Morrie's law is a special
trigonometric identity
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvi ...
. Its name is due to the physicist
Richard Feynman
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superf ...
, 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.
Identity and generalisation
:
It is a
special case
In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of . A limiting case ...
of the more general identity
:
with ''n'' = 3 and α = 20° and the fact that
:
since
:
Similar identities
A similar identity for the sine function also holds:
:
Moreover, dividing the second identity by the first, the following identity is evident:
:
Proof
Geometric proof of Morrie's law

Consider a regular
nonagon
In geometry, a nonagon () or enneagon () is a nine-sided polygon or 9-gon.
The name ''nonagon'' is a prefix hybrid formation, from Latin (''nonus'', "ninth" + ''gonon''), used equivalently, attested already in the 16th century in French ''nonogo ...
with side length
and let
be the midpoint of
,
the midpoint
and
the midpoint of
. The inner angles of the nonagon equal
and furthermore
,
and
(see graphic). Applying the
cosinus definition in the
right angle triangle
A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right ...
s
,
and
then yields the proof for Morrie's law:
[ Samuel G. Moreno, Esther M. García-Caballero: "'A Geometric Proof of Morrie's Law". In: ''American Mathematical Monthly'', vol. 122, no. 2 (February 2015), p. 168]
JSTOR
:
Algebraic proof of the generalised identity
Recall the double angle formula for the sine function
:
Solve for
:
It follows that:
:
Multiplying all of these expressions together yields:
:
The intermediate numerators and denominators cancel leaving only the first denominator, a power of 2 and the final numerator. Note that there are ''n'' terms in both sides of the expression. Thus,
:
which is equivalent to the generalization of Morrie's law.
References
Further reading
* Glen Van Brummelen: ''Trigonometry: A Very Short Introduction''. Oxford University Press, 2020, , pp. 79–83
* Ernest C. Anderson: ''Morrie's Law and Experimental Mathematics''. In: ''Journal of recreational mathematics'', 1998
External links
* {{MathWorld, title=Morrie's Law, urlname=MorriesLaw
Mathematical identities
Trigonometry
Articles containing proofs