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

题目

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

答案

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