证明 SIN²A+SIN²B-SIN²C=2SINASINBCOSC

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  • A, B, C是三角形的内角吗?

    = = = = = = = = =

    证明:设ΔABC的外接圆半径为R.

    由正弦定理,

    a /sin A =b /sin B =c /sin C =2R,

    所以 a =2R sin A,

    b =2R sin B,

    c =2R sin C.

    由余弦定理,

    cos C =(a^2 +b^2 -c^2) /(2ab)

    =[ (4R^2) (sin A)^2 +(4R^2) (sin B)^2 -(4R^2) (sin C)^2 ] / (8R^2 sin A sin B)

    =[ (sin A)^2 +(sin B)^2 -(sin C)^2 ] / (2 sin A sin B).

    所以 (sin A)^2 +(sin B)^2 -(sin C)^2 =2 sin A sin B cos C.

    = = = = = = = = =

    下次提问时,选好分类.