Talk:List of trigonometric identities/Archive 1

message: What's with the tagging? I don't think such tags make sense for basic science articles, while i agree that even for common knowledge some reference math books could/should be listed, but it often makes absolutely no sense to work with quotes/inline citation or footnotes.--Kmhkmh 16:32, 22 September 2007 (UTC)
thread_title:  Tagging 

message: Where did the Sum-to-product identities(such as sin(x)+sin(y)) go?
Was there any reason to delete them?
It looks like the page has undergone some massive editing this year and imho it has become worse.
Imho the primary purpose of the article should be to give a correct and somewhat comprehensive list of all
trigometric identities out there, which are of interest to various groups (mathematicians,engineers,
laymen,students). This also means identities should not be removed because they are "just" a special case of
a more general case or the can "easily" derived from another. Please keep in mind the different purposes and
groups using this page as a lookup/reference.
I don't mind adding additional infos/improving the format but that should not result in identities getting deleted.
--Kmhkmh 16:47, 24 May 2007 (UTC)
thread_title:  Missing/Deleted identities ? 

message: where arctan(y, x) is the generalization of arctan(y/x) which covers the entire circular range (see also
 the account of this same identity in "symmetry, periodicity, and shifts" above for this generalization of arctan).
I believe this may be referencing section 3.3 "Shifts"; however the correlation does not appear to be explicit -- please expound either section as necessary and clarify the notation arctan(b,a) which does not seem to appear anywhere else in the article.
--Eibwen 08:23, 13 April 2006 (UTC)
replies: text: Just a few weeks ago when I found this page, this section was very clear about what to use for the phase (something like: if a > 0 it's one thing, and if a <= 0 it's another).  Now I'm not sure how to interpret it, because there now seems to be a discrepancy between what is stated in the section Other sums of trigonometric functions and what is stated in Periodicity, symmetry, and shifts.  I lack the confidence to determine which is correct, so I'm not going to touch it, but I would genuinely like to see this resolved. --Qrystal 12:44, 15 November 2006 (UTC)
thread_title:  arctan(y, x) 

message: Trigononmetric identities should not be called "equations".  "Equation" means something like "3x + 5 = 17", which is to be solved for x.  "Identity" means something like (x + 1)2 = x2 + 2x + 1, which is true of all values of x, and is not to be solved for x, but rather is to be proved, if one is to contemplate something to be done with it.  "Equality" is a more general term that includes both equations and identities. Michael Hardy 19:18 27 May 2003 (UTC)
:I am the one who changed the first sentence slightly because I thought we usually identities as a set so to me identities are bahaha sounds more natural. The current version seems fine to me. -- Taku 20:18 27 May 2003 (UTC)
::I hope you will all excuse my intrusion (I have yet to regularly log in before posting.) It strikes me that there are two trivial symmetry relations missing from this page, namely sin(x + Pi) = -sin(x) and cos(x + Pi) = -cos(x), of course, the plus signs should be plus-minus signs.  --Dwee
::: dear dwee, I agree with you. sin(x + Pi) seems missing. But from the text above the symmetry section it looks like we were only going to look at where reflections ..resulted in another trig function.. ala cos( Pi/2 - theta) = sin theta. That's all that is justified from the sentence above the table. I wish that sentence said ..here are some fun reflections.. then we could have our sin(x + Pi). 68.13.126.138 (talk) 05:53, 1 February 2010 (UTC)
thread_title:  Identity vs Equation 

