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】
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