Question

有关直线,已知正四面体ABCD,求AB与面BCD的角的余弦

Answer 1

面BCD中,过B做BE⊥CD于E

面ABE中,过A做AF⊥BE于F

则∠ABF即为AB与面BCD的角

设正四面体棱长为a

BC=a,CE=a/2

则BE=(a√3)/2

BF=2BE/3=(a√3)/3

即AB与面BCD的角的余弦

=BF/AB=[(a√3)/3]/a= (√3)/3