Trigonometric functions of 
for 
an integer cannot be expressed in terms of sums, products,
and finite root extractions on real rational
numbers because 13 is not a Fermat prime. This also
means that the tridecagon is not a constructible
polygon.
However, exact expressions involving roots of complex numbers can still be derived using the multiple-angle formula
 |
(1)
|
where 
is a Chebyshev polynomial of the
first kind. Plugging in 
gives
 |
(2)
|
Letting 
and 
then gives
 |
(3)
|
But this is a sextic equation has a cyclic Galois group, and so 
, and hence 
, can be expressed in terms of radicals (of complex
numbers). The explicit expression is quite complicated, but can be generated in the
Wolfram Language using Developer`TrigToRadicals[Sin[Pi/13]].
The trigonometric functions of 
can be given explicitly as the polynomial
roots
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
From one of the Newton-Girard formulas,
 |
(10)
|
The trigonometric functions of 
also obey the identities
 |  |  |
(11)
|
 |  |  |
(12)
|
(P. Rolli, pers. comm., Dec. 27, 2004).
