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求2+2e^(0.4πi)+e^(1.2πi)的模
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求2+2e^(0.4πi)+e^(1.2πi)的模
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答案和解析
|2+2e^(0.4πi)+e^(1.2πi)|
=√[(2+2cos(2π/5)+cos(6π/5))²+(2sin(2π/5)+sin(6π/5))²]
=√[4+4+1+8cos(2π/5)+4cos(6π/5)+4cos(4π/5)]
=√[9+8cos(2π/5)-8cos(π/5)]
=√[9-16sin(3π/10)sin(π/10)]
=√[9-16sin(36°)sin(72°)]
=√[9-16cos(36°)cos(72°)sin(36°)/sin(36°)]
=√[9-4sin(144°)/sin(36°)]
=√5
=√[(2+2cos(2π/5)+cos(6π/5))²+(2sin(2π/5)+sin(6π/5))²]
=√[4+4+1+8cos(2π/5)+4cos(6π/5)+4cos(4π/5)]
=√[9+8cos(2π/5)-8cos(π/5)]
=√[9-16sin(3π/10)sin(π/10)]
=√[9-16sin(36°)sin(72°)]
=√[9-16cos(36°)cos(72°)sin(36°)/sin(36°)]
=√[9-4sin(144°)/sin(36°)]
=√5
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