設α為銳角,求證:sinα+cosα

設α為銳角,求證:sinα+cosα

α∈(0,π/2)
α+π/4∈(π/4,3π/4)
sinα+cosα
=√2(√2/2*sinα+√2/2*cosα)
=√2(sinαcosπ/4+cosαsinπ/4)
=√2sin(α+π/4)
sin(α+π/4)∈[√2/2,√2]
即sinα+cosα∈[√2/2,√2]
因為√2