要使sinx－√¯3cosx＝﹙m－1﹚／m有意义,则m范围为＿＿?

Question

要使sinx－√¯3 cosx＝﹙m－1﹚／m有意义,则m范围为＿＿?

Answer 1

原式可化为
2(1/2×sinx-√3/2×cosx)=(m-1)/m
即 2sin(x-π/3)=(m-1)/m
易知,sin(x-π/3)∈[-1,1]
∴ (m-1)/m=2sin(x-π/3)∈[-2,2]
则有
{m≠0
{-2≤(m-1)/m≤2……*
①当m＞0时,*式可化为
{-2m≤m-1
{m-1≤2m
即m≥1/3,m≥-1
∴m≥1/3
②当m＜0时,*式可化为
{-2m≥m-1
{m-1≥2m
即m≤1/3,m≤-1
∴m≤-1
综上,m的范围是(-∞,-1]∪[1/3,+∞)