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Beliebt Algebra >

z^3=3-sqrt(3)i

Lösung

z^{3} = 3 - 3 i

Lösung

z = 3263 cos (\frac{π}{18}) - 3263 i sin (\frac{π}{18}), z = 3263 cos (\frac{11 π}{18}) + 3263 i sin (\frac{11 π}{18}), z = 3263 cos (\frac{23 π}{18}) + 3263 i sin (\frac{23 π}{18})

Schritte zur Lösung

z^{3} = 3 - 3 i

Für

z^{n} = a

sind die Lösungen

z_{k} = n ∣ a ∣ (cos (\frac{ar g ( a ) + 2 kπ}{n}) + i sin (\frac{ar g ( a ) + 2 kπ}{n})),

k = 0, 1, \dots, n - 1

Für

n = 3, a = 3 - 3 i

∣ a ∣ = 23

ar g (a) = - \frac{π}{6}

z = 323 (cos (\frac{- \frac{π}{6} + 2 \cdot 0 π}{3}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 0 π}{3})), z = 323 (cos (\frac{- \frac{π}{6} + 2 \cdot 1 π}{3}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 1 π}{3})), z = 323 (cos (\frac{- \frac{π}{6} + 2 \cdot 2 π}{3}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 2 π}{3}))

Vereinfache

323 (cos (\frac{- \frac{π}{6} + 2 \cdot 0 π}{3}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 0 π}{3})) : 3263 cos (\frac{π}{18}) - 3263 i sin (\frac{π}{18})

Vereinfache

323 (cos (\frac{- \frac{π}{6} + 2 \cdot 1 π}{3}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 1 π}{3})) : 3263 cos (\frac{11 π}{18}) + 3263 i sin (\frac{11 π}{18})

Vereinfache

323 (cos (\frac{- \frac{π}{6} + 2 \cdot 2 π}{3}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 2 π}{3})) : 3263 cos (\frac{23 π}{18}) + 3263 i sin (\frac{23 π}{18})

z = 3263 cos (\frac{π}{18}) - 3263 i sin (\frac{π}{18}), z = 3263 cos (\frac{11 π}{18}) + 3263 i sin (\frac{11 π}{18}), z = 3263 cos (\frac{23 π}{18}) + 3263 i sin (\frac{23 π}{18})

Beliebte Beispiele

x^{4} - 15 x^{2} + 44 = 0

4 x^{4} = 12 x^{2}

3x^3-9x^2-54x=0

3 x^{3} - 9 x^{2} - 54 x = 0

x^3=-2-2sqrt(3)i

x^{3} = - 2 - 23 i

x^{3} + x^{2} = 16 x + 16