Pros and Cons of Using $\arcsin$ vs $\sin^{-1} x$

[I like Serena]

I've noticed that everyone has its own preference whether to use $\arcsin x$ or $\sin^{-1} x$.
Let me list the pro's and con's as I see them.

1. $\sin^2 x$ is conventionally used for $(\sin x)^2$, suggesting $\sin^{-1} x$ would mean $\frac1{\sin x}$, which it isn't.
   Con: $\sin^{-1}$ is ambiguous.
2. When we have an inverse $f^{-1}$, we have by axiom the identity $f\circ f^{-1} = f^{-1} \circ f = \text{id}$.
   This does not hold true for $\sin^{-1}$, since it's not really an inverse of $\sin$, which is not invertible in the first place.
   Con: Even though $\sin^{-1}$ suggests it's the inverse of $\sin$, it's not. Treating it as such leads to wrong answers. The name $\arcsin$ suggests no such thing.
3. When you've been taught to use $\sin^{-1}$, it's easiest to keep doing so.
   Pro: It's easiest to do what you've been taught in high school.

When looking what is preferred on wikipedia, I can't even find $\sin^{-1}$ being mentioned (except in a redirect). Apparently the preference is for $\arcsin$.

Googling for the difference didn't find me anything (yet).

What am I missing?

 

[topsquark]

I've noticed that everyone has its own preference whether to use $\arcsin x$ or $\sin^{-1}$.
Let me list the pro's and con's as I see them.

1. $\sin^2 x$ is conventionally used for $(\sin x)^2$, suggesting $\sin^{-1} x$ would mean $\frac1{\sin x}$, which it isn't.
   Con: $\sin^{-1}$ is ambiguous.
2. When we have an inverse $f^{-1}$, we have by axiom the identity $f\circ f^{-1} = f^{-1} \circ f = \text{id}$.
   This does not hold true for $\sin^{-1}$, since it's not really an inverse of $\sin$, which is not invertible in the first place.
   Con: Even though $\sin^{-1}$ suggests it's the inverse of $\sin$, it's not. Treating it as such leads to wrong answers. The name $\arcsin$ suggests no such thing.
3. When you've been taught to use $\sin^{-1}$, it's easiest to keep doing so.
   Pro: It's easiest to do what you've been taught in high school.

When looking what is preferred on wikipedia, I can't even find $\sin^{-1}$ being mentioned (except in a redirect). Apparently the preference is for $\arcsin$.

Googling for the difference didn't find me anything (yet).

What am I missing?

I always use \(\displaystyle sin^{-1}( )\) when writing it down for anyone else, but in my own work I prefer "asn( )" despite the fact I learned \(\displaystyle sin^{-1}( )\) in High School. I don't know why...it just seems more "correct" for some reason.

-Dan

 

[Dethrone]

I just realized most high school math textbooks use $\sin^{-1}$. In fact, I have yet to see a high school math textbook that uses \(\displaystyle arcsin\), well, at least the textbooks that are used in Canada. I think it's just easier to introduce to new students, and it's also the notation used on calculators.

 

[Deveno]

This is my take on it:

If you are looking for a function, $\arcsin(x)$, makes more sense to me. This always returns a value in a specified range.

I always interpret $\sin^{-1}(x)$ to be a pre-image: that is, any real number $y$ such that $\sin(y) = x$ (this is not a function, since $y$ is not unique).

"arcsin(x)" and "sin^(-1)(x)" both represent the inverse sine function, which is also denoted as "sin^(-1)(x)." This function is used to find the angle whose sine equals a given value, and it is the inverse of the sine function "sin(x)."

Here's how it works:

1. "arcsin(x)": This is the traditional notation for the inverse sine function, often used in mathematics and scientific contexts. When you see "arcsin(x)," it means you're finding the angle (in radians or degrees, depending on the context) whose sine is equal to "x." For example, if you have "arcsin(0.5)," you're finding the angle whose sine is 0.5.
2. "sin^(-1)(x)": This is an alternative notation for the same inverse sine function. It's common in mathematics, but it's also used in many scientific calculators and computer programming languages. So, "sin^(-1)(x)" is equivalent to "arcsin(x)." It represents the same concept of finding the angle associated with a particular sine value.

In both notations, it's important to note that the output of the inverse sine function is an angle, not a number. The result is typically given in radians within the range [-π/2, π/2] or in degrees within the range [-90°, 90°], as these are the principal values of the inverse sine function. The choice of radians or degrees depends on the context and conventions of the problem you're working on.

 

[chisigma]

Big advantage of the notation $f^{-1} (*) $ is its applicability in all circumstances. Let's take a concrete example ... we all know what is the function $\text{erfc} (*)$ ... its inverse can express with $\text{erfc}^{-1}(*)$ without the need to strive in questionable 'inventions' ...

