百度作业网已知函数f（x）=2cos方x+2sinxcosx．记函数g(x)=f(x)*f(x+π/4) 求g(x)值域
来源：学生作业帮 编辑：百度作业网作业帮 分类：数学作业 时间：2024/07/31 15:27:53
已知函数f（x）=2cos方x+2sinxcosx．记函数g(x)=f(x)*f(x+π/4) 求g(x)值域
f(x)=2(cosx)^2+2sinxcosx=(cos2x+1)+sin2x=cos2x+sin2x+1
f(x+π/4)=cos(2x+π//2)+sin(2x+π//2)+1=sin2x-cos2x+1
g(x)=f(x)*f(x+π/4)=[cos2x+sin2x+1][sin2x-cos2x+1]=(sin2x+1)^2-(cos2x)^2
=(sin2x+1)^2-[1-(sin2x)^2]=2*(sin2x)^2-2sin2x-1=2(sin2x-1/2)^2-1/2
sin2x∈[-1,1] sin2x-1/2∈[-3/2,1/2]
g(x)∈[-1/2,4] g(x)的值域为[-1/2,4]
f(x+π/4)=cos(2x+π//2)+sin(2x+π//2)+1=sin2x-cos2x+1
g(x)=f(x)*f(x+π/4)=[cos2x+sin2x+1][sin2x-cos2x+1]=(sin2x+1)^2-(cos2x)^2
=(sin2x+1)^2-[1-(sin2x)^2]=2*(sin2x)^2-2sin2x-1=2(sin2x-1/2)^2-1/2
sin2x∈[-1,1] sin2x-1/2∈[-3/2,1/2]
g(x)∈[-1/2,4] g(x)的值域为[-1/2,4]
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