Talk:Hyperbolic functions: Difference between revisions

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== Maclaurin not Taylor ... Laurent?==

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The series presented here are Maclaurin series, not Taylor series. We should either add in an "a" term - the taylor series about "a", or call them Maclaurin series. Gareth [[User:139.80.48.19|139.80.48.19]] 21:00, 14 November 2007 (UTC)

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Maclaurin Series are a special case of Taylor series. It is not incorrect to call them Taylor series. <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/67.244.58.252|67.244.58.252]] ([[User talk:67.244.58.252|talk]]) 15:42, 5 February 2009 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

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My understanding is that the expansions given for coth and csch in the "Taylor series expansions" are actually Laurent series, and as such accommodate singularities in the complex plane. These formulas could be written with a "z" variable (for complex) rather than just an "x" variable (usually denoting real). Just trying to get things right, [[User:I&#39;m your Grandma.|Grandma]] ([[User talk:I&#39;m your Grandma.|talk]]) 15:15, 21 November 2014 (UTC)

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== ??? ==

well, ppl look for this page are generally for a glance of the meaning of cosh. wth u described so many complicated information (not professional tho) but not instead just fukcing tell me how to type cosh in a standard calculator?? <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/203.122.102.163|203.122.102.163]] ([[User talk:203.122.102.163|talk]]) 02:06, August 29, 2007 (UTC)</small><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

:: its there. twice. in big print.

:Of course, if your standard calculator is a TI-83 graphing calculator, you can type in ''anything'' by hitting the CATALOG button and finding it in the list, hyperbolic functions included. I think most scientific calculators don't have it built in, but if they can handle complex numbers (look for an "i" key), then I suppose you could calculate cosh, sinh, and tanh from the exponential-form definition. It would be labour-intensive though.--[[Special:Contributions/24.83.219.223|24.83.219.223]] ([[User talk:24.83.219.223|talk]]) 08:05, 3 October 2008 (UTC)

== Error in the LaTeX and normal images ==

I can't see them, look at the error:

Failed to parse (Can't write to or create math output directory): \sinh(x) = \frac{e^x - e^{-x}}{2} = -i \sin(i x)

Why is this?

:I found something similar, not with this equation, but with \sech . I added braces around ix to get \sec {ix} and that seemed to work. What is strange is that all of the other functions with the same format work well for me.

Garethvaughan 19 Jul 2007 (GMT+12)

== Origin of pronunciations ==

Does anyone know how the pronunciations for sinh, cosh, etc came about? Do they differ between British and American usages? [[User:128.12.20.195|128.12.20.195]] 05:47, 22 January 2006 (UTC)

== ERROR in formula ==

There is a problem with the expansion series for arccosh. There is a switch of sign in the sum (right), but not in the examples of the first terms (left), and the values do not match. I don't know which is correct.

== Suggestion ==

Why are you using sinh^2 x for (sinh x)^2?? Go to the [[inverse function]] article and it says the definition for f^2 x when not in trigonometry. Then, it has the exception and it says [[trigonometric functions]], not [[hyperbolic functions]]. [[User:66.245.25.240|66.245.25.240]] 15:42, 12 Apr 2004 (UTC)

:By convention this form is used with hyperbolic functions as well. -- [[User:Decumanus|Decumanus]] | [[User talk:Decumanus|Talk]] 15:43, 12 Apr 2004 (UTC)

The parenthetical comments about pronounciation could be clearer... what type of ch sound in sech, for example

== Some suggestions ==

What about changing the definition of arccsch(x) to ln(1/x + sqrt(1+x^2)/abs(x))?

Also, why does the definition of arcsech(x) contains the plusminus sign? As far as I know, the definition of arcsech(x) is "the inverse of (sech(x) restricted to [0,+inf))" (Howard Anton et.al., Calculus 7th ed, Chapter 7). In that case we should replace plusminus with plus.

[[user:Agro_r]] 11 Feb 2005 (GMT+7)

:Definitions can be different. In order to find an inverse function of a many-to-one function, you need to restrict the interval in which the values of the input are defined in order to turn it into a one-to-one function (since a function has to be one-to-many or one-to-one by definition). You can use different values of the interval in order to turn it into a one-to-one function. For example, with cos, you could use [0,pi) or [-pi/2,pi/2). They're both right, I assume. [[User:Deskana|Deskana]] <small>[[User_Talk:Deskana|(talk)]]</small> 19:21, 8 February 2006 (UTC)

::Agro_r: The plus-minus sign is essential to show that there're two possible output values to the inverse of the csch. It's only when we want to simplify thing (eg using simplistic approach) that the inverse is limited to have only one output for every input and it's at this moment that we define the function. But I see that the expression to arcsch has already been changed. What a pity!

::Deskana: No, it's false to say that using [0,pi) or [-pi/2,pi/2) for cosine are both right. They don't have the same target domains. I'll let you see why.

::[[User:石庭豐|石庭豐]] ([[User talk:石庭豐|talk]]) 13:00, 9 January 2011 (UTC)

== I need help! ==

I want know aplications, can somebody help me?

:If you are in a spaceship (with no gravity), and want to pretend you are on earth, you can take a piece of string, shape it like the cosh function and take a picture. Then people will think that you are affected by gravity, since a piece of string hanging freely looks like the cosh function. Don't know if they have other uses, I'll leave that to other people who know more, to answer. [[User:Cyp|&#922;&#963;&#965;&#960; ''Cyp'']] [[User talk:Cyp|&nbsp;]] 2004&#24180;10&#26376;27&#26085; (&#27700;) 19:56 (UTC)

sinh and cosh are used heavily in electromagnetics applications, as they appear in solutions of Laplace's equation in Cartesian coordinates.

These functions are used heavily in heat transfer, and as general solutions to differential equations eg: θ"+α*θ=0 has a general solution of θ=C1*Cosh(α*x)+C2*Sinh(α*x) (assuming θ is a function of x).

I am coming across both the hyperbolic secant and the hyperbolic cotangent functions in survival analysis. The hypertabastic function, which uses both, is a good fit for several time-until-death problems, like cancer and bridge collapse.[[User:Svyatoslav|Svyatoslav]] ([[User talk:Svyatoslav|talk]]) 02:50, 18 September 2016 (UTC)

:for [[User:Svyatoslav]] sounds interesting could you make an article for the [[hypertabastic function]]? [[User:WillemienH|WillemienH]] ([[User talk:WillemienH|talk]]) 12:52, 18 September 2016 (UTC)

:: {{ping|Svyatoslav|WillemienH}} Article request added to [[Wikipedia:Requested articles/Mathematics#Uncategorized]] in [[special:diff/740024647]] --[[User:CiaPan|CiaPan]] ([[User talk:CiaPan|talk]]) 16:22, 18 September 2016 (UTC)

== I have this problem ==

As can be seen from plots the -cosh(x) function is concave in x.

