已知△ABC中,(a-c)(sinA+sinC)=(a-b)sinB,(1)求∠C;(2)若△ABC的外接圓半徑為2,試求該三角形面積的最大值.
【答案】
分析:(1)利用正弦定理把題設(shè)中的條件中的角的正弦換成邊,化簡(jiǎn)整理得a
2+b
2-c
2=ab,進(jìn)而利用余弦定理求得cosC,則C可得.
(2)利用三角形面積公式表示出三角形的面積,利用正弦定理把邊換成角的正弦,利用兩角和公式化簡(jiǎn)整理,進(jìn)而利用正弦函數(shù)的性質(zhì)求得三角形面積的最大值.
解答:解:(1)由(a-c)(sinA+sinC)=(a-b)sinB,得(a-c)(a+c)=(a-b)b,
∴a
2-c
2=ab-b
2,∴a
2+b
2-c
2=ab,∴cosC=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180155246469788/SYS201310241801552464697020_DA/0.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180155246469788/SYS201310241801552464697020_DA/1.png)
又∵0°<C<180°,∴C=60°
(2)S=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180155246469788/SYS201310241801552464697020_DA/2.png)
absinC=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180155246469788/SYS201310241801552464697020_DA/3.png)
×
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180155246469788/SYS201310241801552464697020_DA/4.png)
ab=4
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180155246469788/SYS201310241801552464697020_DA/5.png)
sinAsinB=4
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180155246469788/SYS201310241801552464697020_DA/6.png)
sinAsin(120°-A)
=4
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180155246469788/SYS201310241801552464697020_DA/7.png)
sinA(sin120°cosA-cos120°sinA)=6sinAcosA+2
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180155246469788/SYS201310241801552464697020_DA/8.png)
sin
2A
=3sin2A-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180155246469788/SYS201310241801552464697020_DA/9.png)
cos2A+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180155246469788/SYS201310241801552464697020_DA/10.png)
=2
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180155246469788/SYS201310241801552464697020_DA/11.png)
sin(2A-30°)+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024180155246469788/SYS201310241801552464697020_DA/12.png)
∴當(dāng)2A=120°,即A=60°時(shí),S
max=3
點(diǎn)評(píng):本題主要考查了解三角形的實(shí)際應(yīng)用.解題的關(guān)鍵是利用了正弦定理完成了邊角問(wèn)題的互化.