OFFSET
1,1
COMMENTS
Let L be the least x > 0 satisfying x + tan(x) = 0.
Then L is also the least x > 0 satisfying x = (sin(x))(sqrt(1+x^2)).
Consequently, for 0 < x < L, for all p > 0, 1/sqrt(1+x^2) - 1/x^p < sin(x) < 1/sqrt(1+x^2) for 0 < x < L.
The number L also occurs in connection with Du Bois Reymond's constants; see the Finch reference.
For x = L the area of right triangle with vertices (0,0), (x,0) and (x,sin(x)), i.e., the one inscribed into the half-wave curve, is maximal. - Roman Witula, Feb 05 2015
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, page 239.
EXAMPLE
L = 2.02875783811043422357697112473471437610838002...
1/L = 0.4929124517549075741877801898222329769156970132...
MATHEMATICA
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Oct 03 2011
STATUS
approved