已知三角形ABC中,a,b,c分别是角A,B,C的对边,且sin^2B+sin^2C-sinBsinC=sin^2A,a

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  • sin^2B+sin^2C-sinBsinC=sin^2A,由正弦定理得

    b²+c²-bc=a²

    b²+c²-a²=bc=2bccosA

    解得cosA=1/2

    A=π/3

    正弦定理a/2R=sinA

    R=1

    ∴BC=√3

    方法一:向量AB•向量AC=AB·AC·cosA

    cosA为定值

    当AB=AC时值最大,为[√(AB²+AC²)]/2

    向量AB•向量AC=AB·AC·cosA

    =√3×√3×1/2

    =3/2

    方法二:向量AB•向量AC=AB·AC·cosA

    =2RsinC·2RsinB·cosA

    =4R²sinC·sinB·cosA

    4R²是定值,cosA是定值.

    ∴求向量AB•向量AC的最大值就是求sinC·sinB的最大值.

    ∴sinC·sinB=sinC·sin(2π/3-C)

    =sinC(√3/2cosC+1/2sinC)

    =√3/2sinCcosC+1/2sin²C

    =√3/4sin2C+1/2[(1-cos2C)/2]

    =√3/4sin2C+1/4-1/4cos2C

    =√3/4sin2C-1/4cos2C+1/4

    =1/2(√3/2sin2C-1/2cos2C)+1/4

    =1/2(cosπ/6sin2C-sinπ/6cos2C)+1/4

    =1/2sin(2C-π/6)+1/4

    当C=π/3时,sin(2C-π/6)最大,1/2sin(2C-π/6)+1/4最大,为3/4

    向量AB•向量AC=AB·AC·cosA

    =2RsinC·2RsinB·cosA

    =4R²sinC·sinB·cosA

    =4×3/4×1/2

    =3/2