Question

y＝sin²x＋2sinxcosx＋1的值域

Answer 1

y=sin²x+2sinxcosx+1

=[1-cos(2x)]/2 +sin(2x) +1

=sin(2x) -(1/2)cos(2x) +3/2

=√[1²+(-1/2)²]sin(2x -θ) +3/2 其中,tanθ=1/2

=(√5/2)sin(2x-θ) +3/2

-1≤sin(2x-θ)≤1

(3-√5)/2≤(√5/2)sin(2x-θ) +3/2≤(3+√5)/2

(3-√5)/2≤y≤(3+√5)/2

函数的值域为[(3-√5)/2,(3+√5)/2].