A332500 -id:A332500 - cp's OEIS frontend

A332506 Decimal expansion of the number snc(2 Pi), where snc is the sine-normal-to-cosine function; see A332500.

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

Offset: 1

Clark Kimberling, May 05 2020

			snc(2 Pi) = 5.544100173964425835269974678885...

Cf. A332500, A003957, A332523 (numerators of convergents), A332524 (denominators of convergents).

u = u /. FindRoot[ u + Cos[u] == 2 Pi, {u, 0}, WorkingPrecision -> 150]
RealDigits[u][[1]]

snc(2 Pi) = 2 Pi + snc(0), where snc(0) = Dottie number (A003957).

A332504 Decimal expansion of the number u in [0,2 Pi] such that snc(u) = 0, where snc is the sine-normal-to-cosine function; see A332500.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, May 05 2020

			u = 0.4787224241179413703304116508523039...

Cf. A332500, A332505.

u = u /. FindRoot[ u Sec[u] + Sin[u] == 1, {u, 0}, WorkingPrecision -> 150]
RealDigits[u][[1]]

u = x - Pi, where snc(x) = Pi, as in A332505.

A332505 Decimal expansion of the number u in [0,2 Pi] such that snc(u) = Pi, where snc is the sine-normal-to-cosine function; see A332500.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 05 2020

			u = 3.620315077707734608793055034131...

Cf. A332500, A332504.

u = u /. FindRoot[ -1 == -Pi Sec[u] + u Sec[u] + Sin[u], {u, 0}, WorkingPrecision -> 150]
d = RealDigits[u][[1]]

u = Pi + x, where snc(x) = 0 as in A332504.

A086751 Decimal expansion of the solution to x*sqrt(1-x^2) + arcsin(x) = Pi/4, or the length of the line connecting the origin to the center of the chord of a circle, centered at 0 and of radius 1, that divides the circle such that 1/4 of the area is on one side and 3/4 is on the other side.

Original entry on oeis.org

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

Offset: 0

Jonathan R. Anderson (neo__jon(AT)hotmail.com), Jul 30 2003

Decimal expansion of the number sin(u'), where u' is the number 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). See A332500. - Clark Kimberling, May 05 2020

			0.403972753299517...

Robert P. P. McKone, Table of n, a(n) for n = 0..19999

Cf. A003957.

Digits := 240 ; x := 0.4 ; for i from 1 to 8 do f := sin(2.0*x)+2.0*x-Pi/2.0 ; fp := 2*cos(2*x)+2.0 ; x := x-evalf(f/fp) ; printf("%.120f\n",sin(x)) ; od: x := sin(x) ; read("transforms3") ; CONSTTOLIST(x) ; # R. J. Mathar, May 19 2009

digits = 105; Sin[FindRoot[Sin[2*a]/2+a == Pi/4, {a, 1/2}, WorkingPrecision -> digits][[1, 2]]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 21 2014 *)

solve(x=0, 1, x*sqrt(1-x^2) + asin(x) - Pi/4) \\ Michel Marcus, May 05 2020

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

Define k(n+1) as k(n) - (k(n)*sqrt(1-k(n)^2) + arcsin(k(n)) - Pi/4). The sequence is the decimal expansion of lim_{n -> infinity} k(n).

Equals sqrt(2-2*sqrt(1-d^2))/2, where d = A003957 is the Dottie number. - Gleb Koloskov, Jun 16 2021

More terms from Jim Nastos, Sep 05 2003

More digits from R. J. Mathar, May 19 2009

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.

A332507 Decimal expansion of the number u such that the line normal to the graph of y = sin x at (u, sin u) passes through (1,0).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, May 05 2020

			u = 0.5530300788531359553083686485...

Cf. A332500, A332521.

u = u /. FindRoot[0 == - Sec[u] + u Sec[u] + Sin[u], {u, 0}, WorkingPrecision -> 150]
RealDigits[u][[1]]

A332523 Numerators of convergents to 2*Pi + Dottie number (A332506).

Original entry on oeis.org

5, 6, 11, 61, 316, 377, 4463, 4840, 9303, 51355, 60658, 1628463, 3317584, 38121887, 41439471, 79561358, 200562187, 280123545, 760809277, 73317814137, 74078623414, 147396437551, 221475060965, 368871498516, 590346559481, 226471603779739, 1359419969237915

Offset: 0

Clark Kimberling, May 05 2020

Cf. A332524, A332506, A212112, A223957, A332500.

d = FindRoot[Cos[u] == 2 Pi - u, {u, 0, 1}, WorkingPrecision -> 10000][[1, -1]];
Numerator[Convergents[d, 100]]    (* A332523 *)
Denominator[Convergents[d, 100]]  (* A332524 *)

A332524 Denominators of convergents to 2 Pi + Dottie number (A332506).

Original entry on oeis.org

1, 1, 2, 11, 57, 68, 805, 873, 1678, 9263, 10941, 293729, 598399, 6876118, 7474517, 14350635, 36175787, 50526422, 137228631, 13224474998, 13361703629, 26586178627, 39947882256, 66534060883, 106481943139, 40849118283120, 245201191641859, 286050309924979

Offset: 0

Clark Kimberling, May 05 2020

Cf. A332523, A332506, A212112, A223957, A332500.

d = FindRoot[Cos[u] == 2 Pi - u, {u, 0, 1}, WorkingPrecision -> 10000][[1, -1]];
Numerator[Convergents[d, 100]]    (* A332523 *)
Denominator[Convergents[d, 100]]  (* A332524 *)

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

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

A332521 Decimal expansion of the number u such that the line normal to the graph of y = sin x at (u, sin u) passes through (2,0).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 05 2020

			u = 2.483804010294740901473257369...

Cf. A332500, A332507.

u = u /. FindRoot[0 == - 2 Sec[u] + u Sec[u] + Sin[u], {u, 0}, WorkingPrecision -> 150]
RealDigits[u][[1]]

cp's OEIS Frontend

A332506 Decimal expansion of the number snc(2 Pi), where snc is the sine-normal-to-cosine function; see A332500.

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A332504 Decimal expansion of the number u in [0,2 Pi] such that snc(u) = 0, where snc is the sine-normal-to-cosine function; see A332500.

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A332505 Decimal expansion of the number u in [0,2 Pi] such that snc(u) = Pi, where snc is the sine-normal-to-cosine function; see A332500.

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A086751 Decimal expansion of the solution to x*sqrt(1-x^2) + arcsin(x) = Pi/4, or the length of the line connecting the origin to the center of the chord of a circle, centered at 0 and of radius 1, that divides the circle such that 1/4 of the area is on one side and 3/4 is on the other side.

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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.

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A332507 Decimal expansion of the number u such that the line normal to the graph of y = sin x at (u, sin u) passes through (1,0).

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A332523 Numerators of convergents to 2*Pi + Dottie number (A332506).

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A332524 Denominators of convergents to 2 Pi + Dottie number (A332506).

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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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