【答案】
分析:(Ⅰ)先根據(jù)向量的減法運(yùn)算求出
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101225533226803685/SYS201311012255332268036016_DA/0.png)
-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101225533226803685/SYS201311012255332268036016_DA/1.png)
,根據(jù)題中的新定義及平面向量的數(shù)量積的運(yùn)算法則表示出f(x),然后利用二倍角的正弦函數(shù)公式及兩角和的正弦函數(shù)公式化為一個(gè)角的正弦函數(shù),然后利用周期公式T=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101225533226803685/SYS201311012255332268036016_DA/2.png)
即可求出f(x)的最小正周期;
(Ⅱ)根據(jù)f(A)=1,由第一問(wèn)求出的f(x)的解析式,根據(jù)A的范圍,利用特殊角的三角函數(shù)值求出A的度數(shù),再根據(jù)A+B的度數(shù)求出B的度數(shù),由已知的BC,sinA及sinB的值,利用正弦定理即可求出AC的值.
解答:解:(Ⅰ)
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101225533226803685/SYS201311012255332268036016_DA/3.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101225533226803685/SYS201311012255332268036016_DA/4.png)
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101225533226803685/SYS201311012255332268036016_DA/5.png)
;(6分)
(Ⅱ)由f(A)=1得
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101225533226803685/SYS201311012255332268036016_DA/6.png)
,
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101225533226803685/SYS201311012255332268036016_DA/7.png)
且
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101225533226803685/SYS201311012255332268036016_DA/8.png)
,
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101225533226803685/SYS201311012255332268036016_DA/9.png)
,解得
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101225533226803685/SYS201311012255332268036016_DA/10.png)
,
又∵
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101225533226803685/SYS201311012255332268036016_DA/11.png)
,∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101225533226803685/SYS201311012255332268036016_DA/12.png)
,(10分)
在△ABC中,由正弦定理得:
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101225533226803685/SYS201311012255332268036016_DA/13.png)
,
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131101225533226803685/SYS201311012255332268036016_DA/14.png)
.(12分)
點(diǎn)評(píng):此題綜合考查了三角函數(shù)的恒等變換,正弦定理及平面向量的數(shù)量積運(yùn)算.函數(shù)周期的求法是把函數(shù)化為一個(gè)角的三角函數(shù),然后利用周期公式求出.熟練掌握三角函數(shù)公式及平面向量的運(yùn)算法則是解本題的關(guān)鍵.