Trigonometric functions of for 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 using the trigonometric identity
(1) |
(2) |
(3) |
(4) |
(5) |
(6) |
(7) |
(8) |
(9) | |||
(10) |
The Discriminant is then
(11) |
so there are three distinct Real Roots. Finding the first one,
(12) |
Writing
(13) |
(14) |
See also Heptagon
© 1996-9 Eric W. Weisstein