Rational values of the arccosine function

• Autorzy: Juan Varona
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Arcosine; cosine; rational; irrational; 11J72; 33B10

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2006
• Tom: 4
• Numer: 2
• Strony: 319-322

Daty

wydano

2006-06-01

online

2006-06-01

Twórcy

autor

Juan Varona

• Universidad de La Rioja

Bibliografia

• [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.
• [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.
• [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.

Identyfikatory

DOI

10.2478/s11533-006-0011-z