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

%I A197133 #33 Feb 12 2025 15:01:54
%S A197133 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,
%T A197133 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,
%U A197133 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
%N A197133 Decimal expansion of least x>0 having sin(x) = sin(2*x)^2.
%C A197133 The Mathematica program includes a graph.
%C A197133 Guide for least x>0 satisfying sin(b*x) = sin(c*x)^2 for selected numbers b and c:
%C A197133 b.....c.......x
%C A197133 1.....2.......A197133
%C A197133 1.....3.......A197134
%C A197133 1.....4.......A197135
%C A197133 1.....5.......A197251
%C A197133 1.....6.......A197252
%C A197133 1.....7.......A197253
%C A197133 1.....8.......A197254
%C A197133 2.....1.......A105199, x=arctan(2)
%C A197133 2.....3.......A019679, x=Pi/12
%C A197133 2.....4.......A197255
%C A197133 2.....5.......A197256
%C A197133 2.....6.......A197257
%C A197133 2.....7.......A197258
%C A197133 2.....8.......A197259
%C A197133 3.....1.......A197260
%C A197133 3.....2.......A197261
%C A197133 3.....4.......A197262
%C A197133 3.....5.......A197263
%C A197133 3.....6.......A197264
%C A197133 3.....7.......A197265
%C A197133 3.....8.......A197266
%C A197133 4.....1.......A197267
%C A197133 4.....2.......A195693, x=arctan(1/(golden ratio))
%C A197133 4.....3.......A197268
%C A197133 1.....4*Pi....A197522
%C A197133 1.....3*Pi....A197571
%C A197133 1.....2*Pi....A197572
%C A197133 1.....3*Pi/2..A197573
%C A197133 1.....Pi......A197574
%C A197133 1.....Pi/2....A197575
%C A197133 1.....Pi/3....A197326
%C A197133 1.....Pi/4....A197327
%C A197133 1.....Pi/6....A197328
%C A197133 2.....Pi/3....A197329
%C A197133 2.....Pi/4....A197330
%C A197133 2.....Pi/6....A197331
%C A197133 3.....Pi/3....A197332
%C A197133 3.....Pi/6....A197375
%C A197133 3.....Pi/4....A197333
%C A197133 1.....1/2.....A197376
%C A197133 1.....1/3.....A197377
%C A197133 1.....2/3.....A197378
%C A197133 Pi....1.......A197576
%C A197133 Pi....2.......A197577
%C A197133 Pi....3.......A197578
%C A197133 2*Pi..1.......A197585
%C A197133 3*Pi..1.......A197586
%C A197133 4*Pi..1.......A197587
%C A197133 Pi/2..1.......A197579
%C A197133 Pi/2..2.......A197580
%C A197133 Pi/2..1/2.....A197581
%C A197133 Pi/3..Pi/4....A197379
%C A197133 Pi/3..Pi/6....A197380
%C A197133 Pi/4..Pi/3....A197381
%C A197133 Pi/4..Pi/6....A197382
%C A197133 Pi/6..Pi/3....A197383
%C A197133 Pi/6..Pi/4..........., x=1
%C A197133 Pi/3..1.......A197384
%C A197133 Pi/3..2.......A197385
%C A197133 Pi/3..3.......A197386
%C A197133 Pi/3..1/2.....A197387
%C A197133 Pi/3..1/3.....A197388
%C A197133 Pi/3..2/3.....A197389
%C A197133 Pi/4..1.......A197390
%C A197133 Pi/4..2.......A197391
%C A197133 Pi/4..3.......A197392
%C A197133 Pi/4..1/2.....A197393
%C A197133 Pi/4..1/3.....A197394
%C A197133 Pi/4..2/3.....A197411
%C A197133 Pi/4..1/4.....A197412
%C A197133 Pi/6..1.......A197413
%C A197133 Pi/6..2.......A197414
%C A197133 Pi/6..3.......A197415
%C A197133 Pi/6..1/2.....A197416
%C A197133 Pi/6..1/3.....A197417
%C A197133 Pi/6..2/3.....A197418
%C A197133 Cf. A197476 for a similar table for sin(b*x) = sin(c*x)^2.
%H A197133 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A197133 From _Gleb Koloskov_, Sep 15 2021: (Start)
%F A197133 Equals arcsin(2*sin(arcsin(3*sqrt(3)/8)/3)/sqrt(3))
%F A197133      = arcsin(2*sin(arcsin(A333322)/3)/A002194). (End)
%e A197133 0.272971849236824950408616...
%t A197133 b = 1; c = 2; f[x_] := Sin[x]
%t A197133 t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .1, .3}, WorkingPrecision -> 100]
%t A197133 RealDigits[t] (* A197133 *)
%t A197133 Plot[{f[b*x], f[c*x]^2}, {x, 0, Pi}]
%t A197133 (* Second program: *)
%t A197133 RealDigits[ ArcSec[ Root[16 - 16 x^2 + x^6, 3]], 10, 100] // First (* _Jean-François Alcover_, Feb 19 2013 *)
%o A197133 (PARI) asin(2*sin(asin(3*sqrt(3)/8)/3)/sqrt(3)) \\ _Gleb Koloskov_, Sep 15 2021
%o A197133 (PARI) asin(polrootsreal(4*x^3-4*x+1)[2]) \\ _Charles R Greathouse IV_, Feb 12 2025
%Y A197133 Cf. A002194, A197134, A197476 (cos), A333322.
%K A197133 nonn,cons
%O A197133 0,1
%A A197133 _Clark Kimberling_, Oct 12 2011
%E A197133 Edited and a(99) corrected by _Georg Fischer_, Jul 28 2021