在平面直角坐標系
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030700736465.png)
中,以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030700751369.png)
為始邊,角
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030700767310.png)
的終邊與單位圓
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030700782291.png)
的交點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030700782309.png)
在第一象限,已知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030700798570.png)
.
(1)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030700814504.png)
,求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030700829440.png)
的值;
(2)若
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030700782309.png)
點橫坐標為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030700860346.png)
,求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030700876532.png)
.
(1)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030700892612.png)
;(2)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030700907716.png)
.
試題分析:(1)解法一是利用
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030700814504.png)
結合平面向量的數(shù)量積得到
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030700938422.png)
與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030700938431.png)
的等量關系,從而求出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030700829440.png)
的值;解法二是將
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030700814504.png)
轉化為兩直線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030700985383.png)
、
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701001357.png)
的斜率之間的關系,進而求出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030700829440.png)
的值;(2)設
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701032620.png)
,利用三角函數(shù)的定義求出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701048463.png)
和
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701063473.png)
的值,然后利用兩角差的正弦公式求出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701063602.png)
的值,最后利用三角行的面積公式求出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701079505.png)
的面積;解法二是利用平面向量的數(shù)量積計算出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701094606.png)
,然后計算出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701063602.png)
的值,最后利用三角形的面積公式計算出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701079505.png)
的面積.
試題解析:(1)解法1:由題可知:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701141537.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701157733.png)
即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701172636.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701188838.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701204509.png)
,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701219587.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701235715.png)
則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701250596.png)
解法2:由題可知:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701141537.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701157733.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701282509.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701297646.png)
∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701204509.png)
,∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701328583.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701344576.png)
,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701250596.png)
;
(2)解法1:由(1)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701391995.png)
,記
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701032620.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701438728.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701453932.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701469863.png)
∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701500611.png)
,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701516929.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240307015162180.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240307015311992.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701547408.png)
解法2:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701516929.png)
即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701578684.png)
,
即:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701172636.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701609794.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824030701391995.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240307016562217.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240307016561287.png)
則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240307016722143.png)
.
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