Forma polar de $$$- \frac{5228171817}{100000000} - i$$$
Tu entrada
Encuentra la forma polar de $$$- \frac{5228171817}{100000000} - i$$$.
Solución
La forma estándar del número complejo es $$$- \frac{5228171817}{100000000} - i$$$.
Para un número complejo $$$a + b i$$$, la forma polar viene dada por $$$r \left(\cos{\left(\theta \right)} + i \sin{\left(\theta \right)}\right)$$$, donde $$$r = \sqrt{a^{2} + b^{2}}$$$ y $$$\theta = \operatorname{atan}{\left(\frac{b}{a} \right)}$$$.
Tenemos que $$$a = - \frac{5228171817}{100000000}$$$ y $$$b = -1$$$.
Por lo tanto, $$$r = \sqrt{\left(- \frac{5228171817}{100000000}\right)^{2} + \left(-1\right)^{2}} = \frac{\sqrt{27343780548073081489}}{100000000}.$$$
Además, $$$\theta = \operatorname{atan}{\left(\frac{-1}{- \frac{5228171817}{100000000}} \right)} - \pi = - \pi + \operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}.$$$
Por lo tanto, $$$- \frac{5228171817}{100000000} - i = \frac{\sqrt{27343780548073081489}}{100000000} \left(\cos{\left(- \pi + \operatorname{atan}{\left(\frac{100000000}{5228171817} \right)} \right)} + i \sin{\left(- \pi + \operatorname{atan}{\left(\frac{100000000}{5228171817} \right)} \right)}\right).$$$
Respuesta
$$$- \frac{5228171817}{100000000} - i = \frac{\sqrt{27343780548073081489}}{100000000} \left(\cos{\left(- \pi + \operatorname{atan}{\left(\frac{100000000}{5228171817} \right)} \right)} + i \sin{\left(- \pi + \operatorname{atan}{\left(\frac{100000000}{5228171817} \right)} \right)}\right) = \frac{\sqrt{27343780548073081489}}{100000000} \left(\cos{\left(\left(\frac{- 180 \pi + 180 \operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}}{\pi}\right)^{\circ} \right)} + i \sin{\left(\left(\frac{- 180 \pi + 180 \operatorname{atan}{\left(\frac{100000000}{5228171817} \right)}}{\pi}\right)^{\circ} \right)}\right)$$$A