Talk:Proofs of trigonometric identities

To avoid redundancy, let's move all proofs of trigonometric identities from other pages onto this single page.

I've noticed that many articles give trigonometric identities and prove them. But when two or more articles prove the same identity, it is a waste of space (even if they use different proofs). I propose that all proofs of trigonometric identities to be placed on this page, so that there will never be any repetition of proofs. --Ķĩřβȳ♥ŤįɱéØ 10:59, 20 October 2006 (UTC)

Sweet job here man. Good move. ☢ Ҡi∊ff⌇↯ 14:17, 20 October 2006 (UTC)

I like this idea. List of trigonometric identities really needs to be cleaned up, and removing proofs would be a good way to do that. This page might end up getting pretty long, though... after all, many things can be proven in more than one way (e.g. proof of angle sum trig identity via geometric methods or Euler's formula). Whoever is working on this task, I wish you the best of luck in remaining organized and coherent. I will watch this page and help out sporadically. --Qrystal 13:48, 15 November 2006 (UTC)

THIS page in its present form is a horrible horrible mess. List of trigonometric identities, on the other hand, seems reasonable. I think it's reasonable to have separate pages. Also, it would be a good idea to make the bulk of this page deal with general strategies for proving trigonometric identities. I am not confident that whoever wrote most of this page understands those techniques. He or she also does not understand how to avoid the hideously ugly formatting now on this page. I've edited many thousands of Wikiepedia math articles; I know what I'm talking about. Michael Hardy 19:09, 13 December 2006 (UTC)

I agree with Michael that it is reasonable to keep the pages separate. Gaussmarkov 16:54, 23 January 2007 (UTC)

Talk:List of trigonometric identities/removed

I have recently removed a whole lot of stuff from the List of trigonometric identities article, and placed it at Talk:List of trigonometric identities/removed. This included a few proofs and a lot of derivations that might want to be put onto this page. I didnt put them here yet because I didnt understand the article and havent have time to wade through it. Conrad.Irwin 21:46, 8 June 2007 (UTC)

Organization

Let's give names to the sub-types of identities, and to each identity, so the readers can find the ones they are looking for. Examples: Pythagorean, inequality, angle sum, etc. "Identity # 62" does not help much. Also, since the reason for this article is proofs, having the identity as a main point and the proof as a sub-point does not make sense to me. --MathMan64 20:27, 20 October 2006 (UTC)

Well I only gave 1 proof for each identity, but in case of more than one proof, it can be listed as a sub-point of the identity. --Ķĩřβȳ♥ŤįɱéØ 00:48, 21 October 2006 (UTC)

Let's label the headings for multiple proofs with descriptions such as:

=== Cotangent ratio ===

==== from definition ====

{ as is in article }

==== from reciprocal ====

{ new proof plan }

--MathMan64 01:50, 21 October 2006 (UTC)

That's fine with me. But the only problem I forsee is, some of these identities just don't have names. Sure, we have some easy ones like "Law of Sines" or "Law of Cosines", but what would we call something like $\lim _{\phi \to 0}{\frac {\sin k\phi }{p\phi }}={\frac {k}{p}}$ ? I've never seen that named in a textbook, it's just given without a name. --Ķĩřβȳ♥ŤįɱéØ 02:18, 21 October 2006 (UTC)

tangent inequality

The tangent inequality is not true for all angles. The tangent of π is zero. So tan π over π is not greater than one. --MathMan64 18:24, 29 October 2006 (UTC)

Oh, I forgot to restrict the domain. I'll fix it. --Ķĩřβȳ♥ŤįɱéØ 01:48, 30 October 2006 (UTC)

identities involving calculus

The previous proof of the Sine and angle ratio identity is circular as noted in A Circular Argument, by Fred Richman, The College Mathematics Journal © 1993. The result follows immediately from the definition of the length of an arc. So I changed the proof to reflect this. I left out a step, pending any discussion: the longest polygonal path for a given number of points on the arc is the one with equal length chords.

Gaussmarkov 23:45, 10 November 2006 (UTC)

I understand the logic behind this step and I too think that the previous proof was inadequate. However, I think the current proof needs to be elaborated because it doesn't go into details, and to the average reader, it doesn't mean much. CommandoGuard 15:51, 12 December 2006 (UTC)

Hmmmm...I just read "A Circular Argument" and the impression I was given by it is that any geometry-based proof of the limit in question which is correct and not circular is always more of a definition than a proof, no? CommandoGuard 13:52, 13 December 2006 (UTC)

Yes, some sort of limit argument is needed. In my opinion, the arc length needed for the current version should be made explicit and referenced to another location in the Wikipedia. That is what I did previously. In the present form, the angle and the arc length are implicitly equated and there is no justification for this. Gaussmarkov 16:51, 23 January 2007 (UTC)

