Trigonometry Angles--Pi/11

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Trigonometric functions of npi/11 for n an integer cannot be expressed in terms of sums, products, and finite root extractions on real rational numbers because 11 is not a Fermat prime. This also means that the hendecagon 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=11 gives

[画像: sin(11alpha)=sinalpha(11-220sin^2alpha+1232sin^4alpha -2816sin^6alpha+2816sin^8-1024sin^(10)alpha). ]

(2)

Letting alpha=pi/11 and x=sin^2alpha then gives

sinpi=0=11-220x+1232x^2-2816x^3+2816x^4-1024x^5.

(3)

But this quintic equation has a cyclic Galois group, and so x, and hence sin(pi/11), 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 /11]].

The trigonometric functions of pi/11 can be given explicitly as the polynomial roots

[画像:cos(pi/(11))] = (32x^5-16x^4-32x^3+12x^2+6x-1)_5

(4)

[画像:cot(pi/(11))] = (11x^(10)-165x^8+462x^6-330x^4+55x^2-1)_(10)

(5)

[画像:csc(pi/(11))] = (11x^(10)-220x^8+1232x^6-2816x^4+2816x^2-1024)_(10)

(6)

[画像:sec(pi/(11))] = (x^5-6x^4-12x^3+32x^2+16x-32)_3

(7)

[画像:sin(pi/(11))] = (1024x^(10)-2816x^8+2816x^6-1232x^4+220x^2-11)_6

(8)

[画像:tan(pi/(11))] = (x^(10)-55x^8+330x^6-462x^4+165x^2-11)_6.