A197009 -id:A197009 - cp's OEIS frontend

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

Rick L. Shepherd, Jul 11 2010

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.
From Clark Kimberling, Oct 10 2011: (Start)
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. (End)

			.2770979764185215189148330868959389680578745857055262190702831821510113134466...

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), A179377 (for triangle height), A049541; A197009 (radius of orthogonal circle).

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]

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

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

cp's OEIS Frontend

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.

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