Last month I posed question for the math folks out there, namely,
What does sin(x)2 mean, or is it ambiguous notation?
and the answers I got were interesting. When I polled my fellow faculty at Komplexify U, the results were pretty evenly split between two choices: (sin(x))2 or ambiguous. When I polled my students, the results were pretty evenly split between three choices: (sin(x))2, sin(x2), and ambiguous. As for the commentors here, there was a definite bias towards (sin(x))2, although considerable effort was made to point out how it might be misinterpreted, and how to better express it.
So what’s the right answer? Easy: there is no ambiguity. The expression sin(x)2 means the square of the sine of x, or to be more clear, ( sin(x) )2.
Let me first off explain why there is no ambiguity here, and then lay down my plan to make all of trig unambiguous henceforth and forever.
Ambiguity and trig
To explain why sin(x)2 is unambiguous, let me first discuss a trigonometric expression that truly is ambiguous: sin(x+1)2. Someone who sees this can easily interpret this as both
- the square of the sine of the quantity x+1: ( sin(x+1) )2
- the sine of the square of the quantity x+1: sin( (x+1)2 )
with the potential ambiguity arising from the common practice of leaving the parentheses off of argument of a trig function. As a consequence, the parentheses in “sin(x+1)2” can be interpreted in two ways: as indicating the argument of a function, or to indicate an algebraic grouping.
When parentheses are used to indicate the argument of function f, then the expression “f(t)” is treated as a single unit, since it represents the unique output value of the function corresponding to the input value t. For example, if f is a generic function, then f(x+1) represents the output when the function f is evaluated at the input value x+1. Therefore, f(x+1)2 must be the square of this value, since f(x+1) is “atomic,” if you will. Said differently, the letter “f” alone, without its parentheses, is a not an algebraic expression; so viewing f(x+1)2 as “f” by itself followed by the expression “(x+1)2” is meaningless. It makes as much sense as writing “(x+1)2f.” (Attentive readers will note that this is essentially basic “order of operations” stuff we’re talking about here.) Applying this logic here, since sine is just a function, if the parentheses indicate its argument, then sin(x+1)2 must be the first interpretation: the square of the sine of (x+1).
However, if we’re assuming that the argument of sine is not being explicitly indicated, then the usual rule of thumb for determining the meaning of the otherwise meaningless expression “sin stuff” is to assume the leftmost complete factor is the argument of sine. For example, given the meaningless expression “sin 3x2 + 2x + 1″, this convention for assigning the argument means this should be interpreted as “sin(3x2) + 2x + 1.” Applying this to sin(x+1)2, if we assume that sin has been written without its argument made explicit, then “(x+1)2” is the first full factor. Here the parentheses are used for an algebraic grouping, because without them the expression “x+1 2” would be different; it would just mean “x+1″, in which case its leftmost factor would just be x. Hence, if we assume the sine notation is being abused, then sin(x+1)2 must be the second interpretation: the sine of the square of (x+1).
Let’s go back to the original sin(x)2. Arguing as above, the only two reasonable interpretations could be
- the square of the sine of x — that is, ( sin(x) )2
- the sine of the square of x — that is, sin(x2)
with the potential ambiguity arising because of the tradition of writing “sin x” instead of “sin(x).” If the parentheses are used to indicate the functional argument of sine, then we use the first interpretation. Otherwise, the parentheses not being used for the argument, then they must be used to indicate some essential algebraic grouping to define the argument. However, taking that view, we interpret this as the sine function applied to the expression (x)2, so that the parentheses are grouping…. just x? Really? You really need to write (x)2 to indicate x2? More to the point, would you ever write (x)2 to indicate x2?
So in interpreting sin(x)2, your two choices are either
- assume correct function notation, or
- assume function notation is being deliberately abused in order to highlight an utterly unnecessary algebraic grouping.
I think the choice is pretty clear.
And to those who suggested that I could have avoided this whole mess by just writing sin2(x)… I have a better idea. I have…
A call for a revolution!
Trigonometry is hard enough for students without the hassle of all this ambiguity, which as the results of the informal poll above show, can stump even the best of us. Often my colleagues lament this fact, as well as the fact that it’s unfortunately impossible to change.
Screw that!
I for one am working to eliminate all traces of notational ambiguity from trigonometry. I teach the students to write trigonometric functions in one and only one utterly unambiguous form, and I simply do not accept their work unless it is written that way. If all students were simply taught not to use ambiguous notation — and their teachers simply did not accept it — we could utterly stamp out this headache in a generation.
So I am calling on you, my brothers and sisters in the trenches of mathematical education, to come to my aid and banish problematic trigonometric notation. More specifically, I am calling on you to
Banish forever the notation “sin x” and “sin2 x” —
In their place, love live “sin(x)” and “sin(x)2“
There are many reasons for doing this, but let me focus on three:
Reason 1. It’s the right thing to do. I don’t mean this in a touchy-feely, Kumbaya way. I mean it literally: it is the correct thing to do. Sine is function, and there is a completely well-defined, unambiguous notation for working with functions, as mentioned above: if f is the name of a function, then f(x) is the expression representing the output value defined when the function f is evaluated at the input value x.
In algebra, we spend significant time teaching students how to properly use function notation. In an algebraic expression, f must have a set of parentheses to indicate an argument, and the symbol f(argument) is a single, non-divisible value. For a generic function, the symbol f x is just meaningless.
