Morrie's law

Morrie's law is a name, that occasionally is used for the trigonometric identity

cos(20^circ) cdot cos(40^circ) cdot cos(80^circ)=frac{1}{8}.

It is a special case of the more general identity

2^n cdot prod_{k=0}^{n-1} cos(2^k alpha)=frac{sin(2^n alpha)}{sin(alpha)}

with n = 3 and α = 20°. The name is due to the physicist Richard Feynman, who used to refer to the identity under that name. Feynman picked that name, because he learned it during his childhood from a boy with the name Morrie Jacobs and afterwards remembered it for all of his life.

A similar identity for the sine function also holds:

sin(20^circ) cdot sin(40^circ) cdot sin(80^circ)=frac{sqrt 3 }{8}.

Moreover, dividing the second identity by the first, the following identity is evident:

tan(20^circ) cdot tan(40^circ) cdot tan(80^circ)=sqrt 3 =tan(60^circ).


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