α+β=2π/3,则y=sin^2α+cos^2β的最小值

题目

α+β=2π/3,则y=sin^2α+cos^2β的最小值

答案

y=(sinα)^2+(cosβ)^2=(1-cos2α)/2+(1+cos2β)/2=1+(cos2β-cos2α)/2=1-sin(β+α)sin(β-α) (和差化积)=1+sin(α+β)sin(α-β)=1+sin(2π/3)sin(α-β)=1+(根号3)/2*sin(α-β)当sin(α-β)=-1时,y有最小值1-(...