Talk:List of trigonometric identities

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This page, despite seeming to have no active content, utilizes 100% of the core the thread is running on. Google Crome Version 84.0.4147.105 (Official Build) (64-bit) — Preceding unsigned comment added by 75.109.252.140 (talk) 17:11, 4 August 2020 (UTC)

Triple tangent/cotangent identities

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)

Proof of Chebyshev formulae

Just in case it's useful to anyone, the Chebyshev formulae for ${\displaystyle \sin nx}$ and ${\displaystyle \cos nx}$ are most easily proved as the imaginary and real parts (respectively) of the formula:

${\displaystyle \operatorname {cis} nx=2\cdot \cos x\cdot \operatorname {cis} ((n-1)x)-\operatorname {cis} ((n-2)x)}$

where ${\displaystyle \operatorname {cis} x=\cos x+i\sin x=e^{ix},}$ as in Euler's formula. Because this is an exponential function, it has the particularly simple summation function ${\displaystyle \operatorname {cis} (x+y)=\operatorname {cis} x\operatorname {cis} y}$, and ${\displaystyle \operatorname {cis} nx=\operatorname {cis} ^{n}x.}$

Given this, the formula can be simplified by dividing both sides by ${\displaystyle \operatorname {cis} ((n-2)x)}$ to get:

${\displaystyle \operatorname {cis} 2x=2\cdot \cos x\cdot \operatorname {cis} x-1.}$

Which can be proved by expanding and then simplifying the right-hand side:

{\displaystyle {\begin{aligned}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{aligned}}}

71.41.210.146 (talk) 17:06, 28 December 2013 (UTC)

Symmetry

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 ${\displaystyle \theta =0,\theta ={\pi \over 4}}$ and ${\displaystyle \theta ={\pi \over 2}}$ respectively? M.A.Redman (talk) 19:14, 10 April 2014 (UTC)

useful identity

It would be handy to give a solution to the geodesic eqn ${\displaystyle a\sin x+b\cos x=c}$ 67.198.37.16 (talk) 20:01, 30 September 2015 (UTC)

If the article gets too long, how shall we reorganize or split it?

By WP:SIZESPLIT and Wikipedia:Article size pages above 60 kb size should be split List of trigonometric identities is about 71 kb so is in need of a split.

Also the list is ab bit hard to follow so lets have a discussion about how we reorganize this page and maybe if and how how we should split it.

My idea is to split it in two, one with let us call it the high school trigonometry and a second one with more advanced subjects.

Advanced subjects would for me include:

and many more lets discuss before we edit that makes nobody happy. WillemienH (talk) 12:44, 25 February 2016 (UTC)

I think the article is on a coherent single topic and so should not be split up. I don't think it's hard to navigate, since there's a table of contents. With two separate articles, someone might have a hard time finding what they're looking for, if they go to the wrong article. Loraof (talk) 20:31, 5 June 2016 (UTC)
I agree. I don't think splitting up is a good idea. The cited guidelines with the 60kb are more than a crude rule of thumb than an absolute figure. We have plenty of compilation and overview articles that are larger than 60kb and where splitting them up is simply not a good option. In fact many of our excellent articles are > 70kb.--Kmhkmh (talk) 20:53, 5 June 2016 (UTC)
I don't know if splitting is needed but holy crap is this page difficult to read with the current format and organization. — Preceding unsigned comment added by Bofum (talkcontribs) 04:44, 1 October 2018 (UTC)

More than two sinusoids

As in the "Arbitrary phase shift" and "Sine and cosine" cases the expressions for a and θ should be unambiguous. To the best of my knowledge that is

${\displaystyle a={\sqrt {\sum _{i,j}a_{i}a_{j}\cos(\theta _{i}-\theta _{j})}}}$

and

${\displaystyle \theta =\operatorname {atan2} \left(\sum _{i}a_{i}\sin \theta _{i},\sum _{i}a_{i}\cos \theta _{i}\right).}$

— Preceding unsigned comment added by Tpreclik (talkcontribs) 03:39, 3 January 2017 (UTC)

Pythagorean Identity

When I learned the following identities, they were all called "Pythagorean identities" with consistency from teacher to teacher:

${\displaystyle \sin ^{2}x+\cos ^{2}x=1}$
${\displaystyle \tan ^{2}x+1=sec^{2}x}$
${\displaystyle 1+\cot ^{2}x=\csc ^{2}x}$

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)

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)

atan2

I recommend removing all mention of atan2 from the article, as atan2 is a computer programming language function—not a standard trigonometric function.—Anita5192 (talk) 16:35, 10 November 2019 (UTC)

I support that. There are only two sections where atan2 is mentioned:

So neither of these uses of atan2 are actually supported by the sources, at least that I can verify. The first one can be removed without any fuss. The second one can be rewritten in terms of standard arctan based on the Mathworld article. -Apocheir (talk) 23:50, 10 November 2019 (UTC)

Sure, atan2 originates from programming languages, but it is nevertheless a perfectly valid mathematical function. It would be useful for the linear combination section, because

${\displaystyle a\cos x+b\sin x=c\cos(x+\varphi )}$

can be made to work for all a and b by using

${\displaystyle c={\sqrt {a^{2}+b^{2}}},}$
${\displaystyle \varphi =\operatorname {atan2} (-b,a)=\operatorname {arg} (a-bi),}$

with the interpretation that

${\displaystyle 0\cdot {\textrm {undefined}}=0.}$

The way it is now, using atan, leaves an unnecessary singularity at a = 0, and the equations do not hold there. The phase range is also only 180 degrees instead of 360, and amplitudes can take on negative values, which is a bit strange. If it's the programming language origin of atan2 that's the main issue, then you can equivalently use the arg function as above.

Perhaps one of the most common places for this linear combination to occur is in the Fourier series, where one converts

${\displaystyle a_{n}\cos \left({\tfrac {2\pi }{P}}nx\right)+b_{n}\sin \left({\tfrac {2\pi }{P}}nx\right)}$

into

${\displaystyle A_{n}\cdot \cos \left({\tfrac {2\pi }{P}}nx-\varphi _{n}\right).}$

That article defines

${\displaystyle A_{n}\triangleq {\sqrt {a_{n}^{2}+b_{n}^{2}}}}$ and ${\displaystyle \varphi _{n}\triangleq \operatorname {arctan2} (b_{n},a_{n}),}$

unabashedly using the atan2 function. In this prototypical application it would be strange to use a 180 degree phase range and negative amplitudes, which is part of the reason why such a parametrization feels weird to me. More generally, a full circle is the most natural range for angles, and amplitudes are most natural if they're non-negative.

I don't unfortunately have suitable sources at hand, so I'm leaving the article as-is, but if someone finds such then I suggest using arctan2 in the article. You can also use arg, but there's no need to introduce a complex function into a real context, no matter how suited they are for expressing things related to the unit circle. -- StackMoreLayers (talk) 02:13, 12 April 2021 (UTC)

Significant identities

There are a great many existing trigonometric identities. However, this article cannot contain all of them and has space only for the more significant identities. Please do not insert identities that are not in common use and especially do not include a huge list of them.—Anita5192 (talk) 17:49, 17 November 2019 (UTC)

I wholeheartedly agree and would go even further. There are many identities on this page that are not sourced or sourced to unreliable sources. The page seems to be a magnet for people who want to see "their" identities in print and needs to be pruned from time to time to weed these things out. I've also noticed some basic identities (that should be in the list) without citations, so a straightforward weeding would have to be done with care and is probably a job for more than one editor.--Bill Cherowitzo (talk) 23:08, 17 November 2019 (UTC)