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Trigonometry Angles--Pi/9


Trigonometric functions of npi/9 radians for n an integer not divisible by 3 (e.g., 40 degrees and 80 degrees) cannot be expressed in terms of sums, products, and finite root extractions on rational numbers because 9 is not a product of distinct Fermat primes. This also means that the regular nonagon is not a constructible polygon.

However, exact expressions involving roots of complex numbers can still be derived using the trigonometric identity

 sin(3alpha)=3sinalpha-4sin^3alpha.
(1)

Let alpha=pi/9 and x=sinalpha. Then the above identity gives the cubic equation

 4x^3-3x+1/2sqrt(3)=0
(2)
 x^3-3/4x=-1/8sqrt(3).
(3)

This cubic is of the form

 x^3+px=q,
(4)

where

p=-3/4
(5)
q=-1/8sqrt(3).
(6)

The polynomial discriminant is then

 D=(p/3)^3+(q/2)^2=-1/(256)<0.
(7)

There are therefore three real distinct roots, which are approximately -0.9848, 0.3420, and 0.6428. We want the one in the first quadrant, which is approximately 0.3420.

sin(pi/9)=RadicalBox[{-, {{(, {sqrt(, 3, )}, )}, /, {(, 16, )}}, +, {sqrt(, {-, {1, /, {(, 256, )}}}, )}}, 3]+RadicalBox[{-, {{(, {sqrt(, 3, )}, )}, /, {(, 16, )}}, -, {sqrt(, {-, {1, /, {(, 256, )}}}, )}}, 3]
(8)
=RadicalBox[{-, {{(, {sqrt(, 3, )}, )}, /, {(, 16, )}}, +, {1, /, {(, 16, )}}, i}, 3]-RadicalBox[{{{(, {sqrt(, 3, )}, )}, /, {(, 16, )}}, +, {1, /, {(, 16, )}}, i}, 3]
(9)
=2^(-4/3)(RadicalBox[{i, -, {sqrt(, 3, )}}, 3]-RadicalBox[{i, +, {sqrt(, 3, )}}, 3])
(10)
 approx 0.34202.
(11)

Similarly,

cos(pi/9)=2^(-4/3)(RadicalBox[{1, +, i, {sqrt(, 3, )}}, 3]+RadicalBox[{1, -, i, {sqrt(, 3, )}}, 3])
(12)
 approx 0.93969.
(13)

Because of the Vieta's formulas, we have the identities

 sin(pi/9)sin((2pi)/9)sin((4pi)/9)=1/8sqrt(3)
(14)
 cos(pi/9)cos((2pi)/9)cos((4pi)/9)=1/8
(15)
 tan(pi/9)tan((2pi)/9)tan((4pi)/9)=sqrt(3).
(16)

(15) is known as Morrie's law.

Ramanujan found the interesting identity

 cos^(1/3)((2pi)/9)+cos^(1/3)((4pi)/9)-cos^(1/3)(pi/9) 
 =[3/2(3^(2/3)-2)]^(1/3)
(17)

(Borwein and Bailey 2003, p. 77; Trott 2004, p. 64).


See also

Morrie's Law, Nonagon, Nonagram, Trigonometry Angles, Trigonometry

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References

Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, 2004. http://www.mathematicaguidebooks.org/.

Cite this as:

Weisstein, Eric W. "Trigonometry Angles--Pi/9." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TrigonometryAnglesPi9.html

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