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
The polynomial discriminant is then
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.
![画像:RadicalBox[{-, {{(, {sqrt(, 3, )}, )}, /, {(, 16, )}}, +, {sqrt(, {-, {1, /, {(, 256, )}}}, )}}, 3]+RadicalBox[{-, {{(, {sqrt(, 3, )}, )}, /, {(, 16, )}}, -, {sqrt(, {-, {1, /, {(, 256, )}}}, )}}, 3]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/TrigonometryAnglesPi9/Inline16.svg){kind=link}
![画像:RadicalBox[{-, {{(, {sqrt(, 3, )}, )}, /, {(, 16, )}}, +, {1, /, {(, 16, )}}, i}, 3]-RadicalBox[{{{(, {sqrt(, 3, )}, )}, /, {(, 16, )}}, +, {1, /, {(, 16, )}}, i}, 3]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/TrigonometryAnglesPi9/Inline19.svg){kind=link}
Similarly,
Because of the Vieta's formulas, we have the identities
(15) is known as Morrie's law.
Ramanujan found the interesting identity
![画像: cos^(1/3)((2pi)/9)+cos^(1/3)((4pi)/9)-cos^(1/3)(pi/9) =[3/2(3^(2/3)-2)]^(1/3)](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/TrigonometryAnglesPi9/NumberedEquation9.svg){kind=link}
(Borwein and Bailey 2003, p. 77; Trott 2004, p. 64).
See also
Morrie's Law, Nonagon, Nonagram, Trigonometry Angles, TrigonometryExplore with Wolfram|Alpha
More things to try:
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. https://www.mathematicaguidebooks.org/.Referenced on Wolfram|Alpha
Trigonometry Angles--Pi/9Cite this as:
Weisstein, Eric W. "Trigonometry Angles--Pi/9." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TrigonometryAnglesPi9.html