Talk:Inverse trigonometric functions

Simplified?

People come to this page when they don't know what the hell any of this jargon means... They come when they want an explanation as if it were expained to a student... If they wanted countless formulas, there's billions of math websites covering that in a simple Google search. But if they really knew that much about it to need such thing, chances are they'd already know it.

Sine and cosine are the only things I know, and I'm damn sure at least they can be explained in a very simple way, using images or not. Something along the lines of "Gets the offset relative to the angle." with some description on what that means... THEN you can put all other mumbo jumbo afterwards.

I think these different functions all need their own seperate pages, too.

Arc?

What is the etymology of arc? JianLi 13:00, 27 May 2006 (UTC)

This is because the argument of a trigonometric function is an arc, so the image of the inverse trigonometric function is also an arc:

sin(arc)=number --> arcsin(number)=arc

Alternatively sinus means bay or gulf and arcus means hoop or bow in Latin. Therefore, sinus and arcus conjure up the opposite ideas in mind RokasT 14:57, 7 September 2007 (UTC).

Graph

It's hard to believe there isn't a graph of this function very close to the top of the page - Jay

Be bold. Shinobu 12:50, 4 August 2006 (UTC)

Needs a definition

Needs a definition, I really am wondering if the arcsin is a function in its own right or if it is just what you have to do to undo a sine function11:39, 15 August 2006 (UTC)Oxinabox1 11:39, 15 August 2006 (UTC)

Arcsine, arccosine, and arctangent are all functions just as the square root of x (√x ), the inverse of x squared (x²), is a distinct function. Computer Guy 990 (talk) 17:01, 7 April 2008 (UTC)

function for arccotangent??

From my understanding of the table it is saying $\ y=\operatorname {arccot} x$ is the same thing as $\ y=\arctan {\frac {1}{x}}$ under the definitions.

However in the relationships among the functions it says

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

and there is a picture of a graph next to it.

However when graphing $\ y=\arctan {\frac {1}{x}}$ you're negative X values will be different from the picture and the latter equation. So is my understanding of the table a little off or is one of these functions not correct?TungstenWolfram 21:13, 29 November 2006 (UTC)

Well, that's what is meant by "usual principal value" or "principal inverses" in the text, isn't it. The functions as given are correct. Maybe the following helps to your confusion. The trigonometric functions are not injective functions, meaning that there are several x-values for one y-value (for instance $\sin(0)=\sin(2\pi )=0$). So technically there should not exist any inverse function to the trigonometric functions. In order to still obtain something like an inverse function, one restricts the range of arguments of the sine to values between $-\pi /2$ and $\pi /2$. In that range the sine is injective and one can define an inverse function arcsin. That range is called "usual principal value" here. Of course one could have chosen any other range in which the sine is injective.

As for your example of arctan, since tan is a pi-periodic function you might want to write $\arctan x=y\mod \pi$. Was this of any help? -- Bamse 02:01, 30 November 2006 (UTC)

Ah ok. Thanks for clearing that up. TungstenWolfram 22:23, 30 November 2006 (UTC)

Could be worthwhile adding a warning or definition of the "principal value" used in this text (which is different from principal value). Bamse 00:24, 1 December 2006 (UTC)

Thank you for drawing my attention to the fact that the definition of arccotangent was inconsistent. I have fixed it now (I think). Its principal value, like that of arccosine, lies between zero and pi. JRSpriggs 08:45, 2 December 2006 (UTC)

Multiple values of inverse trig functions

In advanced high school trig, some texts emphasize that there are often other useful values of inverse trig functions besides the principal value. For example, arcsin (0.5) is 30 degrees but could also be 150, as well as either of these plus n times 360 (n = any integer). This should be mentioned in the article. If AS, AC, and AT are the principal inverse sin, cos, and tan values, then also each one plus n times 360; and (180 - AS), (-AC), and (180 + AT). [Also similarly of course for the inverse cot, sec, and csc values.) L P Meissner 02:04, 18 December 2006 (UTC)Loren P Meissner

Why do you not create a new section, named say "Non-principal values", and explain what they are and what they are good for. But please use radians instead of degrees for consistency. JRSpriggs 08:41, 18 December 2006 (UTC)

The most useful form would be to express the solutions of sin y = x for y in terms of arcsin, and likewise for the other trigonometric functions. Something like:

sin y = x if and only if y = arcsin x + 2kπ or y = π − arcsin x + 2kπ for some integer k.

