A179378 Decimal expansion of the ratio of the area of the triangle corresponding to a circular segment with area r^2 of a circle with radius r to r^2 itself.

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

Offset

0,1

Comments

In other words, the triangle area is A179378*(r^2). 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.

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) orthogonally; specifically:

xo=0.277097976418521518914833086895...

yo=0.289494183027862650094360757305...

m=1.0447358251025919644670467125044...

|OP|=0.4007370341535820008719293563... See the Mathematica program for a graph. [From Clark Kimberling, Oct 10 2011]

References

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Circular Segment.

Formula

Equals sin(A179373)/2 = sin(A179373/2)*cos(A179373/2) = A179375*A179377/2.

Example

.2770979764185215189148330868959389680578745857055262190702831821510113134466...

Mathematica

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]

Prog

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

Crossrefs

Cf. A179373 (central angle, radians), A179374 (central angle, degrees), A179375 (for chord length), A179376 (for "cap height", height of segment), A179377 (for triangle height), A049541; A197009 (radius of orthogonal circle).

Sequence in context: A286800 A323722 A021365 * A155541 A021787 A011052

Adjacent sequences: A179375 A179376 A179377 * A179379 A179380 A179381

Keyword

cons,nonn

Author

Rick L. Shepherd, Jul 11 2010

Status

approved