Since we’ve been looking at an example of ambiguity in notation, let’s look at a very different one. There is a lot to be confused by in inverse trigonometry! We’ll try to untangle the notations of \(\sin^{-1}\) and \(\arcsin\).
Recall that inverse trig functions are the inverses of trigonometric functions such as sine and cosine. For example, since we know that \(\sin\left(\frac{\pi}{6}\right)=\frac{1}{2}\), the inverse function swaps the input and output, so that \(\sin^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{6}\). (There are additional issues due to the fact that there are many angles that have the same sine; we discussed these ideas in Ranges of Inverse Trig Functions.)
Why isn’t sin-1 the cosecant?
We’ll start with this, from 2003:
Inverse Trigonometric Ratios Why aren't the inverse trigonometric ratios equal to 1/(the ratio), as is the case for numbers? e.g. sin^-1 doesn't equal 1/sin (or cosec) while x^-1 = 1/x and why isn't then, say, sin^-1 equal to cosec? This is the rule followed for all other numbers but it doesn't apply in this situation involving trigonometric ratios.
The cosecant can be defined as \(\csc(x)=\frac{1}{\sin(x)}=(\sin(x))^{-1}\); since \(\sin^2(x)\) means \((\sin(x))^2\), why isn’t \(\sin^{-1}(x)\) the cosecant?? Many students struggle with this.
Doctor Rob answered:
Thanks for writing to Ask Dr. Math, Matthew. Yours is an excellent question. The answer involves the fact that the symbol __^(-1) is used for two different purposes, which are confusingly related. If c is a number, then c^(-1) is defined to be the reciprocal of c, or 1/c.
That is, \(c^{-1}=\frac{1}{c}\). We’ll see more about this later.
If f(x) is a function, then f^(-1) is defined to be the inverse function, that is, the function such that f[f^(-1)(x)] = x. In words, if you apply the inverse function to a value, then apply the function to that result, you get the original value back. In your question, the trigonometric functions are functions, and so the second of these ideas applies.
As a simple example, if we define a function \(f(x)=x+2\) that adds 2 to any input, its inverse function is \(f^{-1}(x)=x-2\), which subtracts 2 from its input, undoing the effect of f. To show this, we see that $$f^{-1}(f(x))=f^{-1}(x+2)=(x+2)-2=x,\text{ and}\\f(f^{-1}(x))=f(x-2)=(x-2)+2=x.$$ Applying f and then its inverse returns the original number, and vice versa.
The relation between these is as follows. If c is a nonzero number, there is a function f_c(x) defined by this: for all numbers x, f_c(x) = c*x What is the inverse function of f_c? It turns out that it is defined by this: for all numbers y, (f_c)^(-1)(y) = y/c = (1/c)*y = c^(-1)*y and the inverse function involves the reciprocal of c.
This time, we have a function that multiplies rather than adds; for example, \(f_3(x)=3x\). Its inverse is the function that divides by the same number: \(f_3^{-1}(x)=\frac{x}{3}\). To show that these are inverses, we observe that $$f_3^{-1}\left(f_3(x)\right)=f_3^{-1}\left(3x\right)=\frac{3x}{3}=x,\text{ and}\\f_3\left(f_3^{-1}(x)\right)=f_3\left(\frac{x}{3}\right)=3\cdot\frac{x}{3}=x.$$
So the multiplicative inverse of a number corresponds to the inverse function of a multiplication: Division by c undoes multiplication by c. That’s why it makes sense to use the same notation. We’ll see a deeper reason next.
Two kinds of inverse
We got a very similar question in 2005:
Inverses and Reciprocals of Functions I'm confused about when a negative one exponent means reciprocal and when it means inverse, particularly with trig functions. For example, x^(-1) means 1/x, but sin^(-1)(x) does not mean 1/sin(x).
Doctor Vogler answered:
Hi Anthony, Thanks for writing to Dr. Math. That's a good question: When you raise a function to the -1 power, what does that mean? The short answer is that it means "the inverse," but, unfortunately, there are at least two meanings for that. See also Inverses http://mathforum.org/library/drmath/view/54597.html
Multiplicative inverse: undoing multiplication
When you write
x^(-1),
or
-1
x ,
that is, x raised to the -1 power, that is the same as the multiplicative inverse of x, or 1/x. So if you write
f(x)^(-1),
or
-1
f(x) ,
then generally you mean 1/f(x), the multiplicative inverse of the number f(x). Similarly, if you write
f(x)^2,
or
2
f(x)
you generally mean the square of the number f(x).
