A195693 -id:A195693 - cp's OEIS frontend

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

Clark Kimberling, Oct 12 2011

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.

			0.272971849236824950408616...

Index entries for transcendental numbers.

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

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 *)

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

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

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)

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

A105199 Decimal expansion of arctan(2).

Original entry on oeis.org

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

Offset: 1

Bryan Jacobs (bryanjj(AT)gmail.com), Apr 12 2005

arctan(2) + A073000 = Pi/2.
arctan(2) is the (minimal) central angle of a regular icosahedron, which is the platonic solid having 20 faces and 12 vertices. The (minimal) central angle is AOB, where A and B are any neighboring pair of vertices and O is the center. To evaluate AOB, it is helpful to start with 12 vertices: (0,c*t,d), (d,0,c*t), (c*t,d,0) where c=(1 or -1) and d=(1 or -1) and t is the golden ratio, (1+sqrt(5))/2. For neighboring vertices, one can select (t,1,0) and (0,t,1). - Clark Kimberling, Feb 10 2009
Lesser interior angle (in radians) of a golden rhombus; i.e., either of the angles bisected by the longer diagonal. A137218 is the greater interior angle. - Rick L. Shepherd, Apr 10 2017
The apex angle in the isosceles triangle that is the triangle with angles A, B and C in which the maximum values of sin(A) + sin(B)*sin(C) is attained. The maximum value is phi (A001622) (Rabinowitz, 2007). - Amiram Eldar, Aug 04 2022
Also <5_1> in Conway et al. (1999). - Eric W. Weisstein, Nov 06 2024

			1.107148717794090503017065460...

G. C. Greubel, Table of n, a(n) for n = 1..5000
G. Boros and V. Moll, Sums of arctangents and some formulas of Ramanujan, Scientia Ser. A Math. Sci 11 (2005) 13-24 [MR2196063] eq. (2.11). [From _R. J. Mathar_, Apr 12 2010]
J. H. Conway, C. Radin, L. and Sadun, On Angles Whose Squared Trigonometric Functions Are Rational. Discr. Computat. Geom., Vol. 22, (1999), pp. 321-332.
Stanley Rabinowitz, Problem 3297, Crux Mathematicorum, Vol. 33, No. 8 (2007), pp. 486 and 488; Solution to Problem 3297 by D. J. Smeenk, ibid., Vol. 34, No. 8 (2008), pp. 499-500.
Eric Weisstein's World of Mathematics, Dehn Invariant.
Eric Weisstein's World of Mathematics, Golden Rhombus.
Index entries for transcendental numbers

Cf. A001622, A195693, A197292, A197376.

Cf. A137218 (larger interior angle of the golden rhombus).

RealDigits[ArcTan[2], 10, 105][[1]] (* Indranil Ghosh, Apr 10 2017 *)

default(realprecision, 120);
atan(2) \\ Rick L. Shepherd, Apr 10 2017

Equals Sum_{k>=1} arctan(8k/(4k^4+5)). [Boros and Moll, from R. J. Mathar, Apr 12 2010]
Equals 2*A195693. - Rick L. Shepherd, Apr 10 2017
Equals arcsin(2/sqrt(5)) = arccos(1/sqrt(5)). - Amiram Eldar, Aug 04 2022
Equals 2 - log(5) + (Integral_{x=0..2} log(1 + x^2) dx)/2. - Vaclav Kotesovec, Oct 06 2023
Equals 3*A197292 = A197376/2. - Hugo Pfoertner, Nov 06 2024

Offset corrected by R. J. Mathar, Apr 12 2010

A300074 Decimal expansion of 1/(2*sin(Pi/5)) = A121570/2.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Mar 01 2018

This is the reciprocal of A182007, and one half of A121570.
This is the ratio of the radius r of the circumscribing circle of a regular pentagon and its side length s: r/s = 1/(2*sin(Pi/5)).
A quartic number of denominator 5 and minimal polynomial 5x^4 - 5x^2 + 1. - Charles R Greathouse IV, Mar 04 2018
Appears at Schur decomposition of A=[1 2; 2 3]. - Donghwi Park, Jun 20 2018

			r/s = 0.850650808352039932181540497063011072240401403764816881836740242377...
2*r/s = A121570.

