解题方法
1 . 如图,在三棱柱
中,棱
的中点分别为
在平面
内的射影为D,
是边长为2的等边三角形,且
,点F在棱
上运动(包括端点).
为棱
的中点,求点
到平面
的距离;
(2)求锐二面角
的余弦值的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d06f8edd1a1f18ca2dae700c6a29ab4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06eea0ba6bb7b06e26f963f5cb13bc50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92535536bd3c2761724fd058427f95a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56f7ba05c54b3de1f4378f7c8eb58328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56f7ba05c54b3de1f4378f7c8eb58328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34be4e71cabf458f17a6cd7f24bc70af.png)
(2)求锐二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86ac0983ef8333a915498585f216860c.png)
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名校
2 . 在三棱台
中,
平面ABC,
,且
,
,M为AC的中点,P是CF上一点,且
,
.
平面PBM;
(2)若直线BC与平面PBM的所成角为
,求平面EFM与平面PBM所成夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8783bc74553bf44b61d999a0e4144bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac0b72906641ed13716cfbce50923282.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/080db3af81b29ed10144a1c2e2a4fb8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b6e4a2df58a236c20df5df0d29a466c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd2e870c95b1ed54b281f93e683578bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01f14839cf7e3ec6e25b60765ca25b33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52d0a3279dab4449136fb53754f78e0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97f30533da2e1d2a958dc906c37eba9d.png)
(2)若直线BC与平面PBM的所成角为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac1a63ab608517bb10aa036783dfb51f.png)
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3 . 如图,在四棱锥
中,底面
是边长为2的正方形,
平面
,
,
是
中点,
是
中点.
平面
;
(2)若
,求平面
与平面
的夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b80ee363635d73f601654339028daec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edcf19a7f0dd0cdf59516ae585025110.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cb26a48f8374885250345659418585c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f747eb5b2d21c9de962cbfd4ec4bb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c5a4dfcf4c24a8ecb210cc4c53db221.png)
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4 . 如图,在直四棱柱
中,
.![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fd4c85bb98a2a0afddd7ed92578ad2e.png)
平面
;
(2)求
与平面
所成的角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96486f5fd1670d138c22b9480635d24e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fd4c85bb98a2a0afddd7ed92578ad2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/638537c0a30676c73fea76c80e0f8bd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b2095691cd6e3e50722061cbc0bb648.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ee8456443402a25b1e25d35ff7e1c98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b2095691cd6e3e50722061cbc0bb648.png)
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5 . 如图,在四棱锥
中,
平面
为等边三角形,
,点
为棱
上的动点.
平面
;
(2)当二面角
的大小为
时,求线段
的长度.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80c753cb1eb73fd8d136d00462970797.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f09ad78d4eccd1a9c9ccd3c4af79c79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9cad98d0e4a7f999143500c62b07e6e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b3d2626c477fbb0b0220ccfd1eab9be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/689c065652544780be8b33ae92cbb6d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/500df0e782bb081e608f4bc1d576afcf.png)
(2)当二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e8ef5efedfdded5f660a01f3b3f7461.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1e5fa72f2878b476bc57f0df12d6555.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
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6 . 如图,在三棱柱
中,平面
平面
,平面
平面
.
平面ABC.
(2)若
,
,求直线BC与平面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85a2e10a5aebe40a9018d5ee3ade7af8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2623e1c4ecbd65ae5e0285c0453ffd0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d25e8fc3dda4f8b45491514b6e22a962.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/080db3af81b29ed10144a1c2e2a4fb8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fba5d34d3edfc929177ef9af82d5d0bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9539f8fb13345b449274b67bbda995db.png)
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解题方法
7 . 如图所示的几何体是由圆锥
与圆柱
组成的组合体,其中圆柱的轴截面
是边长为2的正方形,圆锥的高
,M为圆柱下底面圆周上异于A,B的点.
∥平面
;
(2)若
,求直线
与平面
所成角的正切值的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41a8c34f622f1b979feed5ae6ae5d0e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae0a4f38420bb9215dbc9c875b755838.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a55a6b8045f2d6429ac49997c1124a25.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/defa5b53043ae802bb1af7d14374406d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08452588675f76da2f8d31387b3a8224.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b10c2bc31ab83c89237b93159ae64c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08452588675f76da2f8d31387b3a8224.png)
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8 . 如图,直三棱柱
中,
,点
在线段
上,且
,
.
为
的重心;
(2)若
,求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3570a95f68349fcd9417fcda62e78e7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ee8456443402a25b1e25d35ff7e1c98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9073cce40aa78cc9693c988f3eea90aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47af45fbf1714055d9b414a44a8613fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/166ccfa0ac0388903f3f5960c7fa3660.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e223e3d40ee025f15ead90746c28b8c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/167e4b949f6bda469c5ac4af5a85a0db.png)
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9 . 如图,在三棱柱
中,正方形
的棱长为2,
,点M为AB中点,
.
为直三棱柱;
(2)求直线
与平面
所成角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab3e0dba5705e1d749cfb21ebbb2ed93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c645d1dea7ca8a6eb931635dcf63d6f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4149c3692068ea18563a4ef56d9d1cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19ef601ca1f9c4c031adab4ffed297f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e363082228a653eb5b0dbc3c6161a9c9.png)
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10 . 如图,四棱锥
中,底面
为平行四边形,
,
,
,
.
;
(2)若
,
为
的中点,求直线
与平面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ff29971ccc633d89832ffa9bd54afa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e95dcd0c90de6986df58d5a4bd46251f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0063f3f48e49f2970ec7f097567cef5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37002ada5d194d4d062fa3285d7d9824.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b07e317ffe7859e81b42ef4970e344a.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b926f26192d2a052970fe306ce864da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4739ad948445af72d585fe29c745929b.png)
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