已知在三角形ABC中,向量m=（2cosC/2,-sin(A+B)）,n=（cosC/2,2sin(A+B)）且m垂直于n.,若a^2=b^2+1/2c^2,求sin(A-B)

已知在三角形ABC中,向量m=（2cosC/2,-sin(A+B)）,n=（cosC/2,2sin(A+B)）且m垂直于n.,若a^2=b^2+1/2c^2,求sin(A-B)
已知在三角形ABC中,向量m=（2cosC/2,-sin(A+B)）,n=（cosC/2,2sin(A+B)）且m垂直于n.,若a^2=b^2+1/2c^2,求sin(A-B)

已知在三角形ABC中,向量m=（2cosC/2,-sin(A+B)）,n=（cosC/2,2sin(A+B)）且m垂直于n.,若a^2=b^2+1/2c^2,求sin(A-B)
由向量m垂直于n得
（2cosC/2,-sin(A+B)）*（cosC/2,2sin(A+B)）=0
(cosC/2)^2=sin(A+B)^2=sinC^2
(1+cosC)/2=1-(cosC)^2
解得 cosC=1/2,C=60°
由a^2=b^2+1/2c^2得
c^2=2(a^2-b^2)
( sinC)^2=2(( sinA)^2-( sinB)^2)
( sinC)^2=2sin(A+B)sin(A-B)
sinC=2sin(A-B)
所以 sin(A-B)=√3/4