z^4=-7+24i

解

z^{4} = - 7 + 24 i

z = 5 cos (\frac{- arctan ( \frac{24}{7} ) + π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + π}{4}), z = 5 cos (\frac{- arctan ( \frac{24}{7} ) + 3 π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + 3 π}{4}), z = 5 cos (\frac{- arctan ( \frac{24}{7} ) + 5 π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + 5 π}{4}), z = 5 cos (\frac{- arctan ( \frac{24}{7} ) + 7 π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + 7 π}{4})

解答ステップ

z^{4} = - 7 + 24 i

z^{n} = a

の場合, 解は

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

以下のため:

n = 4, a = - 7 + 24 i

∣ a ∣ = 25

ar g (a) = - arctan (\frac{24}{7}) + π

z = 425 (cos (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 0 π}{4}) + i sin (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 0 π}{4})), z = 425 (cos (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 1 π}{4}) + i sin (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 1 π}{4})), z = 425 (cos (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 2 π}{4}) + i sin (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 2 π}{4})), z = 425 (cos (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 3 π}{4}) + i sin (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 3 π}{4}))

簡素化

425 (cos (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 0 π}{4}) + i sin (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 0 π}{4})) : 5 cos (\frac{- arctan ( \frac{24}{7} ) + π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + π}{4})

簡素化

425 (cos (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 1 π}{4}) + i sin (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 1 π}{4})) : 5 cos (\frac{- arctan ( \frac{24}{7} ) + 3 π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + 3 π}{4})

簡素化

425 (cos (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 2 π}{4}) + i sin (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 2 π}{4})) : 5 cos (\frac{- arctan ( \frac{24}{7} ) + 5 π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + 5 π}{4})

簡素化

425 (cos (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 3 π}{4}) + i sin (\frac{- arctan ( \frac{24}{7} ) + π + 2 \cdot 3 π}{4})) : 5 cos (\frac{- arctan ( \frac{24}{7} ) + 7 π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + 7 π}{4})

z = 5 cos (\frac{- arctan ( \frac{24}{7} ) + π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + π}{4}), z = 5 cos (\frac{- arctan ( \frac{24}{7} ) + 3 π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + 3 π}{4}), z = 5 cos (\frac{- arctan ( \frac{24}{7} ) + 5 π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + 5 π}{4}), z = 5 cos (\frac{- arctan ( \frac{24}{7} ) + 7 π}{4}) + 5 i sin (\frac{- arctan ( \frac{24}{7} ) + 7 π}{4})