当 θ∈(0,π/2)时,求证sinθ +cosθ <π/2如题

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当 θ∈(0,π/2)时,求证sinθ +cosθ <π/2如题
当 θ∈(0,π/2)时,求证sinθ +cosθ <π/2
如题

当 θ∈(0,π/2)时,求证sinθ +cosθ <π/2如题
sinθ+cosθ
=√2(√2/2*sinθ+√2/2cosθ)
=√2(sinθcosπ/4+cosθsinπ/4)
=√2sin(θ+π/4)
θ∈(0,π/2)
π/4

sinθ+cosθ=√2*sin〔θ+(π/4)〕
∵ θ∈(0,π/2)
∴θ+(π/4)∈(π/4,3π/4)
∴sin〔θ+(π/4)〕∈(√2/2,1〕
∴sinθ+cosθ=√2*sin〔θ+(π/4)〕∈(1,√2〕
∴max[sinθ+cosθ]√2 <π/2
∴sinθ+cosθ<π/2