已知:tanA+1/tanA=5/2,A∈(π/4,π/2),求cos2A和sin(2A+π/4)的值

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已知:tanA+1/tanA=5/2,A∈(π/4,π/2),求cos2A和sin(2A+π/4)的值

已知:tanA+1/tanA=5/2,A∈(π/4,π/2),求cos2A和sin(2A+π/4)的值
已知:tanA+1/tanA=5/2,A∈(π/4,π/2),求cos2A和sin(2A+π/4)的值

已知:tanA+1/tanA=5/2,A∈(π/4,π/2),求cos2A和sin(2A+π/4)的值
∵tanA=sinA/cosA
∴1/tanA=cosA/sinA
∴tanA+1/tanA=sinA/cosA+cosA/sinA=(sin²A+cos²A)/(sinAcosA)
=1/(1/2sin2A)
=2/sin2A=5/2
∴sin2A=4/5
∵A∈(π/4,π/2)
∴2A∈(π/2,π)
∴cos2A<0
∴cos²2A=1-sin²2A
解得,cos2A=-3/5
∵sin2A=4/5
∴sin(2A+π/4)=sin2Acosπ/4+cos2Asinπ/4
=4/5×(✔2)/2-3/5(✔2)/2
=1/5×(✔2)/2
=(✔2)/10