2sinθ+cosθ/sinθ-3cosθ=-5, - 知识问答

2sinθ+cosθ/sinθ-3cosθ=-5,求cos2θ+4sinθ

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已知(2sinθ+cosθ)/(sinθ－3cosθ)＝－5,求3cos2θ+4sin2θ的值
∵(2sinθ+cosθ)/(sinθ－3cosθ)＝－5
∴(2tanθ+1)/(tanθ－3)＝－5,解得tanθ＝2
故3cos2θ＋4sin2θ＝3×[(1－tan²θ)/(1+tan²θ)]＋4×[(2tanθ)/(1+tan²θ)]
＝3×[(1－2²)/(1+2²)]＋4×[(2×2)/(1+2²)]
＝7/5.

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