62问答网 > 设x∈[0,π⼀3],求函数y=cos(2x-π⼀3)+2sin(x-π⼀6)的最值

设x∈[0,π⼀3],求函数y=cos(2x-π⼀3)+2sin(x-π⼀6)的最值

2025-06-28 23:33:17

推荐回答（1个）

回答1：

y=cos(2x-π/3)+2sin(x-π/6)=cos^2（x-π/6）-sin^2（x-π/6）+2sin(x-π/6)=1-2sin^2（x-π/6）+2sin(x-π/6)=-2[sin^2（x-π/6）-sin(x-π/6)-1/2]=-2{[sin(x-π/6)-1/2]^2-3/4}
=-2[sin(x-π/6)-1/2]^2+3/2
因为sin(x-π/6)∈[-1/2，1/2] x∈[0，π/3]
所以-2[sin(x-π/6)-1/2]^2∈[-2,0]

所以y=cos(2x-π/3)+2sin(x-π/6)∈[-1/2,3/2]