【答案】
分析:先求出cos(α+β)=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213429572757682/SYS201310232134295727576015_DA/0.png)
,sinβ=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213429572757682/SYS201310232134295727576015_DA/1.png)
.利用同角三角函數(shù)關(guān)系求值時(shí)要判斷角的終邊所在的象限,來(lái)確定三角函數(shù)值的符號(hào),此是正確求值的關(guān)鍵,由于α=α+β-β,故sinα=sin[(α+β)-β]=sin(α+β)cosβ-cos(α+β)sinβ,將各角的三角函數(shù)值代入求sinα.
解答:解:∵β∈(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213429572757682/SYS201310232134295727576015_DA/2.png)
,π),cosβ=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213429572757682/SYS201310232134295727576015_DA/3.png)
,∴sinβ=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213429572757682/SYS201310232134295727576015_DA/4.png)
.
又∵0<α<
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213429572757682/SYS201310232134295727576015_DA/5.png)
,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213429572757682/SYS201310232134295727576015_DA/6.png)
<β<π,
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213429572757682/SYS201310232134295727576015_DA/7.png)
<α+β<
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213429572757682/SYS201310232134295727576015_DA/8.png)
,又sin(α+β)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213429572757682/SYS201310232134295727576015_DA/9.png)
,
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213429572757682/SYS201310232134295727576015_DA/10.png)
<α+β<π,
cos(α+β)=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213429572757682/SYS201310232134295727576015_DA/11.png)
=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213429572757682/SYS201310232134295727576015_DA/12.png)
=-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213429572757682/SYS201310232134295727576015_DA/13.png)
,
∴sinα=sin[(α+β)-β]
=sin(α+β)cosβ-cos(α+β)sinβ
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213429572757682/SYS201310232134295727576015_DA/14.png)
•(-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213429572757682/SYS201310232134295727576015_DA/15.png)
)-(-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213429572757682/SYS201310232134295727576015_DA/16.png)
)•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213429572757682/SYS201310232134295727576015_DA/17.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131023213429572757682/SYS201310232134295727576015_DA/18.png)
.
點(diǎn)評(píng):本題考查兩角和與差的正弦公式,此類(lèi)題求值時(shí)一般要先進(jìn)行角的變換,把要求角用已知三角函數(shù)值的角表示出來(lái),用公式展開(kāi)求出三角函數(shù)值,求解時(shí)要用到同角三角函數(shù)的基本關(guān)系,故判斷角的終邊在那個(gè)象限是做對(duì)此類(lèi)題的保證.