Talk:Proofs of trigonometric identities

message: 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)
replies: text: Sweet job here man. Good move. ☢ Ҡi∊ff⌇↯ 14:17, 20 October 2006 (UTC)
replies: text: 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)
thread_title:  To avoid redundancy, let's move all proofs of trigonometric identities from other pages onto this single page. 

message: 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)
thread_title: [[Talk:List of trigonometric identities/removed]]

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

message: 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)
replies: text: 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)
replies: text: 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)

text: 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)

text: 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)
thread_title:  identities involving calculus 

message: :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)
replies: text:  Certainly it's far better than it was then. Michael Hardy (talk) 02:38, 20 January 2008 (UTC)
replies: text:  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)
thread_title:  messy no more 

message: 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)
replies: text: 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)
replies: text: 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)

text: 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)
thread_title:  Proof not holeproof 

message: 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)
thread_title:  Good point 

message: 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)
thread_title:  Circularity is a drag 

message: 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)
thread_title:  Sum of sines and cosines with arguments in arithmetic progression 

message: 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)
thread_title:  Tangent of an average 

message: 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)
thread_title:  Possible error 

message: * 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)
thread_title:  Comments 

message: 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)
thread_title:  Triple tangent identity--probable violation of citation rules. 

message: 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)
thread_title:  No idea 

message: 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)
thread_title:  Proof of sum laws 

message: 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)
replies: text: 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)

text: ← 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)
thread_title:  This is not a proof of addition and difference formulas