Talk:Inverse trigonometric functions/Archive 1

This is an archive of past discussions about Inverse trigonometric functions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

Archive 1

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 • contribs)

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 • contribs)

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)

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. —Preceding unsigned comment added by 94.196.200.70 (talk) 23:34, 12 July 2010 (UTC)

Practical considerations

How is arctan well-conditioned near pi/2? Arcsine and arccosine flatten near pi/2 and 0, respectively. Arctan2 is insensitive to large changes in y-value near pi/2. Why is that better? Isn't it a matter of choosing one's poison? Rounding errors in calculations to obtain values for plugging into arccosine or arctan2 may be a different matter, but that doesn't seem like a reason to disparage the arccosine or arcsine functions for having a different problem (flattening) than the arctan function (insensitive to large changes in y-value). Worse, the plain arctan function blows up (and returns no answer) in cases where both arcsin and arccosine can return pi/2. -- Ac44ck (talk) 03:14, 13 October 2010 (UTC)

Well conditioned means you don't get large changes in the output for small changes in the input. Calculating an angle via arccosine for instance a small inaccuracy in the input when near 1 could lead to a large difference in the angle calculated (and in fact you have to be careful not to have a number greater than 1 in the input). By arctangent blowing up I think you must be thinking of the calculation of y/x which is not part of the arctangent function itself. However even so if you actually try this out in for instance Java which implements IEEE 754 floating point you will find it copes with the infinity quite well! - not that I'd depend on that normally or even use arctangent instead of atan2. Arctangent is not well conditioned near pi/2 because it has a discontinuity there so a small change to the input can lead to a large change in the output. Even atan2 has a discontinuity with its output jumping from +pi to -pi but that doesn't normally cause problems unless you're doing something like looking at changes in an angle. Dmcq (talk)

The problem with arcsine as it relates to computer programs is easily fixed. Programming languages include an atan2 function to deal with a lack in the usual atan function. A 'safe" arcsine function can deal with the problem of the usual arcsine being ill-conditioned near 90 degrees. A bloated rendering of the function in True BASIC follows:

FUNCTION asin_safe_degrees(x)
    OPTION ANGLE degrees
    IF x < 0.707 then
       LET asin_safe_degrees = asin(x)
    ELSE
       LET a = sqr(1-x*x)
       LET a = 90 - asin(a)
       IF x < 0 then
          LET a = -a
       END IF
       LET asin_safe_degrees = a
    END IF
END FUNCTION

-Ac44ck (talk) 06:01, 15 October 2010 (UTC)

This does not really solve the problem because the square-root function is ill-conditioned near zero, i.e. when x² is near 1. JRSpriggs (talk) 08:44, 15 October 2010 (UTC)

Arccot principal value

The article says Arccot goes from 0 to π, however most computer implementations plus Mathworld go from -π/2 to π/2 leaving 0 out. I think we should go with that as the more common one nowadays otherwise it can lead to confusion when people actually calculate it. Any thoughts before I go ahead with that? Dmcq (talk) 15:47, 10 January 2011 (UTC)

Where does the article say that? Perhaps you are confusing Arctan with Arccot? JRSpriggs (talk) 02:15, 11 January 2011 (UTC)

Oh dear. Yes you're quite right don't know why I did that. I've amended what I said above to Arccot. I've just had a read through the Mathworld article and though they go from -π/2 to π/2 in Mathematica they also describe the alternative convention which we have here of going from 0 to π. Each convention has its advantages just I think I'd go with the one used in maths libraries on computers first. We obviously need to describe both conventions though. Dmcq (talk) 09:30, 11 January 2011 (UTC)

And I see a question like this has arisen above in #function for arccotangent?? I think it may have gone from what I'm proposing to what it currently is then. Dmcq (talk) 09:34, 11 January 2011 (UTC)

We have a choice between having the principal value be continuous on the domain (0,π) or discontinuous (and undefined) at 0 on (−π/2,π/2]. I think that the first option is obviously superior. Also, I believe that my reference books support the first option. Why introduce a cut in the domain where one is not needed? JRSpriggs (talk) 14:31, 11 January 2011 (UTC)

Revised introduction

As a user of Wikipedia, I thought the introduction of "Inverse trigonometric functions" could use some work. Looking on the talk page I see that there were two comments on 22 October 2009 and 12 July 2010 about making the introduction and the whole article more accessible to the general public by simplifying the language. There does not seem to be any changes to the introduction since then. What follows is my attempt to improve this part.

