Talk:Law of tangents

Expansion

Can someone improve this page, it is too short and confusing
— Preceding unsigned comment added by 218.101.75.206 (talk) 21:20, 26 May 2006 (UTC)

Certainly this can be expanded. Maybe tomorrow.... Michael Hardy 21:54, 27 May 2006 (UTC)

--shenron 00:38, 15 January 2007 (UTC) More changes, I added the Example Section and changed a couple of minor things in the introduction. I added a little bit about why it's usefull, but there still needs to be information about how it was discovered

--shenron 23:01, 14 January 2007 (UTC) I added a proof to the page today from http://planetmath.org/encyclopedia/ProofOfTangentsLaw.html

I think it would be nice if someone would expand the page by explaining something the law is used for. Perhaps why it was formulated to begin with. If neither of these apply, perhaps indicating this would help.
— Preceding unsigned comment added by Guardian of Light (talk • contribs) 15:57, 31 May 2006 (UTC)

error in example

This article says:

${-\tan {\alpha +60 \over 2}}={5}{\tan {\alpha -60 \over 2}}.$

Take the inverse tangent of both sides:

${-\left({\alpha +60 \over 2}\right)}={{5}\left({\alpha -60 \over 2}\right)}.$

This is a gross error. Taking the inverse tangent of both sides would yield

${-\left({\alpha +60 \over 2}\right)}=\arctan \left(5\tan \left({\alpha -60 \over 2}\right)\right).$

Somehow this got changed to

${-\left({\alpha +60 \over 2}\right)}=5\arctan \left(\tan \left({\alpha -60 \over 2}\right)\right),$

so that "arctan" and "tan" cancel to give

${-\left({\alpha +60 \over 2}\right)}={{5}\left({\alpha -60 \over 2}\right)}.$

That certainly is not correct. Michael Hardy 02:14, 30 January 2007 (UTC)

... and now I've deleted that section. Michael Hardy 02:16, 30 January 2007 (UTC)

• This can be prooved without all of these unnecessary equations. Check out this website, you'll be able to figure out in your own ,mind how to think about these things without having to memorize the unnecessry equations, and then be able to figure out where the equations came from. —Preceding unsigned comment added by 70.192.58.16 (talk • contribs)

• The exemple was: Find $\gamma$, given that Triangle ABC has sides b = 3 and a = 2, as well as angle $\beta$ = 60 degrees. This exercise belongs to the Law of sines (one side and the opposite angle are known). When trying to use the Law of tangents, one obtains a transcendant equation and nothing useful can be done with it. The main error was in the question, that should have been "[...] and $\gamma =\pi /3$". Pldx1 (talk) 11:49, 24 October 2013 (UTC)

Explanation of Symbol

I don't think too many people are familiar with the mathematical operation (what is it called?) that has what looks like an underlined + symbol. (see last equation called "alternative"). This article needs an explanation of that symbol is and how you use it in calculations. JettaMann (talk) 17:24, 13 November 2008 (UTC)

That is just the plus-or-minus symbol that you see in the formula for solving quadratic equations: if

$ax^{2}+bx+c=0{\text{ and }}a\neq 0\,$

then

$x={\frac {-b\pm {\sqrt {b^{2}-4ac\,{}}}}{2a}}.$

In this case, the meaning is that if the first one is "+" then so is the second, and if the first is "−" then so is the second. I.e.

$\tan \left({\frac {\alpha +\beta }{2}}\right)={\frac {\sin \alpha +\sin \beta }{\cos \alpha +\cos \beta }}$

and

$\tan \left({\frac {\alpha -\beta }{2}}\right)={\frac {\sin \alpha -\sin \beta }{\cos \alpha +\cos \beta }}.$

I think very large numbers of people have seen this symbol because solving quadratic equations is taught to virtually everyone including those whose only reason for taking a math course is that they need it to graduate from high school. Michael Hardy (talk) 18:05, 13 November 2008 (UTC)

I should add that the places where this symbols is most frequently seen in trigonometric identities are the following:

${\begin{aligned}\sin(\alpha \pm \beta )&=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\\cos(\alpha \pm \beta )&=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta \end{aligned}}$

The second line indicates that when the first operation is "+" then the second is "−" and vice-versa. Michael Hardy (talk) 18:20, 13 November 2008 (UTC)

Black square

Why does the proof end with a black square? --Ye Olde Luke (talk) 21:56, 15 November 2008 (UTC)

Somewhat conventional, if far from universal. Michael Hardy (talk) 00:00, 16 November 2008 (UTC)

Coordinate geometry?

