已知函数f(x)=负根号3sin^2x+sinxcosx,设a属于(0,π),f(a/2)=(1/4)-(根号3/2),求sina的值
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![已知函数f(x)=负根号3sin^2x+sinxcosx,设a属于(0,π),f(a/2)=(1/4)-(根号3/2),求sina的值](/uploads/image/z/2523956-68-6.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%3D%E8%B4%9F%E6%A0%B9%E5%8F%B73sin%5E2x%2Bsinxcosx%2C%E8%AE%BEa%E5%B1%9E%E4%BA%8E%EF%BC%880%2C%CF%80%EF%BC%89%2Cf%28a%2F2%29%3D%EF%BC%881%2F4%EF%BC%89-%EF%BC%88%E6%A0%B9%E5%8F%B73%2F2%EF%BC%89%2C%E6%B1%82sina%E7%9A%84%E5%80%BC)
已知函数f(x)=负根号3sin^2x+sinxcosx,设a属于(0,π),f(a/2)=(1/4)-(根号3/2),求sina的值
已知函数f(x)=负根号3sin^2x+sinxcosx,
设a属于(0,π),f(a/2)=(1/4)-(根号3/2),求sina的值
已知函数f(x)=负根号3sin^2x+sinxcosx,设a属于(0,π),f(a/2)=(1/4)-(根号3/2),求sina的值
将x=a/2带入f(x)=负根号3sin^2x+sinxcosx中,
得,-根号3sin^2a/2+sina/2cosa/2=1/4-根号3/2
∵2sina/2cosa/2=sina
1-2sin^a/2=cosa
∴ (根号3-2根号3sin^2a/2)+(2sina/2cosa/2=1/2-根号3+根号3
cosa=(1/2-sina)÷根号3
根据:cos^2a+sin^2a=1 (1)
将cosa=(1/2-sina)÷根号3带入(1)中得
sina= (1-3根号5)÷8
sina=(1+3根号5)÷8
∵sina>0
∴sina=(1+3根号5)÷8
已知函数f(x)=-(√3)sin²x+sinxcosx,设α∈(0,π),f(a/2)=(1/4)-(√3/2),求sina的值
f(x)=-(√3/2)(1-cos2x)+(1/2)sin2x=sin2xcos(π/3)+cos2xsin(π/3)-√3/2=sin(2x+π/3)-√3/2
f(α/2)=sin(α-π/3)-√3/2=(1/4)-(√3/2)
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已知函数f(x)=-(√3)sin²x+sinxcosx,设α∈(0,π),f(a/2)=(1/4)-(√3/2),求sina的值
f(x)=-(√3/2)(1-cos2x)+(1/2)sin2x=sin2xcos(π/3)+cos2xsin(π/3)-√3/2=sin(2x+π/3)-√3/2
f(α/2)=sin(α-π/3)-√3/2=(1/4)-(√3/2)
故sin(α-π/3)=1/4,α=π/3+arcsin(1/4),
∴sinα=sin[π/3+arcsin(1/4)]=sin(π/3)cosarcsin(1/4)+cos(π/3)sinarcsin(1/4)
=(√3/2)√(1-1/16)+(1/2)(1/4)=√45/8+1/8=(3√5+1)/8
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