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인기 있는 대수학 >

z^6+1=sqrt(3)i

해법

z^{6} + 1 = 3 i

해법

z = 62 cos (\frac{π}{9}) + 62 i sin (\frac{π}{9}), z = 62 cos (\frac{4 π}{9}) + 62 i sin (\frac{4 π}{9}), z = 62 cos (\frac{7 π}{9}) + 62 i sin (\frac{7 π}{9}), z = 62 cos (\frac{10 π}{9}) + 62 i sin (\frac{10 π}{9}), z = 62 cos (\frac{13 π}{9}) + 62 i sin (\frac{13 π}{9}), z = 62 cos (\frac{16 π}{9}) + 62 i sin (\frac{16 π}{9})

솔루션 단계

z^{6} + 1 = 3 i

1

를 오른쪽으로 이동

z^{6} = 3 i - 1

위한

z^{n} = a

해결책은

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

n = 6, a = 3 i - 1

위한

∣ a ∣ = 2

ar g (a) = - \frac{π}{3} + π

z = 62 (cos (\frac{- \frac{π}{3} + π + 2 \cdot 0 π}{6}) + i sin (\frac{- \frac{π}{3} + π + 2 \cdot 0 π}{6})), z = 62 (cos (\frac{- \frac{π}{3} + π + 2 \cdot 1 π}{6}) + i sin (\frac{- \frac{π}{3} + π + 2 \cdot 1 π}{6})), z = 62 (cos (\frac{- \frac{π}{3} + π + 2 \cdot 2 π}{6}) + i sin (\frac{- \frac{π}{3} + π + 2 \cdot 2 π}{6})), z = 62 (cos (\frac{- \frac{π}{3} + π + 2 \cdot 3 π}{6}) + i sin (\frac{- \frac{π}{3} + π + 2 \cdot 3 π}{6})), z = 62 (cos (\frac{- \frac{π}{3} + π + 2 \cdot 4 π}{6}) + i sin (\frac{- \frac{π}{3} + π + 2 \cdot 4 π}{6})), z = 62 (cos (\frac{- \frac{π}{3} + π + 2 \cdot 5 π}{6}) + i sin (\frac{- \frac{π}{3} + π + 2 \cdot 5 π}{6}))

62 (cos (\frac{- \frac{π}{3} + π + 2 \cdot 0 π}{6}) + i sin (\frac{- \frac{π}{3} + π + 2 \cdot 0 π}{6}))

단순화하세요:

62 cos (\frac{π}{9}) + 62 i sin (\frac{π}{9})

62 (cos (\frac{- \frac{π}{3} + π + 2 \cdot 1 π}{6}) + i sin (\frac{- \frac{π}{3} + π + 2 \cdot 1 π}{6}))

단순화하세요:

62 cos (\frac{4 π}{9}) + 62 i sin (\frac{4 π}{9})

62 (cos (\frac{- \frac{π}{3} + π + 2 \cdot 2 π}{6}) + i sin (\frac{- \frac{π}{3} + π + 2 \cdot 2 π}{6}))

단순화하세요:

62 cos (\frac{7 π}{9}) + 62 i sin (\frac{7 π}{9})

62 (cos (\frac{- \frac{π}{3} + π + 2 \cdot 3 π}{6}) + i sin (\frac{- \frac{π}{3} + π + 2 \cdot 3 π}{6}))

단순화하세요:

62 cos (\frac{10 π}{9}) + 62 i sin (\frac{10 π}{9})

62 (cos (\frac{- \frac{π}{3} + π + 2 \cdot 4 π}{6}) + i sin (\frac{- \frac{π}{3} + π + 2 \cdot 4 π}{6}))

단순화하세요:

62 cos (\frac{13 π}{9}) + 62 i sin (\frac{13 π}{9})

62 (cos (\frac{- \frac{π}{3} + π + 2 \cdot 5 π}{6}) + i sin (\frac{- \frac{π}{3} + π + 2 \cdot 5 π}{6}))

단순화하세요:

62 cos (\frac{16 π}{9}) + 62 i sin (\frac{16 π}{9})

z = 62 cos (\frac{π}{9}) + 62 i sin (\frac{π}{9}), z = 62 cos (\frac{4 π}{9}) + 62 i sin (\frac{4 π}{9}), z = 62 cos (\frac{7 π}{9}) + 62 i sin (\frac{7 π}{9}), z = 62 cos (\frac{10 π}{9}) + 62 i sin (\frac{10 π}{9}), z = 62 cos (\frac{13 π}{9}) + 62 i sin (\frac{13 π}{9}), z = 62 cos (\frac{16 π}{9}) + 62 i sin (\frac{16 π}{9})

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