The notation
is sometimes also used (Gellert et al. 1989, p. 222; Gradshteyn and Ryzhik
2000, p. xxix). Note that the cosecant does not appear to be in consistent widespread
use in Europe, although it does appear explicitly in various German and Russian handbooks
(e.g., Gellert et al. 1989, p. 222; Gradshteyn and Ryzhik 2000, pp. xxix
and p. 43). Interestingly, while is treated on par with the other trigonometric functions
in some tabulations (Gellert et al. 1989, p. 222), it is not in others
(Gradshteyn and Ryzhik 2000, who do not list it in their table of "basic functional
relations" on p. 28, but do give identities involving it on p. 43).
Harris and Stocker (1998, p. 300) call secant and cosecant "rarely used functions," but then devote an entire section to them. Because these functions
do seem to be in widespread use in the United States (e.g., Abramowitz and
Stegun 1972, p. 72), reports of their demise seem to be a bit premature.
The positive integer values of giving incrementally largest values of are given by 1, 3, 22, 333, 355, 103993, ... (OEIS A046947), which are precisely the numerators of
the convergents of
and correspond to the values 1.1884, 7.08617, 112.978, 113.364, 33173.7, ....