Řešení binomické rovnice v goniometrickém tvaru
Urči kořeny binomické rovnice \( x^{5}+4x-4 i=0 \) v goniometrickém tvaru.
\( \begin{aligned} K= & \left\{\sqrt{3}\left(\cos \frac{3 \pi}{20}+i \sin \frac{3 \pi}{20}\right) ; \sqrt{3}\left(\cos \frac{11 \pi}{20}+i \sin \frac{11 \pi}{20}\right) ;\right. \\ & \sqrt{3}\left(\cos \frac{19 \pi}{20}+i \sin \frac{19 \pi}{20}\right) ; \sqrt{3}\left(\cos \frac{27 \pi}{20}+i \sin \frac{27 \pi}{20}\right) ; \\ & \left.\sqrt{3}\left(\cos \frac{35 \pi}{20}+i \sin \frac{35 \pi}{20}\right)\right\}\end{aligned} \)
\( \begin{aligned} K= & \left\{\sqrt{2}\left(\cos \frac{3 \pi}{20}+i \sin \frac{3 \pi}{20}\right) ; \sqrt{2}\left(\cos \frac{11 \pi}{20}+i \sin \frac{11 \pi}{20}\right) ;\right. \\ & \sqrt{2}\left(\cos \frac{19 \pi}{20}+i \sin \frac{19 \pi}{20}\right) ; \sqrt{2}\left(\cos \frac{27 \pi}{20}+i \sin \frac{27 \pi}{20}\right) ; \\ & \left.\sqrt{2}\left(\cos \frac{35 \pi}{20}+i \sin \frac{35 \pi}{20}\right)\right\}\end{aligned} \)
\( \begin{aligned} K= & \left\{\sqrt{2}\left(\cos \frac{3 \pi}{18}+i \sin \frac{3 \pi}{18}\right) ; \sqrt{2}\left(\cos \frac{11 \pi}{18}+i \sin \frac{11 \pi}{18}\right) ;\right. \\ & \sqrt{2}\left(\cos \frac{19 \pi}{18}+i \sin \frac{19 \pi}{18}\right) ; \sqrt{2}\left(\cos \frac{27 \pi}{18}+i \sin \frac{27 \pi}{18}\right) ; \\ & \left.\sqrt{2}\left(\cos \frac{35 \pi}{18}+i \sin \frac{35 \pi}{18}\right)\right\}\end{aligned} \)
\( \begin{aligned} K= & \left\{\sqrt{2}\left(\cos \frac{5 \pi}{20}+i \sin \frac{5 \pi}{20}\right) ; \sqrt{2}\left(\cos \frac{13 \pi}{20}+i \sin \frac{13 \pi}{20}\right) ;\right. \\ & \sqrt{2}\left(\cos \frac{21 \pi}{20}+i \sin \frac{21 \pi}{20}\right) ; \sqrt{2}\left(\cos \frac{29 \pi}{20}+i \sin \frac{29 \pi}{20}\right) ; \\ & \left.\sqrt{2}\left(\cos \frac{37 \pi}{20}+i \sin \frac{37 \pi}{20}\right)\right\}\end{aligned} \)