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liczby zepsolone rownanie
: 10 mar 2010, 23:34
autor: tenisista
z^4 +1=0 bardzo bede wdzieczny za pomoc w rozwiazaniu
: 11 mar 2010, 19:03
autor: irena
\(z^4+1=0\\z=\sqrt[4]{-1}\\z^4=-1+0i\\cos\alpha=-1\ \wedge sin\alpha=0 \Rightarrow \alpha=\pi\\|z^4|=1\\\alpha_k=\frac{\pi+2k\pi}{4}=\frac{\pi}{4}+k\cdot\frac{\pi}{2};\ k=0,\ 1,\ 2,\ 3\\\alpha_1=\frac{\pi}{4}\\\alpha_2=\frac{3}{4}\pi\\\alpha_3=\frac{5}{4}\pi\\\alpha_4=\frac{7}{4}\pi\\z_k=\sqrt[4]{1}(cos\alpha_k+isin\alpha_k}\\z_1=\frac{\sqrt{2}}{2}(1+i)\ \vee \ z_2=\frac{\sqrt{2}}{2}(-1+i)\ \vee \ z_3=\frac{\sqrt{2}}{2}(-1-i)\ \vee \ z_4=\frac{\sqrt{2}}{2}(1-i)\)