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.
@article{bwmeta1.element.doi-10_2478_s11533-006-0011-z,
author = {Juan Varona},
title = {Rational values of the arccosine function},
journal = {Open Mathematics},
volume = {4},
year = {2006},
pages = {319-322},
zbl = {1130.11037},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-006-0011-z}
}
Juan Varona. Rational values of the arccosine function. Open Mathematics, Tome 4 (2006) pp. 319-322. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-006-0011-z/
[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. | Zbl 59.0946.01
[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. | Zbl 0742.11001
[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.