Muniru A Asiru, Table of n, a(n) for n = 0..2000
Wikipedia, Schur decomposition.
Index entries for algebraic numbers, degree 4.

Cf. A001622, A090773, A121570, A182007, A195693, A195723.

RealDigits[1/(2 Sin[Pi/5]), 10, 111][[1]] (* Robert G. Wilson v, Jul 15 2018 *)

1/(2*sin(Pi/5)) \\ Charles R Greathouse IV, Mar 04 2018

sqrt((5+sqrt(5))/10) \\ Charles R Greathouse IV, Mar 04 2018

r/s = 1/A182007 = A121570/2 = (2*phi - 1)*sqrt(2 + phi)/5, with the golden ratio phi = (1 + sqrt(5))/2 = A001622.
From Amiram Eldar, Feb 08 2022: (Start)
Equals cos(arccot(phi)) = cos(arctan(1/phi)) = cos(A195693).
Equals sin(arctan(phi)) = sin(arccot(1/phi)) = sin(A195723). (End)
Equals Product_{k>=1} (1 + (-1)^k/A090773(k)). - Amiram Eldar, Nov 23 2024

A188593 Decimal expansion of (diagonal)/(shortest side) of a golden rectangle.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 04 2011

A rectangle of length L and width W is a golden rectangle if L/W = r = (1+sqrt(5))/2. The diagonal has length D = sqrt(L^2+W^2), so D/W = sqrt(r^2+1) = sqrt(r+2).
Largest root of x^4 - 5x^2 + 5. - Charles R Greathouse IV, May 07 2011
This is the case n=10 of (Gamma(1/n)/Gamma(2/n))*(Gamma((n-1)/n)/Gamma((n-2)/n)) = 2*cos(Pi/n). - Bruno Berselli, Dec 13 2012
Edge length of a pentagram (regular star pentagon) with unit circumradius. - Stanislav Sykora, May 07 2014
The ratio diagonal/side of the shortest diagonal in a regular 10-gon. - Mohammed Yaseen, Nov 04 2020

			1.902113032590307144232878666758764286811397268251...

Chai Wah Wu, Table of n, a(n) for n = 1..10001
Michael Penn, On the fifth root of the identity matrix, YouTube video, 2022.
Eric Weisstein's World of Mathematics, Golden Rectangle.
Eric Weisstein's World of Mathematics, Pentagram.
Index entries for algebraic numbers, degree 4.

Cf. A001622 (decimal expansion of the golden ratio), A019881.
Cf. A188594 (D/W for the silver rectangle, r=1+sqrt(2)).
Cf. A090771, A195693, A195723, A296184.

SetDefaultRealField(RealField(100)); Sqrt((5+Sqrt(5))/2); // G. C. Greubel, Nov 02 2018

r = (1 + 5^(1/2))/2; RealDigits[(2 + r)^(1/2), 10, 130][[1]]
RealDigits[Sqrt[GoldenRatio+2],10,130][[1]] (* Harvey P. Dale, Oct 27 2023 *)

PARI
```
sqrt((5+sqrt(5))/2)
```

Equals 2*A019881. - Mohammed Yaseen, Nov 04 2020
Equals csc(A195693) = sec(A195723). - Amiram Eldar, May 28 2021
Equals i^(1/5) + i^(-1/5). - Gary W. Adamson, Jul 08 2022
Equals sqrt(2 + phi) = sqrt(A296184), with phi = A001622. - Wolfdieter Lang, Aug 28 2022
Equals Product_{k>=0} ((10*k + 2)*(10*k + 8))/((10*k + 1)*(10*k + 9)). - Antonio Graciá Llorente, Feb 24 2024
Equals Product_{k>=1} (1 - (-1)^k/A090771(k)). - Amiram Eldar, Nov 23 2024

A195723 Decimal expansion of arctan(golden ratio).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 23 2011

The polar angle, in radians, of the cone circumscribed to a regular icosahedron from one of its vertices. - Stanislav Sykora, Feb 15 2014

The angle between the diagonal and the shorter side of a golden rectangle. - Amiram Eldar, May 18 2021

			arctan((1+sqrt(5))/2) = 1.0172219678978513677227...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Index entries for transcendental numbers

Cf. A001622, A195693, A243445.

