Kořeny binomické rovnice v goniometrickém tvaru
Urči kořeny binomické rovnice v goniometrickém tvaru:\( x^{6}+144 i=0 \)
\( \begin{aligned} K= & \left\{\sqrt[3]{12}\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right) ; \sqrt[3]{12}\left(\cos \frac{3 \pi}{12}+i \sin \frac{3 \pi}{12}\right) ;\right. \\ & \sqrt[3]{12}\left(\cos \frac{6 \pi}{12}+i \sin \frac{6 \pi}{12}\right) ; \sqrt[3]{12}\left(\cos \frac{15 \pi}{12}+i \sin \frac{15 \pi}{12}\right) ; \\ & \left.\sqrt[3]{12}\left(\cos \frac{18 \pi}{12}+i \sin \frac{18 \pi}{12}\right) ; \sqrt[3]{12}\left(\cos \frac{20 \pi}{12}+i \sin \frac{20 \pi}{12}\right)\right\}\end{aligned} \)
\( \begin{aligned} K= & \left\{\sqrt[3]{12}\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right) ; \sqrt[3]{12}\left(\cos \frac{2 \pi}{12}+i \sin \frac{2 \pi}{12}\right) ;\right. \\ & \sqrt[3]{12}\left(\cos \frac{8 \pi}{12}+i \sin \frac{8 \pi}{12}\right) ; \sqrt[3]{12}\left(\cos \frac{10 \pi}{12}+i \sin \frac{10 \pi}{12}\right) ; \\ & \left.\sqrt[3]{12}\left(\cos \frac{14 \pi}{12}+i \sin \frac{14 \pi}{12}\right) ; \sqrt[3]{12}\left(\cos \frac{22 \pi}{12}+i \sin \frac{22 \pi}{12}\right)\right\}\end{aligned} \)
\( \begin{aligned} K= & \left\{\sqrt[3]{12}\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right) ; \sqrt[3]{12}\left(\cos \frac{5 \pi}{12}+i \sin \frac{5 \pi}{12}\right) ;\right. \\ & \sqrt[3]{12}\left(\cos \frac{9 \pi}{12}+i \sin \frac{9 \pi}{12}\right) ; \sqrt[3]{12}\left(\cos \frac{13 \pi}{12}+i \sin \frac{13 \pi}{12}\right) ; \\ & \left.\sqrt[3]{12}\left(\cos \frac{17 \pi}{12}+i \sin \frac{17 \pi}{12}\right) ; \sqrt[3]{12}\left(\cos \frac{21 \pi}{12}+i \sin \frac{21 \pi}{12}\right)\right\}\end{aligned} \)
\( \begin{aligned} K= & \left\{\sqrt[3]{12}\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right) ; \sqrt[3]{12}\left(\cos \frac{7 \pi}{12}+i \sin \frac{7 \pi}{12}\right) ;\right. \\ & \sqrt[3]{12}\left(\cos \frac{11 \pi}{12}+i \sin \frac{11 \pi}{12}\right) ; \sqrt[3]{12}\left(\cos \frac{5 \pi}{4}+i \sin \frac{5 \pi}{4}\right) ; \\ & \left.\sqrt[3]{12}\left(\cos \frac{19 \pi}{12}+i \sin \frac{19 \pi}{12}\right) ; \sqrt[3]{12}\left(\cos \frac{23 \pi}{12}+i \sin \frac{23 \pi}{12}\right)\right\}\end{aligned} \)