So, if we have w positive then exp{-wcosh(x)} should alos be convex in x, right? Apparantly not always ... according to this article I have this is only valid iff w{sinh(x)}^2<cosh(x), does anybody see why??

== Arc{hyperbolic function} is a misnomer ==

Arc{hyperbolic function} is a misnomer

The article states correctly that the parameter t represents an ''area'', rather than a (circular) angle. Note also that t does not represent an arc length. As such, it is actually a misnomer to refer to the inverse hyperbolic functions using the prefix "arc".

OTOH, for the inverses of the trigonometric functions, the prefix "arc" is not a misnomer, since their values may indeed be thought of as representing arc lengths (or circular angles).

Arc{hyperbolic function} probably originated due to a ''false'' analogy with trigonometric functions. In any event, it is sometimes used nowadays. But IMO its usage should be deprecated in favor of one of several better alternatives. I prefer the simplest alternative: a{hyperbolic function}, in which "a" may correctly be taken to represent "area".

What do other people think? Should we change the article's currect arc{hyperbolic function} notation to my preference, or some other alternative, or leave the current notation as is?

--[[User:David W. Cantrell|David W. Cantrell]] 07:23, 31 Dec 2004 (UTC)

:Hi, I think it the prefix for inverse hyperbolic functions used to be ''Sect'', as in "sector". Why exactly I'm not sure, but that's what I grew up with. [[User:Orzetto|Orzetto]] 09:14, 16 Mar 2005 (UTC)

:: Probably because ''sect'' was easily confused with ''sec'' as in secant. Especially if the angle was t, e.g. ''sectsint'' would be very ambiguous. [[User:218.102.221.84|218.102.221.84]] 07:04, 30 December 2005 (UTC)

:The prefix for inverse hyperbolic functions is ''ar''. The original latin names are ''area sinus hyperbolicus'', ''area cosinus hyperbolicus'' and the respective functions are named ''arsinh'', ''arcosh'' etc. In the US, this knowledge seems to have been lost, so I'm not sure what we should opt for. --[[User:Tob|Tob]] 08:39, 7 December 2005 (UTC)

: The prefix should be ''ar'' not ''arc'' for hyperbolic functions. My opinion is that ''arcsinh'' ect. is wrong and misleading. [[User:SKvalen|SKvalen]] 18:41, 11 December 2005 (UTC)

:: It is wrong, but in the US they all use ''arc''. [[User:218.102.221.84|218.102.221.84]] 07:04, 30 December 2005 (UTC)

''asinh'' etc. is computer science use and there seems to be a consensus that ''arc'' is wrong. Changing to ''ar''. --[[User:Tob|Tob]] 14:00, 4 January 2006 (UTC)

::: An old (American) calculus textbook (Spivak's) has the following to say: "The functions sinh and tanh are one-one; their inverses, denoted by arg sinh and arg tanh (the "argument" of the hyperbolic sine and tangent) ... If cosh is restricted to [0,∞), it has an inverse, denoted by arg cosh...". This seems to confuse the matter further, and a brief Google search suggests that hardly anyone uses this, at least, not on the Internet. [[User:82.12.108.243|82.12.108.243]] 15:05, 13 February 2007 (UTC)

== Use of exponents on function names==

Hi, I have seen that there is some use of the notation <math>\cosh^2()</math> to indicate <math>\cosh(\cosh())</math>. This should not be done as in some countries <math>\cosh^2()</math> actually means <math>[\cosh()]^2</math>, and this may be a source of confusion. It's evil. [[User:Orzetto|Orzetto]] 09:14, 16 MarMarch 2005 (UTC)

:Is this actually true of non-kiddie level math in those countries: [[Function composition]] [[User:TomJF|TomJF]] 04:09, 12 April 2006 (UTC)

:For trigonometric functions, <math>function^2()</math> is synonymous with <math>function()* \cdot function()</math>. For example, <math>\cos^2(t)+\sin^2(t)=1</math>. Out of curiosity, when would you ever nest trigonometric functions? You'd run into unit problems, wouldn't you? Your basic trigonometric functions (cosine, sine, and tangent) have domains in radians, degrees, or gradians, yet have ranges in unit lengths. The cosine of the cosine of an angle would be meaningless. [[User:Sobeita|Sobeita]] ([[User talk:Sobeita|talk]]) 02:36, 16 December 2010 (UTC)

::Almost nine years late, but why not? You might want to find the fixed point of cos(x). [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 04:24, 9 December 2019 (UTC)

:: {{ping|Sobeita}} I have replaced the multiplication symbol in your entry with a mid-dot. We often use the star glyph for a multiplication symbol in a plain text, because ASCII did not contain a mid-dot '''&middot;''' or a times symbol '''×'''. However, in some contexts the star symbol <math>*</math> can be confused with the [[convolution]] operator, so it's better to avoid the star whenever more apropriate symbols are available (like in MathJax/LaTeX). --[[User:CiaPan|CiaPan]] ([[User talk:CiaPan|talk]]) 18:46, 13 December 2019 (UTC)

== No Period? ==

"However, the hyperbolic functions are not periodic." I'm not really an expert in this, but someone told me that the hyperbolic functions do have a period, but it is imaginary. Is this true? --[[User:Dragglebaggle|Dragglebaggle]] 03:09, 20 September 2005 (UTC)

I have fixed symbols like <math>cosh()</math> to render as <math>\cosh().</math> --[[User:CiaPan|CiaPan]] ([[User talk:CiaPan|talk]]) 18:30, 13 December 2019 (UTC)

:Yes, 2πi. --[[User:Macrakis|Macrakis]] 15:52, 20 September 2005 (UTC)

== ArcTanh(adj,opp) ==

I needed a hyperbolic analogue of ArcTan[adjacent,opposite], and "Klueless" provided a relationship that I coded as arcTanh[a_,0]:=If[a==0,∞,0,0];

arcTanh[a_,o_]:=If[Chop[a^2-o^2]==0,∞,Log[(a+o)/SQRT[a^2-o^2]] in Mathematica (ref). (The formula is not in the Mathematica functions library) This allowed me to show that, in the algebra with the Klein 4-group as multiplication table, there is a hyperbolic dual of the Argand diagram, with {a,o} <=> {u=Sqrt[a^2-o^2],theta=arcTanh[a,o]} in which "ulnae" (u) multiply and hyperbolic angles (theta) add on multiplication. The hyperbolic plane is covered by two pairs of hyperbolae. Would a referenced write-up of this be acceptable? It seems to be a significant generalization of hyperbolic functions.

:(ref) http://library.wolfram.com/infocenter/MathSource/4894

Roger Beresford.