A superscript following a number (including the number represented by a function evaluation!) is an exponent, raising that number to a power. For example, if \(f(x)=x+2\), then $$f(3)=3+2=5\\f(3)^{-1}=5^{-1}=\frac{1}{5}\\f(3)^2=5^2=25.$$
And a negative exponent represents division because, for example, $$x^{-1}\cdot x^1=x^{-1+1}=x^0=1,$$ so $$x^{-1}=\frac{1}{x}.$$
Function inverse: undoing composition
By contrast, it is more common for the inverse function to be written f^(-1)(x), or -1 f (x), with the exponent on the f and before the parentheses. The inverse function is different from the multiplicative inverse and has nothing to do with 1/f(x).
Here we are inverting the function f, not the value \(f(x)\). So, as we saw above, if \(f(x)=x+2\), then $$f(1)=1+2=3\\f^{-1}(3)=3-2=1.$$
So why do they use a power of -1 to mean inverse function? The main reason is that when people deal with *iterating* a function, they often write f^n or n f to mean f *composed* with itself n times (rather than f *multiplied* by itself n times). So then 2 f (x) = f(f(x)) and 3 f (x) = f(f(f(x))) and so on. And we also say that f^0(x) is the identity function x. This notation is convenient because it satisfies many of the well-known properties of exponents, such as n m n+m f (f (x)) = f (x). In this notation, it makes perfect sense to write f^-1(x) for the inverse function, because then -1 f (f(x)) = x is exactly the formula I wrote above, with n = -1 and m = 1.
A (positive integer) superscript on a function name itself means to iterate the function, applying the function that many times, just as a superscript on a number means to exponentiate the number, multiplying by it that many times. And iteration follows the same rules as exponentiation.
You don’t see \(f^0\) written very often, but the idea is that if you start with \(x\) and apply function \(f\) 0 times, you’ve done nothing, and are left with \(x\). This is the identity function, which does nothing: \(f^0(x)=x\).
And just as we extend exponents to allow negative exponents, we can extend iteration. As the -1 power of a number “undoes” a multiplication (that is, divides), the -1 “power” of a function “undoes” the function.
Doctor Vogler gave a similar explanation of this in 2013, which you can find here:
Function Exponentiation Convention
A collision with trigonometry notation
But there’s a special notation in trigonometry.
Finally, we come to the trig functions. Here, we run into a problem. You see, when you want to write
2
f(x)
where f(x) = sin x, then
2
sin x
looks more like
f(x^2).
So we have two choices. We could write
sin(x)^2,
and sometimes you see this (such as in computer math programs), or we could write
2
sin x,
and sometimes you see this, such as in the Pythagorean trig identity
2 2
sin x + cos x = 1.
The trouble with this notation is that it means the square of the sin of x, when it looks more like the iterated
sin sin x,
so it can be confusing. It is especially confusing when someone uses the same notation to write
-1
sin x
to mean the inverse sine of x, because this means the function inverse (-1 iterations of sine) not the multiplicative inverse (sin x raised to the -1 power). It is because of this inconsistency that my personal preference is always to use "arcsin" to mean the inverse sin, instead of the -1 exponent. But not everyone follows this convention.
We have here a collision of several notational concepts and conventional shortcuts.
If we always used parentheses around function arguments (as is done in most computer programming languages), we could happily write \(\sin(x)^2\) for squaring, and there would be no confusion. But because trigonometric functions (and also logarithms) were introduced before general function notation, and were traditionally written without parentheses unless absolutely necessary, permission to omit parentheses was “grandfathered in”: We can write \(\sin x\) rather than having to write \(\sin(x)\) as we would for other functions.
And because of that, putting the exponent on the function became standard in trigonometry: We traditionally write \(\sin^2x\) rather than \((\sin x)^2\). This notation developed at the same time as general function notation, in which an exponent more naturally meant function iteration. Yet the -1 notation for inverses, which came from function notation, also came to be applied to trig functions, despite the contradiction. (More below.)
Giving the inverse function its own name, arcsin, is a great solution to this problem … except that the other notation remains very common, in line with inverses of other functions.
What a mess we’re in!
So the bottom line is that a sine (or cosine, etc.) raised to the -1 power probably means the function inverse and NOT the multiplicative inverse, while a sine (or cosine, etc.) raised to the power 2 (or any other positive integer) probably means the number raised to that power, and NOT the iterated function. In any case, you can usually tell by the context.
