A197388 -id:A197388

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A197133

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

+10
94

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

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.

LINKS

Table of n, a(n) for n=0..100.

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)

EXAMPLE

x=0.272971849236824950408616...

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 *)

PROG

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

CROSSREFS

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

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 12 2011

EXTENSIONS

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

STATUS

approved

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