已知(sinα-sinβ)/[sin(α-β)]=a,(cosα-cosβ)/[sin(α+β)]=b,求sin(α-β

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  • (sinα-sinβ)/[sin(α-β)]=a,.(1)

    (cosα-cosβ)/[sin(α+β)]=b,.(2)

    (1)*(2)得

    [sinαcosα-sinαcosβ-cosαsinβ+sinβcosβ]/sin(α-β)sin(α+β)=ab

    [1/2(sin2α+sin2β)-sin(α+β)]/sin(α-β)sin(α+β)=ab

    [sin(α+β)cos(α-β)-sin(α+β)]/sin(α-β)sin(α+β)=ab

    sin(α+β)[cos(α-β)-1]/sin(α-β)sin(α+β)=ab

    [cos(α-β)-1]/sin(α-β)=ab

    cos(α-β)-1=absin(α-β)

    cos^2(α-β)=[1+absin(α-β)]^2

    1-sin^2(α-β)=1+2absin(α-β)+(ab)^2sin^2(α-β)

    sin(α-β){[(1+(ab)^2]sin(α-β)+2ab}=0

    sin(α-β)不等于0

    [1+(ab)^2]sin(α-β)+2ab=0

    sin(α-β)= -2ab/[1+(ab)^2]