已知-∏/6≤B≤∏/4,3sin^2 a-2sin^2 b=2sina,求sin^2 b-1/2sina的最小值.

∵-π/6≤b≤π/4
∴-1/2≤sinb≤ √2/2
3sin² a-2sin² b=2sina
∴sin²b=3sin²a /2 - sina
=(3/2)(sina-1/3)²-1/6∈[-1/2,√2/2]
∴2/3≤sina≤1或-1/3≤sina≤0
sin² b-1/2sina=3sin²a /2 - sina-1/2sina
=3sin²a/2-3sina/2
=3/2(sina-1/2)²-3/8
∴-1/3≤3/2(sina-1/2)²-3/8≤0或0≤3/2(sina-1/2)²-3/8≤2/3
即：-1/3≤sin² b-1/2sina≤0或0≤sin² b-1/2sina≤2/3
∴sin² b-1/2sina的最小值为-1/3.