SetDefaultRealField(RealField(100)); Arctan((1+Sqrt(5))/2); // G. C. Greubel, Aug 20 2018

r=GoldenRatio; N[ArcTan[r],100]
RealDigits[%] (* A195723 *)

atan((1+sqrt(5))/2) \\ G. C. Greubel, Aug 20 2018

Equals arccos(sqrt((5-sqrt(5))/10)). - Stanislav Sykora, Feb 15 2014

Equals Pi/2 - A195693. - Amiram Eldar, May 18 2021

A195692 Decimal expansion of arccos(1/phi), where phi = (1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 22 2011

Every cyclic quadrilateral all of whose angles are greater than arccos((sqrt(5)-1)/2) admits a 3 × 1 grid dissection into three cyclic quadrilaterals [Thm. 2.3 in Choi et al. p. 2]. - Michel Marcus, Aug 13 2019
The base angle of the isosceles triangle of smallest perimeter which circumscribes a semicircle (DeTemple, 1992). - Amiram Eldar, Jan 22 2022
Smallest positive root of the equation sin(x) = cot(x). - Wolfe Padawer, Apr 11 2023

			arccos(1/phi) = 0.904556894302381364127316795661958721...
cos(0.904556894302381364127316795661958721...) = 1/(golden ratio) = 0.618...
sec(0.904556894302381364127316795661958721...) = (golden ratio) = 1.618...

Erica Choi, Dan Ismailescu, Jiho Lee and Joonsoo Lee, Grid dissections of tangential quadrilaterals, arXiv:1908.02251 [math.MG], 2019.
Duane W. DeTemple, The Triangle of Smallest Perimeter which Circumscribes a Semicircle, The Fibonacci Quarterly, Vol. 30, No. 3 (1992), p. 274.
Index entries for transcendental numbers

Cf. A001622, A019669, A139339, A175288, A195693, A195694, A197762.

r = 1/GoldenRatio;
N[ArcCos[r], 100]
RealDigits[%]

acos(2/(sqrt(5)+1)) \\ Charles R Greathouse IV, Nov 21 2024

From Amiram Eldar, Feb 07 2022: (Start)
Equals Pi/2 - A175288.
Equals arcsin(1/sqrt(phi)).
Equals arctan(sqrt(phi)). (End)

Terms replaced with intended terms by Rick L. Shepherd, Jan 30 2013

A195694 Decimal expansion of arccos(-1/r), where r = (1+sqrt(5))/2 (the golden ratio).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 22 2011

Let c1 and c2 be the constants at A195692 and A195694; then c1+c2 = Pi.

			arccos(-1/r) = 2.2370357592874118743353265876175441...

Index entries for transcendental numbers

Cf. A001622, A175288, A195692, A195693.

(See A195692.)
RealDigits[ArcCos[-1/GoldenRatio],10,120][[1]] (* Harvey P. Dale, Jan 15 2012 *)

acos(-2/(sqrt(5)+1)) \\ Charles R Greathouse IV, Nov 21 2024

cp's OEIS Frontend

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

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A105199 Decimal expansion of arctan(2).

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A300074 Decimal expansion of 1/(2*sin(Pi/5)) = A121570/2.

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A188593 Decimal expansion of (diagonal)/(shortest side) of a golden rectangle.

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A195723 Decimal expansion of arctan(golden ratio).

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A195692 Decimal expansion of arccos(1/phi), where phi = (1+sqrt(5))/2 (the golden ratio).

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