message: The proof is still not correct, for the same reasons I said at Talk:Trigonometric function. I will come back later and try to repair it. One obvious new mistake -- f + g = h + j, does not imply f = h or f = j, this doesn't work even with numbers, 1 + 1 = 0 + 2.
Here's a sketch of the way it could go, if you want to try this:
:Let E(t) = C(t) + iS(t) be the parametric equation describing the motion of a particle on the unit circle in the complex plane (and yes, I do mean E(t), NOT E(it)). Here, t is a real variable, E is a complex function, and C and S are real functions (the real and imaginary parts of E). We assume that the motion of the particle is parametrised by the arc length of the circle. This conforms to our geometric notion of cosine and sine as the x- and y-coordinates of a point on the unit circle. (The only difference is, here, it's being traced out over time t). From this, we can deduce that the particle is moving at unit speed. To see why this is so, note that the ratio of the distance between 2 points on the circle, one measured along a straight line, the other along the circular arc connecting them, tends to 1 as the points tend to each other. (Draw a picture with a right triangle of very small angle, basically this amounts to sin x is roughtly = x for small x.) This is more or less where the "limit" argument comes in. What about the direction of velocity vector? It's not hard to see this must be always at a 90 degree angle to the position vector, just by the geometry of the situation (here's where geometry comes in). So, the velocity vector always has equal length (1) to position vector, and is rotated 90 degrees CCW. These two changes correspond to multiplying by i. So, dE/dt = iE, or dC/dt + i(dS/dt) = iE = −S + iC. Rquating real and imaginary parts, dS/dt = C and dC/dt = −S. The second-order equation with initial values for sine and cosine is encapsulated in the single equation d2E/dE2 + E = 0, with E(0) = 1 and E'(0) = i. Physically, this is just an expression of centripetal force, and you can define sine and cosine to be the imaginary and real parts of the unique solution of this equation. This second-order equation can be solved just as the first-order above, because it can be decoupled into 2 first-order equations which are essentially identical to solve.
:Revolver 21:14, 28 Jun 2004 (UTC)
replies: text: No problem...that's what the editing process is for. I did read the new change, it's still not quite right; here, the problem is that the set of solutions is all linear combinations of sines and cosines, not just multiples of sine or cosine. (Geometrically, you've taken a 2-dim soln space and reduced it to the union of 2 lines). I'll come up with something in the near future along the lines of what I have above -- the physics way of seeing it is what's important, I'll try to mention that. I'll try to keep what you already have, but don't take it personally if I rearrange things quite a bit. Revolver 05:12, 29 Jun 2004 (UTC)
thread_title:  Linear DEs proof of d(sin x)/dx <nowiki>=</nowiki> cos x 

message: Regarding:
One procedure that can be used to obtain the elements of this table is as follows:...Third, solve this equation for φ(arcψ(x)) and that is the answer to the question.
:There's nothing wrong with this, but the way I usually teach it is not algebraically but geometrically. Draw a right triangle (imagined to be inscribed in the first quadrant of unit circle). Label the unknown angle θ or something. Say you want to find cos(arcsin(x)). Then θ is the angle which gives you a sine of x, so put an x on the vertical leg and a 1 on the hypotenuse. The cosine of this angle θ is then the horizontal leg over the hypotenuse. Use the pythagorean theorem to get the horizontal leg equal to sqrt(1-x^2). This has the advantage that the only identity to remember is the pythagorean theorem (the others are just different versions, anyhow) and drawing a picture seems easier to visualise what's going on. Of course, to explain this would require pictures, which I don't know how to create or upload. Revolver 23:33, 25 Sep 2004 (UTC)
thread_title: Relating trig functions

message: The notation sin-1 is not rigorous because the function sin is not invertible.
So if there is no objection, i will erase it .
Mr Spok.
sin-1 is generally used in most modern text books to mean arcsine, many students are taught to use sin-1 this page could be confusing for them without this.
:sin-1 is standard notation, and Wikipedia should follow it. Charles Matthews 21:33, 12 Apr 2005 (UTC)
replies: text: Except that it should be written as sin−1 rather than sin-1, i.e. with a proper minus sign rather than a stubby little hyphen (or in TeX as \sin^{-1}\,\!.  Especially in subscripts and superscripts, a hyphen can be hard to see. Michael Hardy 21:51, 12 Apr 2005 (UTC)
replies: text: sin−1 certainly is standard notation, but it does create confusion with 1/sin. I consider safer to use arcsin to avoid any misunderstanding. --Doctor C 20:00, 4 August 2006 (UTC)
thread_title: sin-1

message: sin−1 does not equal 1/sin−1
They are fundamentaly different
sin−1 is the inverse function not the recipricol.
This is fundamental!
::Nor is sin−1 equal to 1/sin. Michael Hardy 20:45, 4 August 2006 (UTC)
thread_title: sin<sup>&minus;1</sup>

message: Are they definitely correct?
I thought 1/(cosx)^2 was secx, not cscx. The same goes for 1/(sinx)^2 being cscx, not secx.
:1/cos2(x) is sec2(x), not just sec(x).  Similarly, 1/sin2(x) = csc2(s), not  csc(x).  The formulas look correct to me.  How did you conclude that anyone was mistaking 1/cos for csc? Michael Hardy 21:34, 9 June 2006 (UTC)
thread_title:  Pythagorean identities 

