函数y=sinxcosx-sinx-cosx的值域是＿＿＿＿？

来源：百度知道 编辑：UC知道 时间：2024/06/22 06:29:50
要详细过程

y=sinxcosx-sinx-cosx
= (1/2)* [(sinx + cosx)^2 - (sinx)^2 - (cosx)^2] - (sinx + cosx)
= (1/2)* [ (sinx + cosx)^2 - 1] - (sinx + cosx)
= (1/2)* (sinx + cosx)^2 - (sinx + cosx) - 1/2
= (1/2) * [(sinx + cosx)^2 - 2(sinx + cosx) + 1 - 1] - 1/2
= (1/2) * [ (sinx + cosx -1)^2 -1] -1/2
= (1/2)* (sinx + cosx -1)^2 -1

其中
sinx + cosx
= √2*[(√2/2)*sinx + (√2/2)*cosx]
=√2 *(sinx * cos45 + cosx *sin45)
= √2 * sin(x+45)
∈[-√2, √2]

sinx + cosx -1 ∈ [-√2 -1 , √2 -1]
(sinx + cosx -1)^2 ∈ [ 0, 3 + 2√2]

原式 ∈ (1/2)* [ 0, 3 + 2√2] -1
= [-1, (1+2√2)/2]

假设sinx+cosx=√2sin(x+∏/4)=t,-√2≤t≤√2
sinxcosx=(t^2-1)/2
y=1/2*(t-1)^2-1
ymin=-1,(t=1)
ymax=(1+2√2)/2,(t=-√2)
-1≤y≤(1+2√2)/2