A086751 -id:A086751 - cp's OEIS frontend

A332500 Decimal expansion of the maximal normal distance between sine and cosine; see Comments.

1, 0, 9, 4, 9, 9, 8, 9, 8, 4, 3, 7, 0, 8, 7, 2, 4, 2, 8, 6, 5, 0, 4, 0, 8, 3, 0, 0, 7, 1, 5, 5, 2, 4, 6, 7, 1, 2, 9, 1, 0, 5, 1, 4, 0, 6, 0, 7, 0, 5, 4, 3, 6, 0, 2, 0, 6, 5, 8, 0, 3, 3, 4, 2, 9, 5, 5, 1, 8, 7, 5, 4, 4, 9, 6, 2, 2, 1, 4, 0, 5, 4, 1, 3, 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.
For each (v, cos v) on C, let C(u) be the line normal to C at (v, cos v), and let (cns v, sin(cns v)) be the point of intersection of C(u) and S. Let e(v) be the distance from (v, cos v) to (cns v, sin(cns v)). We call d(v) the v-normal distance from C to S and note that there exists a unique number v' that maximizes e, and e(v') = d(u'). We call cns the cosine-normal-to-sine function. The numbers u' and v' are given in A332501 and A332503.
Note that the maximal normal distance (see Example) exceeds the normal distance from (Pi/2,1) in sine to (Pi/2,0) in cosine - possibly a surprise!

			2.72573705679992524967463858129656... = the number u in [0,2 Pi] such that the line normal to S at (u, sin u) passes through the point (3 Pi/4,0); cf. A332501.
0.4039727532995172093189617400663... = sin u; cf. A086751.
1.0949989843708724286504083007155... = maximal normal distance between sine and cosine.
1.9866519235847646080193264936226... = snc u; cf A332503.

Index entries for transcendental numbers.

Cf. A086751, A332501, A332502, A332503, A332504, A332505, A332506, A332507, A003957.

Plot[{Sin[x], Cos[x]}, {x, -Pi, 3 Pi}, AspectRatio -> Automatic,
ImageSize -> 600, PlotLabel -> "sine and cosine"]
t = Table[x = x /. FindRoot[Cos[x] == -x Sec[u] + u Sec[u] + Sin[u], {x, 0}], {u, -2 Pi, 2 Pi, Pi/101}];
ListPlot[t, PlotLabel -> "y \[Equal] snc(x)"]
ListPlot[Cos[t], PlotLabel -> "y \[Equal] cos(snc(x))"]
t = Table[x = x /. FindRoot[Sin[x] == x Csc[u] - u Csc[u] + Cos[u], {x, 0.1}], {u, -2 Pi + .01, 2 Pi - .01, Pi/101}];
ListPlot[t, PlotLabel -> "y \[Equal] cns(x)"]
ListPlot[Sin[t], PlotLabel -> "y \[Equal] sin(cns(x))"]
u = u /. FindRoot[0 == (-3 Pi/4) Sec[u] + u Sec[u] + Sin[u], {u, 1}, WorkingPrecision ->120]  (* A332501 *)
y = Sin[u]  (* A086751 *)
d = 2*Sqrt[(u - 3 Pi/4)^2 + y^2]  (* A332500 *)
RealDigits[u][[1]]  (* A332501 *)
RealDigits[y][[1]]  (* A086751 *)
RealDigits[d][[1]]  (* A332500 *)

my(d=solve(x=0,1,cos(x)-x)); sqrt(d^2+2-2*sqrt(1-d^2)) \\ Gleb Koloskov, Jun 16 2021

d(u') = 2*sqrt((u - 3 Pi/4)^2 + (sin u)^2).

Equals sqrt(d^2+2-2*sqrt(1-d^2)) where d = A003957. - Gleb Koloskov, Jun 16 2021

A332503 Decimal expansion of the number v such that the maximal normal distance between sine and cosine is the distance between (u, sin u) and (v, sin v), where u is the number u' in A332501; see Comments.

Original entry on oeis.org

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

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. See A332500.

			v = 1.96870408298082320586768578622442620...

Cf. A332500, A332501, A086751.

u = u /. FindRoot[u - 3 Pi/4 == Sin[u], {u, 1}, WorkingPrecision -> 120]  (* A332501 *)
v = 3 Pi/2 - u   (* A332503 *)
RealDigits[v][[1]]

v = 3Pi/4 - u, where u is given by A332501.

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).

Original entry on oeis.org

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

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

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

cp's OEIS Frontend

A332500 Decimal expansion of the maximal normal distance between sine and cosine; see Comments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A332503 Decimal expansion of the number v such that the maximal normal distance between sine and cosine is the distance between (u, sin u) and (v, sin v), where u is the number u' in A332501; see Comments.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula