函数f(x)=2sin(x-4/π)-sin2x的值域为

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函数f(x)=2sin(x-4/π)-sin2x的值域为

函数f(x)=2sin(x-4/π)-sin2x的值域为
函数f(x)=2sin(x-4/π)-sin2x的值域为

函数f(x)=2sin(x-4/π)-sin2x的值域为
f(x)=2sin(x-4/π) - sin2x
=2sin（x-π/4）- sin2x
=2sin（x-π/4）- cos(2x-π/2)
=2sin（x-π/4）- cos[2（x-π/4）]
=2sin（x-π/4）- 1+2sin^2（x-π/4）
=2sin^2（x-π/4）+ 2sin（x-π/4）- 1
=2[sin（x-π/4）+1/2]^2-3/2
-1 ≤ sin（x-π/4）≤ 1
-1/2 ≤ sin（x-π/4）+1/2 ≤ 3/2
0 ≤ [sin（x-π/4）+1/2]^2 ≤ 9/4
0 ≤ 2[sin（x-π/4）+1/2]^2 ≤ 9/2
-3/2 ≤ 2[sin（x-π/4）+1/2]^2-3/2 ≤ 3
值域【-3/2,3】

因为1-sin2x=(sinx-cosx）^2=2sin^2(x-
π/4），所以f(x)+1=2sin(x-π/4)-sin2x+1=2（sin(X-π/4)+0.5）^2-0.5
又sin(x-π/4)范围为[-1，1],所以f(x)+1值域为[-0.5，4]所以f(x)值域为[-1.5,3]

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