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

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.