 --Lambiam^Talk 10:16, 18 December 2006 (UTC)

I think you're right and that the "General solutions" section should be for example for the arccos: $\cos(y)=x\leftrightarrow y=\pm \arccos(x)+2k\pi {\text{ }}\forall {\text{ }}k\in \mathbb {Z}$ —Preceding unsigned comment added by 93.144.52.208 (talk) 14:04, 13 March 2009 (UTC)

logarithmic forms wrong?

at least for arctan i guess its: ln(1+ix)-ln(1-ix) instead of ln(1-ix)-ln(1+ix). but maybe my understanding of complex numbers is just messed up. if its wrong someone with a broad understanding should check all the other logarithmic forms too. 89.166.145.17 20:19, 15 January 2007 (UTC)

Yes, I think that some of them may be off by a multiple of pi. But it is not apparent to me how to fix them. JRSpriggs 05:43, 16 January 2007 (UTC)

Image of a right triangle

We need to get a better image of a right triangle for the practical applications section. There should be a theta in the lower right corner. JRSpriggs 05:43, 16 January 2007 (UTC)

Differnt def.s

Can someone put in the difference between arc... and ...^-1
i.e. $\arcsin$ and sin^-1
I know one means the infinite no of answers, and that the other means the lowest applicable positve answer,
I think the former is arc... and the latter is ...^-1, but I'm not certain. User_talk:Englishnerd 19:54, 7 February 2007 (UTC)

The "Handbook of Mathematical Functions" uses "Arcsin" for any of the infinite number of inverses of sine and "arcsin" for the principal value, but I am not aware that there is any standard on this. As far as I know, "sin^-1" could mean either one. Most uses of "arcsin" in this article are intended to be the principal value. JRSpriggs 06:56, 8 February 2007 (UTC)

Yes, it is true that many books use "Arcsin( )", "Arccos( )", "Arctan( )", etc., as you have shown above, and "arcsin( )", "arccos( )", "arctan( )", etc., just as you have shown above. This ought to be the universal standard, and all mathematicians just need to get with the plan!
However, as a man with graduate degrees in both mathematics and electrical engineering, and who has taught both, I have encountered several wacky problems:
A). College students who cannot tell the difference between upper case and lower case letters.
B). College students who could not reliably write the difference between upper case and lower case letters.
C). College students who do not understand what "upper case" and "lower case" mean !
BTW, "upper case" means "capital letters", and "lower case" means "little letters" ! Incredible!
In an electrical engineering course in AC circuits, I got a student who could not write or print his own name correctly. He printed "kAelin" all the time. He then told me that whenever he tried to do differently, he would get all confused and include @ signs. Would you believe "K@elin" for his own name??
In AC circuits courses, it is vital to understand that "V" is a DC voltage, and "v" is a voltage that is a function of time, including all AC voltages, and others, too. [Also, likewise for currents.]

I then told him and all of the other students in these courses that as for this kind of notation for DC and AC quantities, if walking through fire was what it took to learn how to do this, then walk through fire! I told them that is was the way that the textbook did things, and it was the way that their professor did it, and I expected them go get with the plan!
I am convinced now that lots of high school and college students would have a hard time dealing with the notation Arcsin( ) and arcsin( ), etc.98.67.173.16 (talk) 06:46, 5 March 2010 (UTC)

Euler's series for the arctan

At the risk of exposing myself as a total idiot, I'll ask: is the more efficient series for arctangent credited to Euler correct? As I read it, the arctan of 1 (which is pi/4 (~= 0.785398)) would be ${\frac {1}{2}}*({\frac {2}{6}}+{\frac {2}{6}}*{\frac {4}{10}}+{\frac {2}{6}}*{\frac {4}{10}}*{\frac {6}{14}}+{\frac {2}{6}}*{\frac {4}{10}}*{\frac {6}{14}}*{\frac {8}{18}}+...)$ $={\frac {1}{2}}*(.33333+.133333+.0571428+.0253968+...)$ which appears to be converging to far less than pi/4 (and a program implementation puts it near 0.261799 after a couple thousand iterations).

It seems like I'm either (1) completely misreading or misunderstanding the text as written, (2) getting confused by something that wasn't as clear as it could be in the article, or (3) something is wrong in the article. I'm guessing the odds of (1) are much greater than the odds of (2) which are much greater than the odds of (3), but hopefully it doesn't hurt to ask. Thanks. TertX 00:11, 28 March 2007 (UTC)

The formula is:

$\arctan x={\frac {x}{1+x^{2}}}\sum _{n=0}^{\infty }\prod _{k=1}^{n}{\frac {2kx^{2}}{(2k+1)(1+x^{2})}}.$

When n = 0, you get the empty product which is 1. So your sum (using your figures) should be:

${\frac {1}{2}}*(1+{\frac {1}{3}}+{\frac {1}{3}}*{\frac {2}{5}}+{\frac {1}{3}}*{\frac {2}{5}}*{\frac {3}{7}}+{\frac {1}{3}}*{\frac {2}{5}}*{\frac {3}{7}}*{\frac {4}{9}}+...)$ $\approx {\frac {1}{2}}*(1.0000+0.3333+0.1333+0.0571+0.0254*2)$

which is 1.5745 / 2 = 0.78725 . Is that close enough? JRSpriggs 09:13, 28 March 2007 (UTC)

P.S. I should explain that the multiplication of the last term in my approximation by 2 is an attempt to approximate the tail of the series. Since the common ratio approaches 1/2, the tail approaches (1/2 + 1/4 + 1/8 + ...) = 1 of the last term included. So I doubled the last term. JRSpriggs 10:20, 28 March 2007 (UTC)

Ah, I missed the empty product with n = 0. I figured it was much more likely a misunderstanding on my part than a typo in the formula. At least I was right about that. :) Thanks for the explanation. TertX 15:59, 28 March 2007 (UTC)

Two forms need to be presented

I've come to realize that not everyone will associate the arc- forms of the -1 forms, i.e.

${\begin{array}{rl}\arcsin &=\sin ^{-1}\\\arccos &=\cos ^{-1}\\\arctan &=\tan ^{-1}\\\operatorname {arcsec} &=\sec ^{-1}\\\operatorname {arccsc} &=\csc ^{-1}\\\operatorname {arccot} &=\cot ^{-1}\end{array}}$

especially since the -1 forms are used more often in printed material. Shouldn't these be associated together? --JB Adder | Talk 06:16, 30 March 2007 (UTC)

As it correctly says in the lead of the article: "The notations sin⁻¹, cos⁻¹, etc are often used for arcsin, arccos, etc, but this notation sometimes causes confusion between (e.g.) arcsin(x) and 1/sin(x).". Consequently, I think that this issue has already been adequately covered. JRSpriggs 10:59, 30 March 2007 (UTC)

As a side note: I was taught BOTH forms. Is it really that difficult for a teacher to present both notations? — Preceding unsigned comment added by 134.253.26.11 (talk)

My concern here is to present the properties of the functions, not to give redundant coverage in each notation. I choose to avoid using the potentially confusing notation as far as possible. JRSpriggs 08:19, 4 May 2007 (UTC)

Any article involving inverse trig functions should just include a note at the describing the two different notations. I prefer the arc* notation, but as long as you have a note at the top, I wouldn't even be against changing between notations in different sections. However, if you're going to use the *-1 notation, you should avoid 1/* notation like the plague and use the secant functions.

