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### Trigonometry angles--pi/13

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 bederived using the multiple-angle formula(1)where is a Chebyshev polynomial of the first kind. Plugging in gives(2)Letting and then gives(3)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)(5)(6)(7)(8)(9)From one of the Newton-Girard formulas,(10)The trigonometric functions of also obey the identities(11)(12)(P. Rolli,..

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