A003957 -id:A003957 - cp's OEIS frontend

A332501 Decimal expansion of the number u' in [0,2 Pi] such that the line normal to the graph of y = sin x at (u', sin u') passes through the point (3 Pi/4,0).

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

Offset: 1

Clark Kimberling, May 05 2020

Let S and C denote the graphs of y = sin x and y = cos x. For each point (u, sin u) on S, let S(u) be the line normal to S at (u, sin u), and let (snc u, cos(snc u)) be the point of intersection of S(u) and C. Let d(u) be the distance from (u,sin u) to (snc u, cos(snc u)). We call d(u) the u-normal distance from S to C and note that in [0,Pi], there is a unique number u' such that d(u') > d(u) for all real numbers u except those of the form u' + k*Pi. We call d(u') the maximal normal distance between sine and cosine, and we call snc the sine-normal-to-cosine function.

The distance from (u',sin u') to its reflection in (3 Pi/4,0) is the maximal normal distance between sine and cosine. This distance is slightly greater than 1. See A332500.

			2.7257370567999252496746385812...

Cf. A332500, A086751, A332503, A177870, A003957.

Mathematica
```
(* See A332500. *)
```

3/4*Pi+solve(x=0,1,cos(x)-x)/2 \\ Gleb Koloskov, Jun 17 2021

Equals (3/4)*Pi + d/2 = A177870 + A003957/2, where d is the Dottie number. - Gleb Koloskov, Jun 17 2021

A377479 Decimal expansion of the smallest positive real root of the equation cos(x) + x*sin(x) = 0.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Oct 29 2024

The absolute value of the x-coordinate of the tangent point between the cosine graph and the straight line through the origin.

			2.79838604578388713672024890313957...

Index entries for transcendental numbers

Cf. A003957, A115365, A197002, A377480, A377481.

ndigits=100; First[RealDigits[First[x/.NSolve[Cos[x]+x Sin[x]==0,x,ndigits]],10,ndigits]]
(* or *)
RealDigits[BesselJZero[-3/2, 1], 10, 100][[1]] (* Vaclav Kotesovec, Oct 31 2024 *)

\\ Note: besseljzero not guaranteed to work here since -3/2 < 0.
solve(x=2,3, cos(x)+x*sin(x)) \\ Charles R Greathouse IV, Jan 23 2025

A121967 Binary expansion of root of cos x = x.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0

Offset: 1

Ben Branman, Sep 25 2008

Cf. A003957.

BaseForm[FindRoot[Cos[x] == x, {x, {1}}, WorkingPrecision -> 1000], 2]

A125579 Decimal expansion of positive root of x^3 = cos(x).

Original entry on oeis.org

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

Offset: 0

Denton J. Dailey (djd1497(AT)aol.com), Jan 03 2007

			0.865474033101614446620685901186228747792911931818935...

Cf. A003957, A124597.

RealDigits[FindRoot[Cos[x] == x^3, {x, {.7, 1}}, WorkingPrecision -> 120][[1, 2, 1]], 10, 111][[1]]

solve(x=0,1,cos(x)-x^3) \\ Charles R Greathouse IV, Apr 16 2014

x^3 = cos(x)

A125580 Decimal expansion of positive root of x^3 = sin(x).

Original entry on oeis.org

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

Offset: 0

Denton J. Dailey (djd1497(AT)aol.com), Jan 03 2007

Three real roots: X = 0, +-0.9286263087...

			0.928626308731734426029349532702654495005680...

Cf. A003957, A124597.

RealDigits[FindRoot[Sin[x] == x^3, {x, {.8, 1}}, WorkingPrecision -> 120][[1, 2, 1]], 10, 111][[1]]

solve(x=.9,1,sin(x)-x^3) \\ Charles R Greathouse IV, Apr 16 2014

x^3 = sin(x)

A133741 Decimal expansion of offset at which two unit disks overlap by half each's area.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Sep 22 2007

			0.8079455065990344186379234801326308858044719291481968445...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Max Chicky Fang, Closed Form for Half-Area Overlap Offset of 2 Unit Disks, arXiv:2403.10529 [math.GM], 2024.
Eric Weisstein's World of Mathematics, Circle-Circle Intersection

Cf. A003957. Equals twice A086751.

