解题方法
1 . 如图,在直角
中,
,斜边
,
是
中点,现将直角
以直角边
为轴旋转一周得到一个圆锥.点
为圆锥底面圆周上一点,且
.
(2)求直线
与平面
所成的角的正切值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/866b81a8384cce4f24867baca2e6820c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/991c8373be20b4325ba779e4dfdc8b15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54275b7e571660d0a9e0370fbfe5050b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/866b81a8384cce4f24867baca2e6820c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2c3d2cba96f6f03520c0b3f6e4da03e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7cb551de43a9c1967e3f36f79480be6.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa3a310c1f8a5af35dc3328d874e18e.png)
您最近一年使用:0次
2023-01-11更新
|
589次组卷
|
5卷引用:上海市浦东新区2022-2023学年高二上学期期末数学试题
上海市浦东新区2022-2023学年高二上学期期末数学试题(已下线)8.6.2直线与平面垂直的判定定理(第1课时)(精讲)(2)-【精讲精练】2022-2023学年高一数学下学期同步精讲精练(人教A版2019必修第二册)(已下线)13.2.3 直线和平面的位置关系(1)(已下线)专题10 空间角与空间距离的综合(1)-期中期末考点大串讲(已下线)上海市四校(复兴高级中学、松江二中、奉贤中学、金山中学)2024届高三下学期3月联考数学试题变式题17-21
解题方法
2 . 在直三棱柱
中,
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/30/557b9f31-09a7-4977-86d3-b9258964561a.png?resizew=161)
(1)求四棱锥
的体积V;
(2)求直线
与平面
所成角的大小;
(3)求异面直线
与
所成角的大小.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/615fc8790237a1b09af51d6bcad6b595.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e96a6b20a35af7755e5d90789ea862da.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/30/557b9f31-09a7-4977-86d3-b9258964561a.png?resizew=161)
(1)求四棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2d0727de4c16b53b4bb6ab370afde6c.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b470c4e195cf7a07b7a331ce4b436e03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d9a8181f7a7fe7f3fac872ce9534f15.png)
(3)求异面直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b470c4e195cf7a07b7a331ce4b436e03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f1f229274a6e17977cc047814212589.png)
您最近一年使用:0次
名校
3 . 如图,四面体
中,
,
,
,
为
的中点.
(1)证明:平面
平面
;
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/31/09cb7873-7e7a-4566-a503-a02594efb0df.png?resizew=230)
(2)设
,
,点
在
上;
①点
为
中点,求
与
所成的角的大小;
②当
的面积最小时,求
与平面
所成的角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdb2dd10731b99c0f4f89ee957f8a239.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd95dc30c0344788b94289c464a3158e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2aca1bdb9459855415e292e73de50ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
(1)证明:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8f5ba965420dfd5aa4da211682df096.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4eb7e9ad5486cf1c5e506b20c5469e8.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/31/09cb7873-7e7a-4566-a503-a02594efb0df.png?resizew=230)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a05e0ab55e325fb3b85fc8ca9c27c76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26fdd8e57562ba94e10e7f1d770826d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
①点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
②当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36691f0269294ecae8f00b7bce97756c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7abd284f76d9f5769bc189508ce2572b.png)
您最近一年使用:0次
解题方法
4 . 已知正方体
的棱长为2,点
分别是棱
和
的中点.
(1)求
与
所成角的大小;
(2)求
与平面
所成角的大小.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad056c25c0fdcbcc765eb5cbc6093f2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10d8eb4a9f462ca0c1d49c3fe91e720d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ce1b066f8869d0ff4513f7a99745125.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9294c4766531857534a81bc536df57e6.png)
您最近一年使用:0次
5 . 如图所示,有满足下列条件的五边形的彩纸
,其中
,
,
.现将彩纸沿
向内进行折叠.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/1/b7616fed-c78f-417e-952c-7cc82b6117d0.png?resizew=425)
(1)求线段
的长度;
(2)若
是等边三角形,折叠后使
⊥
,求直线
与平面
的所成角的大小;
(3)将折叠后得到的四棱锥记为四棱锥
,求该四棱锥的体积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9142a8490de14a87eda628ffa7e28982.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8beaa6c45dc8fe05380f5a9770080a1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f81aae27dd6c2c3af7756c81ceb03f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb656194a6acbb28ecd6669814085a52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/1/b7616fed-c78f-417e-952c-7cc82b6117d0.png?resizew=425)
(1)求线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e742966e3711cfa53dce04022acf4bcc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fa7bbd7831e9ff4f8cffc8889d34f05.png)
(3)将折叠后得到的四棱锥记为四棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5164a3cc47e266446d49127e2ef10c37.png)
您最近一年使用:0次
名校
解题方法
6 . 如图,
是圆柱
的一条母线,
是底面的一条直径,
是圆![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
上一点,且
,
.
