求直线y= 2sin(2x+π/6)+2sin(2x-π/3)与直线y=根号6的图像在（0,π）上所在交点的坐标
来源：学生作业帮助网 编辑：六六作业网 时间：2024/11/28 00:14:33
求直线y= 2sin(2x+π/6)+2sin(2x-π/3)与直线y=根号6的图像在（0,π）上所在交点的坐标
求直线y= 2sin(2x+π/6)+2sin(2x-π/3)与直线y=根号6的图像在（0,π）上所在交点的坐标
求直线y= 2sin(2x+π/6)+2sin(2x-π/3)与直线y=根号6的图像在（0,π）上所在交点的坐标
y=2sin(2x+π/6)+2sin(2x-π/3)
=2sin(2x-π/3+π/2)+2sin(2x-π/3)
=2cos(2x-π/3)+2sin(2x-π/3)
=2√2[sin(2x-π/3)cos(π/4)+cos(2x-π/3)sin(π/4)]
=2√2sin(2x-π/3+π/4)=√6
所以sin(2x-π/12)=√3/2=sin(π/3)
∵x∈(0,π)
∴2x-π/12∈(-π/12,2π-π/12)
∴2x1-π/12=π/3,2x2-π/12=2π/3,
∴x1=5π/24,x2=9π/24
所以交点有2个,分别是(5π/24,√6),(9π/24,√6)
用和差化积公式sin α+sin β=2sin[(α+β)/2]·cos[(α-β)/2]
y= 2sin(2x+π/6)+2sin(2x-π/3)
=4 sin(2x-π/12) cos π/4
=2√2 sin(2x-π/12)
两条线交点处2√2 sin(2x-π/12)=√6
sin(2x-π/12)=√3 /2
因为x∈(0,π)，故(2...
全部展开
用和差化积公式sin α+sin β=2sin[(α+β)/2]·cos[(α-β)/2]
y= 2sin(2x+π/6)+2sin(2x-π/3)
=4 sin(2x-π/12) cos π/4
=2√2 sin(2x-π/12)
两条线交点处2√2 sin(2x-π/12)=√6
sin(2x-π/12)=√3 /2
因为x∈(0,π)，故(2x-π/12)∈(-π/12,23π/12)，得
2x-π/12=π/3或2π/3
x=5π/24或9π/24
收起
（3，2）