Analysis of information sources in references of the Wikipedia article "Inverse function" in English language version.
α = arcsin m This notation is universally used in Europe and is fast gaining ground in this country. A less desirable symbol, α = sin-1 m, is still found in English and American texts. The notation α = inv sin m is perhaps better still on account of its general applicability. [...] A similar symbolic relation holds for the other trigonometric functions. It is frequently read 'arc-sine m' or 'anti-sine m', since two mutually inverse functions are said each to be the anti-function of the other.
[...] §473. Iterated logarithms [...] We note here the symbolism used by Pringsheim and Molk in their joint Encyclopédie article: "2logb a = logb (logb a), ..., k+1logb a = logb (klogb a)." [...] §533. John Herschel's notation for inverse functions, sin−1 x, tan−1 x, etc., was published by him in the Philosophical Transactions of London, for the year 1813. He says (p. 10): "This notation cos.−1 e must not be understood to signify 1/cos. e, but what is usually written thus, arc (cos.=e)." He admits that some authors use cos.m A for (cos. A)m, but he justifies his own notation by pointing out that since d2 x, Δ3 x, Σ2 x mean dd x, ΔΔΔ x, ΣΣ x, we ought to write sin.2 x for sin. sin. x, log.3 x for log. log. log. x. Just as we write d−n V=∫n V, we may write similarly sin.−1 x=arc (sin.=x), log.−1 x.=cx. Some years later Herschel explained that in 1813 he used fn(x), f−n(x), sin.−1 x, etc., "as he then supposed for the first time. The work of a German Analyst, Burmann, has, however, within these few months come to his knowledge, in which the same is explained at a considerably earlier date. He[Burmann], however, does not seem to have noticed the convenience of applying this idea to the inverse functions tan−1, etc., nor does he appear at all aware of the inverse calculus of functions to which it gives rise." Herschel adds, "The symmetry of this notation and above all the new and most extensive views it opens of the nature of analytical operations seem to authorize its universal adoption."[a] [...] §535. Persistence of rival notations for inverse function.— [...] The use of Herschel's notation underwent a slight change in Benjamin Peirce's books, to remove the chief objection to them; Peirce wrote: "cos[−1] x," "log[−1] x."[b] [...] §537. Powers of trigonometric functions.—Three principal notations have been used to denote, say, the square of sin x, namely, (sin x)2, sin x2, sin2 x. The prevailing notation at present is sin2 x, though the first is least likely to be misinterpreted. In the case of sin2 x two interpretations suggest themselves; first, sin x · sin x; second,[c] sin (sin x). As functions of the last type do not ordinarily present themselves, the danger of misinterpretation is very much less than in case of log2 x, where log x · log x and log (log x) are of frequent occurrence in analysis. [...] The notation sinn x for (sin x)n has been widely used and is now the prevailing one. [...]
{{cite book}}: ISBN / Date incompatibility (help) (xviii+367+1 pages including 1 addenda page) (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, USA, 2013.)