Appendix 4

Some Special Values of sin α

The last four values are related to a regular pentagon. For example, the side of a regular pentagon inscribed in a unit circle is 2 sin 36^°, its diagonal is 2 sin 72^°, and their ratio is 2 sin 54^°. These values are also related to the “golden section”: the ratio in which a line segment must be divided if the whole segment is to the longer part as the longer part is to the shorter. This ratio, denoted by ϕ, is equal to , that is, to 2 sin 54^°.

Repeated use of the half-angle formula for the sine leads to the following expressions, where n = 1, 2, 3,…: