Are $\operatorname{asin}$, $\operatorname{acos}$ and $\operatorname{atan}$ considered acceptable standard notation?

Question

When it comes to writing the inverse of a trigonometric function, there are various ways of doing so. The most common ways I've come across are:

• $\arcsin$, $\arccos$, $\arctan$, and
• $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$.

I'm personally not a big fan of the latter notation, since it looks too much like a reciprocal operation than an inverse. However I've also seen $\operatorname{asin}$, $\operatorname{acos}$ and $\operatorname{atan}$, albeit far less frequently.

My question is:

Is the third style I mentioned considered acceptable as standard notation for the inverse of the trig functions? I favour this notation because it's less verbose than the first, but I'd like to know which is preferred among mathematicians (whether by convention or some other reason). Or does it not matter? I haven't had much luck with Google, since relevant searches seem to bring up results along the lines of "$\operatorname{atan}$ vs $\operatorname{atan2}$" rather than "$\operatorname{atan}$ vs $\arctan$".

Answer 1

All three notations are quite common. You can almost certainly use the notation $\operatorname{atrig}$ (where $\operatorname{trig}$ is any trigonometric function you like, e.g. $\sin$, $\cos$, $\tan$, etc.) freely without causing any confusion or worrying about ambiguity.

For example:

• The question What are "$\tan$" and "$\operatorname{atan}$"? on Math SE asks for an explanation of the $\operatorname{atan}$ function in the context of Java programming. The Java documentation confirms that this is part of the language's syntax.

• Other programming languages also use the abbreviated $\operatorname{atrig}$ syntax:C++, python, FORTRAN, and so on. Indeed, on the Wikipedia page discussing inverse trigonometric functions, it is claimed in the notation section "In computer programming languages the inverse trigonometric functions are usually called by the abbreviated forms asin, acos, atan." This is marked "citation needed", but still appears reliable.

• The calculator I have used since high school uses ASIN, ACOS, and ATAN:

  

  On the other hand, TI and Casio calculators appear to use the $\operatorname{trig}^{-1}$ notation.

That being said, my feeling is that the use of $\operatorname{atrig}$ in purely mathematical writing is somewhat uncommon. I can't recall ever having seen this abbreviated notation in a paper, and I likely wouldn't use it myself, but I can't imagine that you would cause confusion. On the other hand, mathematical typesetting packages make it easy to change notation on the fly. For example using the AMS's LaTeX packages, use the \DeclareMathOperator macro, e.g.

 \DeclareMathOperator{\asin}{arcsin}

in order to use \asin to typeset the text $\DeclareMathOperator{\asin}{arcsin}\asin$. If you don't like that, you could instead use

 \renewcommand{\arcsin}{sin^{-1}}

to get $\renewcommand{\asin}{\sin^{-1}}\asin$. This lets you display a more verbose notation while retaining the more abbreviated notation in your LaTeX source.

Answer 2

Custom notation is always acceptable as long as you are consistent and the notation is clear and unambiguous.