www.142370.com:2017-2018学年高中数学 专题 直线与平面所成的角课堂同步试题 新人教A版

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直线与平面所成的角

高考频度:★★★★☆

难易程度:★★★★☆

如图,在四棱锥 P ? ABCD 中,底面 ABCD是平行四边形,?ADC ? 45 , AD ? AC ? 1,O 为 AC 的中点, PO ? 平面 ABCD , PO ? 2, M 为 PD 的中点.

(1)证明: AD ? 平面 PAC ; (2)求直线 AM 与平面 ABCD 所成角的正切值. 【参考答案】(1)见试题解析;(2) 4 5 .
5

AN ? 1 DO ? 5 , MN ? 1 PO ? 1,? tan ?MAN ? 4 5 .

2

4

2

5

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【解题必备】求直线与平面所成的角的一般步骤是: (1)在斜线上选取恰当的点向平面引垂线,在这一步确定垂足的位置是关键; (2)证明所找到的角为直线与平面所成的角; (3)借助三角形的相关知识求角.
注意:直线与平面所成的角的范围是 [0?,90?] .
学霸推荐

1.已知在三棱柱 ABC ? A1B1C1 中,各棱长相等,侧棱垂直于底面,且点 D 是侧面 BB1C1C 的

中心,则 AD 与平面 BB1C1C 所成角的大小是

A. 30?

B. 45?

C. 60?

D. 90?

2.如图,AA1、BB1、CC1 为圆柱 OO1 的母线,BC 是底面圆 O 的直径,D、E 分别是 AA1、CB1 的中点,DE

⊥平面 CBB1,AB=2,BB1=BC.

(1)证明:DE∥平面 ABC; (2)求直线 CA1 与平面 CBB1 所成角的正弦值.
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2.【解析】(1)如图,连接 EO,OA, ∵E、O 分别为 B1C、BC 的中点,∴EO∥BB1, 又 DA∥BB1,且 DA=EO= BB1, ∴四边形 AOED 为平行四边形,即 DE∥OA, 又 DE?平面 ABC,OA? 平面 ABC,∴DE∥平面 ABC.
(2)由题意知,B1C1 是上底面圆 O1 的直径,如图,连接 A1O1,则 A1O1∥AO, 由(1)知 DE∥AO,又 DE⊥平面 CBB1, ∴AO⊥平面 CBB1C1,∴A1O1⊥平面 CBB1C1,
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