Trigonometric functions of for an integer cannot be expressed in terms of sums, products, and finite root extractions on real rational numbers because 11 is not a Fermat Prime. This also means that the Undecagon is not a Constructible Polygon.
However, exact expressions involving roots of complex numbers can still be derived using the trigonometric identity
(1) |
(2) |
(3) |
(4) |
(5) |
(6) | |
(7) | |
(8) |
(9) |
See also Undecagon
References
Beyer, W. H. ``Trigonometry.'' CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press,
1987.