化简 √2+2sin2x ( "√"代表根号)

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化简 √2+2sin2x ( "√"代表根号)
化简 √2+2sin2x ( "√"代表根号)

化简 √2+2sin2x ( "√"代表根号)

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)|

当-45+360k

√2+2sin2x
=√(2+4sinxcosx)
=√(sin^2x+2sinxcosx+cos^2x+sin^2x+2sinxcosx+cos^2x)
=√[2(sinx+cosx)^2]
=(sinx+cosx)*√2

√(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)|