wyznacz pierwiastki liczby zespolone
: 5 kwie 2015, o 22:11
Wyznacz wszystkie pierwiastki: \(\displaystyle{ \sqrt[6]{-16}}\)
\(\displaystyle{ z=\left[ 16 \left( \cos ( \pi +k2 \pi )+i \sin ( \pi +k2 \pi ) \right) \right] ^{ \frac{1}{6} }= \sqrt[6]{16}\left( \cos ( \frac{\pi}{6}+k\frac{2\pi}{6} )+i \sin ( \frac{\pi}{6} +k\frac{2\pi}{6}) \right)}\)
Jest 6 rozwiązań:
\(\displaystyle{ z _{0}= \sqrt[3]{4}\left( \cos \frac{\pi}{6} + i \sin\frac{\pi}{6} \right)}\)
\(\displaystyle{ z _{1}= \sqrt[3]{4}\left( \cos \frac{3\pi}{6} + i \sin\frac{3\pi}{6} \right)}\)
\(\displaystyle{ z _{2}= \sqrt[3]{4}\left( \cos \frac{5\pi}{6} + i \sin\frac{5\pi}{6} \right)}\)
\(\displaystyle{ z _{3}= \sqrt[3]{4}\left( \cos \frac{7\pi}{6} + i \sin\frac{7\pi}{6} \right)}\)
\(\displaystyle{ z _{4}= \sqrt[3]{4}\left( \cos \frac{9\pi}{6} + i \sin\frac{9\pi}{6} \right)}\)
\(\displaystyle{ z _{5}= \sqrt[3]{4}\left( \cos \frac{11\pi}{6} + i \sin\frac{11\pi}{6} \right)}\)