A179378 -id:A179378 - cp's OEIS frontend

A179373 Decimal expansion of the central angle in radians corresponding to a circular segment with area r^2 of a circle with radius r.

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

Offset: 1

Rick L. Shepherd, Jul 11 2010

The arc length of the circular segment/sector is r*A179373. The area of the circular segment, r^2, is 1/Pi (A049541) times the area of the circle. The area of the sector is (r^2)*(A179373/2) = (r^2)*(1 + A179378). See references and cross-references for other relationships.

			2.5541959528370430378296661737918779361157491714110524381405663643020...

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 7.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Circular Segment.
Index entries for transcendental numbers.

Cf. A179374 (same, in degrees), A179375 (for chord length), A179376 (for "cap height", height of segment), A179377 (for triangle height), A179378 (for triangle area), A049541.

RealDigits[ x /. FindRoot[x - Sin[x] - 2, {x, 2}, WorkingPrecision -> 105]][[1]] (* Jean-François Alcover, Oct 30 2012 *)

PARI
```
solve(x=0, Pi, x-sin(x)-2)
```

Decimal expansion of the solution of sin(x) = x - 2.

A179374 Decimal expansion of the central angle in degrees corresponding to a circular segment with area r^2 of a circle with radius r.

Original entry on oeis.org

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

Offset: 3

Rick L. Shepherd, Jul 11 2010

The arc length of the circular segment/sector is r*A179373. The area of the circular segment, r^2, is 1/Pi (A049541) times the area of the circle. The area of the sector is (r^2)*(A179373/2) = (r^2)*(1 + A179378). See references and cross-references for other relationships.

			146.3446481469584283647321150080224451316690962652634500095885765914885737881...

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 7.

G. C. Greubel, Table of n, a(n) for n = 3..10002
Eric Weisstein's World of Mathematics, Circular Segment.

Cf. A179373 (same, in radians), A179375 (for chord length), A179376 (for "cap height", height of segment), A179377 (for triangle height), A179378 (for triangle area), A049541.

RealDigits[(180/Pi)*(x /.FindRoot[x-Sin[x]-2, {x, 2}, WorkingPrecision -> 200]), 10, 100][[1]] (* G. C. Greubel, Nov 16 2018 *)

PARI
```
(solve(x=0, Pi, x-sin(x)-2))*180/Pi
```

Equals A179373*180/Pi = A179373*A072097.

A179375 Decimal expansion of the ratio of the chord length of a circular segment with area r^2 of a circle with radius r to r itself.

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jul 11 2010

In other words, the chord length is A179375*r. The arc length of the circular segment/sector is r*A179373. The area of the circular segment, r^2, is 1/Pi (A049541) times the area of the circle. The area of the sector is (r^2)*(A179373/2) = (r^2)*(1 + A179378). See references and cross-references for other relationships.

			1.914359546159529939985785242460527899501301180791115677192453168859627644264...

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 7.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Circular Segment.
Index entries for transcendental numbers.

Cf. A179373 (central angle, radians), A179374 (central angle, degrees), A179376 (for "cap height", height of segment), A179377 (for triangle height), A179378 (for triangle area), A049541.

RealDigits[ 2*Sin[x/2] /. FindRoot[x - Sin[x] - 2, {x, 2}, WorkingPrecision -> 106]][[1]] (* Jean-François Alcover, Oct 30 2012 *)

PARI
```
2*sin(solve(x=0, Pi, x-sin(x)-2)/2)
```

Equals 2*sin(A179373/2).

A179376 Decimal expansion of the ratio of the height of a circular segment with area r^2 of a circle with radius r to r itself.

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 11 2010

In other words, the segment height ("cap height" in MathWorld link) is A179376*r. The chord length is A179375*r. The arc length of the circular segment/sector is r*A179373. The area of the circular segment, r^2, is 1/Pi (A049541) times the area of the circle. The area of the sector is (r^2)*(A179373/2) = (r^2)*(1 + A179378). See references and cross-references for other relationships.

			.71050581697213734990563924269484526760618954800103872979253477385591...

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 7.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Circular Segment.

Cf. A179373 (central angle, radians), A179374 (central angle, degrees), A179375 (for chord length), A179377 (for triangle height), A179378 (for triangle area), A133742, A049541.

