化简 √2+2sin2x ( "√"代表根号)

解答：

首先看根号里面

2+2sin2x=2+4sinxcosx=2(sinx^2+cosx^2+2sinxcosx)=2(sinx+cosx)^2

于是根号2+2sin2x=根号2*|sinx+cosx|=根号2*根号2*|sin(x+45度）|
=2*|sin(x+45度）|

则
当-45+360k<=x<=135+360k时 根号2+2sin2x=2sin(x+45度)
当135+360k<=x<=315+360k时 根号2+2sin2x=-2sin(x+45度)

√(2+2sin2x)
=√(2(1+sin2x)))
=√(2(sin²x+2sinxconx+cos²x))
=√(2(sinx+cosx)²)
=√(2(√2(sin²(x+π/4))
=2√sin²(x+π/4)
=2|sin(x+π/4)|

√2+2sin2x

=√(2+4sinxcosx)

=√(sin^2x+2sinxcosx+cos^2x+sin^2x+2sinxcosx+cos^2x)

=√[2(sinx+cosx)^2]

=(sinx+cosx)*√2