Toán 12 Số phức

T

tuyn

cho số phức z thỏa [TEX]z + \frac{1}{z} =2cos \alpha[/TEX] CMR [TEX]z^m + \frac{1}{z^m} =2cosm \alpha[/TEX] càng ngắn gọn càng tốt nha :)>-:)>-
[TEX]gt \Leftrightarrow z^2-2cos{ \alpha} z+1=0 \Leftrightarrow \left[\begin{z=cos{ \alpha}+isin{ \alpha}}\\{z=cos{ \alpha}-isin{ \alpha}}[/TEX]
[TEX]+) TH1: z=cos{ \alpha}+isin{ \alpha} \Rightarrow z^m+ \frac{1}{z^m}=cosm{ \alpha}+isinm{ \alpha}+ \frac{1}{cosm{ \alpha}+isinm{ \alpha}}=2cosm{ \alpha}[/TEX]
[TEX]Note: \frac{1}{cosm{ \alpha}+isinm{ \alpha}}=cosm{ \alpha}-isinm{ \alpha}[/TEX]
[TEX]+) TH2: z=cos{ \alpha}- isin{ \alpha}[/TEX]
làm tương tự ta cũng có: [TEX]z^m+ \frac{1}{z^m}=2cosm{ \alpha}[/TEX]
 
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