I know that different people follow different conventions, but whenever I see $\arcsin(x)$ written as $\sin^{-1}(x)$, I find myself thinking it wrong, since $\sin^{-1}(x)$ should be $\csc(x)$, and not possibly confused with another function.

Does anyone say it's bad practice to write $\sin^{-1}(x)$ for $\arcsin(x)$?

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    $\begingroup$ Why do you object to this but not to $f^{-1}(x)$? Or do you object to that too? $\endgroup$ – joriki Apr 1 '11 at 13:40
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    $\begingroup$ All calculators I had, even soviet one, used $\sin^{-1}$ notation on a keyboard. That's how I easily got used to the notation. $\endgroup$ – Yrogirg May 23 '12 at 12:12
  • $\begingroup$ Whenever I see $sin^{-1}x$ I find myself thinking it wrong. Almost no one in my country will write like that to say arcsin $\endgroup$ – phuclv Sep 16 '14 at 15:21
  • $\begingroup$ Also see meaning of powers on trig functions, in particular the answer math.stackexchange.com/a/920967/139123 $\endgroup$ – David K May 28 '15 at 22:12

The notation for trigonometric functions is "traditional", which is to say that it is not the way we would invent notation today.

  • $\sin^{-1}(x)$ means the inverse sine, as you mentioned, rather than a reciprocal. So $\sin^{-1}(x)$ is not an abbreviation for $(\sin(x))^{-1}$. Instead it's notation for $(\sin^{-1})(x)$, in the same way that $f^{-1}(x)$ means the inverse function of $f$, applied to $x$.

  • But $\sin^2(x)$ means $(\sin(x))^2$, rather than $\sin(\sin(x))$. In other contexts, like dynamical systems, if I have a function $f$, the notation $f^2$ means $f \circ f$. This is compatible with the $f^{-1}$ notation, if we take juxtaposition of functions to mean composition: $f^{-1}f^{3}$ will be $f^{2}$ as desired.

So the traditional notation for sine is actually a mixture of two different systems: $-1$ denotes an inverse, not a power, while positive integer exponents denote powers, not iterated compositions.

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. That doesn't mean that they are incorrect, however, as long as other speakers of the language know what they mean.

Moreover, if you wanted to reform the system, there would be an equally strong argument for changing $\sin^2$ to mean $\sin \circ \sin$. This is already slowly happening with $\log$; I think that the usage of $\log^2(x)$ to mean $(\log(x))^2$ is slowly decreasing, because people tend to confuse it with $\log(\log(x))$. That confusion is less likely with $\sin$ because $\sin(\sin(x))$ arises so rarely in practice, unlike $\log(\log(x))$.

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    $\begingroup$ Sorry, I misunderstood. I don't think it does come up, which is why we can get by with the traditional notation. But $\log(\log(x))$ comes up in computer science often enough to get people confused about $\log^2(x)$. $\endgroup$ – Carl Mummert Apr 1 '11 at 15:17
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    $\begingroup$ Just a note that symbolic mathematics packages such as MuPad consider $\cos^{-1}(x)$ as $\sec(x)$ and require you write $\arccos(x)$ $\endgroup$ – bobobobo Apr 2 '11 at 19:16
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    $\begingroup$ Interesting. Wolfram Alpha treats it as inverse sine. If a student in my class wrote $\cos^{-1}$ for secant I'd deduct some points, so I guess MuPad doesn't get perfect marks. $\endgroup$ – Carl Mummert Apr 2 '11 at 19:32
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    $\begingroup$ One notation I like is the parenthesized-exponent for composition, so that $\log^{(2)}(x)$ is $\log(\log(x))$ and $\sin^{(-1)}(x)$ is $\arcsin(x)$; this shows up in CS now and again just to alleviate the log-log confusion. $\endgroup$ – Steven Stadnicki Oct 19 '11 at 15:04
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    $\begingroup$ @StevenStadnicki That overloads the derivative notation. Though it is certainly not common to put that sort of notation on things like $\log$ or $\sin$, it clearly wouldn't work for a generic function $f$, where we already have $f^{(2)}(x)=\frac{d^2f}{dx^2}$. I think a reasonable notation would be to circle the exponent if it means composition, like this: $f^\textcircled{2}$ (which makes intuitive sense with our current notation for composition). $\endgroup$ – process91 Nov 28 '11 at 23:41

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