# Rational values of the arccosine function

Open Mathematics (2006)

- Volume: 4, Issue: 2, page 319-322
- ISSN: 2391-5455

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topJuan Varona. "Rational values of the arccosine function." Open Mathematics 4.2 (2006): 319-322. <http://eudml.org/doc/269232>.

@article{JuanVarona2006,

abstract = {We give a short proof to characterize the cases when arccos(√r), the arccosine of the squareroot of a rational number r ∈ [0, 1], is a rational multiple of π: This happens exactly if r is an integer multiple of 1/4. The proof relies on the well-known recurrence relation for the Chebyshev polynomials of the first kind.},

author = {Juan Varona},

journal = {Open Mathematics},

keywords = {11J72; 33B10},

language = {eng},

number = {2},

pages = {319-322},

title = {Rational values of the arccosine function},

url = {http://eudml.org/doc/269232},

volume = {4},

year = {2006},

}

TY - JOUR

AU - Juan Varona

TI - Rational values of the arccosine function

JO - Open Mathematics

PY - 2006

VL - 4

IS - 2

SP - 319

EP - 322

AB - We give a short proof to characterize the cases when arccos(√r), the arccosine of the squareroot of a rational number r ∈ [0, 1], is a rational multiple of π: This happens exactly if r is an integer multiple of 1/4. The proof relies on the well-known recurrence relation for the Chebyshev polynomials of the first kind.

LA - eng

KW - 11J72; 33B10

UR - http://eudml.org/doc/269232

ER -

## References

top- [1] M. Aigner and G.M. Ziegler: Proofs from THE BOOK, 3rd ed., Springer, 2004.
- [2] D.H. Lehmer: “A note on trigonometric algebraic numbers”, Amer. Math. Monthly, Vol. 40, (1933), pp. 165–166. Zbl59.0946.01
- [3] I. Niven: Irrational numbers, Carus Monographs, Vol. 11, The Mathematical Association of America (distributed by John Wiley and Sons), 1956.
- [4] I. Niven, H.S. Zuckerman and H.L. Montgomery: An introduction to the theory of numbers, 5th ed., Wiley, 1991. Zbl0742.11001
- [5] N.M. Temme: Special functions: an introduction to the classical functions of mathematical physics, John Wiley and Sons, 1996.
- [6] E. W. Weisstein: “Chebyshev polynomial of the first kind”, In: Math-World, A Wolfram Web Resource, http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html.

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