y=sin(2x-π/6)-3cos(2x-π/6)的值域

题目

y=sin(2x-π/6)-3cos(2x-π/6)的值域

答案

y=sin(2x-π/6)-3cos(2x-π/6)
=√10 [1/√10 sin(2x-π/6)-3/√10 cos(2x-π/6)] 【令cosα=1/√10,则3/√10=sinα】
=√10 [cosα sin(2x-π/6)-sinα cos(2x-π/6)]
=√10 sin(2x-π/6-α)
因为sin(2x-π/6-α)∈[1,1]
所以y=√10sin(2x-π/6-α)∈[-√10,√10]
答案：值域为[-√10,√10]