在銳角三解形ABC中,角A、B、C的對邊分別是a、b、c,若sinA=8cosBcosC
(I)求tanB+tanC的值;
(II)若a=3,求△ABC面積的最大值.
【答案】
分析:(I)在銳角三解形ABC中,由sinA=8cosBcosC 利用兩角和差的正弦公式可得sinBcosC+cosBsinC=8cosBcosC,由此求得tanB+tanC 的值.
(II)若a=3,由正弦定理可得△ABC面積 S=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103100020169017850/SYS201311031000201690178018_DA/0.png)
•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103100020169017850/SYS201311031000201690178018_DA/1.png)
•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103100020169017850/SYS201311031000201690178018_DA/2.png)
•sinA=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103100020169017850/SYS201311031000201690178018_DA/3.png)
•tanBtanC,利用基本不等式求得S的最大值.
解答:解:(I)在銳角三解形ABC中,∵sinA=8cosBcosC,
∴sin(B+C)=8cosBcosC,即sinBcosC+cosBsinC=8cosBcosC.
∴tanB+tanC=8.
(II)若a=3,由正弦定理可得
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103100020169017850/SYS201311031000201690178018_DA/4.png)
,
∴△ABC面積 S=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103100020169017850/SYS201311031000201690178018_DA/5.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103100020169017850/SYS201311031000201690178018_DA/6.png)
•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103100020169017850/SYS201311031000201690178018_DA/7.png)
•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103100020169017850/SYS201311031000201690178018_DA/8.png)
•sinA=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103100020169017850/SYS201311031000201690178018_DA/9.png)
•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103100020169017850/SYS201311031000201690178018_DA/10.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103100020169017850/SYS201311031000201690178018_DA/11.png)
•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103100020169017850/SYS201311031000201690178018_DA/12.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103100020169017850/SYS201311031000201690178018_DA/13.png)
•tanBtanC≤
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103100020169017850/SYS201311031000201690178018_DA/14.png)
•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103100020169017850/SYS201311031000201690178018_DA/15.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131103100020169017850/SYS201311031000201690178018_DA/16.png)
×16=9,當且僅當tanB=tanC,即 B=C時,等號成立.
故△ABC面積 S的最大值為9.
點評:本題主要考查兩角和差的正弦公式,同角三角函數(shù)的基本關(guān)系,正弦定理以及基本不等式的應(yīng)用,屬于中檔題.