已知0<x<π2<y<π且sin(x+y)=513(Ⅰ)若tgx2=12,分别求cosx及cosy的值;(Ⅱ)试比较siny与sin(x+y)的大小,并说明理由.-数学

题目简介

已知0<x<π2<y<π且sin(x+y)=513(Ⅰ)若tgx2=12,分别求cosx及cosy的值;(Ⅱ)试比较siny与sin(x+y)的大小,并说明理由.-数学

题目详情

已知0<x<
π
2
<y<π且sin(x+y)=
5
13

(Ⅰ)若tg
x
2
=
1
2
,分别求cosx及cosy的值;
(Ⅱ)试比较siny与sin(x+y)的大小,并说明理由.
题型:解答题难度:中档来源:不详

答案

(Ⅰ)∵0<x<class="stub"π
2
<y<π,tanclass="stub"x
2
=class="stub"1
2
,且0<class="stub"x
2
class="stub"π
4

∴cos=class="stub"x
2
=class="stub"2
5
,sinclass="stub"x
2
=class="stub"1
5

则cosx=2cos2class="stub"x
2
-1=class="stub"3
5
,sinx=class="stub"4
5

又sin(x+y)=class="stub"5
13
class="stub"π
2
<x+y<class="stub"3π
2

∴cos(x+y)=-class="stub"12
13

∴cosy=cos[(x+y)-x]
=cos(x+y)cosx+sin(x+y)sinx
=-class="stub"12
13
•class="stub"3
5
+class="stub"5
13
•class="stub"4
5
=-class="stub"16
65


(Ⅱ)∵0<x<class="stub"π
2
<y<π,
class="stub"π
2
<x+y<class="stub"3π
2
class="stub"π
2
<y<x+y<class="stub"3π
2

又y=sinx在[class="stub"π
2
class="stub"3π
2
]上为减函数,
∴siny>sin(x+y).

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