]已知f(x)=4cos^4x-2cos^2x-1/tan(π/4+x)sin^2(π/4-x) (1),若tan(π/4+x)=3,求f(x)的值(2)当x∈[0,π/2]时,求g(x)=1/2f(x)+sin2x的最大值
来源:学生作业帮助网 编辑:作业帮 时间:2024/06/28 04:11:28
![]已知f(x)=4cos^4x-2cos^2x-1/tan(π/4+x)sin^2(π/4-x) (1),若tan(π/4+x)=3,求f(x)的值(2)当x∈[0,π/2]时,求g(x)=1/2f(x)+sin2x的最大值](/uploads/image/z/11465516-20-6.jpg?t=%5D%E5%B7%B2%E7%9F%A5f%28x%29%3D4cos%5E4x-2cos%5E2x-1%2Ftan%28%CF%80%2F4%2Bx%29sin%5E2%28%CF%80%2F4-x%29+%281%29%2C%E8%8B%A5tan%28%CF%80%2F4%2Bx%29%3D3%2C%E6%B1%82f%28x%29%E7%9A%84%E5%80%BC%EF%BC%882%EF%BC%89%E5%BD%93x%E2%88%88%5B0%2C%CF%80%2F2%5D%E6%97%B6%2C%E6%B1%82g%28x%29%3D1%2F2f%28x%29%2Bsin2x%E7%9A%84%E6%9C%80%E5%A4%A7%E5%80%BC)
]已知f(x)=4cos^4x-2cos^2x-1/tan(π/4+x)sin^2(π/4-x) (1),若tan(π/4+x)=3,求f(x)的值(2)当x∈[0,π/2]时,求g(x)=1/2f(x)+sin2x的最大值
]已知f(x)=4cos^4x-2cos^2x-1/tan(π/4+x)sin^2(π/4-x) (1),若tan(π/4+x)=3,求f(x)的值
(2)当x∈[0,π/2]时,求g(x)=1/2f(x)+sin2x的最大值
]已知f(x)=4cos^4x-2cos^2x-1/tan(π/4+x)sin^2(π/4-x) (1),若tan(π/4+x)=3,求f(x)的值(2)当x∈[0,π/2]时,求g(x)=1/2f(x)+sin2x的最大值
tan(π/4+x)=3,tan²(π/4+x)=9,cos²(π/4+x)=1/10,sin²(π/4+x)=9/10,cos(π/2+2x)=2*1/10-1=-4/5,sin2x=4/5 ,sin²2x=16/25,4sin²x*cos²x=16/25,sin²x*cos²x=4/25,(1-cos²x)*cos²x=4/25, sin^2(π/4-x)=1/2(1-sin2x)=1/10
f(x)=4cos^4x-2cos^2x-1/tan(π/4+x)sin^2(π/4-x) =2cos^4x-2(1-cos²x)*cos²x-1/3*1/10
=(1-2sin²2x)²-8/25-3/10=49/625-200/625-3/10=-151/625-3/10=-1031/3750
(2)当x∈[0,π/2]时,求g(x)=1/2f(x)+sin2x的最大值
g(x)=1/2f(x)+sin2x=2cos^4x-cos^2x-1/2tan(π/4+x)sin^2(π/4-x)+sin2x