设α为锐角,求证:sinα+cosα

问题描述:

设α为锐角,求证:sinα+cosα

平方
1+sin2a(π/2)方>2 得证

sinα+cosα=√2sin(a+π/4)
α为锐角, π/4√2/21sinα+cosα最大值=√2所以
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]
因为√2

sinα+cosα = √2(sinαcos45°+cosαsin45°) = √2sin(α+45°) ≤√2