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Beliebt Algebra >

z^4+2i=6+3i

Lösung

z^{4} + 2 i = 6 + 3 i

Lösung

z = 837 cos (\frac{arctan ( \frac{1}{6} )}{4}) + 837 i sin (\frac{arctan ( \frac{1}{6} )}{4}), z = 837 cos (\frac{arctan ( \frac{1}{6} ) + 2 π}{4}) + 837 i sin (\frac{arctan ( \frac{1}{6} ) + 2 π}{4}), z = 837 cos (\frac{arctan ( \frac{1}{6} ) + 4 π}{4}) + 837 i sin (\frac{arctan ( \frac{1}{6} ) + 4 π}{4}), z = 837 cos (\frac{arctan ( \frac{1}{6} ) + 6 π}{4}) + 837 i sin (\frac{arctan ( \frac{1}{6} ) + 6 π}{4})

Schritte zur Lösung

z^{4} + 2 i = 6 + 3 i

Verschiebe

2 i

auf die rechte Seite

z^{4} = 6 + 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 = 4, a = 6 + i

∣ a ∣ = 37

ar g (a) = arctan (\frac{1}{6})

z = 437 (cos (\frac{arctan ( \frac{1}{6} ) + 2 \cdot 0 π}{4}) + i sin (\frac{arctan ( \frac{1}{6} ) + 2 \cdot 0 π}{4})), z = 437 (cos (\frac{arctan ( \frac{1}{6} ) + 2 \cdot 1 π}{4}) + i sin (\frac{arctan ( \frac{1}{6} ) + 2 \cdot 1 π}{4})), z = 437 (cos (\frac{arctan ( \frac{1}{6} ) + 2 \cdot 2 π}{4}) + i sin (\frac{arctan ( \frac{1}{6} ) + 2 \cdot 2 π}{4})), z = 437 (cos (\frac{arctan ( \frac{1}{6} ) + 2 \cdot 3 π}{4}) + i sin (\frac{arctan ( \frac{1}{6} ) + 2 \cdot 3 π}{4}))

Vereinfache

437 (cos (\frac{arctan ( \frac{1}{6} ) + 2 \cdot 0 π}{4}) + i sin (\frac{arctan ( \frac{1}{6} ) + 2 \cdot 0 π}{4})) : 837 cos (\frac{arctan ( \frac{1}{6} )}{4}) + 837 i sin (\frac{arctan ( \frac{1}{6} )}{4})

Vereinfache

437 (cos (\frac{arctan ( \frac{1}{6} ) + 2 \cdot 1 π}{4}) + i sin (\frac{arctan ( \frac{1}{6} ) + 2 \cdot 1 π}{4})) : 837 cos (\frac{arctan ( \frac{1}{6} ) + 2 π}{4}) + 837 i sin (\frac{arctan ( \frac{1}{6} ) + 2 π}{4})

Vereinfache

437 (cos (\frac{arctan ( \frac{1}{6} ) + 2 \cdot 2 π}{4}) + i sin (\frac{arctan ( \frac{1}{6} ) + 2 \cdot 2 π}{4})) : 837 cos (\frac{arctan ( \frac{1}{6} ) + 4 π}{4}) + 837 i sin (\frac{arctan ( \frac{1}{6} ) + 4 π}{4})

Vereinfache

437 (cos (\frac{arctan ( \frac{1}{6} ) + 2 \cdot 3 π}{4}) + i sin (\frac{arctan ( \frac{1}{6} ) + 2 \cdot 3 π}{4})) : 837 cos (\frac{arctan ( \frac{1}{6} ) + 6 π}{4}) + 837 i sin (\frac{arctan ( \frac{1}{6} ) + 6 π}{4})

z = 837 cos (\frac{arctan ( \frac{1}{6} )}{4}) + 837 i sin (\frac{arctan ( \frac{1}{6} )}{4}), z = 837 cos (\frac{arctan ( \frac{1}{6} ) + 2 π}{4}) + 837 i sin (\frac{arctan ( \frac{1}{6} ) + 2 π}{4}), z = 837 cos (\frac{arctan ( \frac{1}{6} ) + 4 π}{4}) + 837 i sin (\frac{arctan ( \frac{1}{6} ) + 4 π}{4}), z = 837 cos (\frac{arctan ( \frac{1}{6} ) + 6 π}{4}) + 837 i sin (\frac{arctan ( \frac{1}{6} ) + 6 π}{4})

Beliebte Beispiele

z^{4} = 9 + 8 z^{2}

(1+x/(12))^{12}=1.08

(1 + \frac{x}{12})^{12} = 1.08

u^{3} + 8 = (u + 2)

- 6 + w^{3} = - 9

54 = (6 - p^{4}) 3