已知函数f(x)=(1+1/tanx)sin(x)^2 -2sin(x+π/4)*cos(x+π/4) (1)若tanα=2,求f(α) (2)若x∈[π/12,π/2],求f(x)的取值范围
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![已知函数f(x)=(1+1/tanx)sin(x)^2 -2sin(x+π/4)*cos(x+π/4) (1)若tanα=2,求f(α) (2)若x∈[π/12,π/2],求f(x)的取值范围](/uploads/image/z/2738934-54-4.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%3D%281%2B1%2Ftanx%29sin%28x%29%5E2+-2sin%28x%2B%CF%80%2F4%29%2Acos%28x%2B%CF%80%2F4%29+%281%29%E8%8B%A5tan%CE%B1%3D2%2C%E6%B1%82f%28%CE%B1%29+%282%29%E8%8B%A5x%E2%88%88%5B%CF%80%2F12%2C%CF%80%2F2%5D%2C%E6%B1%82f%28x%29%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4)
已知函数f(x)=(1+1/tanx)sin(x)^2 -2sin(x+π/4)*cos(x+π/4) (1)若tanα=2,求f(α) (2)若x∈[π/12,π/2],求f(x)的取值范围
已知函数f(x)=(1+1/tanx)sin(x)^2 -2sin(x+π/4)*cos(x+π/4)
(1)若tanα=2,求f(α)
(2)若x∈[π/12,π/2],求f(x)的取值范围
已知函数f(x)=(1+1/tanx)sin(x)^2 -2sin(x+π/4)*cos(x+π/4) (1)若tanα=2,求f(α) (2)若x∈[π/12,π/2],求f(x)的取值范围
先用tanx=sinx/cosx、倍角公式、诱导公式化简原函数:
f(x)=sin²x+sinxcosx-sin[2(x+π/4)]=(1-cos2x)/2+1/2sin2x-sin(2x+π/2)=-3/2cos2x+1/2sin2x+1/2
(1)由万能公式,cos2x=(1-tan²x)/(1+tan²x),sin2x=2tanx/(1+tan²x)
f(α)=-3/2*(1-2²)/(1+2²)+1/2*2*2/(1+2²)=13/10
(2)为求f(x)范围,再用辅助角公式化简原函数:
f(x)=根号10/2sin(2x-arctan3)+1/2
若x∈[π/12,π/2]:f(x)最大值为(根号10)/2+1/2
f(x)最小值为f(π/12)=-3/2cos(π/6)+1/2sin(π/6)+1/2=-(3根号3)/4+3/4