Talk:Inverse trigonometric functions

Parameter

In $\tan(\theta )$, I understand that $\theta$ is an angle, but for $\arctan(a)$, what is $a$ called? SDSpivey (talk) 06:11, 23 May 2020 (UTC)

Generally, I've just seen it referred to as the "input", and that works regardless of the geometric interpretation of the numerical values (meaning it works for complex values as well). However, if you interpret it using the usual soh-cah-toa, you could call it the "side ratio" since it's the ratio of the opposite and adjacent sides. If you instead interpret it in the context of a triangle with adjacent side equal to 1, then it's just the opposite side of the triangle, as seen in the relationships between... section. Sorry if this response was too slow to be useful. BlackEyedGhost (talk) 04:02, 28 May 2020 (UTC)

proofs / citations

Consider the section Relationships between trigonometric functions and inverse trigonometric functions, which states useful rules without giving any background.

Ideally the section should lead with an introductory sentence or two that also outlines how these rules are derived (with a citation to any reliable source, even a math professor's website, that performs this derivation in detail).

Even failing this, there still ought at least to be some kind of citation given (even if an explicit proof is inappropriate). Cesiumfrog (talk) 04:21, 25 April 2012 (UTC)

They are easy to derive. For example, to get

$\sin(\arctan x)={\frac {x}{\sqrt {1+x^{2}}}}\,,$

one calculates as follows. Let

$\theta =\arctan x\,,$

then

$x=\tan \theta$

$x^{2}=\tan ^{2}\theta$

$1+x^{2}=1+\tan ^{2}\theta =\sec ^{2}\theta$

${\sqrt {1+x^{2}}}=\vert \sec \theta \vert =\sec \theta$

because θ is in (-π/2, π/2) where secant is positive

${\frac {1}{\sqrt {1+x^{2}}}}=\cos \theta$

${\frac {x}{\sqrt {1+x^{2}}}}=\tan \theta \cdot \cos \theta =\sin \theta \,.$

Substituting for θ, we get the desired formula. JRSpriggs (talk) 07:11, 25 April 2012 (UTC)

Yes a citation to where they are derived is the way to go okay, preferably to something that can be accessed easily. Dmcq (talk) 13:24, 25 April 2012 (UTC)

How about we just draw a triangle for each of the identities? Then there is no need to prove them, and the section benefits from some badly needed visual aids. Sławomir Biały (talk) 01:36, 29 April 2012 (UTC)

Yes. In the case above, you could put: θ in a corner, 1 on the adjacent side, x on the opposite side, and √(1+x²) on the hypotenuse. JRSpriggs (talk) 02:10, 29 April 2012 (UTC)

This doesn't appear to be done yet. Here is a diagram that may answer the above concerns for the section Relationships between trigonometric functions and inverse trigonometric functions?

The angle θ is an inverse trig function of x, or √1 + x², or a ratio of them.

Best, M∧Ŝc²ħεИ_τlk 21:06, 14 April 2013 (UTC)

History of the sin^-1(x) notation?

Does anyone known where that awful sin^-1(x) notation originated? I know it goes back a long way... I have seen in in 19th century textbooks. Tfr000 (talk) 13:28, 13 June 2012 (UTC)

Good question. To do it justice, we'd also need to know the histories of the notation for exponentation and of the notation for inverses of arbitrary functions. Cesiumfrog (talk) 06:45, 14 June 2012 (UTC)

Found it... and added it to the article. Tfr000 (talk) 19:35, 26 May 2015 (UTC)

I disagree it is "awful". It is the notation of an inverse function after all. I also find that students get a better understanding when asked to find the derivative of y=sin^-1(x), they instantly understand x=sin(y) (and then can differentiate implicitly). Jim77742 (talk) 00:11, 20 November 2017 (UTC)

But arcsin(y) is not an actual inverse of general sin(x), but of sin(x) on a specific interval -pi/2, pi/2. A function must be a bijection to posses an inverse. Without specifying an interval, sin(x) is not a bijection. 5.173.127.102 (talk) 00:07, 11 March 2024 (UTC)

Extension to complex plane

I reverted an edit by Syed Wamiq Ahmed Hashmi (talk · contribs) to the section Inverse trigonometric functions#Derivatives of inverse trigonometric functions in which he restricted the domains of the derivatives to certain subsets of the real numbers. The formulas he changed were intended to apply to extensions of the inverse functions to the complex plane (a fact which he overlooked, but is now aware of). This brings my attention to an issue — this article mentions the complex extensions in several places, but never defines them. The article was apparently written with only the real versions in mind and then later patched to include pieces of information about the complex extensions. I think that this calls for a redesign of the whole article to give at least equal treatment to the complex extensions. And as Wamiq noticed, we need to indicate where the cuts in the complex domain are located. JRSpriggs (talk) 15:38, 14 April 2013 (UTC)

The problem I am seeing now is how do we handle the fact that there are multiple sheets in the complex domain for the inverse trigonometric functions. And they interact with the fact that the square-root has two sheets. JRSpriggs (talk) 23:41, 14 April 2013 (UTC)

There are several possible ways of defining the extensions of inverse trigonometric functions to the complex plane. In addition to how the functions are computed, they may differ in where one puts the cuts between different sheets of the functions. I would like to suggest the following:

$\arctan z=\int _{0}^{z}{\frac {dz}{1+z^{2}}}\,$

provided that the contour of integration does not cross the part of the imaginary axis which does not lie strictly between -i and +i;

$\arcsin z=\arctan {\frac {z}{\sqrt {1-z^{2}}}}\,$

where the square-root function has its cut along the negative real axis;

$\arccos z={\frac {\pi }{2}}-\arcsin z\,;$

$\operatorname {arccot} z={\frac {\pi }{2}}-\arctan z\,;$

$\operatorname {arcsec} z=\arccos {\frac {1}{z}}\,;$

$\operatorname {arccsc} z=\arcsin {\frac {1}{z}}\,.$

If we adopt that suggestion, then it may affect the signs of the derivatives of the functions. To verify what those derivatives would be:

${\frac {d\arcsin z}{dz}}={\frac {1}{1+\left({\frac {z}{\sqrt {1-z^{2}}}}\right)^{2}}}\cdot {\frac {{\sqrt {1-z^{2}}}-z\left({\frac {-2z}{2{\sqrt {1-z^{2}}}}}\right)}{1-z^{2}}}={\frac {1}{\sqrt {1-z^{2}}}}\,$

which is as expected;

${\frac {d\operatorname {arcsec} z}{dz}}={\frac {-1}{\sqrt {1-z^{-2}}}}\cdot {\frac {-1}{z^{2}}}={\frac {1}{z^{2}{\sqrt {1-z^{-2}}}}}=\pm {\frac {1}{z{\sqrt {z^{2}-1}}}}\,$

which may have a different sign than the usual expression. Similarly for arccsc. JRSpriggs (talk) 08:28, 17 April 2013 (UTC)

Notation

<< I copied this from my talk page. JRSpriggs (talk) 08:01, 15 April 2013 (UTC) >>

Thanks a lot! That section looks better now ☺. Well, I see an issue with the notation used on Wikipedia for the inverse trigonometric functions, i.e., the convention here is to denote all functions with minuscule letters but to add the word arc with the inverse ones (sin x, arcsin x, etc...) but what we (along with our textbooks) do, is to denote regular functions with minuscule letters, e.g., sin x, cos x, etc., and the inverse functions with the first letter majuscule and a −1 superscript, e.g., Sin⁻¹ x, Cos⁻¹ x, etc., which causes no confusion between the inverse function (Sin⁻¹ x) and the multiplicative inverse (sin⁻¹ x). This notation is nowhere to be found here. I personally find the arc notation a bit odd. Do you find this (capital) notation at least worth mentioning in the article (if the arc notation is popular and cannot be removed)? Hoping to get a reply in the affirmative... Regards,

— Syɛd Шαмiq Aнмɛd Hαsнмi (тαlк) 06:26, 15 April 2013 (UTC)

I would rather not change the notation that way. Superscript minus one could be misinterpreted as the multiplicative inverse rather than the inverse with respect to composition. Please see the archive, Talk:Inverse trigonometric functions/Archive 1, for more discussion of this issue.

When referring to one of our articles or talk pages, please use [[this method]] rather than [http://en.wikipedia.org/wiki/this_method this method] . JRSpriggs (talk) 08:01, 15 April 2013 (UTC)

O.K., Now, I’ve got the things you said. I’ll do them as such. As to the notation, the article Inverse function, too, uses f⁻¹ for the inverse function of f which here, means the compositional inverse (the −1 doesn’t mean multiplicative inverse which would be denoted by (f)⁻¹...) So, are you satisfied as to the use of −1? Moreover, as I have already said, this notation doesn’t clash with that for the multiplicative ones. I have seen the archive and I do not demand replacement now, but just a bare mention (like somthing in the beginning of the article, saying that these notations are also used, which don’t cause confusion; for other people like me who don’t use and are unfamiliar with the arc notation).

— Syɛd Шαмiq Aнмɛd Hαsнмi (тαlк) 10:12, 15 April 2013 (UTC)

ISO80000-2-13 states "(sinx)ⁿ, (cosx)ⁿ, etc., are often written sinⁿx, cosⁿx, etc." But there is no mention of sin⁻¹x for arcsinx. — Preceding unsigned comment added by 83.223.9.100 (talk) 10:46, 12 January 2017 (UTC)

Explanation of "z" in "Expression as definite integrals" section?

The "Expression as definite integrals" section lists expressions of the functions of x with a z^2 and dz on the right hand side, but no explanation of where this "z" comes from. It's been a while since I've had trig in school so I'm guessing I need to be reminded of what it is, and it would be good to have a small explanation at the side or bottom explaining where Z comes from. Or, if it's a typo, we need to change the expressions to have x^2 and dx rather than z. But my money is that I'm the one that needs educating. Could a section be added to clear up what "z" is? 74.10.5.213 (talk) 22:32, 1 October 2013 (UTC)

See definite integral. This is not a matter of trigonometry, but rather of your failure to understand the integral notation. For example, in

$\arccos x=\int _{x}^{1}{\frac {1}{\sqrt {1-z^{2}}}}\,dz,\qquad |x|\leq 1$,

if x=0.5, then

$\arccos 0.5=\int _{0.5}^{1}{\frac {1}{\sqrt {1-z^{2}}}}\,dz=\lim _{n\to \infty }\sum _{k=1}^{n}{\frac {1}{\sqrt {1-\left(0.5+(k-{\frac {1}{2}}){\frac {(1-0.5)}{n}}\right)^{2}}}}{\frac {1-0.5}{n}}$.

In other words, z is replaced by $0.5+(k-{\frac {1}{2}}){\frac {(1-0.5)}{n}}$ and dz is replaced by ${\frac {(1-0.5)}{n}}$. OK? JRSpriggs (talk) 02:11, 2 October 2013 (UTC)

Definition of arccot

Wikipedia defines arccot to be the inversion of cot on the interval

]0,π[.

Whereas NIST (http://dlmf.nist.gov/4.23), as well as wolframalpha (http://www.wolframalpha.com/input/?i=arccotangent) use

]-π/2,π/2[.

Yesterday I just needed to plug arccot in somewhere and got quite confused. I think a caveat on this article would be nice explaining the different choice.

Furthermore: the plot of arccot in the complex plane (https://en.wikipedia.org/wiki/File:Complex_ArcCot.jpg) uses the mathematica/nist definition contradicting the article. Very confusing! — Preceding unsigned comment added by 93.194.241.203 (talk) 14:26, 10 January 2014 (UTC)

File:Arctangent Arccotangent.svg shows the correct version of the arccotangent. One must choose between

$\operatorname {arccot}(x)={\frac {\pi }{2}}-\arctan(x)\,$

and

$\operatorname {arccot}(x)=\arctan \left({\frac {1}{x}}\right)\,.$

I choose the former because it gives a value to arccot (0) and is continuous there whereas the latter definition is discontinuous and undefined at zero. JRSpriggs (talk) 07:20, 11 January 2014 (UTC)