Kind regards

$\chi$ $\sigma$

 

[topsquark]

I just realized most high school math textbooks use $\sin^{-1}$. In fact, I have yet to see a high school math textbook that uses \(\displaystyle arcsin\), well, at least the textbooks that are used in Canada. I think it's just easier to introduce to new students, and it's also the notation used on calculators.

My next door neighbor's son was a Professor at MIT. In my senior year of HS I got a Calc book from him (the text is from the 70's) and it used "asn ( )." I got used to it that way.

My senior year in HS was fun...the Summer before my parents got me a math text for my birthday which, unknown to them, covered the first half year of my senior year Pre-Calc class. So having nothing better to do that Fall I perused the Calc text and had a good grasp of Calculus up to Calc II by Christmas. We did Intro Calc I in the Spring so that was easy too. (I had thought the text was a one semester book. Imagine my surprise when I got to College!)

-Dan

 

[I like Serena]

I always use \(\displaystyle sin^{-1}( )\) when writing it down for anyone else, but in my own work I prefer "asn( )" despite the fact I learned \(\displaystyle sin^{-1}( )\) in High School. I don't know why...it just seems more "correct" for some reason.

-Dan

When helping people, I'll use whatever it is that they are using.
In handwriting I usually abbreviate it to "asin", which I first started using when learning programming. It is how it is named in many programming languages. I guess I might as well start abbreviating it to "asn", which is shorter.

I just realized most high school math textbooks use $\sin^{-1}$. In fact, I have yet to see a high school math textbook that uses \(\displaystyle arcsin\), well, at least the textbooks that are used in Canada. I think it's just easier to introduce to new students, and it's also the notation used on calculators.

I did not know yet that most (if not all) Canadian and presumably US high school math books use $\sin^{-1}$. To be honest, I can't recall how it was taught in my (non-English) high school. Anyway, do they mention $\arcsin$?

And yes, good point, $\sin^{-1}$ is what you typically see on calculators.

This is my take on it:

If you are looking for a function, $\arcsin(x)$, makes more sense to me. This always returns a value in a specified range.

I always interpret $\sin^{-1}(x)$ to be a pre-image: that is, any real number $y$ such that $\sin(y) = x$ (this is not a function, since $y$ is not unique).

Such a pre-image makes sense, which is also what we run into when moving on to complex analysis.
Still, it bother me that $\sin^{-1}\left(\sin \frac{5\pi}2\right) \ne \frac{5\pi}2$, pre-image or not.

Big advantage of the notation $f^{-1} (*) $ is its applicability in all circumstances. Let's take a concrete example ... we all know what is the function $\text{erfc} (*)$ ... its inverse can express with $\text{erfc}^{-1}(*)$ without the need to strive in questionable 'inventions' ...

Kind regards

$\chi$ $\sigma$

Yes, I like to see that $f^{-1}$ is really unambiguous as in your example.
So... which version do you prefer?

My next door neighbor's son was a Professor at MIT. In my senior year of HS I got a Calc book from him (the text is from the 70's) and it used "asn ( )." I got used to it that way.

My senior year in HS was fun...the Summer before my parents got me a math text for my birthday which, unknown to them, covered the first half year of my senior year Pre-Calc class. So having nothing better to do that Fall I perused the Calc text and had a good grasp of Calculus up to Calc II by Christmas. We did Intro Calc I in the Spring so that was easy too. (I had thought the text was a one semester book. Imagine my surprise when I got to College!)

-Dan

Heh. So I guess you are also still writing things, at least for yourself, as you learned it in high school.

In my case my brother got me a linear algebra book well before the semesters started. It gave me something fun to do in the holidays before I was even a student.

 

[DavidCampen]

[*]When we have an inverse $f^{-1}$, we have by axiom the identity $f\circ f^{-1} = f^{-1} \circ f = \text{id}$.
This does not hold true for $\sin^{-1}$, since it's not really an inverse of $\sin$, which is not invertible in the first place.
Con: Even though $\sin^{-1}$ suggests it's the inverse of $\sin$, it's not. Treating it as such leads to wrong answers. The name $\arcsin$ suggests no such thing.

How is the arcsin not the inverse sin function?

Inverse trigonometric functions - Wikipedia, the free encyclopedia
Inverse Trigonometric Functions -- from Wolfram MathWorld
The inverse trigonometric functions are the inverse functions of the trigonometric functions, written $\cos^{-1}z$, $\cot^{-1}$, $\csc^{-1}z$, $\sec^{-1}z$, $\sin^{-1}z$, and $\tan^{-1}z$.

 

[I like Serena]

How is the arcsin not the inverse sin function?

Inverse trigonometric functions - Wikipedia, the free encyclopedia
Inverse Trigonometric Functions -- from Wolfram MathWorld
The inverse trigonometric functions are the inverse functions of the trigonometric functions, written cos^(-1)z, cot^(-1)z, csc^(-1)z, sec^(-1)z, sin^(-1)z, and tan^(-1)z.

The $\arcsin$ gives us the arc length on the unit circle belonging to a sine.
That does not mean that it gives us the original angle.

On the wiki page, you can see that they mention that $\sin(\arcsin x) = x$. Note that they do not mention the reverse ordering. And indeed, generally, $\arcsin(\sin x) \ne x$. Only if we limit the sine in domain to for instance the arbitrary $[-\frac\pi 2, \frac\pi 2]$ can we speak of an inverse function.

Note that at the start of the wiki page, when they first mention "inverse", the clause "with suitably restricted domain" is added.

On the wolfram page only complex analysis is mentioned. In complex analysis the inverse sine can be defined, but it is not an (inverse) function (see e.g. wiki or wolfram on functions). It's what they call a multi-valued function. Apparently Wolfram is abbreviating, since properly it should refer to an inverse trigonometric multi-valued function.

Anyway, when the complex multi-valued function is meant, it makes sense to use a different notation.

 

[DavidCampen]

On the wolfram page only complex analysis is mentioned. In complex analysis the inverse sine can be defined, but it is not an (inverse) function (see .

Then why does the Wolfram page, in the 1st sentence, say:
The inverse trigonometric functions are the inverse functions of the trigonometric functions,

And of course, one needs to restrict the values to the principal values; that is discussed on the Wolfram page, it is not a reason to say that there is no inverse function and neither does the Wolfram page restrict the discussion to complex analysis and even if it did, that is also not a reason to say that there is not an inverse function.

Note that at the start of the wiki page, when they first mention "inverse", the clause "with suitably restricted domain" is added.

Of course, that is why when you specify an inverse trig function you are obviously restricting its domain. That does not make it not an inverse function.

The arcsin gives us the arc length on the unit circle belonging to a sine.
That does not mean that it gives us the original angle.

Also incorrect. The arcsin does return an angle. As the wikipedia page says:
Thus, in the unit circle, "the arc whose cosine is x" is the same as "the angle whose cosine is x", because the measure of the length of the arc of the circle in radii is the same as the measurement of the angle in radians.
emphasis added is mine.

 

[I like Serena]

I only said it is not "really" the inverse function of the sine.
I believe we agree that the sine function, without restrictions, is not invertible.

It typically takes students some time to grasp this, since they have this neat button on their calculator that magically inverts the sine.

 

[Dethrone]

I've also realized that some old texts use "tg" instead of "tan". Now I hope no one learns that one incorrectly ;) In fact, one of the subway advertissments for the the recent movie "Lucy" had a background of random mathematical symbols and equations, and I noticed that they used "tg" there too...

 

[I like Serena]

I've also realized that some old texts use "tg" instead of "tan". Now I hope no one learns that one incorrectly ;) In fact, one of the subway advertissments for the the recent movie "Lucy" had a background of random mathematical symbols and equations, and I noticed that they used "tg" there too...

I have seen both "tg" and "tan" so often that I have no clue when one or the other would be used. Since there is no ambiguity, I'm fine with both. In my own handwriting I use "tg" just because it's shorter. ;)

 

[Dethrone]

I didn't realize "tg" was so known. I've never seen it in my life until just a couple of weeks ago, and still, searching it on the internet doesn't yield much results, if any. About your earlier post, I don't think the textbooks ever mentioned arcsine, and I've browsed through many of them. Anyways, new high school textbooks these days try way too hard to make the concepts and materials approachable to students (i.e Nelson science books) that I wouldn't be surprised if they don't mention it for fear it might confuse students seeing the extra letters "arc".

 

[DavidCampen]

I only said it is not "really" the inverse function of the sine.
I believe we agree that the sine function, without restrictions, is not invertible.

Agreed, that is why when an inverse trig function is specified it is too trivial to mention that of course the domain is restricted.

It typically takes students some time to grasp this, since they have this neat button on their calculator that magically inverts the sine.

Since the discussion was about which notation was best it seemed that this discussion was not directed at students who have just started learning trigonometry.

When I was a student we did not have calculators. We used either slide rules, or if we needed more accuracy, log tables. Back then we used $sin^{-1}$ to denote the inverse sin function and I considered arcsin to be an archaic notation.

 

[Dethrone]

Funny,
The arcsin, arccos and arctan functions

This sites uses capital "Arcsine" to refer to the one that isn't invertible and lower case "arcsine" to refer to the one that is invertible/restricted-domain.

