Trigonometry Angles--Pi/13
Trigonometric functions of npi/13 for n 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
| sin(nalpha)=(-1)^((n-1)/2)T_n(sinalpha), |
(1)
|
where T_n(x) is a Chebyshev polynomial of the first kind. Plugging in n=13 gives
| sin(13alpha)=sinalpha(4096sin^(12)alpha-13312sin^(10)alpha+16640sin^8alpha-9984sin^6alpha+2912sin^4alpha-364sin^2alpha+13). |
(2)
|
Letting alpha=pi/13 and x=sin^2alpha then gives
| sinpi=0=4096x^6-13312x^5+16640x^4-9984x^3+2912x^2-364x+13. |
(3)
|
But this is a sextic equation has a cyclic Galois group, and so x, and hence sin(pi/13), 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 pi/13 can be given explicitly as the polynomial roots
From one of the Newton-Girard formulas,
| sin(pi/(13))sin((2pi)/(13))sin((3pi)/(13))sin((4pi)/(13))sin((5pi)/(13))sin((6pi)/(13))=(sqrt(13))/(64) cos(pi/(13))cos((2pi)/(13))cos((3pi)/(13))cos((4pi)/(13))cos((5pi)/(13))cos((6pi)/(13))=1/(64) tan(pi/(13))tan((2pi)/(13))tan((3pi)/(13))tan((4pi)/(13)) tan((5pi)/(13))tan((6pi)/(13))=sqrt(13). |
(10)
|
The trigonometric functions of pi/13 also obey the identities
(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 Resource. https://mathworld.wolfram.com/TrigonometryAnglesPi13.html