??? Isn't that the definition of a radian?--Kirby♥time 23:26, 4 May 2007 (UTC)

The above proof comes from Courant's Differential and Integral Calculus, vol. I, p. 48. It is sufficiently paraphrased to avoid plagarism, and it presents the proof used by generations of calculus students. Courant's version handles the case where the limiting term is negative a bit more explicitly, but otherwise, I feel this proof is adequate for Wikipedia's purposes. --Moly 14:31, 31 October 2007 (UTC) —Preceding unsigned comment added by Moly (talk • contribs)

messy, messy, messy

This is a really messy article in two ways (at least):

• Formatting, typesetting, etc. See my most recent edit.
• Logical structure. One may prove the identity A = E by writing A = B = C = D = E. That's what I did in my most recent edit. One may also prove A = E by saying A = E if blah, and blah is true if blahblah, and blahblahblah is true if etc.etc., and etc.etc. is a known truth (but one must be sure not to write "If A = E then ...."; the "if...then..." has to go in the right direction.

Michael Hardy 01:55, 17 November 2006 (UTC)

messy no more

Michael - and others, do you still regard this page as "messy"? It seems to me to be now quite clearly organised.

Furthermore, I don't see a present need for the "Cleanup" tag. Please give reasons WHY this page still requires cleanup. If no compelling reasons are forthcoming by 20th February 2008, I propose removing the "Cleanup" tag. yoyo (talk) 01:45, 20 January 2008 (UTC)

Certainly it's far better than it was then. Michael Hardy (talk) 02:38, 20 January 2008 (UTC)

I think it could still use some cleanup. First, some of the terms used for identities do match the terms used for the same identities on the List of trigonometric identities page. Ideally, I think this page should match that one, with the proofs linked between sections. Also, the typesetting of the formulae does not match. Look at the equations in the angle sum section for details. Finally, there is inconsistent layout between sections. For example, the Eulerian proof in the Angle sum section is thrown in to the sum section. It seems to me it should really have its own (at least sub-) header to separate it, as is done for other identities on the page.Chaleur (talk) 07:17, 10 August 2010 (UTC)

This is the first time I've ever come across this article. But it seems to be tidy and informative to me. 86.164.127.54 (talk) 21:12, 7 April 2008 (UTC)

Proof not holeproof

The proofs of the "Angle sum identities" contains statements like: "From the negative argument formulae, ..." Perhaps an earlier version of this page showed "the negative argument formulae"; however, the current version doesn't. Anyone like to repair the deficiency? yoyo (talk) 05:47, 20 January 2008 (UTC)

I'll repair it. It's not by negative argument formulae, whatever that is. It's by symmetry. Will link to the identities page. Chaleur (talk) 21:12, 3 August 2010 (UTC)

Okay, now I'm not sure about that. Thinking. Someone else jump in if you think of it first. Chaleur (talk) 21:21, 3 August 2010 (UTC)

I presume this means the even and odd nature of the cosine and sine respectively. Michael Hardy (talk) 22:59, 3 August 2010 (UTC)

Alright, I figured out how to derive using Symmetry. However, I'd still love to see someone put in a non-symmetry-using geometric proof of this identity. There is a better sum/difference proof here: http://staff.jccc.net/swilson/trig/anglesumidentities.htm that covers both cases. Chaleur (talk) 23:30, 3 August 2010 (UTC)

Someone please provide better proofs!

I have used this page twice for proofs of the trigonometric identities. The first time, I was proving Euler's formula by calculus, which relies on trigonometric differentiation, which in turn relies on the angle sum identities. The only two proofs of the angle sum identities were ones that used Euler's formula, which is obviously not applicable in this case, and a proof by picture that looks like it only works for angles between 0 and pi/2. The second time a came to this page, I was looking for a proof of the Prosthaphaeresis identities as they were applied in an analysis problem I was working on. I came to this page and found nothing but the identities listed, with no proof whatsoever! Since the title of this article is "Proofs of the trigonometric identities," I consider that lame! I noticed at the top of the page that it said, "this page may require cleanup". Damn right it does!