与平面
所成角的大小;
(2)求点
到平面
的距离.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/192f4f9446c954a291f779d963f90257.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
上一点,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59fe75f967e8915c9124a5d4ac420a4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55c24a968c73e960698a572ab01e3698.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7abd284f76d9f5769bc189508ce2572b.png)
(2)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4eb7e9ad5486cf1c5e506b20c5469e8.png)
您最近一年使用:0次
2022-11-26更新
|
2365次组卷
|
7卷引用:上海市曹杨第二中学2020-2021学年高二下学期期末数学试题
上海市曹杨第二中学2020-2021学年高二下学期期末数学试题上海市徐汇中学2021-2022学年高二上学期期中数学试题上海市川沙中学2022-2023学年高二上学期期中数学试题(已下线)第10章 空间直线与平面(单元重点综合测试)-2023-2024学年高二数学单元速记·巧练(沪教版2020必修第三册)上海市实验学校2023-2024学年高三下学期四模数学试题 第六章 立体几何初步(单元基础检测卷)(已下线)8.6.2直线与平面垂直(第2课时) 直线与平面垂直的性质(分层作业)-【上好课】
7 . 如图,在四棱锥
中,
⊥平面
,正方形
的边长为
,
,设
为侧棱
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/24/3b4b1972-387a-4e1f-ba7f-9019f00d6f13.png?resizew=146)
(1)求四棱锥
的体积
;
(2)求直线
与平面
所成角
的大小.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00bab2c27eac56fffa4cd7dbe1dcdf1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/24/3b4b1972-387a-4e1f-ba7f-9019f00d6f13.png?resizew=146)
(1)求四棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80c753cb1eb73fd8d136d00462970797.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be54e84508decfcce6d2fcbe6c8c1a92.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f747eb5b2d21c9de962cbfd4ec4bb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
您最近一年使用:0次
2022-11-23更新
|
537次组卷
|
8卷引用:上海市吴淞中学2021-2022学年高二下学期期末数学试题
名校
8 . 如图,在正三棱柱
中,
,异面直线
与
所成角的大小为
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/9/d9016209-25ae-4175-b147-4d5fcd5b4341.png?resizew=193)
(1)求正三棱柱
的体积;
(2)求直线
与平面
所成角的大小.(结果用反三角函数值表示)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e55a2310cbba5e050488cd9296eb195d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d8772aa893a9c1d40f714cb25701701.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2777840758e70e7dbbc18cef8f3d6d2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d88591679796c52024d11c4de641bdb.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/9/d9016209-25ae-4175-b147-4d5fcd5b4341.png?resizew=193)
(1)求正三棱柱
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d8772aa893a9c1d40f714cb25701701.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ac61c24f99a4e466f1e2ea011893866.png)
您最近一年使用:0次
2022-11-08更新
|
376次组卷
|
10卷引用:上海市金山中学2022-2023学年高二下学期期末数学试题
上海市金山中学2022-2023学年高二下学期期末数学试题(已下线)期末真题必刷压轴60题(22个考点专练)-【满分全攻略】2023-2024学年高二数学同步讲义全优学案(沪教版2020必修第三册)上海市实验学校2022届高三下学期开学考试数学试题上海市徐汇区2022届高三下学期二模数学试题上海市七宝中学2022届高三下学期3月月考数学试题上海市闵行(文绮)中学2024届高三上学期期中数学试题(已下线)专题15 立体几何(模拟练)-2(已下线)第19讲 立体几何初步-1(已下线)第19讲 立体几何初步-1(已下线)专题10立体几何初步必考题型分类训练-2
名校
9 . 如图1,在边长为4的菱形ABCD中,∠DAB=60°,点M,N分别是边BC,CD的中点,
,
.沿MN将
翻折到
的位置,连接PA,PB,PD,得到如图2所示的五棱锥P-ABMND.
平面PAG?证明你的结论;
(2)当四棱锥P-MNDB体积最大时,求直线PB和平面MNDB所成角的正弦值;
(3)在(2)的条件下,在线段PA上是否存在一点Q,使得二面角
的平面角的余弦值为
?若存在,试确定点Q的位置;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b761e4554c4ec2d5e76f1e3ba53176a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68cdb0f2acd33222ffa049f66c2e7ce4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d75eaf17d34e29407f37096d1c36177.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37f6574ef8d30c97fbd69269805fefd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5e8433f8c8a712e6db0b639f326c420.png)
(2)当四棱锥P-MNDB体积最大时,求直线PB和平面MNDB所成角的正弦值;
(3)在(2)的条件下,在线段PA上是否存在一点Q,使得二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/000bad0dfe00561e3a45c6643e524ff1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bc4cbe1fa83a288d069935ef4908a2b.png)
您最近一年使用:0次
2022-10-21更新
|
1923次组卷
|
16卷引用:上海市华东师范大学第二附属中学2023-2024学年高二上学期数学期末考试试卷
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10 . 如图,在长方体
中,
,点E在棱
上运动.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/8/ae4abcb4-4599-44c5-9ad4-c7f55edb8487.png?resizew=223)
(1)证明:
;
(2)当E与A重合时,求直线
与平面
所成角的大小(用反三角函数值表示);
(3)
等于何值时,二面角
的大小为
?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/803de786beb875e4a0f13fb1171c86eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/8/ae4abcb4-4599-44c5-9ad4-c7f55edb8487.png?resizew=223)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3af2fef79a35d44f9e11f9d8c5b3f08b.png)
(2)当E与A重合时,求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec353880a1e680a37ecce3b6fb4896f3.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14f7c5a38ee674e24a83426a17906532.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a8f3a8b0608ec011ad95c522fd2ea4d.png)
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