left(\cos{\left(\theta \right)} + i \sin{\left(\theta \right)}\right)$$$ gegeben, wobei $$$r = \sqrt{a^{2} + b^{2}}$$$ und $$$\theta = \operatorname{atan}{\left(\frac{b}{a} \right)}$$$.

Es gilt, dass $$$a = - \frac{5228171817}{100000000}$$$ und $$$b = -1$$$.

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

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

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

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