A197133 Decimal expansion of least x>0 having sin(x) = sin(2*x)^2.

2, 7, 2, 9, 7, 1, 8, 4, 9, 2, 3, 6, 8, 2, 4, 9, 5, 0, 4, 0, 8, 6, 1, 6, 8, 0, 6, 0, 8, 3, 8, 6, 9, 8, 3, 1, 0, 4, 7, 4, 0, 6, 6, 5, 1, 9, 6, 6, 4, 4, 0, 1, 8, 2, 7, 6, 6, 8, 0, 0, 0, 1, 1, 4, 8, 4, 3, 3, 5, 9, 2, 7, 0, 1, 0, 2, 2, 0, 8, 9, 0, 4, 3, 5, 9, 2, 4, 4, 8, 6, 4, 3, 1, 9, 4, 0, 5, 6, 9, 0, 8

Offset: 0

Author

Clark Kimberling, Oct 12 2011

Keywords

• nonn
• cons

Comments

The Mathematica program includes a graph.

Guide for least x>0 satisfying sin(b*x) = sin(c*x)^2 for selected numbers b and c:

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

1.....2.......A197133

1.....3.......A197134

1.....4.......A197135

1.....5.......A197251

1.....6.......A197252

1.....7.......A197253

1.....8.......A197254

2.....1.......A105199, x=arctan(2)

2.....3.......A019679, x=Pi/12

2.....4.......A197255

2.....5.......A197256

2.....6.......A197257

2.....7.......A197258

2.....8.......A197259

3.....1.......A197260

3.....2.......A197261

3.....4.......A197262

3.....5.......A197263

3.....6.......A197264

3.....7.......A197265

3.....8.......A197266

4.....1.......A197267

4.....2.......A195693, x=arctan(1/(golden ratio))

4.....3.......A197268

1.....4*Pi....A197522

1.....3*Pi....A197571

1.....2*Pi....A197572

1.....3*Pi/2..A197573

1.....Pi......A197574

1.....Pi/2....A197575

1.....Pi/3....A197326

1.....Pi/4....A197327

1.....Pi/6....A197328

2.....Pi/3....A197329

2.....Pi/4....A197330

2.....Pi/6....A197331

3.....Pi/3....A197332

3.....Pi/6....A197375

3.....Pi/4....A197333

1.....1/2.....A197376

1.....1/3.....A197377

1.....2/3.....A197378

Pi....1.......A197576

Pi....2.......A197577

Pi....3.......A197578

2*Pi..1.......A197585

3*Pi..1.......A197586

4*Pi..1.......A197587

Pi/2..1.......A197579

Pi/2..2.......A197580

Pi/2..1/2.....A197581

Pi/3..Pi/4....A197379

Pi/3..Pi/6....A197380

Pi/4..Pi/3....A197381

Pi/4..Pi/6....A197382

Pi/6..Pi/3....A197383

Pi/6..Pi/4..........., x=1

Pi/3..1.......A197384

Pi/3..2.......A197385

Pi/3..3.......A197386

Pi/3..1/2.....A197387

Pi/3..1/3.....A197388

Pi/3..2/3.....A197389

Pi/4..1.......A197390

Pi/4..2.......A197391

Pi/4..3.......A197392

Pi/4..1/2.....A197393

Pi/4..1/3.....A197394

Pi/4..2/3.....A197411

Pi/4..1/4.....A197412

Pi/6..1.......A197413

Pi/6..2.......A197414

Pi/6..3.......A197415

Pi/6..1/2.....A197416

Pi/6..1/3.....A197417

Pi/6..2/3.....A197418

Cf. A197476 for a similar table for sin(b*x) = sin(c*x)^2.

Examples

			0.272971849236824950408616...
		

Links

• Index entries for transcendental numbers.

Crossrefs

Cf. A002194, A197134, A197476 (cos), A333322.

Programs

Mathematica

b = 1; c = 2; f[x_] := Sin[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .1, .3}, WorkingPrecision -> 100]
RealDigits[t] (* A197133 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, Pi}]
(* Second program: *)
RealDigits[ ArcSec[ Root[16 - 16 x^2 + x^6, 3]], 10, 100] // First (* Jean-François Alcover, Feb 19 2013 *)

PARI

asin(2*sin(asin(3*sqrt(3)/8)/3)/sqrt(3)) \\ Gleb Koloskov, Sep 15 2021

PARI

asin(polrootsreal(4*x^3-4*x+1)[2]) \\ Charles R Greathouse IV, Feb 12 2025

Formula

From Gleb Koloskov, Sep 15 2021: (Start)

Equals arcsin(2*sin(arcsin(3*sqrt(3)/8)/3)/sqrt(3))

= arcsin(2*sin(arcsin(A333322)/3)/A002194). (End)

Extensions

Edited and a(99) corrected by Georg Fischer, Jul 28 2021