Good point

The user above me made very good points. The proof of the angle sum identities by Euler's formula is not valid because it creates circular dependency. All 3 proofs of Euler's formula (power series, calculus, differential equations) rely on the derivatives of the trigonometric functions, which in turn rely on the angle sum identities to simplify sin(x+h) and cos(x+h). The picture proof of the angle sum identities is not valid either since as said before, it only proves the case where 0<angle<pi/2. This page should be cleaned up. And if I can find better proofs of the angle sum identities, I will try and post them. —Preceding unsigned comment added by 65.66.217.117 (talk) 03:06, 13 July 2008 (UTC)

Circularity is a drag

Yes! It seems that proving the sum identities from the Eulerian premise is circular. Even if it can be shown not to be circular, it would still prove the less obscure from the far more obscure. So, it isn't helpful. I've seen this all over the web. Moreover, the proof doesn't address the cases in which the angles are outside (0,pi/2). I've also seen this throughout the web. A better discussion is needed. Looking for a better proof. —Preceding unsigned comment added by Alb31416 (talk • contribs) 11:54, 4 September 2008 (UTC)

Sum of sines and cosines with arguments in arithmetic progression

There is no proof for these two identities, and I have been unable to find these identities anywhere else on the web. —Preceding unsigned comment added by 69.136.78.92 (talk) 21:19, 15 November 2008 (UTC)

Tangent of an average

Should there be a proof placed here? There is an algebraic one (using the sin and cos sum-to-product identities) but I have also made a geometric proof that does not require the knowledge of those sum-to-product identities, just the pythagorean ones. ~DoubleAW^[c] 23:20, 2 March 2010 (UTC)

Possible error

In the section: Proof of Compositions of trig and inverse trig functions

Isn't the identity used here?

tan( arctan(x) ) ≡ x — Preceding unsigned comment added by 98.169.62.144 (talk) 20:25, 12 July 2012 (UTC)

Comments

• This article defines sin, cos, tan, etc. only for acute angles. Would be much better to see proofs of trigonometric identities for all angles.
• Even if we restrict to acute angles alpha and beta, the sum of alpha and beta can be obtuse. However, the angle sum identities are only proven for the case where alpha + beta is acute. Would be much better if these proofs can be generalized to all acute angles alpha and beta, rather than restricting alpha + beta to acute.

unkx80 (talk) 19:55, 3 January 2013 (UTC)

Triple tangent identity--probable violation of citation rules.

I recently put in proofs of the tangent and cotangent identities, including a reference to an external web site. This very well may violate various rules about citations. Feel free to remove it if so. I happen to think the derivation given there is stupid; that web reference had been put into the Law of cotangents article by someone else. I'm going to remove the latter, because it doesn't belong on that page. If people want to remove this one (or fix it up), it's fine with me. SamHB (talk) 02:19, 30 October 2013 (UTC)

No idea

Under the section "Complementary angle identities", it says:

"Two angles whose sum is π/2 radians (90 degrees) are complementary. In the diagram, the angles at vertices A and B are complementary, so we can exchange a and b, and change θ to π/2 − θ, obtaining:"

And then it decides to go with (θ-π/2) instead of (π/2−θ) all along the table, and it says:

"$\sin \left(\theta -\pi /2\right)=\cos \theta$"

But since sin(-θ)=-sinθ, maybe it should be:

$\sin({\tfrac {\pi }{2}}-\theta )=\cos \theta$,

whereas for (θ-π/2), it would be:

$\sin(\theta -{\tfrac {\pi }{2}})=-\cos \theta$

and things like that?..

Probably a test to see how bold people are? and probably i failed?85.110.71.246 (talk) 20:25, 25 May 2014 (UTC)

Proof of sum laws

To state the obvious, there is a very elegant proof of the sum laws that uses matrices in SL(2) (or, equivalently, complex numbers). That the geometric proof here is fine, as I expect many high school students know neither linear algebra or complex numbers (which is not a good thing, but that is another story), however it may be a bit tedious for advanced readers. — Preceding unsigned comment added by 124.168.57.81 (talk) 21:20, 25 May 2015 (UTC)

This is not a proof of addition and difference formulas

I can't even count how many times I've encountered this nonsensical notion that the proof supplied in this article (the same or very similar) is in fact a proof of addition and difference formulas for cosine and sine. It's not. It's based on geometrical notions which only work in a particular case when all angels (A, B and A+B) are acute. Otherwise it's worthless and suggesting otherwise is nothing less than promoting ignorance, misconceptions and undisciplined reasoning. There is actually simple, elementary and general proof of this identities.

Take unit circle [a] and draw angels A+B, B, -A. Because this is a unit circle coordinates of the point plotted on circle by angle x, are (cos(x),sin(x)). Moreover, sqrt((cos(x))^2 + (sin(x))^2) = 1, and from this it follows that (cos(x))^2 + (sin(x))^2 = 1 {Eq.1}. Take the distance between the point plotted by angle A+B, and point (1,0). It's: sqrt[[cos(A+B)-1]^2 + [sin(A+B)-0]^2]. Take the distance between the point plotted by angle B, and the point plotted by -A. It's: sqrt[[cos(B)-cos(-A)]^2 + [sin(B)-sin(-A)]^2], which after using simple identities cos(-x) = cos(x) and sin(-x) = -sin(x) gives: sqrt[[cos(B) - cos(A)]^2 + [sin(B) + sin(A)]^2]. This distances are equal (they're chords of arcs of the same length). Square and expand both sides of equality and use Eq.1 to simplify some terms. This gives you addition formula for cosines. You can get difference formula in similar way (distances A to B, and A-B to (1,0). For sines just make use of identity: sin(x) = cos(pi/2 -x).

