集合与不等式

cos^2A+cos^2B+cos^2C=1,

3 = 1 + (sinA)^2 + (sinB)^2 + (sinC)^2,

2 = (sinA)^2 + (sinB)^2 + (sinC)^2 >= 3[sinAsinBsinC]^(2/3),

[sinAsinBsinC]^2 <= (2/3)^3

sinAsinBsinC <= {[sinAsinBsinC]^2}^(1/2) <= [2/3]^(3/2).

当 sinA = sinB = sinC = (2/3)^(1/2)时，
(cosA)^2 + (cosB)^2 + (cosC)^2 = 3[1 - 2/3] = 1

sinAsinBsinC = (2/3)^(3/2).

所以，
sinAsinBsinC 的最大值为 (2/3)^(3/2)。