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O expresie cu numere complexe
Posted: Sun Jan 27, 2008 7:48 pm
by Andrei Velicu
Fie \( z \in \mathbb{C}^* \) astfel incat \( z+\frac{1}{z}=-\sqrt{3} \) si \( E_{n}=z^n+\frac{1}{z^n}, n\in \mathbb{N}^* \). Determinati multimea \( \left{E_{n} | n\in \mathbb{N}^*\right} \).
OLM Constanta 2008, Prof. Gheorghe Andrei
Posted: Mon Jan 28, 2008 2:44 pm
by turcas
Fie \( z = r(\cos{a} + i \sin{a}) \) unde \( r > 0 \) si \( a \in [0;2\pi) \) .
Expresia din ipoteza devine :
\( \cos{a}(r+ \frac{1}{r}) + i \sin{a}(r - \frac{1}{r}) = - \sqrt{3} \) .
de aici rezulta ca \( r = 1 \) (partea imaginara trebuie sa fie 0 ) ;
Apoi , inlocuind in relatia din ipoteza , avem :
\( 2 \cos{a} = - \sqrt{3} \) . De aici se disting doua cazuri .
1.) \( a = \frac{5\pi}{6} \Rightarrow E_n = 2 \cdot \cos{n \frac{5\pi}{6}} \) .
2.) \( a = \frac{7\pi}{6} \Rightarrow E_n = 2 \cdot \cos{n \frac{7\pi}{6}} \) .