[a]: There is no loss of generality in using unit circle because taking different radius will only scale all terms in following equality.

Note: You should try to appreciate this proof without drawing any illustrations - it's easier to see generality of this reasoning when there are no particulars in sight. All statements in this proof are obviously true regardless of what values A and B take. And that's the whole point.— Preceding unsigned comment added by 2A01:119F:245:800:2462:B2C0:1638:ED04 (talk) 20:27, 2 March 2017 (UTC)

You can provide a source for that? Can we have a diagram for this explanation (like one on the right)? AXONOV (talk) ⚑ 11:29, 12 January 2022 (UTC)

…For sines just make use of identity… ← That's not correct. The correct formula would be $\sin \theta ={\sqrt {1-\cos ^{2}\theta }}$ ^[0].--AXONOV (talk) ⚑ 11:39, 12 January 2022 (UTC)

This article must be split apart and the title redirected

This article runs contrary to the ordinary and expected inter- and intra-article organization of every other part of Wikipedia, and not in a good way.

It takes a pile of particular parts (the proofs) of loosely related topics and mashes them together in random order and out of context, with limited explanatory prose and almost entirely without sources.

In my opinion this article must be deleted, with salvaged content sourced and split up to specific pages about each identity. If an identity is not notable enough for a dedicated article or at least top-level section of some existing article, then a proof is not notable enough to be included in Wikipedia.

Having this article here as a garbage heap prevents these identities from getting their own dedicated pages, and prevents the us from including multiple proofs of the same theorem/identity, prevents us from adding enough context to explain the identities' purposes, discussing the history, etc. It would be dramatically better to have 20 separate "start" class articles than this one mishmash. –jacobolus (t) 20:15, 27 February 2024 (UTC)

I support deletion of this article. Here are several further reasons for that:

• We have already Trigonometry and List of trigonometric identities, which contain proofs of the most important identities. This is enough.
• The article is misleading, as suggesting that there is only one proof (or a "main" proof) for each identity
• The article is misleading also, as suggesting that finding these proofs require cleverness, and ignoring that there are standard methods that allow automatic proofs of all these identities.

Indeed, tangent half-angle substitution allows reducing most proofs to the verification of the equality of two rational fractions. Also, the expression of the trigonometric functions in terms of complex exponentials allow reducing the same proofs to equalities of exponential polynomials (although the latter method is less elementary than the former, it leads to much simpler computations). Other proof rely on Taylor series or on differential equations. So, we need an article Proof and simplification in trigonometry, but this is certainly not this article. D.Lazard (talk) 14:51, 28 February 2024 (UTC)

I think it would be fine to have dedicated articles about e.g. Angle sum identities, Multiple-angle identities (Double-angle, triple-angle, half-angle etc.), Sum-to-product identities, ...

We already have Pythagorean trigonometric identity, Sum of angles of a triangle (discussing the identity $\alpha +\beta +\gamma =\pi ,$ which should also discuss derivative identities such as $\tan \alpha +\tan \beta +\tan \gamma ={}$$\tan \alpha \,\tan \beta \,\tan \gamma$ and $\tan {\tfrac {1}{2}}\alpha \,\tan {\tfrac {1}{2}}\beta +{}\!$$\tan {\tfrac {1}{2}}\alpha \,\tan {\tfrac {1}{2}}\gamma +{}\!$$\tan {\tfrac {1}{2}}\beta \,\tan {\tfrac {1}{2}}\gamma =1$), Euler's formula, De Moivre's formula, Tangent half-angle formula, and probably several others.

I think Complementary angle, Supplementary angle, and Conjugate angles could be made into a dedicated article, maybe under the title Combining angle pairs which is currently the section heading where they redirect to.

Any of these articles which is moderately specific can include one or more proofs, so long as the content is sourced. It would be nice to e.g. include some of the many "proofs without words" for these identities which have been regularly published in undergraduate-level mathematics magazines in the US and UK for the past few decades.

As you say, an article consisting largely of prose with a few examples about the general concept of trigonometric proofs would also be fine. –jacobolus (t) 15:44, 28 February 2024 (UTC)

I'd support the deletion as well, for the various identities we already have have articles and Wikipedia isn't really the place for compilations/collection of proofs for a list of theorems, there are Wikibook projects for that or outside of Wikimedia the ProofWiki project. Proofs are optional at best and may be added to individual theorems, where it seems appropriate. And there can be articles on individual on particularly notable proofs, but such collections as the one here are imho off topic for Wikipedia.--Kmhkmh (talk) 16:03, 28 February 2024 (UTC)