Question

函数y=(2-sinx)/(2+cosx)的值域为

Answer 1

y = (2-sinx)/(2+cosx)
2y + ycosx = 2 - sinx
sinx + ycosx = 2 - 2y
√(1+y²)sin(x + L) = 2 - 2y
sin(x + L) = (2 - 2y)/√(1 + y²)
-1 ≤ sin(x + L) ≤ 1
-1 ≤ (2 - 2y)/√(1 + y²) ≤ 1
(1/3)(4 - √7) ≤ y ≤ (1/3)(4 + √7)

∴y的值域为[(1/3)(4 - √7)，(1/3)(4 + √7)]

Answer 2

2y+ycosx=2-sinx
sinx+ycosx=2-2y
(1+y^2)^0.5sin(x+u)=2-2y, u=arctanu
-1<=(2-2y)/(1+y^2)^0.5<=1
4(y-1)^2<=1+y^2
3y^2-8y+3<=0
(4-^7)/3<=y<=(4+^7)/3