cp's OEIS Frontend

A195696 Decimal expansion of arccos(sqrt(1/3)) and of arcsin(sqrt(2/3)) and arctan(sqrt(2)).

%I A195696 #94 Nov 20 2024 23:43:15
%S A195696 9,5,5,3,1,6,6,1,8,1,2,4,5,0,9,2,7,8,1,6,3,8,5,7,1,0,2,5,1,5,7,5,7,7,
%T A195696 5,4,2,4,3,4,1,4,6,9,5,0,1,0,0,0,5,4,9,0,9,5,9,6,9,8,1,2,9,3,2,1,9,1,
%U A195696 2,0,4,5,9,0,3,9,7,6,4,5,5,3,8,7,3,9,1,6,0,2,5,8,5,6,2,8,0,7,3,4
%N A195696 Decimal expansion of arccos(sqrt(1/3)) and of arcsin(sqrt(2/3)) and arctan(sqrt(2)).
%C A195696 Angle (in radians) between an edge and (the normal of) a face of the regular tetrahedron. - _R. J. Mathar_, Feb 23 2012
%C A195696 Also known as magic angle; root of P_2(cos(theta)), with P_2(x) being second-order Legendre polynomial. - _Stanislav Sykora_, May 25 2012
%C A195696 From _Stanislav Sykora_, Nov 14 2013: (Start)
%C A195696 Also the angle between the body diagonal of a cube and an incident edge, and therefore the polar angle of the cone circumscribed to a cube from one of its vertices.
%C A195696 Also half of the tetrahedral angle (A156546).
%C A195696 In nuclear magnetic resonance, the angle, with respect to the direction of the main magnetic field, under which a solid sample needs to be spun in order to average to zero unwanted dipole-dipole spin interactions (the magic angle spinning, or MAS, technique). (End)
%C A195696 Also <3_2> in Conway et al. (1999). - _Eric W. Weisstein_, Nov 06 2024
%H A195696 G. C. Greubel, <a href="/A195696/b195696.txt">Table of n, a(n) for n = 0..10000</a>
%H A195696 John H. Conway, Charles Radin, and Lorenzo Sadun, <a href="https://arxiv.org/abs/ph/9812019">On Angles Whose Squared Trigonometric Functions are Rational</a>, arXiv:math-ph/9812019, 1998. See also <a href="https://doi.org/10.1007/PL00009463">Discr. Computat. Geom.</a> (1999) Vol. 22, 321-332.
%H A195696 H. B. Dwight, <a href="https://archive.org/details/tablesofintegral0000unse/page/120/mode/1up">Tables of Integrals and other Mathematical Data</a>. 507.13, 507.22 in Inverse trigonometric functions. New York: Macmillan Publishing, p. 120, 1961.
%H A195696 Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020-2022; p. 53.
%H A195696 C. O. Horgan and J. G. Murphy, <a href="https://www.ams.org/journals/notices/202201/rnoti-p22.pdf">On an angle with magical properties</a>, Notices Amer. Math. Soc., 69:1 (2022), 22-25.
%H A195696 G. Jacob Martens, <a href="https://arxiv.org/abs/2112.09553">Rational right triangles and the Congruent Number Problem</a>, arXiv:2112.09553 [math.GM], 2021, see section 2.6.2 The Trinity vectors, the magic angle, equation (34).
%H A195696 Stanislav Sýkora, <a href="http://www.ebyte.it/stan/blog12to14.html#14Dec31">Magnetic Resonance on OEIS</a>, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
%H A195696 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DehnInvariant.html">Dehn Invariant</a>.
%H A195696 Wikipedia, <a href="http://en.wikipedia.org/wiki/Tetrahedron">Tetrahedron</a>
%H A195696 Wikipedia, <a href="http://en.wikipedia.org/wiki/Magic_angle">Magic angle</a>
%H A195696 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A195696 Equals i*log(sqrt(1/3) - i*sqrt(2/3)). - _Andrea Pinos_, Nov 03 2023
%F A195696 Equals A156546/2 = 2*A197739. - _Hugo Pfoertner_, Nov 06 2024
%e A195696 0.9553166181245092781638571025157577... (= 54.73561031... degrees).
%t A195696 RealDigits[ArcTan[Sqrt[2]],10,120][[1]] (* _Harvey P. Dale_, Dec 13 2014 *)
%o A195696 (PARI) atan(sqrt(2)) \\ _G. C. Greubel_, Jul 05 2017
%o A195696 (Magma) [Arccos(Sqrt(1/3))]; // _G. C. Greubel_, Nov 18 2017
%Y A195696 Cf. A156546, A195695, A197739, A210974 (in degrees), A243445.
%K A195696 nonn,cons
%O A195696 0,1
%A A195696 _Clark Kimberling_, Sep 23 2011