求函数y=sinx+sin2x-cosx(x∈R)的值域

题目

求函数y=sinx+sin2x-cosx(x∈R)的值域

答案

y=sinx+sin(2x)-cosx
=(sinx-cosx)+cos(π/2 -2x)
=√2sin(x-π/4)+cos[2(x-π/4)] 注意：cos(π/2-2x)=cos(2x-π/2).
=√2sin(x-π/4)+1-2sin²(x-π/4) 后面一项用二倍角公式.
=-2[sin(x-π/4)-(√2/4)]² +5/4 配方
sin(x-π/4)=√2/4时,y有最大值ymax=5/4
sin(x-π/4)=-1时,y有最小值ymin=-1-√2
综上,得-1-√2≤y≤5/4,函数的值域为[-1-√2,5/4].