题目内容
(本小题满分12分)
如图,在长方体
中,![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161217839292.gif)
,
为
的中点,
为
的中点.
(1)证明:
;
(2)求
与平面
所成角的正弦值.
如图,在长方体
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161217823487.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161217839292.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161217870600.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161217885202.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161217901270.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161217917327.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161217932241.gif)
(1)证明:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161217948468.gif)
(2)求
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161217979236.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161217995395.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231612181356576.jpg)
(1)略
(2)
与平面
所成角的正弦值为![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231612182911288.gif)
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161217979236.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161217995395.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231612182911288.gif)
解:(1)以
点为原点,分别以
为
轴的正方向,建立如图所示的空间直角坐标系
…1分
依题意,可得
.………………3分
,
,
∴
,
即
,∴
. ………………6分
(2)设
,且
平面
,则
, 即
,
∴
解得
,
取
,得
,所以
与平面
所成角的正弦值为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161219055662.gif)
. ………………12分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161218322210.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161218447420.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161218463353.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161218541390.gif)
依题意,可得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161218556881.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161218572753.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161218587979.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161218603965.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161218634948.gif)
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161218650497.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161217948468.gif)
(2)设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161218697461.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161218712222.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161218853386.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161218868749.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231612188841120.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161218899810.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161218931615.gif)
取
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161218946234.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161218977521.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161217979236.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161217995395.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823161219055662.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231612182911288.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
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