[[User:195.92.168.164|195.92.168.164]] 08:22, 7 October 2005 (UTC)

:Good stuff, but my impression is that Wikipedia should concentrate on widely accepted definitions, not novel insights. By the way, your formulae might be clearer if you didn't use "o" as a variable name and if you didn't you Mathematica-specific conventions -- I assume that If[a==0,∞,0,0] means "if a==0 then ∞ else 0", but I don't understand why there are two 0's there. --[[User:Macrakis|Macrakis]] 15:10, 8 October 2005 (UTC)

ArcTanh[y,x] is not original research, but plagiarism! (I copied from only one source.) ArcTan[y,x] is accepted as it is needed to cope with different angles in opposing quadrants. ArcTanh[y,x] apparently is not, although ArcTanh[-y,-x] is a different angle. Your veto can stand, though Wiki is poorer for it. [[User:195.92.168.168|195.92.168.168]] 09:50, 12 October 2005 (UTC)

Roger, I have no veto, only my judgement based on the arguments above. ArcTanh2 simply doesn't seem widely enough known or used to merit a mention in this article. --[[User:Macrakis|Macrakis]] 11:37, 12 October 2005 (UTC)

==Unaccesable?==

I thought wikipedia was supposed to be an easy to use encyclopedia, these won't be easy for anyone not studying maths to understand

:I have added some additional explanatory text to the introduction of the article. I hope this helps, though there is no doubt room for more. Beyond that, it is true that the content gets moderately technical, but then this is a technical subject. --[[User:Macrakis|Macrakis]] 20:49, 19 October 2005 (UTC)

::Thanks that has helped somewhat --[[User:Albert Einsteins pipe|Albert Einsteins pipe]]

:::As I understand it, math is a tiered study. You can't understand multiplication without an understanding of addition. This description of hyperbolic functions is not unaccessible provided that you already understand the math leading up to it... and if you just want to understand it intuitively without being able to use it, I think it still does a fairly good job. But remember, Wikipedia gives anything and everything about a topic... it never says you have to understand everything on the page at once. If you wanted to learn addition, you could look it up, but at the age of 5 or 6, you wouldn't be able to understand that it's commutative and associative (let alone the comments about Dedekind cuts used to add irrational numbers.) [[User:Sobeita|Sobeita]] ([[User talk:Sobeita|talk]]) 02:47, 16 December 2010 (UTC)

== Why is it defined so? ==

For

<math>\sinh(x) = \frac{e^x - e^{-x}}{2} = -i \sin(i x)</math>

Could anyone tell me why?

--[[User:HydrogenSu|HydrogenSu]] 07:48, 21 December 2005 (UTC)

:Hyperbolic functions that are defined in terms of <math>e^{x}</math> and <math>e^{-x}</math> that bear similarities to the trigonometric (circular) functions. When plotted parametrically, trigonometric functions can be used to create circles. When plotted parametrically, hyperbolic functions can be used to create hyperbolas. That's the best explanation I can offer, as I've only recently been introduced to them. [[User:Deskana|Deskana]] <small>[[User_Talk:Deskana|(talk)]]</small> 19:25, 8 February 2006 (UTC)

::well,

:<math>\sin(x) := \frac{e^{ix} - e^{-ix}}{2i}</math>

::sanity check: <math>e^{i\pi/2} = i => \sin(\pi/2) = 1</math> okay :) Now the hyperbolic variants are the same except without i. --[[User:MarSch|MarSch]] 12:16, 3 May 2006 (UTC)

I think the "best" explanation is that the "usual" trig functions are all areas and lengths on a unit circle. The hyperbolic functions are all areas and lengths on a unit hyperbola. The fact that these have staying power is due to them being useful in some areas of non-mathematical study (see the above "Why?" question). [[User:Svyatoslav|Svyatoslav]] ([[User talk:Svyatoslav|talk]]) 02:56, 18 September 2016 (UTC)

==Names of inverses==

It appears (from Google, etc.) that asinh etc. are the most common form of the inverse functions, not arsinh or arcsinh. WP doesn't try to prescribe best usage, but record actual usage. Thanks to SKvalen for pointing out that arcsinh is ''not'' the most common form. --[[User:Macrakis|Macrakis]] 02:54, 22 December 2005 (UTC)

: If you take Google as a reference, please note that the reason for having the most hits for ''asinh'' etc. is that this is the way these functions are called in computer libraries. I've never seen a mathematician use ''asinh'' etc. --[[User:Tob|Tob]] 14:02, 4 January 2006 (UTC)

::Tob is right. I was taught it was "arsinh". [[User:Deskana|Deskana]] <small>[[User_Talk:Deskana|(talk)]]</small> 19:26, 8 February 2006 (UTC)

:::who cares? [[User:128.197.127.74|128.197.127.74]]

::::I've never seen the variants with only '''ar''' instead of '''arc''' before. Do we have a reference for this usage? Do we have any reference as to what usage should be preferred/is more prevalent? --[[User:MarSch|MarSch]] 12:09, 3 May 2006 (UTC)

:::::Scanning back up through talk, I see that or the hyperbolic variants '''ar''' shoud be preferred over '''arc'''. It would be good to have a source for this. --[[User:MarSch|MarSch]] 12:21, 3 May 2006 (UTC)

:'''Arc''' prefix is definitely wrong! It applies to trigonometric functions only.

:Trigonometric functions are (or: can be) defined in terms of the unit circle's '''arc''' length as a parameter, so inverse trigonometric functions give the arc length as their value (output); that's why they are '''arc'''–functions, and their names have the '''arc''' prefix.

:On the other hand, hyperbolic functions are (or: can be) defined in terms of some hyperbolic figure '''area''' as a parameter (see [[:Image:Hyperbolic functions.svg]]), so inverse hyperbolic functions give an '''area''' as their value (output); that's why they are '''area'''–functions, and their names have '''ar''' prefix, distinct from trigonometric '''arc'''-functions. ---[[User:CiaPan|CiaPan]] 12:36, 22 October 2007 (UTC)

:: Definatly true. So is there any reason to write the integrals in "sin^-1"-Form? IMHO it'd be preferable to write arsinh etc. to avoid misunderstandings (1/sinh). <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/129.217.132.38|129.217.132.38]] ([[User talk:129.217.132.38|talk]]) 10:35, 16 February 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

== Plot Colors ==

The red and green in the plots are really bad for people who are [[color blindness|color blind]]. Any chance somebody could redo them with more accessible colors?

:Sure. What colors are better? I'm not colorblind, so I can't tell. --[[User:M1ss1ontomars2k4|M1ss1ontom]]<font color="green">[[User:M1ss1ontomars2k4/Esperanza|a]]</font>[[User:M1ss1ontomars2k4|rs2k4]] ^{([[User talk:M1ss1ontomars2k4|T]] | [[Special:Contributions/M1ss1ontomars2k4|C]] | [[Special:Emailuser/M1ss1ontomars2k4|@]])} 01:11, 1 December 2006 (UTC)

::As a rule of thumb, dark and deeply-saturated pure greens are a problem for most red-green people. Moving the hue more towards blue-green and/or desaturating the green are usually pretty effective. That might not cut it for blue-yellow people though. Generally, if your colors all have different brightness and saturation levels, you're probably OK. (Wikipedia should really have a set of recommended colors for diagrams and such.) [[User:Shaunm|Shaunm]] 20:37, 20 February 2007 (UTC)

Upon observation, the graphs not only have color varience, but the lines are dotted in different patterns. This removes the alleged problem. [[User:Inthend9|Inthend9]] ([[User talk:Inthend9|talk]]) 17:57, 31 March 2010 (UTC)

== calculus ==

I use maple and am not sure how to use latex.