message: * First off, why define cis when it simply refers to e^ix?
* Also, why introduce a complex function in an article that pretty much only deals with trig functions in the real case?
* And even more so, why introduce cis when nothing is ever done with it?  Although it is said that it can prove the aforementioned identites, there is no proof provided.
Basically, the entry has no reason for introducing the cis function within an article about real trig identities so I think it should be either removed or it's more familiar form, e^ix, should be used to prove a trig identity (or maybe even just provide a link to a section on euler's formula and include the proof for a trig identity there).
:I was puzzled by this "cis" thing, myself, having never heard of it referred to in this way.  It's a good thing this was mentioned in talk, because I'm going to take the initiative and remove all mention of "cis".  The section they were in (Angle sum and difference identities) already has a link to Euler's formula, as recommended in the comment above; however, there is also a link to Ptolemy's theorem, which I don't see as being relevant at all (except that the identities are used in it) so I am removing the "see also".  
:I do disagree with the statement that complex functions need not be present in this article.  However, I believe the link between trigonometric and exponential forms is sufficient in the section entitled Exponential forms, though I have added a link there to Euler's formula as well.  I support the recommendation that proofs of trig identities using Euler's formula should be included, perhaps at Proofs of trigonometric identities.
: --Qrystal 13:40, 15 November 2006 (UTC)
replies: text: I guess I missed it, as it's been removed already.  I reviewed the history, and found it there... and I also found it interesting, so I tried to see if it was relocated to somewhere else.  The cis article has mention of the use of cis in mathematics, and it refers to Euler's formula, which doesn't mention cis at all.  It would be a shame to see the information from this "new (but removed) section" be lost; perhaps this historical/pedagogical topic warrants its own page? --Qrystal 15:38, 8 December 2006 (UTC)

text: I see cis mentioned there, but that's all.  No one's put back the discussion of it, which explained why one would want to temporarily conceal the fact that it's an exponential function for pedagogical reasons. Michael Hardy (talk) 04:44, 11 November 2009 (UTC)
replies: text:  cis is rather obscure and little known, though it has a certain interest. I don't agree at all with the opinions raised above that one should not admit identities involving complex numbers: why not? I don't see that "trigonometric identity" has to mean "real trigonometric identity". When I checked the article a few minutes ago there was just one passing mention of "cis", with no explanation, which seemed pointless. I have added a few words explaining what it is an abbreviation for, but I have not added any more information about it. I have gone back into the history of the article and read the deleted section on cis. I find it interesting. It was deleted with the edit summary I completely removed cis. This is the consensus I see in the discussion page from over a year ago. However, having read the above discussion I do not see consensus to delete it: I see differing opinions. I wonder about reinstating it. JamesBWatson (talk) 09:46, 12 November 2009 (UTC)
replies: text:  I agree with Derekt75 above; I think the the information about cis is interesting, but should be put into a separate cis_(mathematics) page.  It is pretty off topic for a page about trig identities.  If that page existed, then I would feel better about cis being mentioned in this article (with a link to the cis article).  Otherwise I think the introduction of the cis notation here isn't justified. And4e (talk) 02:41, 19 November 2009 (UTC)
replies: text: Cis is not much used, and I could see some sense in suggesting it be excluded for that reason. However, what little use it has is entirely to do with trigonometric identities, so I don't see why it is off topic. It is used (or at least was within living memory, as I remember it being used when I was at school), whereas I believe that hacovercosine (for example) is long since obsolete, but as far as I know nobody has suggested removing all mention of that. The idea of having a separate article on cis seems to me to be giving it far more prominence than it deserves, and a fairly brief mention here for anyone who wants to know what it means seems to make much more sense. JamesBWatson (talk) 20:55, 19 November 2009 (UTC)
thread_title:  Why introduce cis in the "Angle sum and differences identities" section? 

message: I've just added a new section answering the question raised in the header above. Michael Hardy 00:23, 22 November 2006 (UTC)
replies: text: I guess I missed it, as it's been removed already.  I reviewed the history, and found it there... and I also found it interesting, so I tried to see if it was relocated to somewhere else.  The cis article has mention of the use of cis in mathematics, and it refers to Euler's formula, which doesn't mention cis at all.  It would be a shame to see the information from this "new (but removed) section" be lost; perhaps this historical/pedagogical topic warrants its own page? --Qrystal 15:38, 8 December 2006 (UTC)

