For angles in degrees, cos(20)*cos(40)*cos(80) equals 1/8
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 [ edit ] cos ( 20 ∘ ) ⋅ cos ( 40 ∘ ) ⋅ cos ( 80 ∘ ) = 1 8 . {\displaystyle \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 ⋅ ∏ k = 0 n − 1 cos ( 2 k α ) = sin ( 2 n α ) sin ( α ) {\displaystyle 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° and the fact that
sin ( 160 ∘ ) sin ( 20 ∘ ) = sin ( 180 ∘ − 20 ∘ ) sin ( 20 ∘ ) = 1 , {\displaystyle {\frac {\sin(160^{\circ })}{\sin(20^{\circ })}}={\frac {\sin(180^{\circ }-20^{\circ })}{\sin(20^{\circ })}}=1,} since
sin ( 180 ∘ − x ) = sin ( x ) . {\displaystyle \sin(180^{\circ }-x)=\sin(x).} Similar identities [ edit ] A similar identity for the sine function also holds:
sin ( 20 ∘ ) ⋅ sin ( 40 ∘ ) ⋅ sin ( 80 ∘ ) = 3 8 . {\displaystyle \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 ∘ ) ⋅ tan ( 40 ∘ ) ⋅ tan ( 80 ∘ ) = 3 = tan ( 60 ∘ ) . {\displaystyle \tan(20^{\circ })\cdot \tan(40^{\circ })\cdot \tan(80^{\circ })={\sqrt {3}}=\tan(60^{\circ }).} Geometric proof of Morrie's law [ edit ] Regular nonagon A B C D E F G H I {\displaystyle ABCDEFGHI} with O {\displaystyle O} being the center of its circumcircle . Computing of the angles: 40 ∘ = 360 ∘ 9 70 ∘ = 180 ∘ − 40 ∘ 2 α = 180 ∘ − 90 ∘ − 70 ∘ = 20 ∘ β = 180 ∘ − 90 ∘ − ( 70 ∘ − α ) = 40 ∘ γ = 140 ∘ − β − α = 80 ∘ {\displaystyle {\begin{aligned}40^{\circ }&={\frac {360^{\circ }}{9}}\\70^{\circ }&={\frac {180^{\circ }-40^{\circ }}{2}}\\\alpha &=180^{\circ }-90^{\circ }-70^{\circ }=20^{\circ }\\\beta &=180^{\circ }-90^{\circ }-(70^{\circ }-\alpha )=40^{\circ }\\\gamma &=140^{\circ }-\beta -\alpha =80^{\circ }\end{aligned}}} Consider a regular nonagon A B C D E F G H I {\displaystyle ABCDEFGHI} with side length 1 {\displaystyle 1} and let M {\displaystyle M} be the midpoint of A B {\displaystyle AB} , L {\displaystyle L} the midpoint B F {\displaystyle BF} and J {\displaystyle J} the midpoint of B D {\displaystyle BD} . The inner angles of the nonagon equal 140 ∘ {\displaystyle 140^{\circ }} and furthermore γ = ∠ F B M = 80 ∘ {\displaystyle \gamma =\angle FBM=80^{\circ }} , β = ∠ D B F = 40 ∘ {\displaystyle \beta =\angle DBF=40^{\circ }} and α = ∠ C B D = 20 ∘ {\displaystyle \alpha =\angle CBD=20^{\circ }} (see graphic). Applying the cosinus definition in the right angle triangles △ B F M {\displaystyle \triangle BFM} , △ B D L {\displaystyle \triangle BDL} and △ B C J {\displaystyle \triangle BCJ} then yields the proof for Morrie's law:[2]
