设α为锐角,求证:sinα+cosα
问题描述:
设α为锐角,求证:sinα+cosα
答
平方
1+sin2a(π/2)方>2 得证
答
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]
因为√2
答
sinα+cosα = √2(sinαcos45°+cosαsin45°) = √2sin(α+45°) ≤√2