*Wheaton College*

When I opened the MathFest program in Lexington last summer, I took one look at the first page and nearly yelled out loud “NO! NO! NO!” The inside cover to the program contained an advertisement for an online homework system with the following example:

Find the derivative of

*y*= 2 cos(3

*x*− π) with respect to

*x*.

*x*) represents a

*function*!

I should be clear: My irritation is not directed at this particular homework system as much as at the entire mathematics community for the sloppiness in notation that we tolerate, and even encourage, when dealing with trigonometric functions. You can pick up almost any calculus text, peek into almost any math classroom, or attend any number of talks at various MAA events to find a plethora of examples of trig functions lacking their parentheses.

Why do I think the parentheses matter so much? This is not just a pedantic preference on my part. The lack of parentheses represents an irregularity in notation that obscures the meaning of the mathematics. We often use a space to indicate multiplication, as in or 3 sin(

*x*), so leaving off the parentheses hides the fact that we are using a trigonometric function. The confusion is compounded when we say that the derivative of “sine” is “cosine.” If we were to be consistent, this would lead to applying a distorted product rule to get

*x*but not to the trailing 3? If we always used parentheses to enclose a function’s argument, then there would be no confusion.

An even worse abuse of notation occurs in the location of the exponent when a trig function is raised to a power. I will never write sin

^{2}(

*x*) for sin(

*x*)

^{2}because the first notation leads to ambiguity when discussing the inverse trig functions. Since

*f*

^{− 1}(

*x*) is the standard, consistent notation for the inverse function of

*f*(

*x*) , we also use sin

^{− 1}(

*x*) for arcsin(

*x*). If we were consistent with notation, a perfectly reasonable calculation would be

^{−2}(

*x*) in simplifying the expression—an impossible task! Should it be sin

^{2}(

*x*)

^{-1 }= 1/sin

^{2}(

*x*) or sin

^{-1}(

*x*)

^{2}= arcsin (

*x*)

^{2}? The point is that we should never have to make this choice! We

*should*be taking the derivative of the inverse sine function. This is horrific. The bad notation allows at least three different interpretations of the expression

Therefore, I implore you: The next time you use a trig function, please remember the parentheses, put the exponent on the outside, and never, ever write anything like sin3

*x*

^{2 }cos

^{-2}5

*x*.

Tommy Ratliff is a professor of mathematics at Wheaton College in Massachusetts where he enjoys thinking about voting theory, building new science centers, and being precise in his notation.

I have been a math professor for three years. I have been writing arcsin(x), arccos(x), arctan(x), etc. ever since one of my own undergraduate professors pointed out to me the ambiguity in the sin^{-1}(x) notation. Thank you for pointing out the need to rid ourselves of this unnecessary distraction for our students.

ReplyDeleteApropos Tommy Ratcliff's lament for explicit ( & ) vis-a-vis implicit - - - Back in the pre-transistor days - - - when FORTRAN first came out - - - at the Bell Labs the failure to make explicit in FORTRAN - - - multiplication that wasn't so in the math expression - - - was the major source of user coding errors for some time - - - a few years later - - - when discussing with Jonh Kemeny of Dartmouth the Fortran alternative he was developing - - - BASIC - - - we thrashed over several times this issue of implicit operations in math expressions having most likely to be annoyingly explicit in any programming language - - - Duncan Morrill, Merrimack, NH - - - WV1J@QSL.NET

ReplyDeleteTommy, I've been ranting about this for YEARS. Way to go! I don't know if I agree about \sin^2 and \sin^{-1}, since they are so ubiquitous, but it's way better than the alternative of sin x.

ReplyDeleteSo instead of $\sin x$, I should type $\sin \left( x \right)$? And although I prefer $\sin^2 x$, I encourage my students to write $\left( \sin x \right)^2$, now you're telling me to write $\left( \sin \left( x \right) \right)^2$. What we need is a definitive guide from a typesetter to let us know what's proper. We all have peeves (inverse notation for me), but we need a consistent style guide for both written and typeset work.

ReplyDelete