求函数y=sin²x+2sinxcosx+3cos²x的值域

求函数y=sin²x+2sinxcosx+3cos²x的值域
求函数y=sin²x+2sinxcosx+3cos²x的值域

求函数y=sin²x+2sinxcosx+3cos²x的值域
y=sin²x+cos8x+2sinxcosx+2cos²x
=1+sin2x+(1+cos2x)
=sin2x+cos2x+2
=√2sin(2x+π/4)+2
-1

y=sin²x+cos8x+2sinxcosx+2cos²x
=sin2x+cos2x+2
=√2sin(2x+π/4)+2
所以值域[-√2+2，√2+2]