Question

能使等式cosθ-根号3sinθ=(4m-6)/(4-m)成立的实数m的取值范围

Answer 1

cosθ-根号3sinθ
=2cos(θ+π/3）
-2<=2cos(θ+π/3）<=2
所以-2<=(4m-6)/(4-m)<=2

-2<=(4m-6)/(4-m)
(2m-3)/(4-m)+1>=0
(m+1)/(4-m)>=0
所以(m+1)(m-4)<=0
分母不等于0
-1<=m<4

(4m-6)/(4-m)<=2
(2m-3)/(4-m)-1<=0
(3m-7)/(4-m)<=0
(3m-7)(m-4)>=0
m<=7/3,m>4

综上
-1<=m<=7/3

Answer 2

cosθ-根号3sinθ=(4m-6)/(4-m)
2(sinπ/6cosθ-cosπ/6sinθ)=(4m-6)/(4-m)
2sin(π/6-θ)=(4m-6)/(4-m)
sin(π/6-θ)=(4m-6)/2(4-m)
所以：-1≤(4m-6)/2(4-m)≤1
-2≤(4m-6)/(m-4)≤2
解得：-1≤m≤7/3