本试题主要是考查了面面垂直和线面角的求解的综合运用。
(1)第一问中要证明面面垂直关键是证明线面垂直,然后利用判定定理得到。
(2)第二问先根据线面角的定义,作出线面角,然后利用直角三角形的边角的关系求解的得到。
解:(1)∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056090470.png)
与平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232055716526.png)
所成角的正切值依次
是
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056137206.png)
和
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056293333.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056387468.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056824647.png)
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232055856394.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232055716526.png)
,底面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232055716526.png)
是矩形
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057073416.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057089441.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057104533.png)
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057120318.png)
是
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057136365.png)
的中点 ∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057167557.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056496755.png)
…………………………7分
(2)解法一:∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232055856394.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232055716526.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057370548.png)
,又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057385529.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057401429.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056543439.png)
,取
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057448367.png)
中点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057463316.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057494405.png)
中点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057510303.png)
,联结
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057526643.png)
,
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057541769.png)
且
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057572748.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057588567.png)
是平行四边形,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057604515.png)
即为直线
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056512416.png)
与平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056543439.png)
所成的角. 在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057666639.png)
中,,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057682569.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232320577131228.png)
,
∴直线
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056512416.png)
与平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056543439.png)
所成角的正弦值为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056699460.png)
.
解法二:分别以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057791589.png)
为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057806266.png)
轴、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057822310.png)
轴、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057853231.png)
轴建立空间直角坐标系,依题意,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057869634.png)
,则各点坐标分别是
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057900539.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057916571.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057947642.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057962587.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057994577.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232058009565.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232058025567.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232058056710.png)
,
又∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232058072409.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056543439.png)
,
∴平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056543439.png)
的法向量为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232058134787.png)
,
设直线
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056512416.png)
与平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056543439.png)
所成的角为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232058196310.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232320582281432.png)
,
∴直线
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056512416.png)
与平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056543439.png)
所成角的正弦值为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056699460.png)
. …………………………15分
解:(1)∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056090470.png)
与平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232055716526.png)
所成角的正切值依次
是
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056137206.png)
和
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056293333.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056387468.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056824647.png)
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232055856394.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232055716526.png)
,底面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232055716526.png)
是矩形
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057073416.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057089441.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057104533.png)
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057120318.png)
是
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057136365.png)
的中点 ∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057167557.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056496755.png)
…………………………7分
(2)解法一:∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232055856394.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232055716526.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057370548.png)
,又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057385529.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057401429.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056543439.png)
,取
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057448367.png)
中点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057463316.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057494405.png)
中点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057510303.png)
,联结
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057526643.png)
,
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057541769.png)
且
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057572748.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057588567.png)
是平行四边形,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057604515.png)
即为直线
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056512416.png)
与平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056543439.png)
所成的角. 在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057666639.png)
中,,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057682569.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232320577131228.png)
,
∴直线
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056512416.png)
与平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056543439.png)
所成角的正弦值为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056699460.png)
.
解法二:分别以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057791589.png)
为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057806266.png)
轴、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057822310.png)
轴、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057853231.png)
轴建立空间直角坐标系,依题意,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057869634.png)
,则各点坐标分别是
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057900539.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057916571.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057947642.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057962587.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232057994577.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232058009565.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232058025567.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232058056710.png)
,
又∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232058072409.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056543439.png)
,
∴平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056543439.png)
的法向量为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232058134787.png)
,
设直线
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056512416.png)
与平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056543439.png)
所成的角为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232058196310.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232320582281432.png)
,
∴直线
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056512416.png)
与平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056543439.png)
所成角的正弦值为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823232056699460.png)
. …………………………15分