Trigonometry Angles--Pi/7
Trigonometric functions of npi/7 for n an integer cannot be expressed in terms of sums, products, and finite root extractions on real rational numbers because 7 is not a Fermat prime. This also means that the heptagon is not a constructible polygon.
However, exact expressions involving roots of complex numbers can still be derived either using the trigonometric identity
| sin(nalpha)=2sin[(n-1)alpha]cosalpha-sin[(n-2)alpha] |
(1)
|
with n=7 or by expressing sin(pi/7) in terms of complex exponentials and simplifying the resulting expression. Letting (P(x))_n denote the nth root of the polynomial P(x) using the ordering of the Wolfram Language's Root function gives the following algebraic root representations for trigonometric functions with argument pi/7,
with argument 2pi/7,
and with argument 3pi/7,
Root and Galois-minimal expressions can be obtained using Wolfram Language code such as the following:
RootReduce[TrigToRadicals[Sin[Pi/7]]] Developer`TrigToRadicals[Sin[Pi/7]]
Combinations of the functions satisfy
(Bankoff and Garfunkel 1973). A sum identity is given by
Another interesting identity is given by
| [画像: cos^(1/3)((2pi)/7)-[-cos((4pi)/7)]^(1/3)-[-cos((6pi)/7)]^(1/3) =-[1/2(3·7^(1/3)-5)]^(1/3) ] |
(24)
|
![画像: cos^(1/3)((2pi)/7)-[-cos((4pi)/7)]^(1/3)-[-cos((6pi)/7)]^(1/3) =-[1/2(3·7^(1/3)-5)]^(1/3)](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/TrigonometryAnglesPi7/NumberedEquation3.svg){kind=link}
(Borwein and Bailey 2003, p. 77).
See also
Heptagon, Heptagonal Triangle, Silver Constant, Trigonometry Angles, TrigonometryExplore with Wolfram|Alpha
More things to try:
References
Bankoff, L. and Garfunkel, J. "The Heptagonal Triangle." Math. Mag. 46, 7-19, 1973.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Referenced on Wolfram|Alpha
Trigonometry Angles--Pi/7Cite this as:
Weisstein, Eric W. "Trigonometry Angles--Pi/7." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TrigonometryAnglesPi7.html