The article should give the differentials of sinh and cosh, writen in maple as:

diff(sinh(x),x)=cosh(x);

diff(cosh(x),x)=sinh(x);

Sorry I cant edit it myself

:<nowiki> <math> d \cosh(x) = \sinh(x), \; \; d \sinh(x) = \cosh(x) </math> </nowiki> as per [[Latex]]. Maple will convert output into latex code for you; see the help pages.---[[User:Hillman|CH]] 16:14, 10 June 2006 (UTC)

::You possibly meant [[LaTeX]], CH? ;) --[[User:CiaPan|CiaPan]] 09:37, 11 November 2007 (UTC)

== log on inverse functions. ==

log implies that this is to the base 10, when it is actually to the base e so i suggest that this is changed to ln.

:In pure mathematics, log implies base e. But since this topic is relevant for non-mathematicians as well, you have a point. [[User:Fredrik|Fredrik Johansson]] 15:21, 21 August 2006 (UTC)

::Well, not really. log as it's written like this implies it has some '''arbitrary (but valid) base'''. log standing for log₁₀ is a convention while log standing for log_e is '''another''' convention; unless it's written out clearly (eg at the beginning of an article) which implicit base is used.

::It's not a matter when the base isn't important like the expression<br>log (ab) = log a + log b<br>as this is always true whatever base you choose, be it log₂, log₃, log₄ or even log_π! However, it's not the case for our log functions in the article. So either log_e or ln should be used.

::Oh yes, ln standing for log_e is also another convention, but with the difference that this convention is universally accepted and '''without confusion'''.

::[[User:石庭豐|石庭豐]] ([[User talk:石庭豐|talk]]) 17:46, 30 December 2010 (UTC)

== The Imaginary Unit ==

Line 226 ⟶ 38:

::::::P.S. But do make it a proper definition, please. If we say "define", that's what we'd better give the reader. [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 05:02, 19 March 2015 (UTC)

:::::::I already [https://en.wikipedia.org/w/index.php?title=Hyperbolic_function&diff=652030883&oldid=651359806 replaced the phrase "defined by"]. It cannot be defined adequately in half a sentence. I think we should only say "... where ''i'' is the [[imaginary unit]]." My reasoning is that there are many structures that contain elements that conform to his characterization (an infinite number of them in in the quaternions, for example). Once one has pinned it down to the complex numbers, though, this final ambiguity between two points ''is'' resolved by the equivalence: until you choose one, you cannot distinguish which you are working with – just as there is no special point on a given sphere without reference to some other points. Do you agree that we should strip off the half-hearted "definition" and leave it to the link [[Imaginary unit]]? —[[User_talk:Quondum|Quondum]] 14:32, 19 March 2015 (UTC)

What's wrong with defining i as the square root of -1? It seems the objection is that it is better defined as z such that z^2 =-1, where z will have two possible values. But if you define it as the square root of -1, don't you get the implication that it must be that i = -i anyway? (maybe not; I'm asking) If that is true, then the simpler definition is equivalent.-- [[User:Editeur24|editeur24]] ([[User talk:Editeur24|talk]]) 01:23, 30 December 2020 (UTC)

== Error in plot ==

Mmm, I think that there is an error in the image of the iperbola: sinh(a) should refer to the ordinate of the intercepta! <br/>

In fact <math>\cosh^2(a) - \sinh^2(a) = 1</math>, and <math>\cosh(a) \ne 0</math> always. I'm not changing it because I'm new at editing wikipedia, but if this message will not be answered in one week, I'll be back and mod it.

--[[User:Astabada|Astabada]] 22:15, 28 October 2007 (UTC)

:.......and that is exactly what the image shows. Hyp.sine is the length of the vertical red line, that is the ordinate. Hyp.cosine is the length of the horizontal red line, and it is always different from zero — hyperbola does not touch OY axis in any point. In fact it does not approach closer than to x=1, so cosh(''a'')&ge;1 always. --[[User:CiaPan|CiaPan]] 07:34, 29 October 2007 (UTC) <br/>BTW I slightly corrected your LaTeX notation. [[User:CiaPan|CiaPan]]

::.......yes, I'm sorry, it can be viewed in both ways... I would have written <math>\sinh(a)</math> as near as possible to the Y axis, but I recognize it may depend on habits or conventions used in countries different from mine...

::--[[User:Astabada|Astabada]] 11:41, 29 October 2007 (UTC)

:::Right, that's a matter of habits. You can write the symbol at the axis if you want to emphasize the co-ordinate, or at the construction line if you want to emphasize the specific length. Tha latter is useful if some lengths are not perpendicular to any axis or the line does not start from an axis (see [[:Image:Circle-trig6.svg]] for example).

:::In the case we discuss here both methods would be equivalent. --[[User:CiaPan|CiaPan]] 10:41, 30 October 2007 (UTC)

== cosh in nature ==

Someone should write a section about cosh as a solution to a differential equation which naturally appears in a common physics problem. If you take both ends of a rope and keep them at a distance to each other, the rope will hang in a bow like shape which happens to be the curve <math>y = cosh(a\cdot x)/a</math>, where 1/a is the bend radius at the bottom of the curve. The differential equation which appears is <math>y'' = a\sqrt{1+(y')^2}</math>, and when a = 1, y(0) = 1, y'(0) = 0, you can easily se that y = cosh(x) solves the system, by using substitution. This is a top-down solution, I don't know how to make it the other way though. --[[User:Kri|Kri]] ([[User talk:Kri|talk]]) 12:09, 3 November 2008 (UTC)

:Perhaps you didn't notice that in the second paragraph of the article, it says:

::Hyperbolic functions are also useful because they occur in the solutions of some important linear differential equations, notably that defining the shape of a hanging cable, the [[catenary]],...

::::--[[User:Macrakis|macrakis]] ([[User talk:Macrakis|talk]]) 14:38, 3 November 2008 (UTC)

== What does this mean? ==

This phase was added a short time ago: "One important point that must be made regarding Hyperbolic functions is that the derivatives are never the sum equals of the square of the function divided by two." What on earth does this mean??? [[User:Cojoco|cojoco]] ([[User talk:Cojoco|talk]]) 23:40, 3 November 2008 (UTC)

== New error in plot ==

The plot currently shows alpha as an angle between -pi/4 and pi/4. I understand, from reading other comments on this talk page, that the argument should be the shaded area. The animated plot contains the same error. I'm new to editing Wikipedia, and my understanding of the correct interpretation is based only on what I read here, so I won't correct it, but clearly cosh and sinh are defined for all real numbers, but the graphic doesn't illustrate that.

