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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 be derived 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, pers. comm., Dec. 27, 2004).

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## Cite this as:

Weisstein, Eric W. "Trigonometry Angles--Pi/13." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TrigonometryAnglesPi13.html