Talk:Inverse trigonometric function

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1 Arc?
2 Graph
3 Needs a definition
4 function for arccotangent??
5 Multiple values of inverse trig functions
6 logarithmic forms wrong?
7 Image of a right triangle
8 Differnt def.s
9 Euler's series for the arctan
10 Two forms need to be presented
11 atan2 function
12 Order of relations
13 continued fractions
14 Addition formulas
15 The proof of the arcsin formula
16 Graphs of Arcsecant and Arccosecant
17 Arcosh / Arccos
18 Derivatives of inverse trigonometric functions

[edit] 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).

[edit] 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)

[edit] 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)

[edit] function for arccotangent??

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

However in the relationships among the functions it says

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

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(2π) = 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

- π / 2

and

π / 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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.
MHDIV ɪŋglɪʃnɜː(r)d(Suggestion?|wanna chat?) 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)

[edit] 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:

$\arctan x = \frac{x}{1+x^2} \sum_{n=0}^\infty \prod_{k=1}^n \frac{2k x^2}{(2k+1)(1+x^2)}.$

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)

[edit] 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.

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

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 not 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

[edit] 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).

[edit] 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)

[edit] 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

[edit] Addition formulas

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

$\arctan\theta_1 + \arctan\theta_2 = \arctan\frac{\theta_1 + \theta_2}{1 - \theta_1 \theta_2}$

$\arccot\theta_1 + \arccot\theta_2 = \arccot\frac{1 - \theta_1 \theta_2}{\theta_1 + \theta_2}$

$\arcsin\theta_1 + \arcsin\theta_2 = \arcsin(\theta_1 \sqrt{1-\theta_2^2} + \sqrt{1-\theta_1^2}\theta_2)$

$\arccos\theta_1 + \arccos\theta_2 = \arccos(\theta_1 \theta_2 - \sqrt{(1-\theta_1)(1-\theta_2)})$

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

[edit] 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 $i w + \sqrt{1 - w^{2}} \in \mathbb{C}_{-}$ . --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)

[edit] 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)

[edit] 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)

[edit] 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)

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