RealDigits[1-x /. FindRoot[x == Cos[1+x*Sqrt[1-x^2]], {x, 0}, WorkingPrecision -> 120]][[1]] (* Jean-François Alcover, Oct 06 2011 *)

PARI
```
1 - cos(solve(x=0, Pi, x-sin(x)-2)/2)
```

Equals 1 - cos(A179373/2) = 1 - A179377.

A179377 Decimal expansion of the ratio of the height of the triangle corresponding to a circular segment with area r^2 of a circle with radius r to r itself.

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 11 2010

In other words, the triangle height is A179377*r. The segment height ("cap height" in MathWorld link) is A179376*r. The chord length is A179375*r. The arc length of the circular segment/sector is r*A179373. The area of the circular segment, r^2, is 1/Pi (A049541) times the area of the circle. The area of the sector is (r^2)*(A179373/2) = (r^2)*(1 + A179378). See references and cross-references for other relationships.

			.2894941830278626500943607573051547323938104519989612702074652261440891212633...

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 7.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Circular Segment.

Cf. A179373 (central angle, radians), A179374 (central angle, degrees), A179375 (for chord length), A179376 (for "cap height", height of segment), A179378 (for triangle area), A049541.

RealDigits[ Cos[x/2] /. FindRoot[x - Sin[x] - 2, {x, 1}, WorkingPrecision -> 106]][[1]] (* Jean-François Alcover, Oct 30 2012 *)

PARI
```
cos(solve(x=0, Pi, x-sin(x)-2)/2)
```

Equals cos(A179373/2) = 1 - A179376.

A197009 Decimal expansion of the slope of the line y=mx which meets the curve y=cos(x+1) orthogonally over the interval [0, 2*Pi] (as in A197006).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 10 2011

See the Mathematica program for a graph.
xo=0.277097976418521518914833086895...
yo=0.289494183027862650094360757305...
m=1.0447358251025919644670467125044...
|OP|=0.4007370341535820008719293563...

			1.044735825102591964467046712504404113048658932805059578874283182084650....

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A179378, A197002, A196996, A197000.

c = 1;
xo = x /. FindRoot[x == Sin[x + c] Cos[x + c], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A179378 *)
m = 1/Sin[xo + c]
RealDigits[m]  (* A197009 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Cos[x + c], yo - (1/m) (x - xo)}, {x, -Pi/4, Pi/2}],
ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 1}], PlotRange -> All,
AspectRatio -> Automatic, AxesOrigin -> Automatic]

default(realprecision, 100); 1/sin(1 + solve(x=0, 2, x-sin(x+1)*cos(x+1))) \\ G. C. Greubel, Nov 16 2018

A197010 Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) in which a line y=mx meets the curve y=cos(x+1/2) orthogonally.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 10 2011

See the Mathematica program for a graph.
xo=0.4672816053760121337816307268...
yo=0.5675398046001583628839615011...
m=1.21455627200105698029988016754...
|OP|=0.73515544514637791501789646...

Cf. A179378, A197011, A197002, A196996, A197000.

c = 1/2;
xo = x /. FindRoot[x == Sin[x + c] Cos[x + c], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A197010 *)
m = 1/Sin[xo + c]
RealDigits[m]  (* A197011 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Cos[x + c], yo - (1/m) (x - xo)}, {x, -Pi/4, Pi/2}],
 ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 1}], PlotRange -> All,
 AspectRatio -> Automatic, AxesOrigin -> Automatic]

cp's OEIS Frontend

A179373 Decimal expansion of the central angle in radians corresponding to a circular segment with area r^2 of a circle with radius r.

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A179374 Decimal expansion of the central angle in degrees corresponding to a circular segment with area r^2 of a circle with radius r.

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A179375 Decimal expansion of the ratio of the chord length of a circular segment with area r^2 of a circle with radius r to r itself.

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A179376 Decimal expansion of the ratio of the height of a circular segment with area r^2 of a circle with radius r to r itself.

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Keywords

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Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A179377 Decimal expansion of the ratio of the height of the triangle corresponding to a circular segment with area r^2 of a circle with radius r to r itself.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A197009 Decimal expansion of the slope of the line y=mx which meets the curve y=cos(x+1) orthogonally over the interval [0, 2*Pi] (as in A197006).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A197010 Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) in which a line y=mx meets the curve y=cos(x+1/2) orthogonally.

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