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

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, 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, 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

Offset: 1

Author

Clark Kimberling, Oct 15 2011

Keywords

• nonn
• cons

Comments

The Mathematica program includes a graph. Guide for least x>0 satisfying cos(b*x) = cos(c*x)^2, for selected b and c:

b.....c......x

1.....2.......A197476

1.....3.......A197477

1.....4.......A197478

2.....1.......A000796, Pi

2.....3.......A197479

2.....4.......A197480

3.....1.......A019669, Pi/2

3.....2.......A197482

3.....4.......A197483

4.....1.......A168229, arctan(sqrt(7))

4.....2.......A019669, Pi/2

4.....3.......A019679, Pi/12

4.....6.......A197485

4.....8.......A197486

6.....2.......A003881, Pi/4

6.....3.......A019670, Pi/3, tangency point

6.....4.......A197488

6.....8.......A197489

1.....4*Pi....A197334

1.....3*Pi....A197335

1.....2*Pi....A197490

1.....3*Pi/2..A197491

1.....Pi......A197492

1.....Pi/2....A197493

1.....Pi/3....A197494

1.....Pi/4....A197495

1.....2*Pi/3..A197506

2.....3*Pi....A197507

2.....3*Pi/2..A197508

2.....2*Pi....A197509

2.....Pi......A197510

2.....Pi/2....A197511

2.....Pi/3....A197512

2.....Pi/4....A197513

2.....Pi/6....A197514

Pi....1.......A197515

Pi....2.......A197516

Pi....1/2.....A197517

2*Pi..1.......A197518

2*Pi..2.......A197519

2*Pi..3.......A197520

Pi/2..Pi/3....A197521

Pi/2..Pi/6....3

Pi/3..1.......A197582

Pi/3..2.......A197583

Pi/3..1/3.....A197584

See A197133 for a guide for least x>0 satisfying sin(b*x) = sin(c*x)^2 for selected b and c.

Examples

			1.137743932905455557789449860055008349584...
		

Links

• Index entries for transcendental numbers.

Crossrefs

Cf. A197133.

Programs

Mathematica

b = 1; c = 2; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, 1.1, 1.3}, WorkingPrecision -> 200]
RealDigits[t] (* A197476 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 2}]
(* or *)
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 *)

Extensions

Edited by Georg Fischer, Jul 28 2021