Question

已知实数x,y满足x^2+y^2=1,则x+y的取值范围是___;x-√3y的取值范围是___

Answer 1

令x=cosa y=sina
x+y=cosa+sina=√2sin(x+π/4)
-1≤sin(x+π/4)≤1
-√2≤x+y≤√2

x-√3y=cosa-√3sina=2[cosasin(π/6)-cos(π/6)sina]=2sin(π/6-a)
-1≤sin(π/6-a)≤1
-2≤x-√3y≤2

已知实数x,y满足x^2+y^2=1,则x+y的取值范围是( [-√2，√2] )；
x-√3y的取值范围是( [-2，2] )。

Answer 2

0<=(x+y)²<=2(x²+y²)=2
所以
-√2<=x+y<=√2
所以
x+y的取值范围是 [-√2,√2]

设 x=Sint y=cost
x-√3y
=sint-√3cost
=2sin(t-π/3)
所以
x-√3y的取值范围是 [-2,2]

Answer 3

已知实数x,y满足x^2+y^2=1,则x+y的取值范围是_-根号2到+根号2__;x-√3y的取值范围是___