1 = | A B | = 2 ⋅ | M B | = 2 ⋅ | B F | ⋅ cos ( γ ) = 2 2 | B L | cos ( γ ) = 2 2 ⋅ | B D | ⋅ cos ( γ ) ⋅ cos ( β ) = 2 3 ⋅ | B J | ⋅ cos ( γ ) ⋅ cos ( β ) = 2 3 ⋅ | B C | ⋅ cos ( γ ) ⋅ cos ( β ) ⋅ cos ( α ) = 2 3 ⋅ 1 ⋅ cos ( γ ) ⋅ cos ( β ) ⋅ cos ( α ) = 8 ⋅ cos ( 80 ∘ ) ⋅ cos ( 40 ∘ ) ⋅ cos ( 20 ∘ ) {\displaystyle {\begin{aligned}1&=|AB|\\&=2\cdot |MB|\\&=2\cdot |BF|\cdot \cos(\gamma )\\&=2^{2}|BL|\cos(\gamma )\\&=2^{2}\cdot |BD|\cdot \cos(\gamma )\cdot \cos(\beta )\\&=2^{3}\cdot |BJ|\cdot \cos(\gamma )\cdot \cos(\beta )\\&=2^{3}\cdot |BC|\cdot \cos(\gamma )\cdot \cos(\beta )\cdot \cos(\alpha )\\&=2^{3}\cdot 1\cdot \cos(\gamma )\cdot \cos(\beta )\cdot \cos(\alpha )\\&=8\cdot \cos(80^{\circ })\cdot \cos(40^{\circ })\cdot \cos(20^{\circ })\end{aligned}}} Algebraic proof of the generalised identity [ edit ] Recall the double angle formula for the sine function
sin ( 2 α ) = 2 sin ( α ) cos ( α ) . {\displaystyle \sin(2\alpha )=2\sin(\alpha )\cos(\alpha ).} Solve for cos ( α ) {\displaystyle \cos(\alpha )}
cos ( α ) = sin ( 2 α ) 2 sin ( α ) . {\displaystyle \cos(\alpha )={\frac {\sin(2\alpha )}{2\sin(\alpha )}}.} It follows that:
cos ( 2 α ) = sin ( 4 α ) 2 sin ( 2 α ) cos ( 4 α ) = sin ( 8 α ) 2 sin ( 4 α ) ⋮ cos ( 2 n − 1 α ) = sin ( 2 n α ) 2 sin ( 2 n − 1 α ) . {\displaystyle {\begin{aligned}\cos(2\alpha )&={\frac {\sin(4\alpha )}{2\sin(2\alpha )}}\\[6pt]\cos(4\alpha )&={\frac {\sin(8\alpha )}{2\sin(4\alpha )}}\\&\,\,\,\vdots \\\cos \left(2^{n-1}\alpha \right)&={\frac {\sin \left(2^{n}\alpha \right)}{2\sin \left(2^{n-1}\alpha \right)}}.\end{aligned}}} Multiplying all of these expressions together yields:
cos ( α ) cos ( 2 α ) cos ( 4 α ) ⋯ cos ( 2 n − 1 α ) = sin ( 2 α ) 2 sin ( α ) ⋅ sin ( 4 α ) 2 sin ( 2 α ) ⋅ sin ( 8 α ) 2 sin ( 4 α ) ⋯ sin ( 2 n α ) 2 sin ( 2 n − 1 α ) . {\displaystyle \cos(\alpha )\cos(2\alpha )\cos(4\alpha )\cdots \cos \left(2^{n-1}\alpha \right)={\frac {\sin(2\alpha )}{2\sin(\alpha )}}\cdot {\frac {\sin(4\alpha )}{2\sin(2\alpha )}}\cdot {\frac {\sin(8\alpha )}{2\sin(4\alpha )}}\cdots {\frac {\sin \left(2^{n}\alpha \right)}{2\sin \left(2^{n-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,
∏ k = 0 n − 1 cos ( 2 k α ) = sin ( 2 n α ) 2 n sin ( α ) , {\displaystyle \prod _{k=0}^{n-1}\cos \left(2^{k}\alpha \right)={\frac {\sin \left(2^{n}\alpha \right)}{2^{n}\sin(\alpha )}},} which is equivalent to the generalization of Morrie's law.
References [ edit ] ^ 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 ) ^ 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 ) Further reading [ edit ] Glen Van Brummelen: Trigonometry: A Very Short Introduction . Oxford University Press, 2020, ISBN 9780192545466 , pp. 79–83 Ernest C. Anderson: Morrie's Law and Experimental Mathematics . In: Journal of recreational mathematics , 1998 External links [ edit ]