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.
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
KEYWORD
cons,nonn
AUTHOR
Rick L. Shepherd, Jul 11 2010
STATUS
approved