d0 = d /. FindRoot[ 2*ArcCos[d/2] - d/2*Sqrt[4 - d^2] == Pi/2, {d, 1}, WorkingPrecision -> 110]; RealDigits[d0][[1]][[1 ;; 105]] (* Jean-François Alcover, Oct 26 2012, after Eric W. Weisstein *)

default(realprecision, 100); solve(x=0,1, 2*acos(x/2) - (x/2)*sqrt(4-x^2) - Pi/2) \\ G. C. Greubel, Nov 16 2018

d=solve(x=0,1,cos(x)-x);sqrt(2-2*sqrt(1-d^2)) \\ Gleb Koloskov, Feb 27 2021

Equals sqrt(1+A003957) - sqrt(1-A003957) = sqrt(2-2*sqrt(1-A003957^2)) = 2*A086751. - Gleb Koloskov, Feb 26 2021

A138284 Decimal expansion of the negative real part of z0, the smallest second-quadrant solution of z = Cos(z).

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Mar 12 2008

z0 is a repelling fixed point of Cos(z). The only fixed point on the real axis is 0.73908... (A003957), which is an attracting fixed point.

			2.486885698908560230696957...

Cf. A138285 (imaginary part).

z0 = FindRoot[{Re[Cos[x+I*y]]==x, Im[Cos[x+I*y]]==y}, {{x,-2},{y,2}}, WorkingPrecision->150]; RealDigits[z0[[1,2]]]

A138285 Decimal expansion of the imaginary part of z0, the smallest second-quadrant solution of z = Cos(z).

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Mar 12 2008

z0 is a repelling fixed point of Cos(z). The only fixed point on the real axis is 0.73908... (A003957), which is an attracting fixed point.

			1.809361341295703319016276...

Cf. A138284 (real part).

z0 = FindRoot[{Re[Cos[x+I*y]]==x, Im[Cos[x+I*y]]==y}, {{x,-2},{y,2}}, WorkingPrecision->150]; RealDigits[z0[[2,2]]]

A184952 High water marks in A177413.

Original entry on oeis.org

0, 1, 2, 4, 40, 52, 82, 4839, 5813, 8366, 11153, 46254, 1040968, 12925493

Offset: 1

Ben Branman, Dec 21 2011

12925493 does not appear in A177413 until n=292558.

			The first few terms of the continued fraction of the Dottie number are 0, 1, 2, 1, 4, 1, 40 of which the high water marks are 0, 1, 2, 4, 40...

Eric Weisstein's World of Mathematics, Dottie Number

Cf. A003957, A177413, A185185.

z = x /. FindRoot[x == Cos[x], {x, 0},
   WorkingPrecision -> 100000]; data = ContinuedFraction[z];
g[list_] :=
Delete[list,
  Transpose[{DeleteCases[
     Table[If[list[[n]] < list[[n - 1]], n, no], {n, 2,
       Length[list]}], no]}]];

Offset changed by Andrew Howroyd, Aug 10 2024

A185185 Indices of record values in A177413.

Original entry on oeis.org

0, 1, 2, 4, 6, 19, 29, 53, 2035, 3995, 5328, 10141, 14675, 292557

Offset: 1

Ben Branman, Dec 22 2011

Eric Weisstein's World of Mathematics, Dottie Number.

Cf. A184952, A177413, A003957.

z = x /. FindRoot[x == Cos[x], {x, 0},
   WorkingPrecision -> 1000000]; data = ContinuedFraction[z];
g[list_] :=
Delete[list,
  Transpose[{DeleteCases[
     Table[If[list[[n]] < list[[n - 1]], n, no], {n, 2,
       Length[list]}], no]}]]; p =
FixedPoint[g, DeleteDuplicates[data]]; Flatten[
Map[Position[data, #, 1, 1] &, p, {1}]]-1

A177413(a(n)) = A184952(n).

Offset changed by Andrew Howroyd, Aug 10 2024

cp's OEIS Frontend

A332501 Decimal expansion of the number u' in [0,2 Pi] such that the line normal to the graph of y = sin x at (u', sin u') passes through the point (3 Pi/4,0).

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A377479 Decimal expansion of the smallest positive real root of the equation cos(x) + x*sin(x) = 0.

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A121967 Binary expansion of root of cos x = x.

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A125579 Decimal expansion of positive root of x^3 = cos(x).

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A125580 Decimal expansion of positive root of x^3 = sin(x).

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A133741 Decimal expansion of offset at which two unit disks overlap by half each's area.

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A138284 Decimal expansion of the negative real part of z0, the smallest second-quadrant solution of z = Cos(z).

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Mathematica

A138285 Decimal expansion of the imaginary part of z0, the smallest second-quadrant solution of z = Cos(z).

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