已知向量a=(1+sin2x,sinx-cosx),b=(1,sinx+cosx)设函数f(x)=a·b(1)若x∈[π/4,π/2],求f(x)的最大值与最小值(2)若f(θ)=8/5,求sin4θ的值
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![已知向量a=(1+sin2x,sinx-cosx),b=(1,sinx+cosx)设函数f(x)=a·b(1)若x∈[π/4,π/2],求f(x)的最大值与最小值(2)若f(θ)=8/5,求sin4θ的值](/uploads/image/z/615983-23-3.jpg?t=%E5%B7%B2%E7%9F%A5%E5%90%91%E9%87%8Fa%3D%281%2Bsin2x%2Csinx-cosx%29%2Cb%3D%281%2Csinx%2Bcosx%29%E8%AE%BE%E5%87%BD%E6%95%B0f%28x%29%3Da%C2%B7b%281%29%E8%8B%A5x%E2%88%88%5B%CF%80%2F4%2C%CF%80%2F2%5D%2C%E6%B1%82f%28x%29%E7%9A%84%E6%9C%80%E5%A4%A7%E5%80%BC%E4%B8%8E%E6%9C%80%E5%B0%8F%E5%80%BC%282%29%E8%8B%A5f%28%CE%B8%29%3D8%2F5%2C%E6%B1%82sin4%CE%B8%E7%9A%84%E5%80%BC)
已知向量a=(1+sin2x,sinx-cosx),b=(1,sinx+cosx)设函数f(x)=a·b(1)若x∈[π/4,π/2],求f(x)的最大值与最小值(2)若f(θ)=8/5,求sin4θ的值
已知向量a=(1+sin2x,sinx-cosx),b=(1,sinx+cosx)
设函数f(x)=a·b
(1)若x∈[π/4,π/2],求f(x)的最大值与最小值
(2)若f(θ)=8/5,求sin4θ的值
已知向量a=(1+sin2x,sinx-cosx),b=(1,sinx+cosx)设函数f(x)=a·b(1)若x∈[π/4,π/2],求f(x)的最大值与最小值(2)若f(θ)=8/5,求sin4θ的值
f(x)=向量a.向量b.
=(1+sin2x)*1+(sinx-cosx)*(sinx+cosx).
=1+sin2x-(cos^2x-sin^2x).
=1+sin2x-cos2x.
=1+√2sin(2x-π/4).
∴f(x)=√2sin(2x-π/4)+1.
(1)∵x∈[π/4,π/2],∴(2x-π/4)∈[π/4,3π/4]
令2x-π/4=π/2,即x=3π/8时,sin(2x-π/4)=1,f(x)取得最大值,即f(x)max=1+√2.
当2x-π/4=π/4或2x-π/4=3π/4时,sin(2x-π/4)=√2/2,f(x)取得最小值,即f(x)min=2.
(2) 若f(θ)=8/5时,求sin4θ的值.
f(θ)=√2sin(2θ-π/4)+1=8/5.
√2sin(2θ-π/4)=3/5.
√2sin2θcos(π/4)-√2cos2θsin(π/4)=3/5.
sin2θ-cos2θ=3/5.
(sin2θ-cos2θ)^2=(3/5)^2.
sin^2(2θ)-2sin2θcos2θ+cos^2(2θ)=9/25.
1-sin4θ=9/25.
sin4θ=1-9/25.
∴sin4θ=16/25.