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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