【答案】
分析:(1)將已知等式左右兩邊平方,利用同角三角函數(shù)間的基本關(guān)系化簡,求出2sinxcosx的值小于0,由x的范圍得到sinx大于0,cosx小于0,利用完全平方公式求出sinx-cosx的值,與已知等式聯(lián)立求出sinx與cosx的值,即可確定出tanx的值;
(2)由α的范圍及cosα的值,利用同角三角函數(shù)間的基本關(guān)系求出sinα的值,由sin(α+β)的值大于0,及α與β的范圍,求出α+β的范圍,利用同角三角函數(shù)間的基本關(guān)系求出cos(α+β)的值,將cosβ變形為cos[(α+β)-α],利用兩角和與差的余弦函數(shù)公式化簡后,把各自的值代入即可求出值.
解答:解:(1)將sinx+cosx=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103101221045793809/SYS201311031012210457938018_DA/0.png)
②兩邊平方得:(sinx+cosx)
2=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103101221045793809/SYS201311031012210457938018_DA/1.png)
,
∴1+2sinxcosx=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103101221045793809/SYS201311031012210457938018_DA/2.png)
,即2sinxcosx=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103101221045793809/SYS201311031012210457938018_DA/3.png)
<0,
∵x∈(0,π),∴sinx>0,cosx<0,
∴(sinx-cosx)
2=1-2sinxcosx=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103101221045793809/SYS201311031012210457938018_DA/4.png)
,
∴sinx-cosx=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103101221045793809/SYS201311031012210457938018_DA/5.png)
②,
聯(lián)立①②解得:sinx=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103101221045793809/SYS201311031012210457938018_DA/6.png)
,cosx=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103101221045793809/SYS201311031012210457938018_DA/7.png)
,
則tanx=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103101221045793809/SYS201311031012210457938018_DA/8.png)
=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103101221045793809/SYS201311031012210457938018_DA/9.png)
;
(2)∵0<α<
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103101221045793809/SYS201311031012210457938018_DA/10.png)
<β<π,且sin(α+β)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103101221045793809/SYS201311031012210457938018_DA/11.png)
>0,cosα=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103101221045793809/SYS201311031012210457938018_DA/12.png)
,
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103101221045793809/SYS201311031012210457938018_DA/13.png)
<α+β<π,
∴cos(α+β)=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103101221045793809/SYS201311031012210457938018_DA/14.png)
=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103101221045793809/SYS201311031012210457938018_DA/15.png)
,sinα=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103101221045793809/SYS201311031012210457938018_DA/16.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103101221045793809/SYS201311031012210457938018_DA/17.png)
,
則cosβ=cos[(α+β)-α]=cos(α+β)cosα+sin(α+β)sinα=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103101221045793809/SYS201311031012210457938018_DA/18.png)
×
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103101221045793809/SYS201311031012210457938018_DA/19.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103101221045793809/SYS201311031012210457938018_DA/20.png)
×
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103101221045793809/SYS201311031012210457938018_DA/21.png)
=-
點評:此題考查了兩角和與差的余弦函數(shù)公式,同角三角函數(shù)間的基本關(guān)系,以及完全平方公式的運用,熟練掌握公式是解本題的關(guān)鍵.