This should be fixed. Wikipedia must use common math definitions, esp. standard ones, not introduce new. Sergey B Kirpichev (talk) 11:52, 11 April 2016 (UTC)

Can you show evidence that there is a standard definition, agreed by virtually all authors, other than the one I gave? I do not think so. JRSpriggs (talk) 18:11, 11 April 2016 (UTC)

NIST is a kinda of standards. What's much more important - virtually all CAS (and related mathematical software in general) agreed on that definition, both open and closed-source (i.e. Maxima, Mathematica, Maple, Matlab). I believe, this definition should be first (but we should mention others, if you have refs to quote). Last but not least, mentioned above inconsistent plots are in place. Sergey B Kirpichev (talk) 00:29, 12 April 2016 (UTC)

My reference is to the "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables" by the National Bureau of Standards (which is what NIST was called before its name was changed), issued June 1964, fifth printing August 1966 with corrections. See page 79, section 4.4 Inverse Circular Functions definitions. Formula 4.4.3 says:

$\arctan z\,=\,\int _{0}^{z}{\frac {\mathrm {d} t}{1+t^{2}}}\,=\,{\frac {\pi }{2}}-\operatorname {arccot} z\,.$

"The path of integration must not cross ... the imaginary axis in the case of 4.4.3 except possibly inside the unit circle." JRSpriggs (talk) 14:49, 13 April 2016 (UTC)

So wait, this is still left ambiguous. Have we all collectively decided to uphold the standards from that book? Or should we uphold the standards from the NIST website? Math Machine 4 (talk) 19:35, 15 December 2020 (UTC)

Furthermore, could someone who owns that book please verify if that claim is accurate? Because, I'm pretty doubtful that they'd change the international standard for something and only announce that new standard in one single book, but fail to update their websites to account for said update. Math Machine 4 (talk) 23:42, 19 December 2020 (UTC)

Actually, come to think of it, since that book came out in 1964 (before the Internet), and the current standard displayed on the NIST website is different, wouldn't that mean that the definition given in that book is outdated? Unless the NIST stopped being the international standard some time between 1966 and whenever they got a website, how is this even up for debate? Math Machine 4 (talk) 23:54, 19 December 2020 (UTC)

NIST is not an international standard; it's a national standard (hence the N). The ISO (which is international, hence the I) uses 0 < θ < π, or at least it did in 2009. See page 25 of the PDF at https://people.engr.ncsu.edu/jwilson/files/mathsigns.pdf (this is a scan of the 2009 standard, since you have to pay over $100 for the current standard, but I don't believe that it has changed). This is also in several American trigonometry textbooks that I've used, so I can cite them if you want American sources (to go with NIST and Wolfram, which are also American). Toby Bartels (talk) 01:50, 11 March 2021 (UTC)

So, we're just going to let this whole Wikipedia article contradict itself, constantly swapping between one definition and the other, just because no internet user on the entire Earth is willing to pay $160 to see what the current international standard is? So we'll just say "do them both?". Why would this even cost money, shouldn't an international standard be in the public domain? If they're so important, why don't they get funding from somewhere else? Or better yet, instead of charging a week worth of rent for a silly article (it depresses me to know that that's only a week), why not charge something reasonable, like $20? I'd pay $20 to read that article. You have to realize, there are 12 parts. If someone tried to pay for all 12 parts, that'd be $1200 and 2 of the parts would be missing.

In any case, let's give the benefit of the doubt and say the standard hasn't changed. Because it probably hasn't. There still isn't anything on the inverse trig functions for complex numbers, which I find completely ridiculous. We either say we can't take inverse trig functions of complex numbers (which is a blatant lie), or that the NIST is the standard for complex numbers, while the ISO is the standard for real numbers. Which doesn't make any sense because real numbers are to complex numbers what thumbs are to fingers. And also, I see NO REASON WHATSOEVER for NIST and ISO to have different standards. It completely DEFEATS THE PURPOSE of a standard: for things to be unambiguous. One answer. That's what upsets me. That and the fact it was even considered okay to cite a book nobody has access to as a source, but that's pretty much irrelevant at this point.

