Trigonometry Angles--Pi/11
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 hendecagon 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 quintic 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/11]].
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)
|
 |
(11)
|
 |
(12)
|
The trigonometric functions of 
also obey
the identity
 |
(13)
|
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A4 root lattice