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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

cos(pi/(13))=(64x^6-32x^5-80x^4+32x^3+24x^2-6x-1)_6
(4)
cot(pi/(13))=(13x^(12)-286x^(10)+1287x^8-1716x^6+715x^4-78x^2+1)_(12)
(5)
csc(pi/(13))=(3x^(12)-364x^(10)+2912x^8-9984x^6+16640x^4-13312x^2+4096)_(12)
(6)
sec(pi/(13))=(x^6+6x^5-24x^4-32x^3+80x^2+32x-64)_4
(7)
sin(pi/(13))=(4096x^(12)-13312x^(10)+16640x^8-9984x^6+2912x^4-364x^2+13)_7
(8)
tan(pi/(13))=(x^(12)-78x^(10)+715x^8-1716x^6+1287x^4-286x^2+13)_7.
(9)

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

cos^2(pi/(13))+cos^2((3pi)/(13))+cos^2((4pi)/(13))=1/8(11+sqrt(13))
(11)
sin(pi/(13))+sin((3pi)/(13))+sin((4pi)/(13))=sqrt(1/8(13+3sqrt(13)))
(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

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