Question

证明 SIN²A+SIN²B-SIN²C=2SINASINBCOSC

Answer 1

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。