Lösung 3.3:1c

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The calculation follows a fairly set pattern. We write the number
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<center> [[Image:3_3_1c.gif]] </center>
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<math>4\sqrt{3}-4i</math>
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in polar form and then use de Moivre's formula.
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[[Image:3_3_1_c.gif]] [[Image:3_3_1_c_text.gif]]
[[Image:3_3_1_c.gif]] [[Image:3_3_1_c_text.gif]]
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This gives
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<math>4\sqrt{3}-4i=8\left( \cos \left( -\frac{\pi }{6} \right)+i\sin \left( -\frac{\pi }{6} \right) \right)</math>
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and then we get, on using de Moivre's formula,
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<math>\begin{align}
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& \left( 4\sqrt{3}-4i \right)^{22}=8^{22}\left( \cos \left( 22\centerdot \left( -\frac{\pi }{6} \right) \right)+i\sin \left( 22\centerdot \left( -\frac{\pi }{6} \right) \right) \right) \\
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& =\left( 2^{3} \right)^{22}\left( \cos \left( -\frac{11\pi }{3} \right)+i\sin \left( -\frac{11\pi }{3} \right) \right) \\
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& =2^{3\centerdot 22}\left( \cos \left( -\frac{12\pi -\pi }{3} \right)+i\sin \left( -\frac{12\pi -\pi }{3} \right) \right) \\
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& =2^{66}\left( \cos \left( -4\pi +\frac{\pi }{3} \right)+i\sin \left( -4\pi +\frac{\pi }{3} \right) \right) \\
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& =2^{66}\left( \cos \left( \frac{\pi }{3} \right)+i\sin \left( \frac{\pi }{3} \right) \right) \\
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& =2^{66}\left( \frac{1}{2}+i\frac{\sqrt{3}}{2} \right)=2^{65}\left( 1+i\sqrt{3} \right) \\
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\end{align}</math>

Version vom 07:28, 24. Okt. 2008

The calculation follows a fairly set pattern. We write the number \displaystyle 4\sqrt{3}-4i in polar form and then use de Moivre's formula.


Image:3_3_1_c.gif Image:3_3_1_c_text.gif


This gives


\displaystyle 4\sqrt{3}-4i=8\left( \cos \left( -\frac{\pi }{6} \right)+i\sin \left( -\frac{\pi }{6} \right) \right)


and then we get, on using de Moivre's formula,


\displaystyle \begin{align} & \left( 4\sqrt{3}-4i \right)^{22}=8^{22}\left( \cos \left( 22\centerdot \left( -\frac{\pi }{6} \right) \right)+i\sin \left( 22\centerdot \left( -\frac{\pi }{6} \right) \right) \right) \\ & =\left( 2^{3} \right)^{22}\left( \cos \left( -\frac{11\pi }{3} \right)+i\sin \left( -\frac{11\pi }{3} \right) \right) \\ & =2^{3\centerdot 22}\left( \cos \left( -\frac{12\pi -\pi }{3} \right)+i\sin \left( -\frac{12\pi -\pi }{3} \right) \right) \\ & =2^{66}\left( \cos \left( -4\pi +\frac{\pi }{3} \right)+i\sin \left( -4\pi +\frac{\pi }{3} \right) \right) \\ & =2^{66}\left( \cos \left( \frac{\pi }{3} \right)+i\sin \left( \frac{\pi }{3} \right) \right) \\ & =2^{66}\left( \frac{1}{2}+i\frac{\sqrt{3}}{2} \right)=2^{65}\left( 1+i\sqrt{3} \right) \\ \end{align}

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