求证数学题,在三角形ABC中,求证sin^2(A)+sin^2(B)+sin^2(C)_数学解答

求证数学题,在三角形ABC中,求证sin^2(A)+sin^2(B)+sin^2(C)

人气：149 ℃　时间：2020-09-11 20:24:47

解答

sin^2A+sin^2B+sin^2C
=(1-cosA)/2 +(1-cosB)/2 +(1-cos^2C)
=2-cos(A+B)cos(A-B)-cos^2C
=2+cosCsoc(A-B)-cos^2C≤2+|cosC|-cos^2C=-(|cosC|-1/2)^2+9/4
当cosC=1/2时,（即A=B=C=60°）有最大值9/4
∴(sinA)^2+(sinB)^2+(sinc)^2≤9/4