已知sin(A+派/4)=7根号2/10 A属于(0,派/4) 求函数f(x)=cos2x+5cosAcosx+1的值域
来源:学生作业帮助网 编辑:作业帮 时间:2024/07/17 12:02:33
![已知sin(A+派/4)=7根号2/10 A属于(0,派/4) 求函数f(x)=cos2x+5cosAcosx+1的值域](/uploads/image/z/13388563-19-3.jpg?t=%E5%B7%B2%E7%9F%A5sin%28A%2B%E6%B4%BE%2F4%29%3D7%E6%A0%B9%E5%8F%B72%2F10+A%E5%B1%9E%E4%BA%8E%EF%BC%880%2C%E6%B4%BE%2F4%EF%BC%89+%E6%B1%82%E5%87%BD%E6%95%B0f%28x%29%3Dcos2x%2B5cosAcosx%2B1%E7%9A%84%E5%80%BC%E5%9F%9F)
已知sin(A+派/4)=7根号2/10 A属于(0,派/4) 求函数f(x)=cos2x+5cosAcosx+1的值域
已知sin(A+派/4)=7根号2/10 A属于(0,派/4) 求函数f(x)=cos2x+5cosAcosx+1的值域
已知sin(A+派/4)=7根号2/10 A属于(0,派/4) 求函数f(x)=cos2x+5cosAcosx+1的值域
∵sin(A+派/4)=7根号2/10
∴sinA+cosA=7/5
2sinA*cosA=(sinA+cosA)²-1=49/25-1=24/25,sinA*cosA=12/25
联立上面两个式子,又A属于(0,派/4),所以cosA>sinA
cosA=4/5,sinA=3/5代入函数
f(x)=cos2x+5cosAcosx+1=cos2x+4cosx+1=2(cosx)²+4cosx=2cosx(cosx+2)
当cosx=-1时,取得最小值-2,当cosx=1时,取得最大值6,
所以,函数f(x)=cos2x+5cosAcosx+1的值域为[-2,6]
A属于(0,派/4) A+派/4属于(派/4,派/2)
sin(A+派/4)=7根号2/10 cos(A+派/4)=根号2/10
cosA=cos[(A+派/4)-派/4]
=cos(A+派/4)cos派/4+sin(A+派/4)sin派/4
=根号2/2(7根号2/10+根号2/10 )
=4/5
f(x)=cos2x+...
全部展开
A属于(0,派/4) A+派/4属于(派/4,派/2)
sin(A+派/4)=7根号2/10 cos(A+派/4)=根号2/10
cosA=cos[(A+派/4)-派/4]
=cos(A+派/4)cos派/4+sin(A+派/4)sin派/4
=根号2/2(7根号2/10+根号2/10 )
=4/5
f(x)=cos2x+5cosAcosx+1
=2cos^2x-1+4cosx+1
=2cos^2x+4cosx
=2(cos^2x+2cosx+1)-2
=2(cosx+1)^2-2
当cosx=-1时 fmin=-2
当cosx=1时 fmax=6
值域[-2,6]
收起