Let's just be honest on the article: "In order to take an arc cotangent, first, ask yourself: are you in America, or somewhere else in the world? If you answered America, you first take the reciprocal, then take the arctangent. If you're in some other country, take the arctangent, and subtract it from π/2." Yeah, it sounds silly, doesn't it? Well, too bad it's true. And again, too bad they didn't define it for complex numbers. You could say "Well, obviously, for complex numbers, you just do the same thing." Except no. The definition they list implies both the input and output are real numbers. They said nothing about subtracting anything from π/2, all they said was "the answer to cot(y)=x over the interval (0,π)" (paraphrased). I don't even know who I'm arguing with at this point, I'm just ticked off at how stupid this is.

And also, if anyone wants to pay $160 to see that article, here ya go: https://www.iso.org/standard/64973.html. PS, it only takes Swiss Francs, for some reason. Not even Euros, but Francs. If you're a gambling man, and want it at a discount, here ya go: https://www.docuarea.org/home/19852-ISO-80000-2-2019.html, but I cannot guarantee it's not a scam, nor can I guarantee it's being legally hosted. All I can say is it exists. And P.P.S., if someone does buy it, and they're legally allowed to, could they please host a picture of the arccotangent part on imgur and post the link here? Thanks. Math Machine 4 (talk) 23:28, 12 June 2022 (UTC)

Radian… and category

see also
(in Russian) ru:Обсуждение:Тригонометрические функции#Радианы, & ru:Обсуждение:Безразмерная величина#Единица измерения (?!?) безразмерной величины… --De Riban5 (talk) 09:03, 17 January 2015 (UTC)

You don't need to include the category to the WP article in another language. You can click on the language on the left side. M∧Ŝc²ħεИ_τlk 11:28, 17 January 2015 (UTC)

Inverse cosecant formatting

In the "Relationships between trigonometric functions and inverse trigonometric functions" table we can see the arccsc(x) function has a white background. Any ideas on how we can fix that? It sticks out like a sore thumb.

--Jason B. (talk) 21:01, 13 April 2016 (UTC)

Roman vs italic

Looking back at the old history of this article, it seems as if the $e$'s, the $i$'s, and the $dx$'s were all typeset in their more standard italic version, in contrast to how it is now. I'd like to put things back, but according to WP:MOSMATH, one should not go through and make these sorts of changes wholesale. However, I think it's warranted here, as this must have been done at some point in the article's past anyway, and this change would simply put it back to its original form. This is especially important for the $i$'s and the $e$'s, as roman versions are extremely nonstandard. Any thoughts/objections? Deacon Vorbis (talk) 21:23, 11 February 2017 (UTC)

I'm going to go ahead and make the changes. If anyone seriously objects, let me know here. Deacon Vorbis (talk) 14:51, 24 February 2017 (UTC)

There's no need to change the article now, but upright dx, e, and i are all standard according to the ISO. See pages 22&23, 24, and 27 (respectively) of the PDF file at https://people.engr.ncsu.edu/jwilson/files/mathsigns.pdf (this is a scan of the 2009 standard, since you have to pay over $100 for the current standard, but I don't believe that it has changed). Toby Bartels (talk) 01:53, 11 March 2021 (UTC)

Exponential Form

Hello all! I wondered if there is any standard expression of the inverse trigonometric functions in exponential form. I believe that the standard trigonometric functions can be expressed in exponential form, and that this form is of sufficient import to sometimes be used as the definition of these functions. I was reminded of this when I saw the infinite series section, because the exponential function can be represented and defined as an infinite series. If there exist such a representation, could it be mentioned here? Thanks for reading! JonathanHopeThisIsUnique (talk) 05:03, 30 November 2017 (UTC)

See Inverse trigonometric functions#Logarithmic forms. JRSpriggs (talk) 22:29, 30 November 2017 (UTC)

