A197739 -id:A197739 - cp's OEIS frontend

A195696 Decimal expansion of arccos(sqrt(1/3)) and of arcsin(sqrt(2/3)) and arctan(sqrt(2)).

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

Offset: 0

Clark Kimberling, Sep 23 2011

Angle (in radians) between an edge and (the normal of) a face of the regular tetrahedron. - R. J. Mathar, Feb 23 2012
Also known as magic angle; root of P_2(cos(theta)), with P_2(x) being second-order Legendre polynomial. - Stanislav Sykora, May 25 2012
From Stanislav Sykora, Nov 14 2013: (Start)
Also the angle between the body diagonal of a cube and an incident edge, and therefore the polar angle of the cone circumscribed to a cube from one of its vertices.
Also half of the tetrahedral angle (A156546).
In nuclear magnetic resonance, the angle, with respect to the direction of the main magnetic field, under which a solid sample needs to be spun in order to average to zero unwanted dipole-dipole spin interactions (the magic angle spinning, or MAS, technique). (End)
Also <3_2> in Conway et al. (1999). - Eric W. Weisstein, Nov 06 2024

			0.9553166181245092781638571025157577... (= 54.73561031... degrees).

G. C. Greubel, Table of n, a(n) for n = 0..10000
John H. Conway, Charles Radin, and Lorenzo Sadun, On Angles Whose Squared Trigonometric Functions are Rational, arXiv:math-ph/9812019, 1998. See also Discr. Computat. Geom. (1999) Vol. 22, 321-332.
H. B. Dwight, Tables of Integrals and other Mathematical Data. 507.13, 507.22 in Inverse trigonometric functions. New York: Macmillan Publishing, p. 120, 1961.
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2022; p. 53.
C. O. Horgan and J. G. Murphy, On an angle with magical properties, Notices Amer. Math. Soc., 69:1 (2022), 22-25.
G. Jacob Martens, Rational right triangles and the Congruent Number Problem, arXiv:2112.09553 [math.GM], 2021, see section 2.6.2 The Trinity vectors, the magic angle, equation (34).
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Eric Weisstein's World of Mathematics, Dehn Invariant.
Wikipedia, Tetrahedron
Wikipedia, Magic angle
Index entries for transcendental numbers

Cf. A156546, A195695, A197739, A210974 (in degrees), A243445.

[Arccos(Sqrt(1/3))]; // G. C. Greubel, Nov 18 2017

RealDigits[ArcTan[Sqrt[2]],10,120][[1]] (* Harvey P. Dale, Dec 13 2014 *)

atan(sqrt(2)) \\ G. C. Greubel, Jul 05 2017

Equals i*log(sqrt(1/3) - i*sqrt(2/3)). - Andrea Pinos, Nov 03 2023

Equals A156546/2 = 2*A197739. - Hugo Pfoertner, Nov 06 2024

A197833 Decimal expansion of least x > 0 having sin(2x) = 3Pisin(3Pi*x).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 19 2011

For a discussion and guide to related sequences, see A197739.

			x=0.16484394670494001260057035619088988930523218...

Cf. A197739.

b = 1; c = 3*Pi;
f[x_] := Cos[b*x]^2; g[x_] := Sin[c*x]^2; s[x_] := f[x] + g[x];
r = x /. FindRoot[b*Sin[2 b*x] == c*Sin[2 c*x], {x, .16, .17}, WorkingPrecision -> 110]
RealDigits[r](* A197833 *)
m = s[r]
RealDigits[m](* A197834 *)
Plot[{b*Sin[2 b*x], c*Sin[2 c*x]}, {x, 0, .6}]
d = m/2; t = x /. FindRoot[s[x] == d, {x, .4, .42}, WorkingPrecision -> 110]
RealDigits[t] (* A197835 *)
Plot[{s[x], d}, {x, 0, .7}, AxesOrigin -> {0, 0}]
d = m/3; t = x /. FindRoot[s[x] == d, {x, .91, .92}, WorkingPrecision -> 110]
RealDigits[t](* A197836 *)
Plot[{s[x], d}, {x, 0, Pi/2}, AxesOrigin -> {0, 0}]
d = 1; t = x /. FindRoot[s[x] == d, {x, .4, .5}, WorkingPrecision -> 110]
RealDigits[t] (* A197837 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1/2; t = x /. FindRoot[s[x] == d, {x, .95, .96}, WorkingPrecision -> 110]
RealDigits[t] (* A197838 *)
Plot[{s[x], d}, {x, 0, 1}, AxesOrigin -> {0, 0}]

A197821 Decimal expansion of least x > 0 having sin(2x) = Pisin(2Pix).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 19 2011

For a discussion and guide to related sequences, see A197739.

			0.45929534126210755105483751035805264919200...