So why do we persist in allowing “sin x” if it is so problematic? Tradition, more than anything else. The notation “sin x” is archaic, denoting nothing more that an abbreviation for the expression “sinus rectus arcus x,” which (very, very loosely) means “the vertical perpendicular component measured from the arc x.” It was written in much the same spirit as writing “ae” as an abbreviation for the word “aequis” (latin for “equals”), which was the preferred method to connect two sides of a equation before Robert Recorde introduced the = symbol in his 1557 text The Whetstone of Witte.
It’s from a point in mathematical history in which the function concept was still not well defined: in fact, this archaic notion of “sin x” has always been ambiguous, for its numerical value could not be assigned until a radial length was given. It makes as much sense today to use this notation in mathematics as it does to continue the Olde Englishe practise of adding silent “e”s to the end of everything (as in the full title of Recorde’s work, The Whetstone of Witte, whiche is the seconde parte of Arthmeteke: containing the extraction of rootes; the cosske practise, with the rule of equation; and the workes of Surde Nombers.)
This old idea served its purpose, but it is not how we work with sine today. Today, we define the radial length to be 1. Today, sine is an honest-to-God function, with a well-defined, utterly unambiguous rule of assignment of values. As a result befitting its modern function interpretation, it makes sense to use its modern function notation, namely, sin(x).
As a corollary, if you’re willing to grant this, then there is also reason to continue the doubly lamentable symbol “sin2x,” since sin(x)2 serves exactly the same purpose. Now I know what some of you are saying right now: “Why not compromise, and use sin2(x)?” The reason is simple and two-fold.
- For any other function f mapping a set X into itself, the symbol f n already has a meaning: it is the n-fold composition of f with itself, i.e. f(f(f(…f(x))…))) with f repeated n times. In particular, f 2(x) means f(f(x)). (And if you think about it, that’s really what sin2 x says: sin sin x.) Indeed, the very idea that compositions of functions of arbitrary sets had an underlying arithmetic — namely, f n
f m = f n+m — was one of the bases for modern abstract algebra. Extending this a bit more, if we (quite reasonably) set f 0 to be the identity function, then the notation of “f -1” for an inverse function is patently obvious.
- Speaking of inverses, students who get accustomed over weeks of exposure to the idea that sinn x means ( sin x )n are therefore at a loss to understand why, all of a sudden, sin-1(x) is suddenly not ( sin x )-1, better known as csc x. The concept of an inverse function is difficult enough as it is without adding yet another layer of notational ambiguity to it. If we simply consistently apply standard function notation to sine, then sin-1(x)* can only mean the arcsine function, whereas sin(x)-1 can only mean the cosecant function and sin(x-1) can only mean the function sin(1/x) (a favorite counterexample function in calculus).
* As a side issue, when discussing the arcsine function, I myself actually only use the arcsin(x) notation rather than the sin-1(x) notation, partly to avoid the aforementioned notational ambiguity, but mostly because there is NO INVERSE to the sine function, which (being periodic) fails to be one-to-one in a particularly egregious manner. Given that the so-called “inverse trigonometric” functions are actually only local inverses designed to preserve acute radian angle measures — that is, arcs — I prefer the arcsine name and notation better.
Reason 2. It’s the right thing to do. Now I mean it in a touchy-feely, Kumbaya way. Trigonometry is a difficult subject as it is: it’s students’ first real, in-depth exposure to transcendental functions. It is the first time students need to work with expressions and formulas for which they have little algebraic intuition, a subject that includes a memorizing a large amount of information (e.g. six functions times five reference angles times four quadrants equals 120 basic trig evaluations) by the end of the first three days.
For many students, “trigonometry” seems like an entirely separate aspect of mathematics with its own indecipherable rules. Persisting in using notation that violates what they’ve been previous taught (or will be taught in a later algebra class) not only validates this viewpoint, but also certainly harms whatever understanding they might have had about the function concept in general. Really, what’s a better use of students’ — and teachers’ — time:
- Cementing over and over the common core notion of function, together with the standard order of operations, or
- Weakening student understanding of the function concept while validating the “trig is different” mentality by spending hours fostering ambiguous and confusing special cases for trig, only to spend further hours trying to decipher the ambiguous notation taught?
Let’s give ’em a break.
Reason 3. It’s what your calculator does. This is not as compelling a reason as the other two, but I think that in an age of handheld symbolic calculators and campus-wide access to computer algebra systems, it’s not entirely irrelevant. Calculators like the TI-89 and programs like Maple already use this convention… that is to say, already use only correct function notation when dealing with trigonometry. If using correct notation in class can also help bridge the syntactic difficulty in using a CAS, I’m all for it.
I suppose one argument against all of this, of course, is what’s the point? Every calculus book abuses trig notation, so why waste time trying to teach students to use it correctly? First off, if that’s your attitude, perhaps teaching is not really your calling. Might I suggest “paper weight” or “wind sock” or some other task that doesn’t require any energy exerted on your part?
Beyond that, as a teacher you can use poor notation as a learning experience over and over again. In my classes, students are taught to only use correct function notation for everything, including trigonometric functions and logarithms. In fact, the act of converting outdated notation from the textbook into proper function notation is always the first step in any problem solving algorithm I share with students. Moreover, to emphasize its importance (and encourage students to get into the habit of using it), students in my classes who make simple, bone-headed mistakes based directly on misinterpreting or misusing trigonometric notation sacrifice all partial credit on a problem. I may not be able stamp out poor notation from the calculus text, but I sure as hell do it in my classes!
So what say you, O teachers of mathematics? Will you banish obsolete trigonometric notation and the endless ambiguities it causes? Will you promote true functional notation, and help make trig just that little bit more accessible to your students?
Who’s with me?
