三角函数套公式题20.把下列各式化成y=Asin(ωx+Φ)或y=Acos(ωx+Φ)的形式（1）y=(sinx+cosx)^2+2(cos^2)x(2)y=sinx(sinx+√3cos)(x∈R)（3）f(x)=sin(2x+[π/3])-√3(sin^2)x+sinxcosx+(√3/2)

Question

三角函数套公式题
20.把下列各式化成y=Asin(ωx+Φ)或y=Acos(ωx+Φ)的形式
（1）y=(sinx+cosx)^2+2(cos^2)x
(2)y=sinx(sinx+√3cos)(x∈R)
（3）f(x)=sin(2x+[π/3])-√3(sin^2)x+sinxcosx+(√3/2)

Answer 1

20.把下列各式化成y=Asin(ωx+Φ)或y=Acos(ωx+Φ)的形式
（1）y=(sinx+cosx)^2+2(cos^2)x
=sin²x+cos²x+2sinxcosx+2cos²x
=1+sin2x+1+cos2x
=2+√2sin(2x+π/4)
(2)y=sinx(sinx+√3cos)(x∈R)
=sin²x+√3sinxcosx
=1/2*(1-cos2x)+√3/2*sin2x
=√3/2sin2x-1/2*cos2x+1/2
=sin(2x-π/6)+1/2
（3）f(x)=sin(2x+[π/3])-√3(sin^2)x+sinxcosx+(√3/2)
=sin2xcosπ/3+cos2xsinπ/3+1/2sin2x-√3/2(1-cos2x)+√3/2
=1/2*sin2x+√3/2*cos2x+1/2sin2x-√3/21+√3/2 cos2x+√3/2
=sin2x+√3cos2=2(1/2sin2x+√3/2*cos2x)
=2sin(2x+π/3)