w^3=-2-2isqrt(3)

解

w^{3} = - 2 - 2 i 3

w = 34 cos (\frac{2 π}{9}) - 34 i sin (\frac{2 π}{9}), w = 34 cos (\frac{4 π}{9}) + 34 i sin (\frac{4 π}{9}), w = 34 cos (\frac{10 π}{9}) + 34 i sin (\frac{10 π}{9})

解答ステップ

w^{3} = - 2 - 2 i 3

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 = 3, a = - 2 - 2 i 3

∣ a ∣ = 4

ar g (a) = \frac{π}{3} - π

w = 34 (cos (\frac{\frac{π}{3} - π + 2 \cdot 0 π}{3}) + i sin (\frac{\frac{π}{3} - π + 2 \cdot 0 π}{3})), w = 34 (cos (\frac{\frac{π}{3} - π + 2 \cdot 1 π}{3}) + i sin (\frac{\frac{π}{3} - π + 2 \cdot 1 π}{3})), w = 34 (cos (\frac{\frac{π}{3} - π + 2 \cdot 2 π}{3}) + i sin (\frac{\frac{π}{3} - π + 2 \cdot 2 π}{3}))

簡素化

34 (cos (\frac{\frac{π}{3} - π + 2 \cdot 0 π}{3}) + i sin (\frac{\frac{π}{3} - π + 2 \cdot 0 π}{3})) : 34 cos (\frac{2 π}{9}) - 34 i sin (\frac{2 π}{9})

簡素化

34 (cos (\frac{\frac{π}{3} - π + 2 \cdot 1 π}{3}) + i sin (\frac{\frac{π}{3} - π + 2 \cdot 1 π}{3})) : 34 cos (\frac{4 π}{9}) + 34 i sin (\frac{4 π}{9})

簡素化

34 (cos (\frac{\frac{π}{3} - π + 2 \cdot 2 π}{3}) + i sin (\frac{\frac{π}{3} - π + 2 \cdot 2 π}{3})) : 34 cos (\frac{10 π}{9}) + 34 i sin (\frac{10 π}{9})

w = 34 cos (\frac{2 π}{9}) - 34 i sin (\frac{2 π}{9}), w = 34 cos (\frac{4 π}{9}) + 34 i sin (\frac{4 π}{9}), w = 34 cos (\frac{10 π}{9}) + 34 i sin (\frac{10 π}{9})