High Quality Content by WIKIPEDIA articles! In mathematics, the polar sine of a vertex angle of a polytope is defined as follows. Let v1, ..., vn, n 2, be non-zero vectors from the vertex in the directions of the edges. The absolute value of the polar sine does not change if all of the vectors v1, ..., vn are multiplied by positive constants. In case n = 2, the polar sine is the ordinary sine of the angle between the two vectors.Polar sines were investigated by Euler in the 18th century.