Řešení binomické rovnice v goniometrickém tvaru
Urči kořeny binomické rovnice v goniometrickém tvaru:
\( x^{9}-8=(2 i)^{3} \)
\( \begin{aligned} K= & \left\{\sqrt[18]{2^{7}}\left(\cos \frac{8 \pi}{36}+i \sin \frac{8 \pi}{36}\right) ; \sqrt[18]{2^{7}}\left(\cos \frac{6 \pi}{12}+i \sin \frac{6 \pi}{12}\right)\right. \\ & \sqrt[18]{2^{7}}\left(\cos \frac{24 \pi}{36}+i \sin \frac{24 \pi}{36}\right) ; \sqrt[18]{2^{7}}\left(\cos \frac{32 \pi}{36}+i \sin \frac{32 \pi}{36}\right) ; \\ & \sqrt[18]{2^{7}}\left(\cos \frac{14 \pi}{12}+i \sin \frac{14 \pi}{12}\right) ; \sqrt[18]{2^{7}}\left(\cos \frac{48 \pi}{36}+i \sin \frac{48 \pi}{36}\right) ; \\ & \sqrt[18]{2^{7}}\left(\cos \frac{56 \pi}{36}+i \sin \frac{56 \pi}{36}\right) ; \sqrt[18]{2^{7}}\left(\cos \frac{72 \pi}{4}+i \sin \frac{8 \pi}{4}\right) ;\end{aligned} \)
\( \begin{aligned} K= & \left\{\sqrt[18]{2^{8}}\left(\cos \frac{7 \pi}{36}+i \sin \frac{7 \pi}{36}\right) ; \sqrt[18]{2^{8}}\left(\cos \frac{5 \pi}{12}+i \sin \frac{5 \pi}{12}\right)\right. \\ & \sqrt[18]{2^{8}}\left(\cos \frac{23 \pi}{36}+i \sin \frac{23 \pi}{36}\right) ; \sqrt[18]{2^{8}}\left(\cos \frac{31 \pi}{36}+i \sin \frac{31 \pi}{36}\right) ; \\ & \sqrt[18]{2^{8}}\left(\cos \frac{13 \pi}{12}+i \sin \frac{13 \pi}{12}\right) ; \sqrt[18]{2^{8}}\left(\cos \frac{47 \pi}{36}+i \sin \frac{47 \pi}{36}\right) ; \\ & \sqrt[18]{2^{8}}\left(\cos \frac{55 \pi}{36}+i \sin \frac{55 \pi}{36}\right) ; \sqrt[18]{2^{8}}\left(\cos \frac{71 \pi}{4}+i \sin \frac{7 \pi}{4}\right) ;\end{aligned} \)
\( \begin{aligned} K= & \left\{\sqrt[18]{2^{7}}\left(\cos \frac{7 \pi}{36}+i \sin \frac{7 \pi}{36}\right) ; \sqrt[18]{2^{7}}\left(\cos \frac{5 \pi}{12}+i \sin \frac{5 \pi}{12}\right)\right. \\ & \sqrt[18]{2^{7}}\left(\cos \frac{23 \pi}{36}+i \sin \frac{23 \pi}{36}\right) ; \sqrt[18]{2^{7}}\left(\cos \frac{31 \pi}{36}+i \sin \frac{31 \pi}{36}\right) ; \\ & \sqrt[18]{2^{7}}\left(\cos \frac{13 \pi}{12}+i \sin \frac{13 \pi}{12}\right) ; \sqrt[18]{2^{7}}\left(\cos \frac{47 \pi}{36}+i \sin \frac{47 \pi}{36}\right) ; \\ & \sqrt[18]{2^{7}}\left(\cos \frac{55 \pi}{36}+i \sin \frac{55 \pi}{36}\right) ; \sqrt[18]{2^{7}}\left(\cos \frac{71 \pi}{4}+i \sin \frac{7 \pi}{4}\right) ;\end{aligned} \)
\( \begin{aligned} K= & \left\{\sqrt[18]{2^{6}}\left(\cos \frac{7 \pi}{36}+i \sin \frac{7 \pi}{36}\right) ; \sqrt[18]{2^{6}}\left(\cos \frac{5 \pi}{12}+i \sin \frac{5 \pi}{12}\right)\right. \\ & \sqrt[18]{2^{6}}\left(\cos \frac{23 \pi}{36}+i \sin \frac{23 \pi}{36}\right) ; \sqrt[18]{2^{6}}\left(\cos \frac{31 \pi}{36}+i \sin \frac{31 \pi}{36}\right) ; \\ & \sqrt[18]{2^{6}}\left(\cos \frac{13 \pi}{12}+i \sin \frac{13 \pi}{12}\right) ; \sqrt[18]{2^{6}}\left(\cos \frac{47 \pi}{36}+i \sin \frac{47 \pi}{36}\right) ; \\ & \sqrt[18]{2^{6}}\left(\cos \frac{55 \pi}{36}+i \sin \frac{55 \pi}{36}\right) ; \sqrt[18]{2^{6}}\left(\cos \frac{71 \pi}{4}+i \sin \frac{7 \pi}{4}\right) ;\end{aligned} \)