设AC与BD交于O,A1C1与B1D1交于O1.
∵正方体AC1,
∴AC⊥BD,AA1⊥底面BD,
∴A1O⊥BD.易知OO1⊥BD.
∴∠A1OO1是二面角A1-BD-O1的平面角.
设AB=2,则AO=√2,A1O=√6,
∴sin∠A1OO1=(√2）/√6=（√3）/3,为所求.