【答案】
分析:把f(x)解析式中的被减数中的角度
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/0.png)
-x变形为π-(x+
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/1.png)
)后,利用诱导公式变形,提取2后,再利用两角差的正弦函数公式及特殊角的三角函数值化为一个角的正弦函数,根据正弦函数的值域即可求出f(x)的最大值;找出ω的值,利用周期公式即可求出f(x)的周期,令k=0即可求出函数图象的一个对称点
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/2.png)
,即可得到正确的选项.
解答:解:
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/3.png)
=sin[π-(x+
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/4.png)
)]-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/5.png)
cos(x+
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/6.png)
)
=sin(x+
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/7.png)
)-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/8.png)
cos(x+
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/9.png)
)
=2[
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/10.png)
sin(x+
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/11.png)
)-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/12.png)
cos(x+
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/13.png)
)]
=2sin[(x+
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/14.png)
)-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/15.png)
]
=2sin(x-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/16.png)
),
∵x∈R,∴x-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/17.png)
∈R,
∴-1≤sin(x-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/18.png)
)≤1,
则f(x)的最大值为2;
∵ω=1,∴周期T=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/19.png)
=2π;
当x-
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/20.png)
=kπ(k∈Z)时,f(x)图象关于某一点对称,
∴当k=0,求出x=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/21.png)
,即f(x)图象关于x=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131025125146852224154/SYS201310251251468522241010_DA/22.png)
对称,
故选B
点评:此题考查了三角函数的恒等变形,三角函数的周期性及其求法,正弦函数的对称性以及三角函数的最值,灵活运用三角函数的恒等变形把f(x)化为一个角的正弦函数是本题的突破点.