Trigonometric functions of for an integer cannot be expressed in terms of sums, products, and finite root extractions on real rational numbers because 13 is not a Fermat prime. This also means that the tridecagon is not a constructible polygon.
However, exact expressions involving roots of complex numbers can still be derived using the multiple-angle formula
(1)
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where is a Chebyshev polynomial of the first kind. Plugging in gives
(2)
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Letting and then gives
(3)
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But this is a sextic 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/13]].
The trigonometric functions of can be given explicitly as the polynomial roots
(4)
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(5)
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(6)
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(7)
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(8)
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(9)
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From one of the Newton-Girard formulas,
(10)
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The trigonometric functions of also obey the identities
(11)
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(12)
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(P. Rolli, pers. comm., Dec. 27, 2004).