text: I see cis mentioned there, but that's all.  No one's put back the discussion of it, which explained why one would want to temporarily conceal the fact that it's an exponential function for pedagogical reasons. Michael Hardy (talk) 04:44, 11 November 2009 (UTC)
replies: text:  cis is rather obscure and little known, though it has a certain interest. I don't agree at all with the opinions raised above that one should not admit identities involving complex numbers: why not? I don't see that "trigonometric identity" has to mean "real trigonometric identity". When I checked the article a few minutes ago there was just one passing mention of "cis", with no explanation, which seemed pointless. I have added a few words explaining what it is an abbreviation for, but I have not added any more information about it. I have gone back into the history of the article and read the deleted section on cis. I find it interesting. It was deleted with the edit summary I completely removed cis. This is the consensus I see in the discussion page from over a year ago. However, having read the above discussion I do not see consensus to delete it: I see differing opinions. I wonder about reinstating it. JamesBWatson (talk) 09:46, 12 November 2009 (UTC)
replies: text:  I agree with Derekt75 above; I think the the information about cis is interesting, but should be put into a separate cis_(mathematics) page.  It is pretty off topic for a page about trig identities.  If that page existed, then I would feel better about cis being mentioned in this article (with a link to the cis article).  Otherwise I think the introduction of the cis notation here isn't justified. And4e (talk) 02:41, 19 November 2009 (UTC)
replies: text: Cis is not much used, and I could see some sense in suggesting it be excluded for that reason. However, what little use it has is entirely to do with trigonometric identities, so I don't see why it is off topic. It is used (or at least was within living memory, as I remember it being used when I was at school), whereas I believe that hacovercosine (for example) is long since obsolete, but as far as I know nobody has suggested removing all mention of that. The idea of having a separate article on cis seems to me to be giving it far more prominence than it deserves, and a fairly brief mention here for anyone who wants to know what it means seems to make much more sense. JamesBWatson (talk) 20:55, 19 November 2009 (UTC)
thread_title:  New section: why "cis"? 

message: If these are identities, shouldn't the dominant symbol in the math formulae be the identity symbol, rather than the equals sign? 82.12.107.150 16:28, 1 October 2006 (UTC)
thread_title:  Identity symbol 

message: There's no need to clutter up this already haphazard article with proofs. --Ķĩřβȳ♥ŤįɱéØ 10:50, 20 October 2006 (UTC)
thread_title:  Merge the proofs of the identities to [[Proofs of trigonometric identities]] 

message: sin(arctan(x)) = x/(1+x^2) 
The table shows x/(1-x^2)
I havn't checked for further errors
Reference wolfram's mathworld under InverseTrigonometricFunctions
::I think I've cleaned up the table.  Please note:  Entries in any column, not in any row, are the six functions of one angle. Michael Hardy 23:39, 21 November 2006 (UTC)
replies: text: You could eliminate the confusion by placing phi to the left of the table, and psi on top of it.  That, combined with the phi(arcpsi(x)) should make it clear whether or not you look in the row or column first.  Would the table still work if the functions on the top row all had arc- added to the beginning of them? —The preceding unsigned comment was added by 134.253.26.11 (talk) 22:41, 2 May 2007 (UTC)
thread_title:  Trigonometric conversions table errors 

message: A common practice in Mathematics is to write sin x instead of sin (x) and sin 2x instead of sin (2x). Shouldn't the formulas in this article be written without the unnecessary parentheses? — Telking 09:57, 27 February 2007 (UTC)
replies: text: That's a common practice, but in many cases I, for one, prefer to write these with parentheses. Michael Hardy 20:08, 27 February 2007 (UTC)

text: I prefer leaving out parens myself, but other peple (such as Mr Hardy, here) may find the other easier to write and read.  OneWeirdDude 23:18, 5 March 2007 (UTC)

text: Considering the common use of powers and the very formal, reference-like tone of the article, parentheses should be used, especially because these are functions.  I rarely see f(x) written as fx is a formal setting. —The preceding unsigned comment was added by 134.253.26.11 (talk) 22:28, 2 May 2007 (UTC)
thread_title:  Redundant parentheses 

message: Shouldn't "trig identity" and "trigonometric identities" redirect here? Ketsuekigata 14:59, 23 April 2007 (UTC)
replies: text: It appears that those two pages have redirected to this page since May 2006. Michael Hardy 15:46, 23 April 2007 (UTC)
replies: text: Ah, how odd. It didn't do that the first time for me. Ketsuekigata 19:42, 23 April 2007 (UTC)
thread_title:  Redirect 

message: Ive always thought that the tangent line began at the point 1,0 as opposed to where it is now. Can someone explain it to me? —The preceding unsigned comment was added by 124.190.128.160 (talk) 12:50, 27 April 2007 (UTC)
replies: text: There are different ways of doing it.  I prefer the more conventional way to the one in this picture.  Maybe I'll put together a different one at some point. Michael Hardy 18:09, 4 May 2007 (UTC)
thread_title:  circular function picture 

message: This page is a sprawling mess of different forms of equations and identities and I think that the best way to help this article is to simplify and shorten it. I feel that, as a list, it should not contain long proofs;and that, as a trig article it should not contain any hyperbolic functions. Maybe we should create a parallel article List of hyperbolic identities. Any help would be appreciated. Conrad.Irwin 23:08, 30 April 2007 (UTC)
thread_title:  Page needs help! 

