OFFSET
0,1
COMMENTS
Decimal expansion of the number sin(u'), where u' is the number in [0,2 Pi] such that the line normal to the graph of y = sin x at (u', sin u') passes through the point (3 Pi/4,0). See A332500. - Clark Kimberling, May 05 2020
LINKS
Robert P. P. McKone, Table of n, a(n) for n = 0..19999
FORMULA
Define k(n+1) as k(n) - (k(n)*sqrt(1-k(n)^2) + arcsin(k(n)) - Pi/4). The sequence is the decimal expansion of lim_{n -> infinity} k(n).
Equals sqrt(2-2*sqrt(1-d^2))/2, where d = A003957 is the Dottie number. - Gleb Koloskov, Jun 16 2021
EXAMPLE
0.403972753299517...
MAPLE
Digits := 240 ; x := 0.4 ; for i from 1 to 8 do f := sin(2.0*x)+2.0*x-Pi/2.0 ; fp := 2*cos(2*x)+2.0 ; x := x-evalf(f/fp) ; printf("%.120f\n", sin(x)) ; od: x := sin(x) ; read("transforms3") ; CONSTTOLIST(x) ; # R. J. Mathar, May 19 2009
MATHEMATICA
digits = 105; Sin[FindRoot[Sin[2*a]/2+a == Pi/4, {a, 1/2}, WorkingPrecision -> digits][[1, 2]]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 21 2014 *)
PROG
(PARI) solve(x=0, 1, x*sqrt(1-x^2) + asin(x) - Pi/4) \\ Michel Marcus, May 05 2020
(PARI) my(d=solve(x=0, 1, cos(x)-x)); sqrt(2-2*sqrt(1-d^2))/2 \\ Gleb Koloskov, Jun 16 2021
CROSSREFS
KEYWORD
AUTHOR
Jonathan R. Anderson (neo__jon(AT)hotmail.com), Jul 30 2003
EXTENSIONS
More terms from Jim Nastos, Sep 05 2003
More digits from R. J. Mathar, May 19 2009
STATUS
approved