 

[I like Serena]

When I was a student we did not have calculators. We used either slide rules, or if we needed more accuracy, log tables. Back then we used $sin^{-1}$ to denote the inverse sin function and I considered arcsin to be an archaic notation.

Now that you mention it, I recall that when I saw $\arcsin$ in my father's books, I remember assuming it was archaic as well. So I started out on high school with $\sin^{-1}$.

But I do not consider it archaic any more.
Now I think it's only an attempt to simplify math that tends to keep $\arcsin$ out of view.
Actually, I'm surprised to see that wikipedia consistently uses $\arcsin$.

 

[topsquark]

Now that you mention it, I recall that when I saw $\arcsin$ in my father's books, I remember assuming it was archaic as well. So I started out on high school with $\sin^{-1}$.

But I do not consider it archaic any more.
Now I think it's only an attempt to simplify math that tends to keep $\arcsin$ out of view.
Actually, I'm surprised to see that wikipedia consistently uses $\arcsin$.

Okay, now you need to ask the question: Who remembers "versine" and "haversine?"

-Dan

 

[I like Serena]

Funny,
The arcsin, arccos and arctan functions

This sites uses capital "Arcsine" to refer to the one that isn't invertible and lower case "arcsine" to refer to the one that is invertible/restricted-domain.

Yep. That is odd.
And indeed, here:
Inverse Sine -- from Wolfram MathWorld
you can see that Wolfram refers to $\sin^{-1}$ or $\arcsin$ as the multi-valued version, and to $\text{Sin}^{-1}$ or $\text{Arcsin}$ as the principal values.
This is what I expect.

 

[I like Serena]

Okay, now you need to ask the question: Who remembers "versine" and "haversine?"

-Dan

I've never learned them, never used them, and know only of their existence from hear-say.
Can we consider them archaic please?

 

[topsquark]

I've never learned them, never used them, and know only of their existence from hear-say.
Can we consider them archaic please?

"arc-haversine"? (Giggle)

-Dan

 

[Deveno]

(snip)Such a pre-image makes sense, which is also what we run into when moving on to complex analysis.
Still, it bother me that $\sin^{-1}\left(\sin \frac{5\pi}2\right) \ne \frac{5\pi}2$, pre-image or not.
(end snip)

Well this is sort of an awkward thing with functions:

Say we have a function $f:A \to B$. If $X$ is any subset of $A$, we get another function $X \to B$, which we typically ALSO call "$f$". This is a problem, especially in analysis, where what we may be actually integrating is some restriction of a given function.

For example, if I ask you to "unsquare" a number I squared, and I give you the number I got after squaring, you OUGHT to ask: well, what set did your original number come from? The functions:

$f: \Bbb R \to \Bbb R,\ f(x) = x^2$ and:

$g: \Bbb R^+ \to \Bbb R,\ g(x) = x^2$ are TWO different functions, but it's highly likely both of them will simply be referred to as: "$x^2$".

We can "unsquare" $g$ unambiguously (take the square root). No can do with $f$. Domains and co-domains matter (often, we are "too focused on $f$" to remember this).

I think part of this, is just how our brains work: we prefer to link concepts bijectively. You often see examples of this made clear in criminal cases:

"Mr. Jones was found standing over the victim's body with a bloody knife in his hand, therefore-he must be guilty".

While it IS true, that if Mr. Jones DID commit this heinous crime, that would be ONE way he would wind up standing over said body with above-mentioned bloody knife; it is not the ONLY way such an event might come to pass. And yet we leap to such conclusions all the time. We're just not "good" at reasoning about many-to-one correspondences.

 

[DavidCampen]

A nice thing about math as compared to the sciences is that while the notation may change; what was true a 100, 200 or even 2,000 years ago is still true today. Compare that to a 100 year old chemistry, physics, biology etc. textbook.

Speaking of classic texts, you can buy a very nicely made, hardcopy edition of Heath's English language translation of Euclid's Elements for less than $30. I highly recommend it.

Euclid's Elements: AU Euclid, Dana Densmore, Thomas L. Heath: 9781888009187: Amazon.com: Books

 

[chisigma]

I would like to report a profit in this interesting discussion the following thread ...

http://mathhelpboards.com/advanced-applied-mathematics-16/angle-change-rate-function-instanteneous-angle-11611.html#post54321

It is a fine example of what it means a multivalued function​​. Here $\theta(0)$ can take any value, not necessarily between $- \frac{\pi}{2}$ and $+ \frac{\pi}{2}$ , so that the solution will be the branch of the function that includes $\theta(0)$...

Kind regards

$\chi$ $\sigma$