Why is this text on the talk page and not in the article itself? Well, I made the mistake of making a small addition to sister page "Trigonometry". I was accused of "subversive vandalism" and told that my contribution was "absolutely ridiculous" by an unnamed guardian of that page. I have reason to believe this page has a kinder and gentler community. (I made some minor changes 3 weeks ago and there was no cannon fire shot at me.) But just in case someone has some objections, I would prefer to work them out here in the "back room" rather than on the "front porch".

(As a PS and relative to the immediately preceding thread, I have struggled with this arccot issue and address it my edit. I started out in the camp of the "heavy hitters", but eventually came around to the view of the "majority" and a continuous arccot. So, I agree with JRSpriggs. The discontinuous arccot does meet the minimum absolute value criterion; but the continuous arccot is more symmetrical with the other functions. The differences in fundamental identities is a matter of adding or subtracting π radians or 180° to or from two of the equations. I did check this out against two spreadsheets I have, CRC Handbook and other printed tables, and documentation for two programming languages [programs not installed on current computer]. All old stuff--I'm an old guy. I found nothing conclusive. Most don't deal with negative arguments for any of the functions, leaving that to the user. And besides, we would have to change that nice arccot graph!)

Inverse trigonometric functions

In mathematics, the inverse trigonometric functions (otherwise known as cyclometric functions) are six functions that are the reverse of the basic trigonometric functions of sine, cosine, tangent, cotangent, secant, cosecant. Like these basic trigonometric functions, the six reverse functions are primarily used to study triangles and the relationships among the angles and sides of triangles. They are also used to model periodic (wave) phenomena in engineering and physics.

In this article, these reverse functions are called:

arcsine, arccosine, arctangent, arccotangent, arcsecant, arccosecant.

They are commonly abbreviated as:

arcsin, arccos, arctan, arccot, arcsec, arccsc.

(In many programming languages, electronic spreadsheets, and pocket calculators, the "arc" prefix is shortened to just "a".) In many other venues, the notation "−1" is used in place of "arc"; however this "−1" notation is ambiguous and confusing since it could indicate exponentiation, multiplication, or even subtraction. If z is the function, an exponent of −1 means reciprocal (1/z); a multiplier of −1 means negative (−z); and subtraction means a unit reduction (z−1). None of these operations represents how the inverse trigonometric functions work and "−1" is not used in this article to indicate arc functions.

The basic trigonometric functions take an angle as the input argument and generate a ratio of two triangle sides as the output value. The inverse trigonometric functions reverse this process−they take a ratio of two triangle sides as the input argument and generate the value of the angle as output. The following lists illustrate the relationship between the basic functions and the arc functions and show that they are completely reversible, in both directions:

Basic functions	Inverse functions	Inverse of basic function	Basic function of inverse
sin(angle) = O/H	arcsin(O/H) = angle	arcsin(sin(angle)) = angle	sin(arcsin(O/H)) = O/H
cos(angle) = A/H	arccos(A/H) = angle	arccos(cos(angle)) = angle	cos(arccos(A/H)) = A/H
tan(angle) = O/A	arctan(O/A) = angle	arctan(tan(angle)) = angle	tan(arctan(O/A)) = O/A
cot(angle) = A/O	arccot(A/O) = angle	arccot(cot(angle)) = angle	cot(arccot(A/O)) = O/A
sec(angle) = H/A	arcsec(H/A) = angle	arcsec(sec(angle)) = angle	sec(arcsec(H/A)) = H/A
csc(angle) = H/O	arccsc(H/O) = angle	arccsc(csc(angle)) = angle	csc(arccsc(H/O)) = H/O