[[User:Dobrojoe|Dobrojoe]] ([[User talk:Dobrojoe|talk]]) 21:47, 5 December 2009 (UTC)

I just looked again, and don't see a problem with the animated plot after all. I don't know what I thought I was seeing. [[User:Dobrojoe|Dobrojoe]] ([[User talk:Dobrojoe|talk]]) 07:59, 22 December 2009 (UTC)

== Tanh x ==

Under hyperbolic functions tanhx is said to equal (e^x-e^-x)/ (e^x+e^-x)...correct

The writer then goes on to say, thus

tanhx = (e^2x -1)/(e^2x +1)

I dont think thats right

(----Mike B) <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/41.241.108.243|41.241.108.243]] ([[User talk:41.241.108.243|talk]]) 16:43, 12 March 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

:Multiply top and bottom by ''e^x''. --<span style="border:1px solid #006000">[[User:Dr Greg|<font style="color:#FFFF80;background:#006000">'''&nbsp;Dr&nbsp;Greg&nbsp;'''</font>]][[User talk:Dr Greg|<font style="color:#006000;background:#FFFF80">&nbsp;<small>talk</small>&nbsp;</font>]]</span> 18:34, 12 March 2010 (UTC)

== The Maloney Formula ==

There's a reference to "The MaloneyFormula" sinh(x)-r*cosh(x)=x. But I couldn't find anything like it anywhere else? Any ideas? <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/78.56.23.241|78.56.23.241]] ([[User talk:78.56.23.241|talk]]) 21:57, 9 April 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

:Looks more like the baloney formula. [[User:Fredrik|Fredrik Johansson]] 15:12, 13 April 2010 (UTC)

::Yeah, it's obviously false, for example for r=0, removed. It was added by 128.46.111.18, and constitutes his only contribution. [[User:Sergiacid|Sergiacid]] ([[User talk:Sergiacid|talk]]) 08:15, 26 April 2010 (UTC)

== Readability through visualization ==

''Regarding the excerpt,'' "For contrast, in the terminology of topological groups, B forms a compact group while A is non-compact since it is unbounded."

I happened to notice that the unbounded set had unbounded functions giving its coordinates. Am I right in thinking one causes the other? I compared the hyperbolic and trigonometric functions and noticed the functions' (un)boundedness, having not known already. Especially by someone who knows what connections to make, matching images to text could be a useful guide for elucidation. ᛭ [[User:LokiClock|LokiClock]] ([[User talk:LokiClock|talk]]) 12:16, 11 July 2011 (UTC)

== Survey: New Images and Sech(x) ==

Hi all,

I think we ought to have a graph of [[Sine|sin/cos]]/sinh/cosh in complex 4-space (''x''_Re, ''x''_Im, ''y''_Re, ''y''_Im) that shows the relationship between the four. Do you think this will make a good picture for this article? Please vote below!

Doesn't the Hyperbolic Secant curve look like the [[Gaussian distribution]] (bell) curve?

Thanks,

[[User talk:68.173.113.106|The Doctahedron]], 04:30, 18 December 2011 (UTC)

: Is there someplace we can already see this graph? I've noticed the similarity before. Many shapes look similar, but when attached to physics are apparently quite different. The [[Cauchy distribution]] also looks like the Gaussian distribution, and parabolas look like [[catenaries]], but somehow I'd never mistake a slack chain for a bouncing ball. It's how the image implies forces producing the curve, the mechanical distinction. For that same reason we better sort Cauchy vs. Gaussian by their sources. Plus, sech has a sensual slouch, where the Gaussian is quite binary in the way it inflects at the base. ᛭ [[User:LokiClock|LokiClock]] ([[User talk:LokiClock|talk]]) 23:20, 19 December 2011 (UTC)

::In the ''Mathematical Gazette'', problem 83B starts with the premise (based on March 1997 note 81.10 by Tony Robin) that <math>f(x) = \frac{1}{2}\lambda \text{sech}^2(\lambda x)</math> gives an approximation to the standard normal density function (with mean 0 and variance 1). The problem is to show that the variance of the approximation is the same as the actual variance (=1) if <math>\lambda = \pi / \sqrt{12}</math>. ''Math. Gaz.'' 83, November 1999, p. 515 gives a proof. I doubt that this belongs in this article, though. [[User:Duoduoduo|Duoduoduo]] ([[User talk:Duoduoduo|talk]]) 17:00, 13 January 2012 (UTC)

::: I don't know about that. Address the perception, but come back with information about the similarity instead of the difference. That, I think, is truly encyclopedic. ᛭ [[User:LokiClock|LokiClock]] ([[User talk:LokiClock|talk]]) 15:41, 29 January 2012 (UTC)

== question about the diagram of circular and hyperbolic angle ==

This is a discussion on [[Talk:Hyperbolic_angle]].

There might be a mistake in the diagram of circular and hyperbolic angle: File:Circular and hyperbolic angle.svg.

[[File:Circular and hyperbolic angle.svg|thumb|This diagram show the use of [[hyperbolic sector#Hyperbolic triangle|a "hyperbolic triangle"]] to define the hyperbolic functions]]

There are two angles in the diagram. One is the hyperbolic angle related to the hyperbola xy=1, and let's call this hyperbolic angle u_hyp. The other is the circular angle related to the circle xx+yy=2, and let's call this circular angle u_cir. Right now in this diagram, these two angles are shown to be indentical, i.e. u_hyp=u_cir=u. I think such indentity should not exit in general, i.e. u_hyp is usually not equal to u_cir. By equations, u_hyp=ArcTanh[Tan[u_cir]]>=u_cir, and u_cir=ArcTan[Tanh[u_hyp]]<=u_hyp, where the equals sign holds only when u_hyp=u_cir=0. In other words, u_hyp is equal to the area of the yellow and red regions, while u_cir is equal to the area of yellow region only.

