在正方体ABCD-A1B1C1D1中,E为DD1的中点,求二面角E-AC-D的正切值

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在正方体ABCD-A1B1C1D1中,E为DD1的中点,求二面角E-AC-D的正切值

在正方体ABCD-A1B1C1D1中,E为DD1的中点,求二面角E-AC-D的正切值
在正方体ABCD-A1B1C1D1中,E为DD1的中点,求二面角E-AC-D的正切值

在正方体ABCD-A1B1C1D1中,E为DD1的中点,求二面角E-AC-D的正切值
连接E与AC的中点 O
设正方体边长为1
因为DD'垂直平面ABCD
DO垂直AC
则 AC垂直平面DEO
则 二面角E-AC-D即为∠EOD
E为DD1的中点 则 ED=1/2
DO=(根号2)/2
则 tan∠EOD=DE/DO=(1/2)/[(根号2)/2]=(根号2)/2

ED 垂直平面ABCD 即ED 垂直平面ACD
连接 AC BD 交予O 连接OE
因为 ED 垂直平面ACD
所以 ED 垂直OD
那么角EOD 就是二面角E-AC-D的夹角
若AB 为1 那么OD=√2/2 DE=1/2 OE=√3/2
tanEOD=DE/OD=√2/2

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