Morrie's law explained

Morrie's law is a special trigonometric identity. Its 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.[1]

Identity and generalisation

\cos(20\circ)\cos(40\circ)\cos(80\circ)=

1
8

.

It is a special case of the more general identity

2n

n-1
\prod
k=0

\cos(2k\alpha)=

\sin(2n\alpha)
\sin(\alpha)

with n = 3 and α = 20° and the fact that

\sin(160\circ)
\sin(20\circ)

=

\sin(180\circ-20\circ)
\sin(20\circ)

=1,

since

\sin(180\circ-x)=\sin(x).

Similar identities

A similar identity for the sine function also holds:

\sin(20\circ)\sin(40\circ)\sin(80\circ)=

\sqrt3
8

.

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

\tan(20\circ)\tan(40\circ)\tan(80\circ)=\sqrt3=\tan(60\circ).

Proof

Geometric proof of Morrie's law

ABCDEFGHI

with side length

1

and let

M

be the midpoint of

AB

,

L

the midpoint

BF

and

J

the midpoint of

BD

. The inner angles of the nonagon equal

140\circ

and furthermore

\gamma=\angleFBM=80\circ

,

\beta=\angleDBF=40\circ

and

\alpha=\angleCBD=20\circ

(see graphic). Applying the cosinus definition in the right angle triangles

\triangleBFM

,

\triangleBDL

and

\triangleBCJ

then yields the proof for Morrie's law:[2]

\begin{align} 1&=|AB|\ &=2 ⋅ |MB|\\ &=2 ⋅ |BF|\cos(\gamma)\\ &=22|BL|\cos(\gamma)\\ &=22 ⋅ |BD|\cos(\gamma)\cos(\beta)\\ &=23 ⋅ |BJ|\cos(\gamma)\cos(\beta)\\ &=23 ⋅ |BC|\cos(\gamma)\cos(\beta)\cos(\alpha)\\ &=23 ⋅ 1\cos(\gamma)\cos(\beta)\cos(\alpha)\\ &=8 ⋅ \cos(80\circ)\cos(40\circ)\cos(20\circ) \end{align}

Algebraic proof of the generalised identity

Recall the double angle formula for the sine function

\sin(2\alpha)=2\sin(\alpha)\cos(\alpha).

Solve for

\cos(\alpha)

\cos(\alpha)=\sin(2\alpha)
2\sin(\alpha)

.

It follows that:

\begin{align} \cos(2\alpha)&=

\sin(4\alpha)
2\sin(2\alpha)

\\[6pt] \cos(4\alpha)&=

\sin(8\alpha)
2\sin(4\alpha)

\\ &\vdots\\ \cos\left(2n-1\alpha\right) &=

\sin\left(2n\alpha\right)
2\sin\left(2n-1\alpha\right)

. \end{align}

Multiplying all of these expressions together yields:

\cos(\alpha)\cos(2\alpha)\cos(4\alpha)\cos\left(2n-1\alpha\right)=

\sin(2\alpha)
2\sin(\alpha)

\sin(4\alpha)
2\sin(2\alpha)

\sin(8\alpha)
2\sin(4\alpha)

\sin\left(2n\alpha\right)
2\sin\left(2n-1\alpha\right)

.

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,

n-1
\prod
k=0

\cos\left(2k\alpha\right)=

\sin\left(2n\alpha\right)
2n\sin(\alpha)

,

which is equivalent to the generalization of Morrie's law.

See also

\alpha=2-nx

on Morrie's law

References

Further reading

Notes and References

  1. W. A. Beyer, J. D. Louck, and D. Zeilberger, A Generalization of a Curiosity that Feynman Remembered All His Life, Math. Mag. 69, 43–44, 1996. (JSTOR)
  2. 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)