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)
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Let and . Then the above identity gives the cubic equation
(2)
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(3)
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This cubic is of the form
(4)
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where
(5)
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(6)
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The polynomial discriminant is then
(7)
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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)
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(9)
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(10)
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(11)
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Similarly,
(12)
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(13)
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Because of the Vieta's formulas, we have the identities
(14)
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(15)
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(16)
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(15) is known as Morrie's law.
Ramanujan found the interesting identity
(17)
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(Borwein and Bailey 2003, p. 77; Trott 2004, p. 64).