Cf. A197739.

b = 1; c = Pi;
f[x_] := Cos[b*x]^2; g[x_] := Sin[c*x]^2; s[x_] := f[x] + g[x];
r = x /. FindRoot[b*Sin[2 b*x] == c*Sin[2 c*x], {x, .45, .46}, WorkingPrecision -> 110]
RealDigits[r]  (* A197821 *)
m = s[r]
RealDigits[m]  (* A197822 *)
Plot[{b*Sin[2 b*x], c*Sin[2 c*x]}, {x, 0, Pi}]
d = m/2; t = x /. FindRoot[s[x] == d, {x, .7, .8}, WorkingPrecision -> 110]
RealDigits[t]   (* A197823 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = m/3; t = x /. FindRoot[s[x] == d, {x, .8, .9}, WorkingPrecision -> 110]
RealDigits[t]  (* A197824 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1; t = x /. FindRoot[s[x] == d, {x, .7, .8}, WorkingPrecision -> 110]
RealDigits[t]  (* A197726 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1/2; t = x /. FindRoot[s[x] == d, {x, .89, 9.1}, WorkingPrecision -> 110]
RealDigits[t]  (* A197826 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]

Definition corrected by Georg Fischer, Aug 10 2021

A197827 Decimal expansion of least x > 0 having sin(2x) = 2Pisin(4Pi*x).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 19 2011

For a discussion and guide to related sequences, see A197739.

			0.24405505512124668685356429784849535656...

Cf. A197739.

b = 1; c = 2 Pi;
f[x_] := Cos[b*x]^2; g[x_] := Sin[c*x]^2; s[x_] := f[x] + g[x];
r = x /. FindRoot[b*Sin[2 b*x] == c*Sin[2 c*x], {x, .24, .25}, WorkingPrecision -> 110]
RealDigits[r]  (* A197827 *)
m = s[r]
RealDigits[m]  (* A197828 *)
Plot[{b*Sin[2 b*x], c*Sin[2 c*x]}, {x, 0, Pi}]
d = m/2; t = x /. FindRoot[s[x] == d, {x, .4, .42}, WorkingPrecision -> 110]
RealDigits[t] (* A197829 *)
Plot[{s[x], d}, {x, 0, .7}, AxesOrigin -> {0, 0}]
d = m/3; t = x /. FindRoot[s[x] == d, {x, .91, .92}, WorkingPrecision -> 110]
RealDigits[t] (* A197830 *)
Plot[{s[x], d}, {x, 0, Pi/2}, AxesOrigin -> {0, 0}]
d = 1; t = x /. FindRoot[s[x] == d, {x, .4, .5}, WorkingPrecision -> 110]
RealDigits[t] (* A197700 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1/2; t = x /. FindRoot[s[x] == d, {x, .93, .94}, WorkingPrecision -> 110]
RealDigits[t] (* A197832 *)
Plot[{s[x], d}, {x, 0, 1}, AxesOrigin -> {0, 0}]

Definition corrected by Georg Fischer, Aug 10 2021

A197488 Decimal expansion of least x > 0 having cos(6x) = (cos 4x)^2.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 15 2011

The Mathematica program includes a graph. See A197476 for a guide for the least x > 0 satisfying cos(b*x) = (cos(c*x))^2 for selected b and c.

Also the solution of the least x > 0 satisfying (cos(x))^2 + (sin(3x))^2 = 1/2. See A197739. - Clark Kimberling, Oct 19 2011

			0.9218840880158607848199692488618106365729956...

Index entries for transcendental numbers.

Cf. A197476.

b = 6; c = 4; f[x_] := Cos[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .92, .93}, WorkingPrecision -> 100]
RealDigits[t] (* A197488 *)
Plot[{f[b*x], f[c*x]^2}, {x, 0, 1}]
RealDigits[ ArcCos[ Root[ -2 + 8#^2 - 6#^4 + #^6 & , 5]/2], 10, 99] // First (* Jean-François Alcover, Feb 19 2013 *)

Digits from a(92) on corrected by Jean-François Alcover, Feb 19 2013

A197758 Decimal expansion of least x>0 having sin(2x)=4*sin(8x).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 19 2011

For a discussion and guide to related sequences, see A197739.

			x=0.37145829403348635250583272851951240980...