[[User:Armeria wiki|Armeria wiki]] ([[User talk:Armeria wiki|talk]]) 05:30, 10 June 2013 (UTC)

:Two lines that meet form an angle, but the measure of that angle depends on whether it is viewed as a circular angle or a hyperbolic angle. These measures are very different functions. As you note u_cir and u_hyp are different measures of the "same angle" formed by two lines that meet. The measure of this angle is the yellow area when the angle is circular, and the measure is the red area when it is hyperbolic.[[User:Rgdboer|Rgdboer]] ([[User talk:Rgdboer|talk]]) 21:28, 10 June 2013 (UTC)

::The measure of angle is the red area plus the yellow area when the angle is hyperbolic.[[User:Armeria wiki|Armeria wiki]] ([[User talk:Armeria wiki|talk]]) 23:50, 10 June 2013 (UTC)

The problem is the 'hyperbolic angle' is NOT AN [[Angle|ANGLE]], but rather AN AREA OF A [[Hyperbolic sector|HYPERBOLIC SECTOR]]. When you say 'hyperbolic angle' you should understand 'hyperbolic argument', and drop the geometric meaning of the word 'angle'. The hyperbolic functions have similar names to trigonometric functions and they fit similar relationships, but they are themselves NOT SIMILAR to trigonometric functions, and their argument is no way similar to angle. Period. --[[User:CiaPan|CiaPan]] ([[User talk:CiaPan|talk]]) 19:34, 11 May 2014 (UTC)

:Yes, the use of "angle" in this context is confusing since "hyperbolic angle" does not have the uniform density seen with circular angle. But usage is our guide in WP so we must describe it. The usage as [[argument of a function]] for these hyperbolic functions supports the "angle" usage since they parallel the [[circular functions]]. Anything we can do to imply "hyperbolic measure" (as Klein says), and to provide clarity to readers on this important juncture in mathematics, is encouraged. Currently the article says:

::The hyperbolic functions take real values for a real argument called a [[hyperbolic angle]]. The size of a hyperbolic angle is the area of its [[hyperbolic sector]]. The hyperbolic functions may be defined in terms of the [[hyperbolic sector#Hyperbolic triangle|legs of a right triangle]] covering this sector.

:While the above discussion with Ameria was concluded last year, suggestions are welcome,[[User:Rgdboer|Rgdboer]] ([[User talk:Rgdboer|talk]]) 20:54, 11 May 2014 (UTC)

:Circular angles stay the same under rotation. Hyperbolic angles are preserved by [[squeeze mapping]]. One of the ways that the two types of "angle" are gathered together is as [[invariant measure]]s. Comparison of circle and hyperbola is necessary to give hyperbolic functions a basis for comprehension.[[User:Rgdboer|Rgdboer]] ([[User talk:Rgdboer|talk]]) 00:58, 17 May 2014 (UTC)

== Fanciful pronuciations not referenced ==

The following pronunciations given in the article are ridiculous baby-talk:

:(/ˈsɪntʃ/ or /ˈʃaɪn/) (/ˈtæntʃ/ or /ˈθæn/) (/ˈʃɛk/ or /ˈsɛtʃ/) (/ˈkoʊθ/ or /ˈkɒθ/) (/ˈkɒʃ/) (/ˈkoʊʃɛk/ or /ˈkoʊsɛtʃ/)

I can imagine a professor babbling while writing on a blackboard (i.e. paying attention to what he is writing, rather than what he is saying,) but there is nothing meaningful or encyclopedic about such pronunciations. If it is necessary in the article, by all means give the correct pronunciation for the full words, but this is absurd for what is supposed to be an encyclopedia article, and completely lacking reliable references as well. [[Special:Contributions/75.164.231.22|75.164.231.22]] ([[User talk:75.164.231.22|talk]]) 23:48, 2 December 2014 (UTC)

Undoing reversion by "I'm your Grandma." This is not a joke. The joke is that somebody thinks these pronunciations are legitimate. Say "hyperbolic sine" instead of "cinch" or "shine" if you want your speech to be understood and you don't want people to groan when they hear you. Furthermore, find a '''reliable reference''' to these "pronunciations" if you want them to stay in the article. (Same person, different IP) [[Special:Contributions/71.222.77.91|71.222.77.91]] ([[User talk:71.222.77.91|talk]]) 06:28, 5 December 2014 (UTC)

:Anonymous IP person, I apologize, and I now see what you are getting at. Should we have pronunciation guides for these words? When I was a kid (long ago) we learned that certain symbols stood for certain sounds. The symbols that Wiki uses, however, are not familiar to me. Still, I occasionally see the pronunciation symbols in some articles, never with a citation for reference, just used. I also note that the use of the pronunciation symbols (whatever they are called) is not common in mathematics articles (but see Fourier analysis where the pronunciation symbols for the name "Fourier" are shown). I'm not sure that any of this is too big of a deal, but other editors might have an opinion. Still ticking, [[User:I&#39;m your Grandma.|Grandma]] ([[User talk:I&#39;m your Grandma.|talk]]) 14:59, 5 December 2014 (UTC)

::These pronunciations are standard, probably used by all professional mathematicians. I've added reliable references. --<span style="border:1px solid #006000">[[User:Dr Greg|<font style="color:#FFFF80;background:#006000">'''&nbsp;Dr&nbsp;Greg&nbsp;'''</font>]][[User talk:Dr Greg|<font style="color:#006000;background:#FFFF80">&nbsp;<small>talk</small>&nbsp;</font>]]</span> 19:17, 5 December 2014 (UTC)

:::Thank you, Grandma and Dr. Greg. I'm satisfied that there is now a reference. The abbreviations themselves (sinh, cosh, tanh, coth, sech, csch) are very standard, but I wasn't aware that their pronunciations were so standardized. [[Special:Contributions/71.222.77.91|71.222.77.91]] ([[User talk:71.222.77.91|talk]]) 20:38, 5 December 2014 (UTC)

== Arc is not a misnomer ==

Contrary to the unreferenced assertions made above, the terms arcsinh, arccosh, etc., are not misnomers. The fact that the hyperbolic angle is equal to twice the area described in a unit hyberbola does not mean or even imply that it is not an ''arc''. After all, even a circular angle is equal to twice the the area described in a unit circle, and nobody says that therefore it is not an arc. Nobody says "arsin" or "arcos" or "area sine" or "area cosine", but one rather says arcsin, arccos, arc sine, and arc cosine.

Line 425 ⟶ 126:

::I do not have the slightest doubt that the historic/linguistic roots in Latin (''arcus'' (bow) and ''area'' (ground), perseus.tufts.edu) pale besides the prevalent prefixes (a-, arc-, ar-). Nevertheless, the use of ''arc-'' in connection with inverse hyperbolics '''is a misnomer'''. It is not the first, and will not be the last misnomer that becomes a '''general habit''' in referring to notions. I do not agree to encyclopedias having the task to avoid hurt feelings for using such made explicit misnomers, but rather to have the noble task of passing on the evidenced true roots of naming conventions. Reporting the updates in contemporary prevalence is a newly acquired advantage of electronic encyclopedias. My preference for the prefix "a-", for both inverses of "circular" and "hyperbolic" trigs, may be obvious from the above, but is no guidance for WP ("arg-" would be tedious, and ^(-1) is too mathy). Please, do not conceal that "arc-" for inverse hyperbolics results from a misnomer. [[User:Purgy Purgatorio|Purgy]] ([[User talk:Purgy Purgatorio|talk]]) 08:51, 21 December 2017 (UTC)

:::It is '''not a misnomer''' because inverse hyperbolic functions do represent an arc, which is imaginary. Think of hyperbolic functions as trigonometric functions with imaginary arguments and you'll understand: ''θ'' = arccosh ''x'' means that the corresponding arc length is ''iθ''. [[User:Flora Canou|Flora Canou]] ([[User talk:Flora Canou|talk]]) 06:56, 12 December 2018 (UTC)

== Move discussion in progress ==

::::As an old IT boy, I tell you that the abbreviation of the scientific names of trigonometric and hyperbolic functions was determined by the fact that the names of the functions in [[Fortran IV]] could be at most 8 characters and two characters were reserved for the type of function (integer, real, double), respectively the type of argument. Function names longer than 6 characters have been truncated. Many programmers wrote the names of the functions in mathematical texts as they knew them from programming. --[[User:Turbojet|Turbojet]] ([[User talk:Turbojet|talk]]) 08:43, 29 March 2021 (UTC)

There is a move discussion in progress on [[Talk:Sech (disambiguation)#Requested move 18 September 2018 |Talk:Sech (disambiguation)]] which affects this page. Please participate on that page and not in this talk page section. Thank you. <!-- Talk:Sech (disambiguation) crosspost --> —[[User:RMCD bot|RMCD bot]] 09:06, 18 September 2018 (UTC)

:::It is not just an imaginary arclength, but an actual one if you look at 1+1 Minkowski space with the "unit circle" (actually a hyperbola) embedded in it. You have to use the Minkowski metric to get the arclength along the "unit circle". See my comment at [[Talk:Inverse hyperbolic functions]], section "Arc interpretation for inverse hyperbolic functions". [[Special:Contributions/2001:171B:2274:7C21:59C0:D11E:8871:EC52|2001:171B:2274:7C21:59C0:D11E:8871:EC52]] ([[User talk:2001:171B:2274:7C21:59C0:D11E:8871:EC52|talk]]) 22:22, 10 May 2022 (UTC)

== Proposed change to the section "Relationship to the exponential function" ==

== Proposed change to the section "Relationship to the exponential function" ==

Hi all! I am completely new to editing Wikipedia, and I have an idea for a change. From what I've read, it seems I should propose the change here.

Line 504 ⟶ 206:

#My intent was specifically to emphasize not only the uniqueness, but also that the standard definitions for <math>\cosh(x)</math> and <math>\sinh(x)</math> are just particular instances of the general formulas for <math>f_\text{e}</math> and <math>f_\text{o}</math>. If nothing else, it makes the definitions really easy to remember. Thoughts?

#I'm not sure what you mean about duplicate links. I see even and odd properties mentioned, but I don't see any mention or link for general even-odd decomposition formulas in the current version of the hyperbolic functions article. Did I miss something?

#Regarding the blog as a source, this seems like a tricky issue. I've seen proofs presented on Wikipedia without references before (e.g. I just checked and the Mean Value Theorem is an example of this), so is it better to present the proof with no reference than to provide one with a blog reference? I wouldn't put a proof here, but perhaps the short elementary proof could fit in the article on even and odd functions. I appreciate you helping me learn the ropes! <!-- Template:Unsigned --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Greg at Higher Math Help|Greg at Higher Math Help]] ([[User talk:Greg at Higher Math Help#top|talk]] • [[Special:Contributions/Greg at Higher Math Help|contribs]]) 15:51, 30 November 2018 (UTC)</small> <!--Autosigned by SineBot-->

:::The unique decomposition of a function as the sum of an even and an odd functions was stated in [[Even and odd functions]], but one should know that it was there for finding it. Thus I have restructured this article for making it visible, and I have create the redirects {{no redirect|Even–odd decomposition}}, {{no redirect|Even part of a function}} and {{no redirect|Odd part of a function}}. Thus your suggestion is now reduced to simply add somewhere in this article: {{tq|Hyperbolic cosine and sine can also be defined as the [[Even–odd decomposition|even and odd parts]] of the [[exponential function]].}} [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 17:56, 30 November 2018 (UTC)

::::Perfect! I like how you structured it, and the proof you provide is concise. I've edited the Definitions section for the hyperbolic functions to include very brief parenthetical remarks linking to your [[Even and odd functions#Even–odd decomposition|Even–odd decomposition section]]. I also fixed a couple minor typos in that section and added an indication that <math>f_\text{e}</math> is even and <math>f_\text{o}</math> is odd (it's implied but I think it's more clear this way). <!-- Template:Unsigned --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Greg at Higher Math Help|Greg at Higher Math Help]] ([[User talk:Greg at Higher Math Help#top|talk]] • [[Special:Contributions/Greg at Higher Math Help|contribs]]) 13:23, 1 December 2018 (UTC)</small> <!--Autosigned by SineBot-->

== Short description ==

Before January 27, the short description (imported from Wikidata) was {{tqq|analog of the ordinary trigonometric function}}. Because "analog" is confusing here, I have changed if to {{tqq|Mathematical function related with trigonometric functions}}. {{u|Macrakis}} changed it to {{tqq|Mathematical functions on hyperbolas similar to trigonometric functions on circles}} with edit summary "also better grammar".

I disagree with this new description for two reasons. Firstly, it is wrong or at least confusing, since "on" after "function" generally specifies the ___domain, being an abbreviation of "defined on" (function on a curve, function on a manifold, function on an algebraic variety, ...). Secondly, the relationship between hyperbolic functions and hyperbolas is unclear for many readers, as most applications are not related to geometry. As short descriptions are aimed for easier navigation and searching, this seem a bad idea to mention in the short description some relatively minor facts that are ignored by most readers.

About the grammar: As the article title is singular, it should refered as such in the short description. So, the plural in Makrakis' version may be confusing. IMO, this article title and [[Trigonometric function]] should be moved to plural per [[WP:PLURAL#Exceptions]], but this is another question. (This move has been requested and done. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 14:39, 28 January 2020 (UTC))

Do someone have a (short) formulation that is fine for everybody? [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 10:59, 28 January 2020 (UTC)

:Happy to work with you on a better short description!

:"related with" is [https://books.google.com/ngrams/graph?content=related+with%3Aeng_gb_2012%2F%28related+to%3Aeng_gb_2012%2Brelated+with%3Aeng_gb_2012%29%2Crelated+with%3Aeng_us_2012%2F%28related+to%3Aeng_us_2012%2Brelated+with%3Aeng_us_2012%29&year_start=1800&year_end=2008&corpus=15&smoothing=3&share=&direct_url=t1%3B%2C%28related%20with%3Aeng_gb_2012%20/%20%28related%20to%3Aeng_gb_2012%20%2B%20related%20with%3Aeng_gb_2012%29%29%3B%2Cc0%3B.t1%3B%2C%28related%20with%3Aeng_us_2012%20/%20%28related%20to%3Aeng_us_2012%20%2B%20related%20with%3Aeng_us_2012%29%29%3B%2Cc0 rare and unidiomatic]; "to" is the usual construction.

