Trigonometric functions of 
radians for 
an integer not divisible by 3 (e.g., 
and 
) cannot be expressed in terms of sums, products,
and finite root extractions on rational
numbers because 9 is not a product of distinct Fermat
primes. This also means that the regular nonagon
is not a constructible polygon.
However, exact expressions involving roots of complex numbers can still be derived using the trigonometric identity
 |
(1)
|
Let 
and 
.
Then the above identity gives the cubic equation
 |
(2)
|
 |
(3)
|
This cubic is of the form
 |
(4)
|
where
 |  |  |
(5)
|
 |  |  |
(6)
|
The polynomial discriminant is then
 |
(7)
|
There are therefore three real distinct roots, which are approximately 
, 0.3420, and 0.6428. We want the one in the first quadrant, which is approximately 0.3420.
 |  | ![RadicalBox[{-, {{(, {sqrt(, 3, )}, )}, /, {(, 16, )}}, +, {sqrt(, {-, {1, /, {(, 256, )}}}, )}}, 3]+RadicalBox[{-, {{(, {sqrt(, 3, )}, )}, /, {(, 16, )}}, -, {sqrt(, {-, {1, /, {(, 256, )}}}, )}}, 3]](/images/equations/TrigonometryAnglesPi9/Inline16.svg) |
(8)
|
 |  | ![RadicalBox[{-, {{(, {sqrt(, 3, )}, )}, /, {(, 16, )}}, +, {1, /, {(, 16, )}}, i}, 3]-RadicalBox[{{{(, {sqrt(, 3, )}, )}, /, {(, 16, )}}, +, {1, /, {(, 16, )}}, i}, 3]](/images/equations/TrigonometryAnglesPi9/Inline19.svg) |
(9)
|
 |  | ![2^(-4/3)(RadicalBox[{i, -, {sqrt(, 3, )}}, 3]-RadicalBox[{i, +, {sqrt(, 3, )}}, 3])](/images/equations/TrigonometryAnglesPi9/Inline22.svg) |
(10)
|
 |  |  |
(11)
|
Similarly,
 |  | ![2^(-4/3)(RadicalBox[{1, +, i, {sqrt(, 3, )}}, 3]+RadicalBox[{1, -, i, {sqrt(, 3, )}}, 3])](/images/equations/TrigonometryAnglesPi9/Inline28.svg) |
(12)
|
 |  |  |
(13)
|
Because of the Vieta's formulas, we have the identities
 |
(14)
|
 |
(15)
|
 |
(16)
|
(15) is known as Morrie's law.
Ramanujan found the interesting identity
![cos^(1/3)((2pi)/9)+cos^(1/3)((4pi)/9)-cos^(1/3)(pi/9)
=[3/2(3^(2/3)-2)]^(1/3)](/images/equations/TrigonometryAnglesPi9/NumberedEquation9.svg) |
(17)
|
(Borwein and Bailey 2003, p. 77; Trott 2004, p. 64).
