函数f(x)=2sin(x-4/π)-sin2x的值域为

题目

函数f(x)=2sin(x-4/π)-sin2x的值域为

答案

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】