message: Instead of providing proofs, why not just list the identities organized by type (half-angle, power-reduction, etc.).  All I really want to see when searching for "trig identities" is a bunch of identities that aren't conjested with proofs.  Links to proofs would make the page more compact and a better reference piece.  Perhaps separate pages titled "Table of trigonometric function identities" and "Proofs and derivations of..." is required. —The preceding unsigned comment was added by 134.253.26.11 (talk) 22:25, 2 May 2007 (UTC)
thread_title:  Hard to find specific identities 

message: I can find the formula \sin^2 x + \cos^2 x = 1 in the page but i can't find the formula \cot^2 x + 1 = \csc^2 x . —The preceding unsigned comment was added by Novwik (talk • contribs) 11:01, 6 May 2007 (UTC)
thread_title:  square identities 

message: Some of the derivations on the page seem to use \theta as a "number", and use x as an angle. This means you end up taking the arc-whatever of an angle, to get an number. The trigonometric functions take an angle as their argument and return a "number" (for want of a better word), and the inverse trig functions take a "number" as their argument and return an angle. It seems odd, when everywhere else, \theta is the angle, and x is the "number". Why is \theta  not being used in its conventional meaning? Am I missing something here? If nobody objects, I'll change this. Alternatively, if I forget to change it, which is quite likely, feel free to change it.
Tim Goodwyn 17:17, 19 May 2007 (UTC)
replies: text: Are you referring only to the identities in the section titled "exponential definitions"? Michael Hardy 20:10, 19 May 2007 (UTC)
thread_title:  Derivations 

message: This article is not as good as it once was.
Why are the two identities in "Basic relationships" grouped together into a single section containing nothing else?  It doesn't make sense. Michael Hardy 22:25, 24 May 2007 (UTC)
replies: text: Probably derivations should be on a separate page.  Also, I'd consider excluding identities that are about hyperbolic functions only. Michael Hardy 20:40, 25 May 2007 (UTC)
thread_title:  reorganize 

message: I have removed a lot of the content from this article because it didnt fit. What I have removed can be found at Talk:List of trigonometric identities/removed with reasons for why it doesnt fit. I can forsee there being a lot of disagreement about this, but I felt that without some drastic action, this article would take too much effort to tidy up. Conrad.Irwin 22:16, 7 June 2007 (UTC)
replies: text: Whew - way to be bold!  This looks like a well-executed, healthy trim of the content.  Nice work. Doctormatt 22:46, 7 June 2007 (UTC)

text: I think it was a good idea to make the move. Having a separate apge for proofs or other things is good idea. Though it might be justified to add 1 or 2 things again, i'd overall it looks clearly better now.--Kmhkmh 14:55, 8 June 2007 (UTC)
thread_title:  Removed Content 

message: This article is very printer unfriendly.  Many of the equations get cut off, and the PNG images are printed as solid black squares. ✈ James C. (talk) 20:23, 9 February 2008 (UTC)
thread_title:  printability 

message: There is no proof of these identities on wikipedia or anywhere else on the web. Can anyone verify/prove them?  —Preceding unsigned comment added by 69.136.78.92 (talk) 21:21, 15 November 2008 (UTC)
thread_title:  Sum of sines and cosines with arguments in arithmetic progression 

message: I'm rather annoyed that Pharaoh of the Wizards is reverting my edits, here and elsewhere.
Currently under "Matrix form" there's a main article link to matrix multiplication.  This implies that matrix multiplication is a page filling in the details about the matrix form of the sum and difference formula.  Rot.  It's related, nothing more.  Call off your dogs.
150.203.35.113 (talk) 08:26, 18 November 2008 (UTC)
thread_title:  Not registered != vandalism 

