复数计算 (1-根号3i)^10-(1+根号3i)^10=?
来源:学生作业帮 编辑:作业帮 分类:数学作业 时间:2024/08/27 07:40:29
复数计算 (1-根号3i)^10-(1+根号3i)^10=?
解析:
(1-根号3i)^10
=2^10 ×(1/2 -根号3/2 ×i)^10
=2^10 ×[cos(-π/3)+sin(-π/3)×i]^10
=2^10 ×[cos(-10π/3)+sin(-10π/3)×i]
=2^10 ×[cos(2π/3)+sin(2π/3)×i]
=2^10 ×(-1/2+根号3/2×i )
而 (1+根号3i)^10
=2^10 ×(1/2 +根号3/2 ×i)^10
=2^10 ×[cos(π/3)+sin(π/3)×i]^10
=2^10 ×[cos(10π/3)+sin(10π/3)×i]
=2^10 ×[cos(4π/3)+sin(4π/3)×i]
=2^10 ×[-cos(π/3)-sin(π/3)×i]
=2^10 ×(-1/2-根号3/2×i )
所以:
(1-根号3i)^10-(1+根号3i)^10
=2^10 ×(-1/2+根号3/2×i )-2^10 ×(-1/2-根号3/2×i )
=2^10 ×[(-1/2+根号3/2×i )-(-1/2-根号3/2×i )]
=2^10 ×根号3×i
=1024根号3×i
(1-根号3i)^10
=2^10 ×(1/2 -根号3/2 ×i)^10
=2^10 ×[cos(-π/3)+sin(-π/3)×i]^10
=2^10 ×[cos(-10π/3)+sin(-10π/3)×i]
=2^10 ×[cos(2π/3)+sin(2π/3)×i]
=2^10 ×(-1/2+根号3/2×i )
而 (1+根号3i)^10
=2^10 ×(1/2 +根号3/2 ×i)^10
=2^10 ×[cos(π/3)+sin(π/3)×i]^10
=2^10 ×[cos(10π/3)+sin(10π/3)×i]
=2^10 ×[cos(4π/3)+sin(4π/3)×i]
=2^10 ×[-cos(π/3)-sin(π/3)×i]
=2^10 ×(-1/2-根号3/2×i )
所以:
(1-根号3i)^10-(1+根号3i)^10
=2^10 ×(-1/2+根号3/2×i )-2^10 ×(-1/2-根号3/2×i )
=2^10 ×[(-1/2+根号3/2×i )-(-1/2-根号3/2×i )]
=2^10 ×根号3×i
=1024根号3×i
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