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Trigonometry Angles--Pi/23


Trigonometric functions of pi/p for p prime have an especially complicated Galois-minimal representation. In particular, the case cos(pi/23) requires approximately 500 MB of space using the Wolfram Language command Developer`TrigToRadicals[Cos[Pi/23]]. However, they can be expressed concisely as algebraic numbers. For example, letting (P(x))_n denote the nth root of the polynomial P(x) using the ordering of the Wolfram Language's Root function, sin(pi/23) is given by

 sin(pi/(23))=(4194304x^(22)-24117248x^(20)+60293120x^(18)-85917696x^(16)+76873728x^(14)-44843008x^(12)+17145856x^(10)-4209920x^8+631488x^6-52624x^4+2024x^2-23)_(12),

and cos(pi/23) by

 cos(pi/(23))=(2048x^(11)-1024x^(10)-5120x^9+2304x^8+4608x^7-1792x^6-1792x^5+560x^4+280x^3-60x^2-12x+1)_(12).

See also

Trigonometry Angles

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Cite this as:

Weisstein, Eric W. "Trigonometry Angles--Pi/23." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TrigonometryAnglesPi23.html

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