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受欢迎的代数 >

ix^6-1=0

解答

i x^{6} - 1 = 0

解答

x = \frac{2 ( 1 + 3 )}{4} + i \frac{2 ( 1 - 3 )}{4}, x = \frac{2}{2} + i \frac{2}{2}, x = \frac{2 ( 1 - 3 )}{4} + i \frac{2 ( 1 + 3 )}{4}, x = \frac{2 ( - 1 - 3 )}{4} + i \frac{2 ( - 1 + 3 )}{4}, x = - \frac{2}{2} - i \frac{2}{2}, x = \frac{2 ( - 1 + 3 )}{4} + i \frac{2 ( - 1 - 3 )}{4}

求解步骤

i x^{6} - 1 = 0

将

1

到右边

i x^{6} = 1

两边除以

i

x^{6} = - i

For

z^{n} = a

the solutions are

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

对于

n = 6, a = - i

∣ a ∣ = 1

ar g (a) = - \frac{π}{2}

x = 61 (cos (\frac{- \frac{π}{2} + 2 \cdot 0 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 0 π}{6})), x = 61 (cos (\frac{- \frac{π}{2} + 2 \cdot 1 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 1 π}{6})), x = 61 (cos (\frac{- \frac{π}{2} + 2 \cdot 2 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 2 π}{6})), x = 61 (cos (\frac{- \frac{π}{2} + 2 \cdot 3 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 3 π}{6})), x = 61 (cos (\frac{- \frac{π}{2} + 2 \cdot 4 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 4 π}{6})), x = 61 (cos (\frac{- \frac{π}{2} + 2 \cdot 5 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 5 π}{6}))

化简

61 (cos (\frac{- \frac{π}{2} + 2 \cdot 0 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 0 π}{6})) : \frac{2 ( 1 + 3 )}{4} + i \frac{2 ( 1 - 3 )}{4}

化简

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

化简

61 (cos (\frac{- \frac{π}{2} + 2 \cdot 2 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 2 π}{6})) : \frac{2 ( 1 - 3 )}{4} + i \frac{2 ( 1 + 3 )}{4}

化简

61 (cos (\frac{- \frac{π}{2} + 2 \cdot 3 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 3 π}{6})) : \frac{2 ( - 1 - 3 )}{4} + i \frac{2 ( - 1 + 3 )}{4}

化简

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

化简

61 (cos (\frac{- \frac{π}{2} + 2 \cdot 5 π}{6}) + i sin (\frac{- \frac{π}{2} + 2 \cdot 5 π}{6})) : \frac{2 ( - 1 + 3 )}{4} + i \frac{2 ( - 1 - 3 )}{4}

x = \frac{2 ( 1 + 3 )}{4} + i \frac{2 ( 1 - 3 )}{4}, x = \frac{2}{2} + i \frac{2}{2}, x = \frac{2 ( 1 - 3 )}{4} + i \frac{2 ( 1 + 3 )}{4}, x = \frac{2 ( - 1 - 3 )}{4} + i \frac{2 ( - 1 + 3 )}{4}, x = - \frac{2}{2} - i \frac{2}{2}, x = \frac{2 ( - 1 + 3 )}{4} + i \frac{2 ( - 1 - 3 )}{4}

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