:"Related to trigonometric functions" seems rather vague. They are both also related to the exponential function.

:How about "The hyperbolic functions are to hyperbolas what the trigonometric functions are to circles." (Avoiding the word analog -- not quite sure why you object to it.)

:Alternatively, we can emphasize their role in DE's: "The hyperbolic functions are solutions to many important differential equations."

:It is, after all, a ''short'' description, so can't mention all the important characteristics.

:Thoughts? --[[User:Macrakis|Macrakis]] ([[User talk:Macrakis|talk]]) 15:17, 28 January 2020 (UTC)

::What about "Main solutions of the differential equation y{{''}} {{=}} y "? (Apparently italics are impossible in short description.) [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 16:31, 28 January 2020 (UTC)

:::Too technical for a short description. --[[User:Macrakis|Macrakis]] ([[User talk:Macrakis|talk]]) 17:35, 28 January 2020 (UTC)

== 'Osborn's rule' ==

The 'Useful relations' section mentions 'Osborn's rule', citing a 'mnemonic' in a 1902 paper which neither proves it nor states in the way the article does. It even mentions it fails for fourth terms (?). This needs a better reference or removal. [[User:Kranix|Kranix]] ([[User talk:Kranix|talk]] &#124; [[Special:Contributions/Kranix|contribs]]) 17:03, 27 May 2020 (UTC)

: {{re|Kranix}} Fixed: [[special:diff/959218432]]. --[[User:CiaPan|CiaPan]] ([[User talk:CiaPan|talk]]) 17:58, 27 May 2020 (UTC)

: The error about '2, 6, 10, 14... sinh's' has been introduced in this edit [[special:diff/253404134]] in 2008. The error about 'any identity' was even older. --[[User:CiaPan|CiaPan]] ([[User talk:CiaPan|talk]]) 18:06, 27 May 2020 (UTC)

::{{re|CiaPan}} As it is now, I don't see what <math>\theta, 2\theta, \varphi,</math> etc. could refer to. The exponent bit was more or less right, but wasn't explained well. As far as sourcing, it's fine to include the original, but we should include something else, especially for establishing that this is actually referred to as "Osborne's rule". I'll take a look a bit later if it's still hanging around. &ndash;[[User:Deacon Vorbis|Deacon Vorbis]]&nbsp;([[User Talk:Deacon Vorbis|carbon]]&nbsp;&bull;&nbsp;[[Special:Contributions/Deacon Vorbis|videos]]) 18:21, 27 May 2020 (UTC)

::: {{re|Deacon Vorbis}} Hopefully some of these would qualify as RS?

:::* https://undergroundmathematics.org/glossary/osborns-rule

:::* https://archive.uea.ac.uk/jtm/4/dg4p1.pdf

:::* https://www.cambridge.org/core/journals/mathematical-gazette/article/9739-fibonometry/F708D0F6A464669A928835FC16FE856D

::: This one contains a clear proof of the rule

:::* https://math.stackexchange.com/questions/138842/proof-of-osbornes-rule

::: alas, as a user-generated content, it's not reliable enough. :( --[[User:CiaPan|CiaPan]] ([[User talk:CiaPan|talk]]) 19:43, 27 May 2020 (UTC)

::::{{re|Deacon Vorbis}} I think this would be more readable if <math>\theta</math> and <math>\varphi</math> were changed to ''x'' and ''y'', respectively. Then they would correspond to the identities which follow.—[[User:Anita5192|Anita5192]] ([[User talk:Anita5192|talk]]) 20:17, 27 May 2020 (UTC)

::::: The 1902 article seems like a fine reference. The section does need amending, though, because (a) Osborne's method needs an example to be easily understood, and (b) this section goes on to things unrelated to Osborne's method, without any transition. I don't know trig functions well enough to do it myself. Also, Osborne's example in his 1902 article is too hard to understand-- pick something simple. --[[User:Editeur24|editeur24]] ([[User talk:Editeur24|talk]]) 02:05, 30 December 2020 (UTC)

== "Hypersine" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] ==

[[File:Information.svg|25px|alt=Information icon]] A discussion is taking place to address the redirect [[:Hypersine]]. The discussion will occur at [[Wikipedia:Redirects for discussion/Log/2020 July 3#Hypersine]] until a consensus is reached, and readers of this page are welcome to contribute to the discussion. <!-- from Template:RFDNote --> [[User:1234qwer1234qwer4|1234qwer1234qwer4]] ([[User talk:1234qwer1234qwer4|talk]]) 17:48, 3 July 2020 (UTC)

== Editing for concision. ==

The article begins, “In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle.” Is “ordinary” redundant? Later in the same paragraf, it says, “Also, similarly to how the derivatives…” Should “similarly to” be shortened to “just as”? [[User:Solomonfromfinland|Solomonfromfinland]] ([[User talk:Solomonfromfinland|talk]]) 15:52, 29 September 2024 (UTC)

:No, ordinary is not redundant. The derivatives should not be mentioned in the lead section at all. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 03:08, 30 September 2024 (UTC)

== Inequalities ==

I would like to add some useful inequalities for cosh/sinh to the page, since these are often useful when analyzing these functions, and generally hard to find online.

I tried to make an edit here: [[special:diff/1264119498]]

But it was reverted. If anyone has suggestions for improvement, I'd be happy to take them. [[User:Thomasda|Thomasda]] ([[User talk:Thomasda#top|talk]]) 22:24, 21 December 2024 (UTC)

: I think [[user:D.Lazard|D.Lazard]] may have been concerned that you might be trying to promote the work of the cited author, Zhu Ling, which might involve a conflict of interest (hence the edit summary "... Apparent COI and self-promoting"). The material added here also consisted entirely of decontextualized symbolic formulas whose purpose and significance is not made clear. If these formulas are "often useful", is there some secondary survey source explaining when and how, and how they fit in with other scholarship? If you can convey some of that, an addition might be easier to defend. {{pb}} Part of the problem is a broader one: this article is currently extremely incomplete, not particularly well organized, not very well sourced or illustrated, etc. So there might also be a bit of an [[WP:UNDUE|"undue weight"]] concern. There are at least many dozens of more important and relevant formulas to include in a generic article about the hyperbolic functions, and these inequalities seem like a bit of a recent niche topic. Making them prominent here may imply more significance than justified. Personally I don't think this is usually enough reason to remove material (unless it's way out of scope / clearly belongs at a different page), but the only real way to address such concerns is to put a whole lot more work into improving the core parts of an article, which nobody seems to be in a hurry to do here. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 22:56, 21 December 2024 (UTC)

== Logistic function ==

The article on the [[logistic function]] discusses the hyperbolic tangent. It doesn't have to be much in this article, but I am thinking that we should at least let the reader know that there is a relationship between the hyperbolic functions and the (standard) logistic function. What do you say? —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 21:04, 1 May 2025 (UTC)