曲线y=2sin(x+π4)cos(x-π4)和直线y=12在y轴右侧的交点按横坐标从小到大依次记为P1,P2,P3,…,则|P2P4|等于______.-数学

题目简介

曲线y=2sin(x+π4)cos(x-π4)和直线y=12在y轴右侧的交点按横坐标从小到大依次记为P1,P2,P3,…,则|P2P4|等于______.-数学

题目详情

曲线y=2sin(x+
π
4
)cos(x-
π
4
)和直线y=
1
2
在y轴右侧的交点按横坐标从小到大依次记为P1,P2,P3,…,则|P2P4|等于______.
题型:填空题难度:中档来源:不详

答案

∵y=2sin(x+class="stub"π
4
)cos(x-class="stub"π
4

=2sin(x-class="stub"π
4
+class="stub"π
2
)cos(x-class="stub"π
4

=2cos(x-class="stub"π
4
)cos(x-class="stub"π
4

=cos[2(x-class="stub"π
4
)]+1
=cos(2x-class="stub"π
2
)+1
=sin(2x)+1
若y=2sin(x+class="stub"π
4
)cos(x-class="stub"π
4
)=class="stub"1
2

则2x=2kπ+class="stub"3π
2
±class="stub"π
3
(k∈N)
x=kπ+class="stub"3π
4
±class="stub"π
6
(k∈N)
故|P2P4|=π
故答案为:π

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