在△ABC中,角A满足sinA^4-cosA^4≤cosA-sinA,则角A的取值范围是?

在△ABC中,角A满足sinA^4-cosA^4≤cosA-sinA,则角A的取值范围是?
在△ABC中,角A满足sinA^4-cosA^4≤cosA-sinA,则角A的取值范围是?

在△ABC中,角A满足sinA^4-cosA^4≤cosA-sinA,则角A的取值范围是?
(sinA)^4-(cosA)^4≤cosA-sinA
(sin²A+cos²A)(sin²A-cos²A)≤cosA-sinA
sin²A-cos²A≤cosA-sinA
(sinA+cosA)(sinA-cosA)≤-(sinA-cosA)
(sinA-cosA)(sinA+cosA+1)≤0 ①
而sinA+cosA+1=√2sin(A+π/4)+1
当0

sinA^4-cosA^4≤cosA-sinA
(sin²A-cos²A)(sin²A+cos²A)≤cosA-sinA
sin²A-cos²A≤cosA-sinA
(sinA-cosA)(sinA+cosA)+(sinA-cosA)≤0
(sinA+cosA+1)(sinA-cosA)≤0

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sinA^4-cosA^4≤cosA-sinA
(sin²A-cos²A)(sin²A+cos²A)≤cosA-sinA
sin²A-cos²A≤cosA-sinA
(sinA-cosA)(sinA+cosA)+(sinA-cosA)≤0
(sinA+cosA+1)(sinA-cosA)≤0
∵ A是三角形内角，所以sinA>0,又1+cosA>0
∴ sinA+cosA+1>0
∴ sinA-cosA≤0
∵ sinA>0, ∴ cosA>0
∴ tanA≤1
∴ A∈ (0,π/4]

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