△ABC中求证sin^A+sin^B+sin^c≤9/4RT

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△ABC中求证sin^A+sin^B+sin^c≤9/4RT

△ABC中求证sin^A+sin^B+sin^c≤9/4RT
△ABC中求证sin^A+sin^B+sin^c≤9/4
RT

△ABC中求证sin^A+sin^B+sin^c≤9/4RT
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