And besides... "arc" is easier to say than "inverse". It has one less syllable. :-P

atan2 function

The 2arctan(...) alternative to atan2 seems to work for all real x, not just x>0 like the article says. Could someone double check this? — Preceding unsigned comment added by 134.253.26.11 (talk)

The condition for the first pair of formulas is "provided that either x > 0 or y ≠ 0". If that condition fails, then x ≤ 0 and y = 0. In which case, the formulas become

$\operatorname {atan2} \,(y,x)=2\arctan {\frac {0}{0}}$

which is undefined when it should be π. JRSpriggs 08:31, 4 May 2007 (UTC)

I understand the arctan of both infinity and indeterminate expression is undefined, but what about the case x < 0 and y ≠ 0? The text doesn't assert whether the 2arctan(...) alternatives is good for that range of values, but it seems to actually work there. I think the 2arctan(...) alternatives are good for all real x and y except for y=x=0. y can be zero as long as x is nonzero, in which case you can use the formula where y is in the numerator. Am I mistaken? I don't think I understand why one would ever use the formular where y is in the denominator. Perhaps you could point me to a proof of these equations (and also put that link in the article).

Order of relations

Changed some relations and the order of them at the beginning, I think is in a more natural now. But the format isn't good any suggestions? Ricardo sandoval 00:11, 9 May 2007 (UTC)

continued fractions

the continued fraction of the arctan is interesting but incomplete, this article is very hard to digest, so I suggest that it is either removed, or the continued fractions of all six inverse functions are added.

i repeat, this article is extremely intimidating

Addition formulas

What about the addition formulas shouldnt they be of some interest?

$\arctan \theta _{1}+\arctan \theta _{2}=\arctan {\frac {\theta _{1}+\theta _{2}}{1-\theta _{1}\theta _{2}}}$

$\operatorname {arccot} \theta _{1}+\operatorname {arccot} \theta _{2}=\operatorname {arccot} {\frac {1-\theta _{1}\theta _{2}}{\theta _{1}+\theta _{2}}}$

$\arcsin \theta _{1}+\arcsin \theta _{2}=\arcsin(\theta _{1}{\sqrt {1-\theta _{2}^{2}}}+{\sqrt {1-\theta _{1}^{2}}}\theta _{2})$

$\arccos \theta _{1}+\arccos \theta _{2}=\arccos(\theta _{1}\theta _{2}-{\sqrt {(1-\theta _{1})(1-\theta _{2})}})$

Or are they uninteresting due to the complex definitions? T.Stokke 11:45, 22 September 2007 (UTC)

The proof of the arcsin formula

is quite incomplete. The main point, i.e. that it is well-defined, is missing completely and is non-trivial. For that one would need to show that $iw+{\sqrt {1-w^{2}}}\in \mathbb {C} _{-}$. --129.132.146.66 17:44, 22 October 2007 (UTC)

• There is another problem. The well-definedness of ${\sqrt {1-w^{2}}}=exp({\frac {1}{2}}log(1-w^{2}))$ is not shown either. In fact it is also wrong, because for $w\in (-\infty ,-1]\cup [1,\infty )$ the logarithm is undefined as $1-w^{2}\in (-\infty ,0]$. -- HelmutGrohne —Preceding unsigned comment added by 212.201.78.242 (talk) 12:52, 8 June 2008 (UTC)

$\log(-x)\!\,=i\pi +\log(x)$ for x > 0. so $\exp({\frac {1}{2}}\log(-x))=\exp({\frac {i\pi }{2}}+{\frac {1}{2}}\log(x))=\exp({\frac {i\pi }{2}})\exp({\frac {1}{2}}\log(x))=i{\sqrt {x}}={\sqrt {-x}}$ And because $Re(\log(x)){\xrightarrow {x\rightarrow 0}}-\infty$, the formula is also true for x=0. So it's true for all $x\in \mathbb {R}$ 84.190.30.97 (talk) 20:53, 10 June 2010 (UTC)

Graphs of Arcsecant and Arccosecant

This page would benefit from having all of the inverse trigonometric graphs. Can someone make ones for Arcsecant and Arccosecant in the same vein as the other four inverse functions that have graphs? M@$+@ Ju ~ ♠ 17:44, 6 November 2007 (UTC)

I added such graph. Bamse 06:03, 7 November 2007 (UTC)

Arcosh / Arccos

Discussing about arcosh and how it has as output the area between the two rays, we saw, that actually the arccos could also be seen as the area instead of the angle, by definig cos as a fonction of the area. Why not? --Saippuakauppias ⇄ 14:07, 22 January 2008 (UTC)

Derivatives of inverse trigonometric functions

How is d/dx arcsin(x) = d/dx 1/sin(x)? I thought arcsin =/= 1/sin. —Preceding unsigned comment added by 12.206.238.206 (talk) 12:23, 29 January 2008 (UTC)

True
98.67.173.16 (talk) 05:49, 5 March 2010 (UTC)

Yes, arcsin is not equal to 1/sin.

And so, as you would expect, their derivatives are also not equal:

${\frac {d}{dx}}{\frac {1}{\sin x}}$$={\frac {d}{dx}}\csc(x)=-{\frac {\cos(x)}{\sin ^{2}(x)}}=-\csc(x)\cot(x)$, which is clearly different from

${\frac {d}{dx}}\arcsin x={\frac {1}{\sqrt {1-x^{2}}}}$.

The section inverse trigonometric functions#Derivatives of inverse trigonometric functions is supposed to be a brief summary of the differentiation of trigonometric functions article.

Currently that section "derives" the derivative of arcsin with this rushed one-liner:

if $\theta =\arcsin x\!$, we get ${\frac {d\arcsin x}{dx}}={\frac {d\theta }{d\sin \theta }}={\frac {1}{\cos \theta }}={\frac {1}{\sqrt {1-\sin ^{2}\theta }}}={\frac {1}{\sqrt {1-x^{2}}}}$.

How can we make this less confusing? (a) add a few steps -- i.e., summarize *less* concisely, perhaps something like:

if $\theta =\arcsin x\!$, we get ${\frac {d}{dx}}\arcsin x={\frac {d}{dx}}\theta ={\frac {1}{{\frac {d}{d\theta }}x}}={\frac {1}{{\frac {d}{d\theta }}\sin \theta }}={\frac {1}{\cos \theta }}={\frac {1}{\sqrt {1-\sin ^{2}\theta }}}={\frac {1}{\sqrt {1-x^{2}}}}$.

Would it be overkill to also point out that ${\frac {1}{{\frac {d}{d\theta }}\sin \theta }}$ is not equal to ${\frac {d}{d\theta }}{\frac {1}{\sin \theta }}$?

Or would it be better to (b) delete the rushed one-liner entirely -- summarize *more* concisely -- and link to the much longer, more relaxed proof over at differentiation of trigonometric functions#Differentiating the inverse sine function ? --68.0.124.33 (talk) 18:13, 12 July 2010 (UTC)

Requested move

The following is a closed discussion of the proposal. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result was Move to Inverse trigonometric functions. EdJohnston (talk) 01:49, 26 June 2008 (UTC)

We have Trigonometric functions, so for coherence, the article should be named similarly, Inverse trigonometric functions. The plural form makes more sense (it's just a couple of notable functions, not really a status, as opposed to, for example, periodic function). Cenarium (talk) 21:17, 16 June 2008 (UTC)

