α,β是锐角,α+β=2π/3,cos(α-β)的取值范围是

来源:学生作业帮助网 编辑:六六作业网 时间:2024/12/26 09:31:50
α,β是锐角,α+β=2π/3,cos(α-β)的取值范围是α,β是锐角,α+β=2π/3,cos(α-β)的取值范围是α,β是锐角,α+β=2π/3,cos(α-β)的取值范围是本题建议楼主使用线性

α,β是锐角,α+β=2π/3,cos(α-β)的取值范围是
α,β是锐角,α+β=2π/3,cos(α-β)的取值范围是

α,β是锐角,α+β=2π/3,cos(α-β)的取值范围是
本题建议楼主使用线性规划来解决

α=2π/3-β
α-β=2π/3-2β
0<β<π/2
0<2β<π
-π<-2β<0
-π/3<2π/3-2β<2π/3
-1/2≤cos(α-β)≤1
所以cos(α-β)的取值范围是:
[-1/2,1]

∵α,β是锐角
∴0<α<π/2,0<β<π/2
∵α+β=2π/3
∴α=2π/3-β
∴0<2π/3-β<π/2
∴-π/2<-β<-π/6
同理可得π/6<α<π/2
∴-π/2+π/6<α-β<π/2-π/6
∴-π/3<α-β<π/3
∴1/2<cos(α-β)≤1

cos(a-b)=cos(2a-2Pi/3)
因为0所以:pi/6由余弦函数图象可知:1/2