(A = adjacent side, O = opposite side, H = hypotenuse. Angles can be radians or degrees, where π radians = 180°)

Domain and range of the inverse functions

The basic trigonometric functions are also called circular functions (literally and figuratively) because many input arguments (the "domain") generate the identical output value. For example, sin(0°) = sin(360°) = sin(720°) = sin(−360°), etc. The basic trigonometric functions are many-to-one relationships. This makes the arc functions inherently multivalued−they are one-to-many functions. In other words, for the arc functions to be the complete reverse of the basic functions, one input argument would have to generate an infinite number of output values−an impossible situation. So the range of output values for the arc functions must be limited by a "branch cut" to a single half-cycle of the input arguments of the basic trigonometric functions. These output values of the arc functions are referred to as the "principal values". Usually the minimum absolute values are selected as the principal values. A near-universal consensus has developed for the principal values of all the arc functions−except as explained in the following paragraph.

The exceptions to the principal-values consensus involve the negative input arguments and returned values of arctan and arccot. Different ranges of principal output values for negative input arguments of arctan are said to exist ^[1] but are impossible to find. However, there are two schools with different ranges of principal output values for negative input arguments of arccot. Those in the minority include the National Institute of Standards and Technology ^[2], MathWorld ^[3], and all of the best on-line trig calculators found on the Internet ^[4] ^[5] ^[6]. The difference in principal values is a matter the quadrant in which those values are found. The graph of arccot used by the minority is discontinuous at x = 0, where the principal values jump π radians (180°). The choice of principal values also affects the equations of "fundamental identities" (see next two sections), making some a little simpler and some a bit more involved, but netting-out to zero in total. But they are different. So, as a practical matter when using electronic spreadsheets, pocket calculators, programming languages, or on-line calculators with negative input arguments for arccot, it is good practice to examine the documentation for the output range of the function. If there is none, then the output range should be tested with trial negative input arguments. As can be seen in the graph at right and in the table of output ranges below, this article follows the majority convention of all positive values and continuous graph for the arccot.

The principal values of the inverse functions are listed here:

Name	Usual notation	Definition	Domain of x for real result	Range of usual principal value (radians)	Range of usual principal value (degrees)
arcsine	y = arcsin(x)	x = sin(y)	−1 ≤ x ≤ +1	−π/2 ≤ y ≤ +π/2	−90° ≤ y ≤ +90°
arccosine	y = arccos(x)	x = cos(y)	−1 ≤ x ≤ +1	+π ≥ y ≥ 0	+180° ≥ y ≥ 0°
arctangent	y = arctan(x)	x = tan(y)	−∞ < x < +∞	−π/2 < y < +π/2	−90° < y < +90°
arccotangent	y = arccot(x)	x = cot(y)	−∞ < x < +∞	+π > y > 0	+180° > y > 0°
arcsecant	y = arcsec(x)	x = sec(y)	−∞ < x ≤ −1 +1 ≤ x < +∞	+π/2 < y ≤ +π 0 ≤ y < +π/2	+90° < y ≤ +180° 0° ≤ y < +90°
arccosecant	y = arccsc(x)	x = csc(y)	−∞ < x ≤ −1 +1 ≤ x < +∞	0 > y ≥ −π/2 +π/2 ≥ y > 0	0° > y ≥ −90° +90° ≥ y > 0°

In this table, the order of the y values corresponds to the order of the x values. The domain of the inverse functions is equal to the range of the basic functions. However, the range of all inverse functions is restricted to π radians (180°) of the domain of the basic functions. If x is allowed to be a complex number, then the range of y applies only to its real part. (∞ is the symbol for infinity.)

References

Colin.campbell.27 (talk) 05:28, 6 March 2011 (UTC)

I appreciate the effort which you put into this. However, I think that it is much too wordy (long winded). We should assume that the reader of this article is already familiar with the trigonometric functions, since one normally learns those before learning their inverses, not simultaneously as you seem to be assuming. JRSpriggs (talk) 19:05, 6 March 2011 (UTC)

Can we negotiate?