message: I'm considering adding the identity
:a\cos x+b\cos(x+\alpha)= c \cos(x+\beta),
where
:
  c = \sqrt{a^2 + b^2 + 2ab\cos \alpha},\,
and
:
  \beta = \arctan \left(\frac{b\sin \alpha}{a + b\cos \alpha}\right) + \begin{cases}
0 & \text{if } a + b\cos \alpha \ge 0, \\
\pi & \text{if } a + b\cos \alpha < 0.
\end{cases}
This is similar to, and derived from, the following already listed identity:
:a\sin x+b\sin(x+\alpha)= c \sin(x+\beta),
with same c and \beta.
I was wondering if there would be any objections to adding it to the article. My derivation was kind of pedestrian thus lengthy so I don't want to include it in the proofs companion article. 
I was wondering if anyone could check my derivation or offer a source to cite or hint whether this is straightforward thus acceptable without proof. Thanks. 128.138.43.211 (talk) 06:46, 5 September 2009 (UTC)
replies: text: Without a source it can't be added. Also to me it doesn't look particularly interesting, but of course that is just an opinion. JamesBWatson (talk) 22:10, 9 September 2009 (UTC)
thread_title:  General linear combination of cosines 

message: Could we simplify the first image in the introduction by removing those functions that are almost never actually used, like exsec and versin? --Apoc2400 (talk) 11:21, 26 November 2009 (UTC)
replies: text: I don't see the point. The whole purpose of the diagram is to make the point that all the trig functions can be so represented. Nobody is expected to make more detailed use of the diagram than that, and besides, if anyone does want to look at it in more detail it is not so complicated as to be too confusing. JamesBWatson (talk) 12:24, 26 November 2009 (UTC)
replies: text: So, what is the point other than to scare off readers? --Apoc2400 (talk) 13:18, 26 November 2009 (UTC)
replies: text: I'm not sure I really expect 'readers' for this page. It is a list of reference data and the illustration gives reference type data. The articles Trigonometry or Trigonometric functions are more where people read an article. Dmcq (talk) 13:59, 26 November 2009 (UTC)

text: I suppose the illustration could say something like 'many of these terms are no longer in common use'. Dmcq (talk) 14:02, 26 November 2009 (UTC)

text: Michael Hardy (talk) 14:03, 27 November 2009 (UTC)
replies: text: Yes, those are perfectly alright. However, the existing diagram shows in one simple step the point that all of the functions can be seen as parts of a single diagram. We could have several different diagrams illustrating different groups of functions, but this would involve looking at a string of diagrams, and would not so clearly make the single point "all of them can be taken from one arrangement". I still do not see that the single diagram is too complex for anyone to see whichever function they like, if they look for it. And are we really to believe that anyone looking for information on trigonometric identities is going to be "scared off" because they see a diagram that they don't like? JamesBWatson (talk) 11:11, 3 December 2009 (UTC)
thread_title:  Introduction image 

message: Currently, in my browser (Firefox 3.6.10) in the Pythagorean section I get, in big red letters:
    Failed to parse (unknown function\cosec): 1 + \tan^2\theta = \sec^2\theta\quad\text{and}\quad 1 + \cot^2\theta = \cosec^2\theta.\!
Is this likely to be my browser being non-compliant, or is the formula incorrectly encoded?
175.45.146.134 (talk) 11:38, 26 September 2010 (UTC)
replies: text: It's not your browser; it's an incorrect edit made by someone else. It's been fixed. Ozob (talk) 12:49, 26 September 2010 (UTC)
thread_title:  cosec rendering 

message: Two new formulae have recently been inserted into the article.  Here's the first:
\sin n\theta = \sum_{k=0 \atop k = 1\, \bmod\, 2}^n (-1)^{\frac{k-1}{2}}\,\binom{n}{k} \,\cos^{n-k}\theta\,\sin^k\theta
I'm not sure why this is an improvement.  In this formula (for sine), what happens when "k" is two, for example?  Apparently we get the square root of negative one on the right-hand side.  Why not restore the previous simpler formula that didn't have any complex or imaginary numbers in it? The notation underneath sigma seems somewhat obscure, and it doesn't become more understandable by consulting Modulo.  The old formula seemed fairly simple and straightforward:
\sin nx = \sum_{k=0}^n \binom{n}{k} \cos^kx\,\sin^{n-k}x\,\sin\left(\frac{1}{2}(n-k)\pi\right)
Moreover, the latter (old) formula seems to be the one favored in reliable sources.
Anythingyouwant (talk) 16:11, 27 August 2011 (UTC)
replies: text: When k is two you don't add the term because you only add the terms for odd k i.e. if k is 1 mod 2. It could be written as odd(k) or even(k) for the other one. The formula does not involve any complex numbers but does show the relationship to their use in de Moivre's formula. The trick using sin to generate 0 1 or -1 depending on k is far more obscure and does not match up in an obvious way to de Moivre's formula. It is not straightforward to generate all the terms and then multiply half of them with zero to remove them again. None of them has a reliable source so I'll try and find one as that should clear up the matter, I think what was there was somebody messing around trying to be clever but just obscuring things. I'm very surprised you have not come across the modulo operation, I believed it was done in most schools nowadays and that children would normally know about it before they come across the trig functions at this level. I'll stick in odd and even instead for the moment. Dmcq (talk) 16:49, 27 August 2011 (UTC)
replies: text: Note that I provided a reliable source at the end of my previous comment.Anythingyouwant (talk) 17:01, 27 August 2011 (UTC)
replies: text: It's not really but it's as good as any so I'll revert to what was there as the ones I've seen with google just give the first few terms and put etc at the end. Dmcq (talk) 18:03, 27 August 2011 (UTC)

