本试题主要是考查了三角函数定义,以及余弦定理和三角恒等变换的综合运用。
(1)由题意,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233647911831.png)
,因为点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233647661313.png)
的坐标为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233647755733.png)
,
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233647973599.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233648004602.png)
,
(2)在△
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233648020450.png)
中,由余弦定理,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232336480351440.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232336480671222.png)
,结合三角函数的性质得到结论。
解:(1)由题意,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233647911831.png)
,因为点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233647661313.png)
的坐标为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233647755733.png)
,
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233647973599.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233648004602.png)
, ………… 3分
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232336481912121.png)
.………… 5分
(2)解法一:在△
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233648020450.png)
中,由余弦定理,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232336480351440.png)
, …… 6分
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232336480671222.png)
. …………8分
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233647677648.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232336483011075.png)
, …………10分
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233648316822.png)
.
因此函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233647801449.png)
的值域是
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233647895595.png)
。 …………12分
解法二:由题意,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233648379933.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233648410719.png)
,……6分
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232336484253696.png)
……………………………………8分
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233647677648.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232336484721014.png)
, ……………10分
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233648316822.png)
.即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233647801449.png)
的值域是
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823233647895595.png)
. …………12分