本题可用体积法.过点A作平面A1BD的垂线AM,垂足为M.过点A作AN⊥A1B于点N.显然sinθ=AM/AN
设AA1=2AB=2a
AN=AA1*AB/A1B=2a*a/√(4a^2+a^2)=2√5a/5
V四棱锥A1-ABD=1/3*1/2*a^2*2a=1/3*S△A1BD*AM
A1D=A1B=√5a,BD=√2a,故S△A1BD=1/2*√2a*√[(√5a)^2-(√2a/2)^2]=3a^2/2
于是1/3*1/2*a^2*2a=1/3*3a^2/2*AM
AM=2a/3
于是sinθ=AM/AN=2a/3/(2/5)=√5/3
故cosθ=√(1-5/9)=2/3