$\sin^{-1} x$ means the inverse of $\sin x$ (it is also often called $\arcsin x$), but it can fairly easily be confused with $\sin(x)^{-1}$. Why is it used, when $\arcsin x$ is easier to type and is not easily confused with other things?
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asked Apr 1, 2015 at 23:42
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4$\begingroup$ The answer is tradition, see MSE thread math.stackexchange.com/questions/30317/arcsin-written-as-sin-1x "This is simply a fact of life, like an irregular conjugation of a verb. As with other languages, the things that we use most often are the ones that are likely to remain irregular" writes Carl Mummert. Gauss already objected to this notation, and yet it is still around. $\endgroup$– ConifoldCommented Apr 2, 2015 at 0:06
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$\begingroup$ @Conifold I only partialy agree. One often use sinus in a context of multiple iterations, so it does make sense to use the inverse notation in this case. arcsin is there for every other case, but then we're back with what you very well explained on languages. $\endgroup$– VicAcheCommented Apr 21, 2015 at 12:37
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$\begingroup$ While I agree that $\sin^{-1}(x)$ and $\sin(x)^{-1}$ can be confused due to fact that they share the symbols, I'm fairly sure that many formulas could be understood easily if we didn't have all those names for similar and related functions. Why do we have a name for $\csc(x)=\frac{1}{\sin(x)}$ but not one for the naturally appearing functions $\log(1-x)$ or $\sin(2\pi\, x)$? $\endgroup$– Nikolaj-KCommented Oct 5, 2015 at 7:42
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1$\begingroup$ @NikolajK My guess is that $csc$ etc. is used far more commonly. $\endgroup$– The Guy with The HatCommented Oct 5, 2015 at 11:02
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5 Answers
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If you want someone to blame blame John Herschel, son of the famous astronomer William Herschel. The father in 1781 gave us the planet Uranus, the son in 1813 gave us the dubious symbol $\sin^{-1}(x)$ for the inverse sine. As precedent, he cited the 'operator powers' notation, like ${d^n}f$ from calculus, so according to Herschel we should write $d^{-1}f$ for $\int f$. "The symmetry of this notation, and above all the new and most extensive views it opens seem to authorize its universal adoption", he wrote. Herschel was aware of the potential confusion with arithmetical reciprocals, but apparently thought that the benefits outweighed the downsides.
First notation for the inverse sine was proposed by Daniel Bernoulli in 1729, it was $AS$, Euler shortened it to $A$ alone. Lagrange was first to introduce $\operatorname{arc}.\!\sin$, and with the dot eventually removed it became the standard in the continental Europe. And US and Britain would have followed suit too... if it wasn't for Herschel Jr.! When he published Traite de la Luminere in French in 1829 he himself used Lagrange's $arc.sin$. Perhaps it was a leftover of the Newton-Leibniz priority dispute and/or cultural rivalry between Britain and the continent, but British often insisted on being notationally different, feet and pounds are another example, as are the long division symbol, $\tan$ vs $\text{tg}$ for tangent, and $\sinh$ vs $\text{sh}$ for hyperbolic sine.
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Here's some detail.
From Herschel's 'On a Remarkable Application of Cotes's Theorem', Philosophical Transactions, 1813:
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Gauss wrote a review of this article in the Göttingsche Gelehrte Anzeigen (reprinted in Werke, vol. 4, p. 361):
I will translate the last sentence:
What the author says about the notation $\cos^2 A$, which some newer mathematical writers use for the square of $\cos A$, totally against all analogy, as accordingly this should mean the cosine of an angle $= \cos A$, has our total approbation.
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$\begingroup$ Ahhh, so Gauss was protesting against the arithmetical notation, not the Herschel's. I am not sure it's "totally against all analogy" though. This issue would arise every time more than one "multiplication" is defined on a set of objects, in principle $f^n$ may refer to any of them. But since arithmetical multiplication is the most elementary, and most frequently used, it makes sense for it to have the benefit of the simplest notation. $\endgroup$– ConifoldCommented Apr 2, 2015 at 20:03
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Florian Cajori has it in A History of Mathematical Notations Vol. II Paragraph 533 that "John Herschel's notation for inverse functions $\sin^{-1}x$,... etc., was published by him in the Philosophical Transactions of London, for the year 1813." Cajori goes on to indicate that Hershel gave specific instructions to not confuse the notation with the multiplicative inverse.
His justification for the notation, revolves around a "symmetry" with other notations like, for example, writing $\Delta\Delta\Delta x$ as $\Delta^3x$, and he lists several others. I refer you to the literature for the full description.
Now in the same, Cajori notes that "Some years later, Hershel explained that in 1813 he used ...[these notations]... 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."
There is much more detail in this reference, in particular explaining the logic of the notation.
answered Apr 2, 2015 at 0:26
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1$\begingroup$ It amuses me to no end that this notation is justified in operator-like terms. $\endgroup$ Commented Oct 17, 2015 at 21:00
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Notations of this kind are determined by high school textbooks. I noticed that $\sin^{-1}$ is common in the English-speaking countries, while in many European countries they prefer $\arcsin$. This is consistent with the remark in the end of Conifold's answer about nationalistic reasons for this difference.
I was educated in Ukraine (then a part of the former Soviet Union) and I have never seen $\sin^{-1}$ before I came to the US.
answered Apr 2, 2015 at 2:12
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$\begingroup$ A third of my students are from the eastern part of Europe and Russia. All of them are oriented toward "arcsinus" and so forth, and none of them are familiar with this notation at first. Usage wise, I can confirm your experience, although one of my students brought his father's old 80's ti calculator back from Macedonia over break, and guess what notation was on the calculator. :). I suppose it can't be wholly unfamiliar to them, but most of my European students are not big calculator users. $\endgroup$ Commented Apr 2, 2015 at 2:41
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$\begingroup$ I suppose the old 80-th calculator was not MADE in Macedonia:-) $\endgroup$ Commented Apr 2, 2015 at 3:30
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$\begingroup$ @J. W. Perry What was the symbol for tangent on that calculator? I bet it was the inauthentic $\tan$ :) – $\endgroup$– ConifoldCommented Apr 2, 2015 at 20:38
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$\begingroup$ I'll ask him to bring it in again. It was an old solar calculator. $\endgroup$ Commented Apr 2, 2015 at 21:09
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First because of Herschel, second because of the designers of pocket calculators. Unfortunately many pocket calculators give $sin^{-1}(x) = arcsin(x)$. That is incorrect. Also Wikipedia has it wrong.
$sin^{-1}(x)$ means $\frac{1}{sin(x)}$ like $sin^{2}(x)$ means $sin(x)\cdot sin(x)$. Otherwise it would be highly inconsistent to write $sin^{2}(x) + cos^{2}(x) = 1$, as also Wikipedia does: $sin^2(x)$ = $(sin(x))^2$.
The inconsistency of Herschel's mis-design is also acknowledged by this Wikipedia article.
Further it would be confusing to write $sin(x)^{n}$, in particular $sinx^{n}$ if the parentheses are not written as is often done.
