若三角形ABC三个内角A,B,C依次成等差数列,且U=cosA+2icos平方C/2,求|U-i|的取值范围

若三角形ABC三个内角A,B,C依次成等差数列,且U=cosA+2icos平方C/2,求|U-i|的取值范围... 若三角形ABC三个内角A,B,C依次成等差数列,且U=cosA+2icos平方C/2,求|U-i|的取值范围 展开
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由已知,设A=B-α,C=B+α,则A+B+C=3B=π
所以:B =π/3
U=cosA+2icos²(C/2)
=cosA+i(1+cosC)
=cosA+i+icosC

|U-i|=|cosA+icosC|
=√(cos²A+cos²C)
=√(2+cos2A+cos2C)/2
=√[1+cos(A+C)cos(A-C)]
=√[1+cos(2π/3)cos(A-C)]
=√[1-(cos2α)/2]
显然-π/3<α<π/3,-1/2<cos2α≤1
1/2≤1-(cos2α)/2<5/4
√2/2≤|U-i|<√5/2
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