In a regular rectangular pyramid SABCD, all edges of which are 1, find the sine of the angle between line BD and plane.

To find the sin between AC and the SBC plane, you need to find the projection of this straight line on the SBC plane.
But for this you need to know that the side faces are 1.
AM is the perpendicular to SB:

(1 ^ 2 – x * (1/2) ^ 2) = V3 / 2 – low angle – AFM.

To find the sine of this angle, you can use the OMC triangle.
AC = V2, and OC = V2 / 2. ОМ = V ((V3 / 2) ^ 2 – (V2 / 2) ^ 2) = 1/2 = 0.5.
MC = AM = V3 / 2.
Then sin OCM = (1/2) / V3 / 2. = 1 / V3 = 0.57735.