Complex logarithmic forms

The change I made on 2020-09-03 was reverted on 2020-09-05. I believe it was valid. I now also have several changes to the Logarithmic Forms that I believe make them valid everywhere for principal values of the functions, not just on the complement of the branch cuts. If anyone wants to review these identities in advance of my posting them, please reply here within the next 48 hours. Rickhev1 (talk) 18:28, 8 September 2020 (UTC)

Inverse trigonometric functions with their OWN WIKIPEDIA PAGE

Today I tried to open a Wikipedia page for the arctangent function. However, my suggestion was declined. According to the reviewer the page is not needed as it is already in Inverse trigonometric functions. This section is interesting. However, it covers everything whatever is needed and whatever is not. For me this page is OK. Let it be there. However, it looks like a textbook for inverse trigonometrical functions. It may be convenient for study of trigonometry of inverse functions, but it may be very inconvenient if one wants to find some information for a specific function like $\arcsin(x),\arcsin(x),\arctan(x)$. Therefore, if some people found it sufficient, there are users who would prefer separate consideration of individual inverse function. I do not understand why sine, cosine and tangent functions deserve their own pages in the Wikipedia while their inverse counterparts like $\arcsin(x),\arcsin(x),\arctan(x)$ do not deserve individual pages in the Wikipedia. There is no logic behind it. And it is very inconvenient to look for a specific information in the section that combines all inverse trigonometric functions together. Wikipedia should follow the convenient style adopted by MathWorld where all inverse functions are not merged together, but considered independently from each other. I hope you will find my suggestion reasonable. For convenience of readers this issue should be resolved in Wikipedia in future. Math&App (talk) 22:31, 6 October 2020 (UTC)

For the record, only the sine has its own page; the rest are all treated at Trigonometric functions (unless I'm missing something obvious). It is a bit odd, and I'd probably prefer that sine redirect to the main page with any content not already present merged in. Personally, I think the inverse functions should all be lumped together in one page as is currently done. There's always a tradeoff between splitting and merging, but here, I think the topics are so closely related that covering them all together works best. –Deacon Vorbis (carbon • videos) 22:50, 6 October 2020 (UTC)

Hi Deacon Vorbis,

Thank you for your feedback! It is very inconvenient to find a particular information about a function when everything is merged together. As I have mentioned, it is OK to keep everything merged. I am not against it. However, there should be independent pages as well for each trigonometric functions. Many popular handbooks in math keep each trig functions in a separate chapter to make convenient to find more specific information without wasting time of a reader. This is a common convention and we should follow this convention. Therefore, there should be more options for the readers. Some readers prefer to read merged version while other readers would be more comfortable for trig functions written in individual pages. Let's make this option for readers. They will decide themselves according to their needs and convenience. For me, reading this merged version is very inconvenient due to extra information that takes lot of my time. I would prefer MathWorld format that adopts individual pages for each function. I am sure there are plenty of people who would prefer individual pages as well. We should be more democratic to allow readers to have a choice between merged and unmerged versions that they find more convenient for their needs rather than to impose the only merged format to everyone. Let's both versions be present! It will be very reasonable to give this choice to the Wikipedia readers. Math&App (talk) 00:52, 7 October 2020 (UTC)

Applications section

This is nicely written subsection. That's where Wikipedia shines. I think we should extend such subsections. --AXONOV (talk) ⚑ 10:00, 7 July 2021 (UTC)

Arc --> Angle

The nice thing about the Arc- names is that they always remind you that the arc functions give back an Angle. Look at some complicated formula involving arctans and arcsines and at least you know the 'arcs' are giving you nothing more complicated than an angle. Acorrector (talk) 11:52, 22 November 2021 (UTC)

Solutions to elementary trigonometric equations (≥ 2 questions for editor 9.51.61.147)

The 4th column in the first table of the section contains typographical errors in rows 4 and 5.
Moreover, in the 4th column of rows 3, 4, 5 it may be questioned whether $k\pi$ is overspecified in the light of the $\iff$ and the rows 1, 2, 6, 7, 8. Nomen4Omen (talk) 15:31, 18 September 2022 (UTC)

India Education Program course assignment

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The above message was substituted from {{IEP assignment}} by PrimeBOT (talk) on 20:02, 1 February 2023 (UTC)