So what does “arc” in arcsin mean?
That reference to “arcsin” as a better name leads us to this question from 2001:
Trigonometry Terminology Dear Dr. Math, Recently in my Pre-Calculus math class, we have been dealing with functions such as sine, cosine, and tangent. We are currently confused with some math terminology, and you may be able to help us. Our question is: Why is the term "arc" used for the inverse of sine, cosine, and tangent instead of just saying the function to the -1 power? Any help that you can provide us is greatly appreciated.
Scott has learned only \(\arcsin\) and not \(\sin^{-1}\). Where does that come from?
I answered:
Hi, Scott. Trigonometry terminology is somewhat archaic in general, and doesn't fit well with algebraic notation. There are some good reasons for it both starting out different, and being kept different, though it's not an ideal situation. As for its origin, "arcsin" simply means "the arc whose sine is ...". This is pretty straightforward, and was especially so before function notation and its "f^-1" was invented. It appears that you have been taught only the "arcsin" form, but in fact "sin^-1" is quite common.
The arc (of a unit circle) is the radian measure of the angle, so this is another way to say “the angle whose sine is …”.
Some books or teachers or cultures prefer one notation or the other; we see both often.
I gave more details on why we don’t just switch over to \(\sin^{-1}\), summarizing what we’ve seen above:
As for its continuation, the use of sin^-1 collides with the tradition of writing sin^2(x) for the square of the sine, another holdover from early usage. One of the two has to give way, unless we keep both, and trust the reader to see whether sin^-1(x) means the inverse sine or the inverse of the sine, the cosecant. I personally prefer arcsin (especially in writing for Dr. Math) for this reason. This notation also retains its popularity (and might even be growing) because in computer programs it is easier to name a function "atan" rather than something clumsy like "tan_inverse." In my mind, it makes good sense to give the inverse functions names of their own, like arcsin, rather than always treating them as mere inverse functions.
This is similar to the fact that we use distinct names for the inverse function pair \(\log(x)\) and \(e^x\), which is sometimes written as \(\exp(x)\). It would be very awkward to write about \(\exp^{-1}(x)\) instead of \(\log(x)\), or \(\log^{-1}(x)\) rather than \(e^x\).
If you study the history of the notation, as in Jeff Miller's Earliest Uses of Symbols for Trigonometric and Hyperbolic Functions http://jeff560.tripod.com/trigonometry.html you find that early forms of "arcsin" arose in the 1700s, and the sin^2 notation came at the same time; sin^-1 was introduced in 1813. I presume that the inverse function notation in general was an immediate precursor to that (or else arose from it), but I can't locate that information. The above page says that arcsin is usual on the continent, while sin^-1 is used in England and America. I don't know whether that is still true.
I find in Cajori (v.2 p.176) that Herschel, who introduced \(\sin^{-1}\) in 1813, recognized the potential confusion with others’ use of \(\sin^2(x)\) as \((\sin(x))^2\), as well as with notation for reciprocals. It appears that this was in fact the introduction of general inverse function notation:
So to some extent the difference in usage relates to national traditions.
But “arc” does not mean inverse!
I’ll close with a question from 2004, taking us to a more advanced set of functions, where we can see the “arc” terminology from a new perspective:
Difference Between arsinh and arcsinh Functions Why isn't there a "c" in arsinh, arcosh, and artanh? These are the equivalents of arcsin, arccos and arctan but they are hyperbolic. I was just curious as to why there is no "c". Is there a reason?
Very likely you have never heard of these “hyperbolic functions”. Whereas the trigonometric functions (sometimes called circular functions) are based on a circle \(x^2+y^2=1\), so that \(\cos^2(\theta)+\sin^2(\theta)=1\), the hyperbolic functions are based on the hyperbola \(x^2-y^2=1\), so that \(\cosh^2(t)-\sinh^2(t)=1\). Specifically, it turns out that $$\cosh(t)=\frac{e^t+e^{-t}}{2}\\\\\sinh(t)=\frac{e^t-e^{-t}}{2}$$
As we’ve seen, the inverse trig functions are named either by attaching a -1 exponent, or by prepending “arc”. Why would it be different for inverse hyperbolic functions? For that matter, is it?
I answered, admitting that this was unfamiliar to me, and showing my gradual discovery:
Hi, Matt. None of us seem to have heard of this; we all use arcsinh, arccosh, and arctanh for the inverse functions. Can you tell me where you see your versions--in a math text, a programming language, or what? Is it the standard British usage? Searching the web for the terms, I see indications that some programming languages use your forms. Perhaps because they were limited to 6 characters in a name. I also seem to see them used in German and Finnish pages, and some in the UK.
