The alpha plane is drawn through the AC side of the regular triangle ABC.

The alpha plane is drawn through the AC side of the regular triangle ABC. The angle between the height BD of the triangle and the plane is 30 degrees. Find the sine of the angle between line AB and the alpha plane.

Let the length of the side of an equilateral triangle ABC be equal to X cm.

Then the length of the height BD = X * √3 / 2 cm.

From the vertex B we construct the perpendicular BE to the plane.

The triangle BED is rectangular, in which the angle BDE = 30, then BE = BD / 2 = X * √3 / 4 cm.

The segment AE is the projection of the side AB onto the plane, then the angle BAE is the angle between the plane and the side AB.

SinBAE = BE / AB = (X * √3 / 4) / X = √3 / 4.

Answer: The sine of the angle between side AB and the plane is √3 / 4.