已知tan
β=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824035028686375.png)
,sin(
α+
β)=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824035028701406.png)
,其中
α,
β∈(0,π),則sin
α的值為________.
依題意得sin
β=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824035028717348.png)
,cos
β=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824035028732368.png)
;注意到sin(
α+
β)=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824035028701406.png)
<sin
β,因此有
α+
β>
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824035028764423.png)
(否則,若
α+
β≤
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824035028764423.png)
,則有0<
β<
α+
β≤
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824035028764423.png)
,0<sin
β<sin(
α+
β),這與“sin(
α+
β)<sin
β”矛盾),則cos(
α+
β)=-
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824035028810430.png)
,sin
α=sin[(
α+
β)-
β]=sin(
α+
β)·cos
β-cos(
α+
β)sin
β=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824035028701481.png)
.
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