text: By the way if you look down further at Mathworld you'll see the formula you removed from the deMoivre article. It has no support elsewhere either and what was in the article looked like a copy so I'm happy for it to be gone. Dmcq (talk) 18:06, 27 August 2011 (UTC)
replies: text: OK, thanks.  I've added some text that hopefully makes it clearer: "In each of these two equations, the first parenthesized term is a binomial coefficient, and the final trigonometric function equals one or minus one or zero so that half the entries in each of the sums are removed."Anythingyouwant (talk) 18:47, 27 August 2011 (UTC)
thread_title: New multiple angle formulae

message: I've had occasion to look up this web page four times in two days and, despite its massive size, on all four occasions none of the trigonometric identities I was looking for was there. They are: \cot x+\tan x=2\mathrm{cosec} 2x,   \cot x-\tan x=2\cot 2x,   1+i\cot x=2/(1-e^{2ix}),   \sec x=2e^{ix}/(1+e^{2ix}),   \frac{\sin y}{\cos x-\cos y}=\cot\frac{1}{2}(y+x)+\cot\frac{1}{2}(y-x),   \sec x=1+\sum_1^\infty\frac{E_kx^{2k}}{(2k)!},   \frac{1}{2}\tan\frac{x}{2}=\sum_{k=0}^\infty\frac{2^{2k+2}-1}{(2k+2)!}B_{k+1}x^{2k+1},   \sin(15)=(\sqrt{6}-\sqrt{2})/4   and an integral formula I won't bore you with. OK, so some of those are a bit esoteric, but all but the last one are assumed prior knowledge on pages 2 to 5 of Hardy(1949) "Divergent Sequences".Mollwollfumble (talk) 21:02, 23 October 2011 (UTC)
thread_title: Incompleteness

message: These formulas have a name (a classical name). Unfortunately I don't remember exactly the spelling. The name is (or approximately is) prostospheratic formulas or prostoferatic formulas. I know it is a not commonly used name these days but it would be nice if someone remembers it/ finds it and finds even why they were called that way. I am asking this not only for making an addition to thee article but also for myself. I want to remember the name and learn the neaning/ origin of it. Thanks.  franklin  03:53, 26 January 2010 (UTC)
replies: text: I suppose that might be mildly interesting, but it seems to me that any such names would be too obscure to be of much encyclopedic interest. JamesBWatson (talk) 08:24, 28 January 2010 (UTC)

text: * Not what I am asking. I am just asking for the name not inviting to any analysis of its encyclopedic value. By the way, the name (the meaning of the Greek origin) gives some geometric interpretation or some application of the formula. But for the moment I just want to remember the name the rest depends on what exactly the name was.  franklin  11:41, 28 January 2010 (UTC)

text: I believe Prosthaphaeresis is what you're looking for. Dmcq (talk) 11:46, 28 January 2010 (UTC)

text: * Thanks so much. I knew the name in Spanish but no way I could have guessed all those h's in the name in English. And it seems to be that the name had EV after all since it already have an article. So, a link from here to that article would be useful.  franklin  11:52, 28 January 2010 (UTC)

text: Hope someone reads this, I do not really post on wiki often, but wanted to get this info out there (I'm sure I'm breaking some rule somewhere but I thought this was important to point out); for the SINX±SINY sum-to-product rule I have that the SIN and COS are switched in the result, not the ± symbols.  The way I read it here suggests that if you have sinx+siny then the 2sin(1/2*x+y)cos(1/2*x-y) and if it is sinx-siny then you get 2sin(1/2*x+y)cos(1/2*x+y), thinking only that the symbol (+ or -) relating the two angles of COS are changed in the result.  I'm posing that if sinx+siny then 2sin(1/2*x+y)cos(1/2*x-y) or if SINx-SINy then 2COS(1/2*x+y)SIN(1/2*x-y), only the sin and cos are switched, everything else stays the same.  Am I totally off?  Info from a calculus MathXL online course (precalc review chapter) problem GR.4.111.  I don't know the html tags to make the time i worte this and all that stuff official :( sorry.   you can delete this or revise it to fit in better, assuming my comment made any sense (or is even valid). nagromltNagromlt (talk) 21:40, 16 March 2011 (UTC)
thread_title:  Product-to-sum and sum-to-product identities 

