作业帮求函数y=cosx(cosx+sinx)的值域
来源：学生作业帮 编辑：作业帮 分类：数学作业 时间：2024/05/15 15:04:46
求函数y=cosx(cosx+sinx)的值域
如题~
如题~
y=cosx(cosx+sinx)
=cos²x+sinxcosx
=(cos2x+1)/2+1/2·sin2x
=1/2·(sin2x+cos2x)+1/2
=1/2·√2(√2/2·sin2x+√2/2·cos2x)+1/2
=1/2·√2(cosπ/4·sin2x+sinπ/4·cos2x)+1/2
=1/2·√2sin(2x+π/4)+1/2
=√2/2·sin(2x+π/4)+1/2
∵sin(2x+π/4)∈【-1,1】
∴1/2-√2/2≤√2/2·sin(2x+π/4)+1/2≤1/2+√2/2
故值域为：【1/2-√2/2,1/2+√2/2】
再问： =cos²x+sinxcosx =(cos2x+1)/2+1/2·sin2x 这步是怎么变的？
再答： 根据cos2x=2cos²x-1 得cos²x=(co2x+1)/2 还有sin2x=2sinxcosx
=cos²x+sinxcosx
=(cos2x+1)/2+1/2·sin2x
=1/2·(sin2x+cos2x)+1/2
=1/2·√2(√2/2·sin2x+√2/2·cos2x)+1/2
=1/2·√2(cosπ/4·sin2x+sinπ/4·cos2x)+1/2
=1/2·√2sin(2x+π/4)+1/2
=√2/2·sin(2x+π/4)+1/2
∵sin(2x+π/4)∈【-1,1】
∴1/2-√2/2≤√2/2·sin(2x+π/4)+1/2≤1/2+√2/2
故值域为：【1/2-√2/2,1/2+√2/2】
再问： =cos²x+sinxcosx =(cos2x+1)/2+1/2·sin2x 这步是怎么变的？
再答： 根据cos2x=2cos²x-1 得cos²x=(co2x+1)/2 还有sin2x=2sinxcosx