• Weak oppose. It would represent an exception to Wikipedia:naming conventions#Prefer singular nouns, and I don't see why in this case the plural form makes more sense as claimed above. Would it be better to rename trigonometric functions, Kelvin functions and others that follow this pattern, to conform to the existing policy? Andrewa (talk) 10:55, 19 June 2008 (UTC)
  • Support (change of vote). Hopefully this strong consensus will be reflected in an explicit naming convention in due course. Andrewa (talk) 10:10, 20 June 2008 (UTC)
• Support, per nom. This seems to come under the 'concerns a set that requires a plural' clause. There are a number of articles titled "foo functions" (in Category:Special functions, for example), and in each case, it's because they're about a specific collection of named functions, not a property that general functions may or may not have. A good rule of thumb might be the natural form of the first sentence: it would sound silly for this article to begin 'an inverse trigonometric function is' or for continuous function to begin 'the continuous functions are'. Algebraist 15:33, 19 June 2008 (UTC)
• Support. The move makes sense, because the articles do talk about several functions. Calling them all together (inverse) trig. functions is standard nomenclature. Jakob.scholbach (talk) 16:34, 19 June 2008 (UTC)
• Support as above. CRGreathouse (t | c) 18:11, 19 June 2008 (UTC)
• Support. The adjective "trigonometric" does not describe an identifiable mathematical property. So there is no way to define a general "trigonometric function". Therefore, what we are left with is a list of a finite and small number of functions that have been given the designation "trigonometric functions". VectorPosse (talk) 18:33, 19 June 2008 (UTC)
• Support as above. This is existing guidance, the reason WP:NC says that titles that "concern a set of objects" should be in the plural. The real test would seem to be: do we have any common reason to link to these in the singular? So complex number, because we often want to speak of a particular complex number. Septentrionalis PMAnderson 22:07, 19 June 2008 (UTC)
• Support. It makes sense that Kelvin functions, trigonometric functions, etc. are plural, whereas bounded function, even function, recursive function, etc., are singular. An article titled elephant is about any elephant, where there is not some finite list containing all of them, whereas here we're dealing with a small specified list of functions. Michael Hardy (talk) 23:10, 19 June 2008 (UTC)
  • ....OK, now I see that the manual now says "or concerns a set". This seems to be such a case. ("Set" of course is not meant in the mathematical sense, but more of an epistemological sense.) Michael Hardy (talk) 23:11, 19 June 2008 (UTC)

    The wording small class suggests itself ;-> Septentrionalis PMAnderson 00:49, 20 June 2008 (UTC)

The above discussion is preserved as an archive of the proposal. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Move of some detail to atan2 article

I intend moving some of the detail from the 'Two argument variant of arctangent' to the atan2 article and put a 'main atan2' at the head of the section here. I'll probably put a short note at the top of the section here saying why on earth anyone would want the function. Dmcq (talk) 13:09, 11 October 2008 (UTC)

On the recommended method of calculation

I am no mathematician, indeed I'm far from being one, but I'm not totally useless with it, so, standing on a middle ground, yet far from the spot of the real deal, I do not understand why the method to find the arcos happens to demand one to find the arcsine first, which itself demands one to find the arctan which, enters into a loop as it requires oen to find another arctan, there might be a way around this loop professionals know about, but I doubt they are the kind that will come to Wikipedia for this info, or am I wrong in some way and there is no recursivity in here? I mean, I came here just to find how to make excel get some surface areas for me... I didn't expected to find something I can only miscomprehend or else comprehend as eternally recusive.Undead Herle King (talk) 21:15, 27 August 2009 (UTC)

I'm going to remove that section entirely. I don't believe the methods described are actually used anywhere and there is no citation. What's there isn't notable. Dmcq (talk) 20:39, 28 August 2009 (UTC)

The methods are precisely those used by MPFR according to the documentation: [1] Fredrik Johansson 08:57, 29 August 2009 (UTC)

Whatever about them being used in some packages they would only be "recommended" in certain circumstances. In other circumstances CORDIC methods or dividing one polynomial by another would be used. It all depends on what the requirements are. The best I can think of that can be said for them are that they are equalities rather than approximations so possibly some other text could be put round them and then the citation could be used too. Recommended method though is just going far too far. Another thing wrong with such a section is that it is a howto. Dmcq (talk) 20:26, 29 August 2009 (UTC)

Trigonometric_function#Computation is much better written I think. Dmcq (talk) 20:45, 29 August 2009 (UTC)

