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 A197133 Decimal expansion of least x>0 having sin(x) = sin(2*x)^2. 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 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. Sequence in context: A170854 A215140 A278419 * A178206 A245976 A245216 Adjacent sequences:  A197130 A197131 A197132 * A197134 A197135 A197136 KEYWORD nonn,cons,changed AUTHOR Clark Kimberling, Oct 12 2011 EXTENSIONS Edited and a(99) corrected by Georg Fischer, Jul 28 2021 STATUS approved

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Last modified September 27 17:05 EDT 2021. Contains 347693 sequences. (Running on oeis4.)