OFFSET
1,1
COMMENTS
Also, the first positive solution of x*sin(x)=(1-cos(x)).
The function hsinc(x) = (1-cos(x))/x is the Hilbert transform of sinc(x) = sin(x)/x. Both functions play a considerable role in various branches of physics, particularly in spectroscopy.
The value of hsinc(a) is in A257452.
The kissing points [x,y] of the two tangents with the smallest nonzero |x|, drawn from the apex [0,1] of the function y = cos(x) to itself, have the coordinates [+a,cos(a)] and [-a,cos(a)], respectively. The angle each of the tangents subtends with the Y axis is theta = atan(1/sin(a)). - Stanislav Sykora, Oct 17 2015
For a curve S in the xy-plane starting at the origin, pointing to the right, turning counterclockwise with constant curvature K, and with an arclength of 1, let Y denote the maximum y-value of any point in S. Then, this constant is equal to the value of K that maximizes Y. - Andrew Slattery, Sep 11 2021
LINKS
Stanislav Sykora, Table of n, a(n) for n = 1..2000
EXAMPLE
2.3311223704144226136678359559171213382690776953861145751...
Added in support of the Oct 17 2015 comment:
cos(a) = -0.689157736645164443889295..., theta = atan(1/sin(a)) = 0.943742927149971739026594... rad = 54.072486671015691988683987... deg.
MATHEMATICA
RealDigits[x/.FindMaximum[(1-Cos[x])/x, x, WorkingPrecision->200] [[-1]]] [[1]] (* Harvey P. Dale, Mar 29 2022 *)
PROG
(PARI) a = solve(x=1, 3, x*sin(x)-1+cos(x))
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Stanislav Sykora, Apr 23 2015
STATUS
approved