TOPICS
Search

Trigonometry Angles--Pi/11


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

 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

cos(pi/(11))=(32x^5-16x^4-32x^3+12x^2+6x-1)_5
(4)
cot(pi/(11))=(11x^(10)-165x^8+462x^6-330x^4+55x^2-1)_(10)
(5)
csc(pi/(11))=(11x^(10)-220x^8+1232x^6-2816x^4+2816x^2-1024)_(10)
(6)
sec(pi/(11))=(x^5-6x^4-12x^3+32x^2+16x-32)_3
(7)
sin(pi/(11))=(1024x^(10)-2816x^8+2816x^6-1232x^4+220x^2-11)_6
(8)
tan(pi/(11))=(x^(10)-55x^8+330x^6-462x^4+165x^2-11)_6.
(9)

From one of the Newton-Girard formulas,

sin(pi/(11))sin((2pi)/(11))sin((3pi)/(11))sin((4pi)/(11))sin((5pi)/(11))=sqrt((11)/(1024))=(sqrt(11))/(32)
(10)
cos(pi/(11))cos((2pi)/(11))cos((3pi)/(11))cos((4pi)/(11))cos((5pi)/(11))=1/(32)
(11)
tan(pi/(11))tan((2pi)/(11))tan((3pi)/(11))tan((4pi)/(11))tan((5pi)/(11))=sqrt(11).
(12)

The trigonometric functions of pi/11 also obey the identity

 tan((3pi)/(11))+4sin((2pi)/(11))=sqrt(11).
(13)

See also

Hendecagon, Trigonometry Angles, Trigonometry

Explore with Wolfram|Alpha

References

Beyer, W. H. "Trigonometry." CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987.

Cite this as:

Weisstein, Eric W. "Trigonometry Angles--Pi/11." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TrigonometryAnglesPi11.html

Subject classifications