That's ironic, your arguments correspond to points 1 and 4 of "Wikipedia is not a manual, guidebook, textbook, or scientific journal" (1.instruction manual & 4. Textbooks and annotated texts) while mine correspond to points 5 and 7 ("Scientific journals and research papers" and "Academic language") however I concede to some point that the nature of this article demands academic language I was asking for it to be clearer to laymen, although I have found excel does solves arcosines (just under a label I did not expect) I still believe it would help if arcosines could be described in relations understandable to those not yet informed of what an arcosine is (relations which must be as much mathematical as they are linguistic, that is, in the case of a cosine it is the relation found in a right-angle triangle between an angle's adjacent side and the hypotenuse with the later's length as the denominator and the former's length as the numerator, while a linguistic relation would simply relate the cosine with tangents and cosines or with trigonometry or even with the length of the sides of a triangle side called the "adjacent side" and the "hypotenuse" but do little else to explain the concept). Now, claiming Wikipedia is not an instruction manual it means it is not there to solve problems where the question is "how to act" on some matter, however as much as it is encyclopedic it must solve questions on "how to conceptualize" something, whether it be a chair or an inverse trigonometric function. With this goal in mine knowing how to reach any inverse trigonometric function with the least number of simplest knowledgeable element is necessary (for a cosine you can reach it by having the adjacent side's length and the hypotenuse's length, if you do not know the angle, I've forsaken my trigonometry but I think you could also use either of these measures and the angle from which you wanted to extract the cosine), my criticism on the article stems from the fact I did not found any such explanation of how to reach an arcosine (instead of it a series of equally complex inverse trigonometric functions had to be solved first and the simplest of them, the only one that did not require the solution of another of them, was recursive).Undead Herle King (talk) 04:17, 30 August 2009 (UTC)

I think what you are saying is that you would like a straight description of the arccos, etc., functions here instead of just saying they are the inverses of the corresponding Trigonometric functions. I'm not too sure it's a great idea, it's only one level of indirection and the title of this article is Inverse trigonometric function so you'd expect a person to look up what's being inverted.

As to the original business about the half-angle arctan formula being recursive that is true but that formula is only applied until x is small enough to use another method, for instance a series. All those formulas that appeared in the "recommended" section still appear in the section at the beginning about relations between the functions so in fact the section was also a repeat and redundant. As you've found out most any halfway decent package with maths capabilities will provide most if not all the functions. The section on series gives explicit ways of calculating them, though that's not how they are actually done in math libraries normally. Dmcq (talk) 09:37, 30 August 2009 (UTC)

too much too soon

The definition of a good introduction is that it briefly itemizes what is in the rest of the article. This introduction has way too much technical jargon and really belongs elsewhere in the article with good explanations of the jargon. 4.249.3.102 (talk) 17:25, 22 October 2009 (UTC)

beautiful graphics

But what in the world do they mean? why do certain colors show up at certain places and how do they relate to the jargon above them? 4.249.3.102 (talk) 17:28, 22 October 2009 (UTC)

Usual notation

Is "arcsin" really the usual notation? It seems to me that "sin⁻¹" is more common. JIMp _talk·cont 10:23, 14 January 2010 (UTC)

The terminology in a paragraph is doubly unfortunate

This statement:

"The two-argument atan2 function computes the arctangent of y/x given y and x",

uses unfortunately (and not really needed) terminology.

This is because it uses the terminology of another common field of applied mathematics, probability, in "the arctangent of y/x given y and x", which is the language of "conditional probability". In conditional probability statements, you have several different variables - in this case three, x, y, and the output of the function - but rather than letting x and y be variable, you take them as "given". This is an important step in a lot of probability calculations.

Here is a much better way of stating the original sentence:

"With the inputs x and y, the two-argument function "atan2" computes the arctangent of y/x ." —Preceding unsigned comment added by 98.67.173.16 (talk) 17:13, 5 March 2010 (UTC)

I fail to follow the reasoning. The usages are the same as far as I can see. There is no need for such distinctions. Dmcq (talk) 23:26, 5 March 2010 (UTC)

FWIW, "computing X given Y" is different to "computing X from Y". The first is indeed a statement of probability, while the second is a function description. OrangeDog (τ • ε) 22:32, 1 June 2010 (UTC)

Inverse trigonometric functions in the complex plane

What do these images show? How were they created? What do they mean? Neither the captions nor the image description pages shed any light on this. OrangeDog (τ • ε) 13:19, 1 June 2010 (UTC)