Cf. A197739

b = 1; c = 4;
f[x_] := Cos[b*x]^2; g[x_] := Sin[c*x]^2; s[x_] := f[x] + g[x];
r = x /. FindRoot[b*Sin[2 b*x] == c*Sin[2 c*x], {x, .37, .38}, WorkingPrecision -> 110]
RealDigits[r]  (* A197758 *)
m = s[r]
RealDigits[m]  (* A197759 *)
Plot[{b*Sin[2 b*x], c*Sin[2 c*x]}, {x, 0, Pi}]
d = m/2; t = x /. FindRoot[s[x] == d, {x, 0.64, 0.65}, WorkingPrecision -> 110]
RealDigits[t]  (* A197760 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = m/3; t = x /. FindRoot[s[x] == d, {x, 0.72, 0.73}, WorkingPrecision -> 110]
RealDigits[t]  (* A197761 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1; t = x /. FindRoot[s[x] == d, {x, 0.6, 0.7}, WorkingPrecision -> 110]
RealDigits[t]  (* A019692, pi/5 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1/2; t = x /. FindRoot[s[x] == d, {x, 0.6, 0.8}, WorkingPrecision -> 110]
RealDigits[t]   (* A003881 *)
Plot[{s[x], d}, {x, 0, 1}, AxesOrigin -> {0, 0}]
RealDigits[ ArcTan[ Sqrt[ Root[17#^3 - 109#^2 + 115# - 15&, 1] ] ], 10, 99] // First (* Jean-François Alcover, Feb 27 2013 *)

A197589 Decimal expansion of least x>0 satisfying f(x)=m/2, where m is the maximal value of the function f(x)=cos(x)^2+sin(2x)^2.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 18 2011

For a discussion and guide to related sequences, see A197739.

			x=1.12868019433775284470060498453346294726...

Cf. A197739, A195700.

b = 1; c = 2;
f[x_] := Cos[b*x]^2; g[x_] := Sin[c*x]^2; s[x_] := f[x] + g[x];
r = x /. FindRoot[b*Sin[2 b*x] == c*Sin[2 c*x], {x, .65, .66}, WorkingPrecision -> 110]
RealDigits[r]  (* A195700, arcsin(sqrt(3/8)) *)
m = s[r]
RealDigits[m]
Rationalize[{m, m/2, m/3, 2 m/3, m/4, 3 m/4}]
Plot[{b*Sin[2 b*x], c*Sin[2 c*x]}, {x, 0, Pi}]
d = m/2; t = x /. FindRoot[s[x] == d, {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[t]  (* A197589 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = m/3; t = x /. FindRoot[s[x] == d, {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[t]  (* A197591 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1; t = x /. FindRoot[s[x] == d, {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[t] (* A019670, pi/3 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
d = 1/2; t = x /. FindRoot[s[x] == d, {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[t]  (* A197592 *)
Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]

A197591 Decimal expansion of least x>0 satisfying f(x)=m/3, where m is the maximal value of f(x)=cos(x)^2+sin(2x)^2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 18 2011

For a discussion and guide to related sequences, see A197739.

			x=1.2253054542403810364502801560817479178...

Cf. A197739, A197589.

Mathematica
```
(See the program at A197589.)
```

A197592 Decimal expansion of least x>0 having (cos(x))^2+(sin(2x))^2=1/2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 18 2011

For a discussion and guide to related sequences, see A197739.

			x=1.2333519040018934291489892079836642261607...

Cf. A197739, A197589.

(See the program at A197589.)
RealDigits[ ArcCos[ Sqrt[ (5-Sqrt[17])/2 ]/2 ], 10, 99] // First (* Jean-François Alcover, Feb 27 2013 *)

A197759 Decimal expansion of the maximum of (cos(x))^2+(sin(4x))^2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 19 2011

For a discussion and guide to related sequences, see A197739.

			x=1.8610480166527243646008234352213209286165...

Cf. A197739.

Mathematica
```
(See the program at A197758.)
```

cp's OEIS Frontend

A195696 Decimal expansion of arccos(sqrt(1/3)) and of arcsin(sqrt(2/3)) and arctan(sqrt(2)).

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A197833 Decimal expansion of least x > 0 having sin(2*x) = 3*Pi*sin(3*Pi*x).

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A197821 Decimal expansion of least x > 0 having sin(2*x) = Pi*sin(2*Pi*x).

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A197827 Decimal expansion of least x > 0 having sin(2*x) = 2*Pi*sin(4*Pi*x).

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A197488 Decimal expansion of least x > 0 having cos(6x) = (cos 4x)^2.

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A197758 Decimal expansion of least x>0 having sin(2x)=4*sin(8x).

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Mathematica

A197589 Decimal expansion of least x>0 satisfying f(x)=m/2, where m is the maximal value of the function f(x)=cos(x)^2+sin(2x)^2.

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Mathematica

A197591 Decimal expansion of least x>0 satisfying f(x)=m/3, where m is the maximal value of f(x)=cos(x)^2+sin(2x)^2.

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A197833 Decimal expansion of least x > 0 having sin(2x) = 3Pisin(3Pi*x).

A197821 Decimal expansion of least x > 0 having sin(2x) = Pisin(2Pix).

A197827 Decimal expansion of least x > 0 having sin(2x) = 2Pisin(4Pi*x).