Trigonometry Angles--Pi/9

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.

			(8)
			(9)
			(10)
			(11)

Similarly,

			(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

(17)

(Borwein and Bailey 2003, p. 77; Trott 2004, p. 64).