message: Perhaps my edit of 04:43, 23 November wasn't clear.  I believe that, if it is acceptable to give these things names, the way I have done it is correct.  I don't understand the objection that "changing the cotangent identity to the sine-double identity (even if accurate) is not [acceptable]".  It wouldn't be accurate, and I never meant to imply such a thing.  It is not my intention to give the equation about tangents any name other than the tangent identity, the one about cotangents any name other than the cotangent identity, or the one about sines any name at all.  I'm not aware that the one about double sines has any name, though if someone comes up with one, that's fine.  SamHB (talk) 16:58, 23 November 2013 (UTC)
thread_title:  Triple tangent/cotangent identities 

message: Just in case it's useful to anyone, the Chebyshev formulae for \sin nx and \cos nx are most easily proved as the imaginary and real parts (respectively) of the formula:
:\operatorname{cis} nx = 2 \cdot \cos x \cdot \operatorname{cis} ((n-1) x) - \operatorname{cis} ((n-2) x)
where \operatorname{cis} x = \cos x + i \sin x = e^{ix}, as in Euler's formula.  Because this is an exponential function, it has the particularly simple summation function \operatorname{cis}(x+y) = \operatorname{cis} x \operatorname{cis} y, and  \operatorname{cis}nx = \operatorname{cis}^n x.
Given this, the formula can be simplified by dividing both sides by \operatorname{cis} ((n-2) x) to get:
:\operatorname{cis} 2x = 2 \cdot \cos x \cdot \operatorname{cis} x - 1.
Which can be proved by expanding and then simplifying the right-hand side:
: 
\begin{align}
2 \cdot \cos x \cdot \operatorname{cis} x - 1
&= 2 \cdot \cos^2 x + 2i \cdot \cos x \cdot \sin x - \cos^2 x - \sin^2 x \\
&= \cos^2 x + 2i \cdot \cos x \cdot \sin x - \sin^2 x \\
&= (\cos x + i\cdot \sin x)^2 \\
&= \operatorname{cis}^2 x \\
&= \operatorname{cis} 2x
\end{align}
71.41.210.146 (talk) 17:06, 28 December 2013 (UTC)
thread_title:  Proof of Chebyshev formulae 

message: Forgive me if I'm being an idiot, but in the table that includes the cofunction identities, should the transformations not be described as reflections in  \theta = 0, \theta = {\pi \over 4} and  \theta = {\pi \over 2}  respectively? M.A.Redman (talk) 19:14, 10 April 2014 (UTC)
thread_title:  Symmetry 

message: It would be handy to give a solution to the geodesic eqn a\sin x+b\cos x=c 67.198.37.16 (talk) 20:01, 30 September 2015 (UTC)
thread_title:  useful identity 

message: As in the "Arbitrary phase shift" and "Sine and cosine" cases the expressions for  and  should be unambiguous. To the best of my knowledge that is
:a=\sqrt{\sum_{i,j}a_i a_j \cos(\theta_i-\theta_j)}
and 
:\theta=\operatorname{atan2} \left( \sum_i a_i \sin\theta_i, \sum_i a_i \cos\theta_i\right).
— Preceding unsigned comment added by Tpreclik  (talk • contribs) 03:39, 3 January 2017 (UTC)
thread_title:  More than two sinusoids 

message: When I learned the following identities, they were all called "Pythagorean identities" with consistency from teacher to teacher:
:\sin^2 x  + \cos^2 x  = 1  
:\tan^2 x + 1 = sec^2 x 
:1 + \cot^2 x = \csc^2 x 
However, online it seems that some people only use the term "Pythagorean identity" to refer to the former.  Therefore, would anybody object to me changing "identity" to "identities"?  Thank you.LakeKayak (talk) 19:18, 18 February 2017 (UTC)
replies: text: As it seems that nobody has an opinion, I am going to make the change and wait for people's responses from there.LakeKayak (talk) 01:28, 21 February 2017 (UTC)
thread_title:  Pythagorean Identity