已知向量a=(sinx,sinx+cosx)b=(2cosx,cosx-sinx),设f(x)=a*b(1)当x∈[0,π/2]时,求函数f(x)的值域;(2)[-π/8,0],f(θ)=2/5,求sin(2θ+3π/4)的值.
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![已知向量a=(sinx,sinx+cosx)b=(2cosx,cosx-sinx),设f(x)=a*b(1)当x∈[0,π/2]时,求函数f(x)的值域;(2)[-π/8,0],f(θ)=2/5,求sin(2θ+3π/4)的值.](/uploads/image/z/2524312-64-2.jpg?t=%E5%B7%B2%E7%9F%A5%E5%90%91%E9%87%8Fa%3D%28sinx%2Csinx%2Bcosx%29b%3D%282cosx%2Ccosx-sinx%29%2C%E8%AE%BEf%28x%29%3Da%2Ab%EF%BC%881%EF%BC%89%E5%BD%93x%E2%88%88%5B0%2C%CF%80%2F2%5D%E6%97%B6%2C%E6%B1%82%E5%87%BD%E6%95%B0f%28x%29%E7%9A%84%E5%80%BC%E5%9F%9F%EF%BC%9B%EF%BC%882%EF%BC%89%5B-%CF%80%2F8%2C0%5D%2Cf%28%CE%B8%29%3D2%2F5%2C%E6%B1%82sin%282%CE%B8%2B3%CF%80%2F4%29%E7%9A%84%E5%80%BC.)
已知向量a=(sinx,sinx+cosx)b=(2cosx,cosx-sinx),设f(x)=a*b(1)当x∈[0,π/2]时,求函数f(x)的值域;(2)[-π/8,0],f(θ)=2/5,求sin(2θ+3π/4)的值.
已知向量a=(sinx,sinx+cosx)b=(2cosx,cosx-sinx),设f(x)=a*b
(1)当x∈[0,π/2]时,求函数f(x)的值域;
(2)[-π/8,0],f(θ)=2/5,求sin(2θ+3π/4)的值.
已知向量a=(sinx,sinx+cosx)b=(2cosx,cosx-sinx),设f(x)=a*b(1)当x∈[0,π/2]时,求函数f(x)的值域;(2)[-π/8,0],f(θ)=2/5,求sin(2θ+3π/4)的值.
f(x)=a*b =2sinxcosx+(sinx+cosx)(cosx-sinx)=sin2x+cos2x=√2sin(2x+π/4)
1)当x∈[0,π/2]时 2x+π/4 ∈[π/4,π/4 +π] 当2x+π/4=π/2 f(x)取到最大值
当2x+π/4=-π/2时 f(x)取到最小值
所以f(x)的值域为 [-√2,√2]
2) θ ∈[-π/8,0] 则 2θ+3π/4∈[π/2,3π/4]
sin(2θ+3π/4)=sin(2θ+π/4+π/2)=cos(2θ+π/4)
又f(θ)=2/5=√2sin(2θ+π/4)
则 sin(2θ+π/4)=√2/5
所以cos(2θ+π/4) =√23/5=sin(2θ+3π/4)
f(x)=sian2x+cos2x=√(2)sin(
x[0,π/2].2x+π/4∈[π/4,5π/4],sin(2θ+3π/4)∈[--(√(2)/2),1],f(x)∈[-1,√(2)]