oblicz:
\(\displaystyle{ (1 + i)^{66}}\)
\(\displaystyle{ (- \sqrt{3} +1 )^{79}}\)
wskazowka: wzor Moivre'a
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- Posty: 12
- Rejestracja: 11 lis 2008, o 17:40
- Płeć: Mężczyzna
- Lokalizacja: Rzeszów
- Pomógł: 5 razy
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\(\displaystyle{ a) \newline ft(1+i\right)^{66} =
ft(\sqrt{2}\left(\frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}}\right)\right)^{66} = ft(\sqrt{2}\left(cos\left(\frac{\pi}{4}\right) + i sin\left(\frac{\pi}{4}\right)\right)\right)^{66} = \sqrt{2}^{66}\left(cos\left(66\frac{\pi}{4}\right) + i sin\left(66\frac{\pi}{4}\right)\right)}\)
\(\displaystyle{ b) \newline ft(-\sqrt{3}+1\right)^{79} = ft(2\left(-\frac{\sqrt{3}}{2}+i\frac{1}{2}\right)\right)^{79} = ft(2\left(cos\left(\frac{5\pi}{6}\right) + i sin\left({\frac{5\pi}{6} \right)\right) \right)^{79} = 2^{79}\left(cos\left(79\frac{5\pi}{6}\right)+isin\left(79\frac{5\pi}{6}\right)\right)}\)
ft(\sqrt{2}\left(\frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}}\right)\right)^{66} = ft(\sqrt{2}\left(cos\left(\frac{\pi}{4}\right) + i sin\left(\frac{\pi}{4}\right)\right)\right)^{66} = \sqrt{2}^{66}\left(cos\left(66\frac{\pi}{4}\right) + i sin\left(66\frac{\pi}{4}\right)\right)}\)
\(\displaystyle{ b) \newline ft(-\sqrt{3}+1\right)^{79} = ft(2\left(-\frac{\sqrt{3}}{2}+i\frac{1}{2}\right)\right)^{79} = ft(2\left(cos\left(\frac{5\pi}{6}\right) + i sin\left({\frac{5\pi}{6} \right)\right) \right)^{79} = 2^{79}\left(cos\left(79\frac{5\pi}{6}\right)+isin\left(79\frac{5\pi}{6}\right)\right)}\)