已知sin(A+π/4)=7根号2/10,A∈(π/4,π/2).求cosA的值,求函数f(x)=cos2x+5/2sinAsinx的值域复制搜索

来源:学生作业帮助网 编辑:作业帮 时间:2024/11/14 11:24:52

已知sin(A+π/4)=7根号2/10,A∈(π/4,π/2).求cosA的值,求函数f(x)=cos2x+5/2sinAsinx的值域复制搜索
已知sin(A+π/4)=7根号2/10,A∈(π/4,π/2).求cosA的值,
求函数f(x)=cos2x+5/2sinAsinx的值域
复制搜索

已知sin(A+π/4)=7根号2/10,A∈(π/4,π/2).求cosA的值,求函数f(x)=cos2x+5/2sinAsinx的值域复制搜索
sin(A+π/4)=7√2/10,
A属于(π/4,π/2)
cos(A+π/4)
=-√2/10
cosA=cos[(A+π/4)-π/4]
=cos(A+π/4)cosπ/4+sin(A+π/4)sinπ/4
=(-√2/10)*(√2/2)+(7√2/10)*(√2/2)=3/5
A∈(π/4,π/2),所以sinA=4/5,
f(x)=cos2x+5/2sinAsinx
= cos2x+2sinx
=1-2sin²x+2sinx
=-2(sinx-1/2)²+3/2
Sinx=1/2时,函数取到最大值3/2.
Sinx=-1时,函数取到最小值-3.
所以函数值域是[-3,3/2].

sin(A+π/4)=7√2/10,
A属于(π/4,π/2)
cos(A+π/4)
=-√2/10
cosA=cos[(A+π/4)-π/4]
=cos(A+π/4)cosπ/4+sin(A+π/4)sinπ/4
=(-√2/10)*(√2/2)+(7√2/10)*(√2/2)=3/5


A∈(π/4,π/2),所以...

全部展开

sin(A+π/4)=7√2/10,
A属于(π/4,π/2)
cos(A+π/4)
=-√2/10
cosA=cos[(A+π/4)-π/4]
=cos(A+π/4)cosπ/4+sin(A+π/4)sinπ/4
=(-√2/10)*(√2/2)+(7√2/10)*(√2/2)=3/5


A∈(π/4,π/2),所以sinA=4/5,
f(x)=cos2x+5/2sinAsinx
= cos2x+2sinx
=1-2sin²x+2sinx
=-2(sinx-1/2)²+3/2
sinx的值域是(-1,1)
Sinx=1/2时,函数取到最大值3/2.
Sinx=-1时,函数取到最小值-3.
所以函数值域是[-3,3/2].

收起