y=(sinx+cosx)^2+2cos^2x怎么变成y=sin(ωx+ρ)的形式?

题目

y=(sinx+cosx)^2+2cos^2x怎么变成y=sin(ωx+ρ)的形式?

答案

y=sin²x+cos²x+2sinxcosx+(2cos²x-1)+1
=1+sin2x+cos2x+1
=√2(√2/2*sin2x+√2/2cos2x)+2
=√2(sin2xcosπ/4+cos2xsinπ/4)+2
=√2sin(2x+π/4)+2