You rejected my proposed introduction and lead section because:

(A) I discussed the basic trigonometric functions too much;

(B) You found my writing style too wordy.

My response:

(A) Ironically, I think the second paragraph of the current introduction and the secondary introductory paragraph ("General solutions") focus too much on the six basic trigonometric functions. I tried to discuss these as little as needed--(1) to explain the inverse relationship and (2) to contrast the inverse functions against them. The only things that need to be known about the basic functions are (1) how these two sets of trig functions are the inverse of each other and (2) because the basic functions are multi-valued it is necessary to select or reduce the range of the inverse functions. I do not think the current two introductory sections do a very good job of keeping these two aspects distinct. One aspect defines the inverse, the other describes how it works. My proposal puts (1) in the introduction and (2) in the first section below the Contents list

(B) Wikipedia articles should be accessible to all readers--from the advanced to the novice. The advanced reader will skip the introduction and proceed directly to indefinite integrals of the complex plane, or some such. The introduction is for the novice--someone who has some familiarity with sin, cos, tan and has vague recollections from high-school of the inverse functions--probably just remembers the superscript -1 notation and thinks inverse means reciprocal. Such a reader needs to be "spoon fed".

"Wordiness" is in the eye of the reader. Math types believe formulas and equations convey a lot of information. The majority of the population prefer words. I do not claim to be the greatest technical writer, but two things I have learned: LEAVE NO GAPS and MOVE SLOWLY--otherwise the writer could lose the reader. I prefer to call my style "semi-redundant"--overlapping steps in different words aimed at keeping the reader on track.

(C) Having said all this, I count 22 lines in my introduction vs. 18 lines in the current one, and I count 10 lines in my "Domain and range" vs. 14 lines in the current "General solutions". [The 10 lines assumes removing the paragraph on the principal values of acrcot, which I know is too dense, but it needs expansion. It is an important issue. Maybe it should have its own section?] Lastly, I could probably squeeze a few more lines out of the remaining text. My proposed introduction and lead section merit reconsideration.

Colin.campbell.27 (talk) 05:23, 14 March 2011 (UTC)

One principle which I have learned is "Brevity is the soul of wit.". Keep it short or else people will not bother to read it. And they may lose sight of the forest for the trees. JRSpriggs (talk) 14:18, 14 March 2011 (UTC)

When logic fails, tell jokes! Review WP:OWN.

Colin.campbell.27 (talk) 04:38, 19 March 2011 (UTC)

Mentioning logic in a discussion on Wikipedia tends to give bad vibes, I'd avoid it. It is generally equated with original research and persistent point of view pushers. Dmcq (talk) 10:06, 19 March 2011 (UTC)

To Colin.campbell: If you had made a reasonable edit and I arbitrarily reverted you (merely because it was not my idea), then that would be "ownership". However, no such thing has happened. You have not even tried to change the article. You merely sought my agreement to your proposed change and I declined to give it because I felt that it was much too wordy. You can still make your change without my prior agreement. If you do so, then you should expect others to change it to suit themselves. JRSpriggs (talk) 15:56, 20 March 2011 (UTC)

Cyclometric function

The term "cyclometric function" seems to only occur as a translation of the German "cyclometrische Function" (now obsolete) or the Dutch "cyclometrische functie". As such I'm not sure that it merits a mention in the article since it's not really used in English. For now I've put the term in a parenthetical remark to indicate that it is used only rarely and the term "inverse trigonometric function" is the accepted standard.--RDBury (talk) 12:45, 19 November 2011 (UTC)

[1] MathWorld inverse tangent page

[2] NIST inverse cotangent page

[3] MathWorld inverse cotangent page

[4] 1728 Software Systems trig calculator

[5] CalculatorSoup "arc-trig" calculator

[6] MathWorld arccot calculator

[1]

[2]

[3]

[4]

[5]

[6]