Calculati \( \left( \matrix{ \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2}} \right)^{2008} \).
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Internet Olympiad Problema 2
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Beniamin Bogosel
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Internet Olympiad Problema 2
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Marius Mainea
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Re: Internet Olympiad Problema 2
Beniamin Bogosel wrote:Calculati \( \left( \matrix{ \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2}} \right)^{2008} \).
Internet Olympiad, Ariel University of Samaria, Israel
\( \left( \matrix{ \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2}} \right)^{2008}=\left( \matrix{ cos a & sin a \\ -sin a & cos a \right)^{2008}=\left(\matrix {cos{2008a}& sin{2008a}\\-sin 2008a & cos2008a}\right)=-\left( \matrix{ \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2}} \right) \)
unde \( a=\frac{\pi}{3} \)