sinα+cosα+3=√2[(√2/2)sinα+(√2/2)cosα]+3
=√2sin(α+π/4)+3
∵-1≤sin(α+π/4)≤1
-√2≤√2sin(α+π/4)≤√2
3-√2≤√2sin(α+π/4)+3≤3+√2
∵3-√2>0
∴sinα+cosα+3>0