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A197476 Decimal expansion of least x>0 having cos(x) = cos(2*x)^2.

%I A197476 #30 Feb 15 2025 08:32:14
%S A197476 1,1,3,7,7,4,3,9,3,2,9,0,5,4,5,5,5,5,7,7,8,9,4,4,9,8,6,0,0,5,5,0,0,8,
%T A197476 3,4,9,5,8,4,8,0,4,2,9,0,3,4,9,5,7,5,2,7,2,0,1,5,1,8,2,5,2,6,7,3,6,0,
%U A197476 9,8,3,4,7,3,4,7,2,7,2,1,7,7,9,8,8,0,3,2,8,0,5,1,7,6,4,4,7,2,7
%N A197476 Decimal expansion of least x>0 having cos(x) = cos(2*x)^2.
%C A197476 The Mathematica program includes a graph.  Guide for least x>0 satisfying cos(b*x) = cos(c*x)^2, for selected b and c:
%C A197476 b.....c......x
%C A197476 1.....2.......A197476
%C A197476 1.....3.......A197477
%C A197476 1.....4.......A197478
%C A197476 2.....1.......A000796, Pi
%C A197476 2.....3.......A197479
%C A197476 2.....4.......A197480
%C A197476 3.....1.......A019669, Pi/2
%C A197476 3.....2.......A197482
%C A197476 3.....4.......A197483
%C A197476 4.....1.......A168229, arctan(sqrt(7))
%C A197476 4.....2.......A019669, Pi/2
%C A197476 4.....3.......A019679, Pi/12
%C A197476 4.....6.......A197485
%C A197476 4.....8.......A197486
%C A197476 6.....2.......A003881, Pi/4
%C A197476 6.....3.......A019670, Pi/3, tangency point
%C A197476 6.....4.......A197488
%C A197476 6.....8.......A197489
%C A197476 1.....4*Pi....A197334
%C A197476 1.....3*Pi....A197335
%C A197476 1.....2*Pi....A197490
%C A197476 1.....3*Pi/2..A197491
%C A197476 1.....Pi......A197492
%C A197476 1.....Pi/2....A197493
%C A197476 1.....Pi/3....A197494
%C A197476 1.....Pi/4....A197495
%C A197476 1.....2*Pi/3..A197506
%C A197476 2.....3*Pi....A197507
%C A197476 2.....3*Pi/2..A197508
%C A197476 2.....2*Pi....A197509
%C A197476 2.....Pi......A197510
%C A197476 2.....Pi/2....A197511
%C A197476 2.....Pi/3....A197512
%C A197476 2.....Pi/4....A197513
%C A197476 2.....Pi/6....A197514
%C A197476 Pi....1.......A197515
%C A197476 Pi....2.......A197516
%C A197476 Pi....1/2.....A197517
%C A197476 2*Pi..1.......A197518
%C A197476 2*Pi..2.......A197519
%C A197476 2*Pi..3.......A197520
%C A197476 Pi/2..Pi/3....A197521
%C A197476 Pi/2..Pi/6....3
%C A197476 Pi/3..1.......A197582
%C A197476 Pi/3..2.......A197583
%C A197476 Pi/3..1/3.....A197584
%C A197476 See A197133 for a guide for least x>0 satisfying sin(b*x) = sin(c*x)^2 for selected b and c.
%H A197476 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A197476 1.137743932905455557789449860055008349584...
%t A197476 b = 1; c = 2; f[x_] := Cos[x]
%t A197476 t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, 1.1, 1.3}, WorkingPrecision -> 200]
%t A197476 RealDigits[t] (* A197476 *)
%t A197476 Plot[{f[b*x], f[c*x]^2}, {x, 0, 2}]
%t A197476 (* or *)
%t A197476 RealDigits[ ArcCos[ ((19 - 3*Sqrt[33])^(1/3) + (19 + 3*Sqrt[33])^(1/3) - 2)/6], 10, 99] // First (* _Jean-François Alcover_, Feb 19 2013 *)
%Y A197476 Cf. A197133.
%K A197476 nonn,cons
%O A197476 1,3
%A A197476 _Clark Kimberling_, Oct 15 2011
%E A197476 Edited by _Georg Fischer_, Jul 28 2021