Today it is easy to find information about these terms; Wikipedia (today) says,
The inverse hyperbolic functions are:
- area hyperbolic sine “arsinh” (also denoted “sinh−1“, “asinh” or sometimes “arcsinh“)
- area hyperbolic cosine “arcosh” (also denoted “cosh−1“, “acosh” or sometimes “arccosh“)
- and so on.
As we’ll see, it didn’t say that then!
I also find a few references to "arsinh" as meaning "area hyperbolic sine"; this seems to be the reason for using "ar" rather than "arc" in some languages. The idea evidently is that whereas the trig functions take an angle (or arc) as their argument, so that the inverse function returns an angle (or arc), the hyperbolic functions actually take an area (which in the case of circular functions is proportional to the arc length, so we don't see the difference). So it really does make more sense to use "area" rather than "arc"; the inverse function returns the "area" whose hyperbolic sine is such and such. I suppose we use "arc" here just because we use it for trig functions without thinking of its meaning, and therefore think of "arc" as if it just meant "inverse". I never thought about this before!
Thinking is a good thing! And that’s why I’m including this question on a page about trig functions.
Just as children learn words they hear by guessing from context, and so sometimes infer the wrong meaning for a word, adults can do the same. We learn that “arcsine” means “inverse sine”, and overgeneralize that so that we take “arc” as a prefix to mean “inverse”. But it does not: It means “the arc/angle whose sine is …”.
Here is an illustration from Wikipedia showing the meaning of the argument:
Similarly, you can see that an arc of the unit circle is twice the area of its sector, so the definitions are perfectly consistent.
I eventually found good sources, among them Wikipedia’s page on hyperbolic functions; I commented:
The latter uses the more usual (at least in my part of the world) "arc" names, though it earlier emphasized that the argument is not an angle but an area. Your "ar" names turn out to be better. I'd love to know the history of this, and how some parts of the world use one set of names while the rest use one that is less meaningful.
Wikipedia, as I noted above, now gives preference to the “ar” names; in 2004, it used only “arc” forms, but by 2006 it added the “ar” forms as an alternative:
The inverse functions are the inverse hyperbolic sine ”arcsinh” (also called ”arsinh” or ”asinh”) and so on.
By 2007 that was reversed:
The inverse functions are the inverse hyperbolic sine “arsinh” (also called “arсsinh” or “asinh”) and so on.
By 2009, “arc” was deprecated:
The inverse hyperbolic function are the area hyperbolic sine “arsinh” (also called “asinh”, or sometimes by the misnomer of “arcsinh”) and so on.
The current article on the inverse functions gives more details, showing that “arc” is not entirely a misnomer:
The earliest and most widely adopted symbols use the prefix arc- (that is: arcsinh, arccosh, arctanh, arcsech, arccsch, arccoth), by analogy with the inverse circular functions (arcsin, etc.). For a unit hyperbola (“Lorentzian circle”) in the Lorentzian plane (pseudo-Euclidean plane of signature (1, 1)) or in the hyperbolic number plane, the hyperbolic angle measure (argument to the hyperbolic functions) is indeed the arc length of a hyperbolic arc.
Also common is the notation \(\sinh^{-1}\), \(\cosh^{-1}\), etc., although care must be taken to avoid misinterpretations of the superscript −1 as an exponent. The standard convention is that \(\sinh^{-1}x\) or \(\sinh^{-1}(x)\) means the inverse function while \((\sinh x)^{-1}\) or \(\sinh(x)^{-1}\) means the reciprocal \(1/\sinh x\). Especially inconsistent is the conventional use of positive integer superscripts to indicate an exponent rather than function composition, e.g. \(\sinh^2x\) conventionally means \((\sinh x)^2\) and not \(\sinh(\sinh x)\).
Because the argument of hyperbolic functions is not the arclength of a hyperbolic arc in the Euclidean plane, some authors have condemned the prefix arc-, arguing that the prefix ar- (for area) or arg- (for argument) should be preferred. Following this recommendation, the ISO 80000-2 standard abbreviations use the prefix ar- (that is: arsinh, arcosh, artanh, arsech, arcsch, arcoth).
This shows some subtlety in the meaning of “area” and “arc” that is beyond my full understanding.
Overall, this is an interesting illustration of how language changes, even in mathematics! The history I wished to know was going on as I wrote!