Why is "coordinate geometry" an appropriate category for this article? I don't see coordinates in it anywhere. Michael Hardy (talk) 04:27, 17 November 2008 (UTC)

Perhaps because it has to do with triangles... on a co-ordinate plane? —Preceding unsigned comment added by 71.163.34.82 (talk) 22:56, 9 December 2008 (UTC)

There's nothing about a coordinate plane in the article. The article is about triangles in a Euclidean plane. One can put a coordinate system on the plane, but there's no need for that in this article and it's not there. Michael Hardy (talk) 23:21, 9 December 2008 (UTC)

Alright, i guess I see the distinction. Maybe you're right. —Preceding unsigned comment added by 71.163.34.82 (talk) 01:40, 10 December 2008 (UTC)

Law of Tangents

Please note, The "Law of Tangents" is plural, not singular. Per the article's triangular drawing the Law of Tangents consist of 6 ratio identity formulas. So why is it, we only see one formula in the article? The article is VERY incomplete.

Sincerely, Mike Brady, mbrady94107@yahoo.com —Preceding unsigned comment added by 169.230.110.161 (talk) 23:55, 3 November 2009 (UTC)

As with the Cosine rule, there is only was Law, rotating the triangle, or swapping sides does not change the maths or algebra. Martin451 (talk) 23:59, 3 November 2009 (UTC)

Uses

The article claims that this formula is just as useful as the sine and cosine formulas, but I cannot see any situation in which the tangent formula would not be a much more difficult method. I therefore believe that the tangent formula is just a curiosity, and not actually useful. I would be interested to know if anyone can show that I am wrong without a contrived example. Dbfirs 17:02, 21 March 2010 (UTC)

Sources

I'm going to try to gather some sources here (feel free to pitch in), with an eye toward eventually improving the historical and technical discussion in the article. Just dumping stuff here now in no particular order; will try to clean up later. –jacobolus (t) 18:51, 15 March 2023 (UTC)

• Bosmans, Henri. "Le Traité des Sinus de Michel Coignet". Annales de la Société scientifique de Bruxelles. 25. Introduction, note 27, pp. 115–119.
• Mathews, R. M. "The Law of Cosines Versus the Law of Tangents". School Science and Mathematics 16, no. 1 (1916): 41-45.
• Bradley, H. C., T. Yamanouti, W. V. Lovitt, and R. C. Archibald. "Discussions: Geometric Proofs of the Law of Tangents." American Mathematical Monthly 28, no. 11/12 (1921): 440-443.
• Wu, Rex H. "Proof Without Words: The Law of Tangents." Mathematics Magazine 74, no. 2 (2001): 161-161.
• Laudano, Francesco. "106.40 The law of tangents and the formulae of Mollweide and Newton." The Mathematical Gazette 106, no. 567 (2022): 516-517.
• Van Brummelen, Glen (2021). The Doctrine of Triangles. Princeton University Press.
• Hinckley, A. (1940). "1460. Formulae for the Solution of Triangles". Mathematical Gazette. 24 (260): 204–206. doi:10.2307/3605713. JSTOR 3605713.
• De Morgan, Augustus (1846). "LXII. On the first introduction of the words tangent and secant". London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 28 (188): 382–387. doi:10.1080/14786444608645439.

new "Alternate Form" section seems off topic

@Smichr I don't understand the relevance of the "Alternate Form" section you added in special:diff/1221826895. It doesn't seem very closely relevant to the identity that is the subject of this page. I find it confusing, and I suspect other readers will as well. If the only connection is that these are both trigonometric identities and both involve the tangent function, then it probably shouldn't be here. –jacobolus (t) 07:13, 2 May 2024 (UTC)

A bit, er, tangential, perhaps, and out of sequence (I’d be inclinded to swap it with History) but I guess the relevance is in the last sentence: apparently it can be considered an augmentation of the LoT proper, so it’s not only superficially similar. In articles about formulae I often see discussion of others that are special cases, generalizations, isomorphic in a different context, used in a notable derivation, &c.—Odysseus1479 08:21, 2 May 2024 (UTC)

When you say "can be considered"... according to whom? I think it seems basically entirely unrelated. Compare:

${\frac {a-b}{a+b}}={\frac {\tan {\tfrac {1}{2}}(\alpha -\beta )}{\tan {\tfrac {1}{2}}(\alpha +\beta )}},\qquad \tan(\alpha )={\frac {\sin(\gamma )}{{\frac {b}{a}}-\cos(\gamma )}}$

The first one is a relation between the half-tangents of the sum/difference of two angles and the sum/difference of two corresponding opposite sides (or in the spherical case, the half-tangents of those). The second one involves two sides and the included angle on one side of the equation and one of the other angles on the other side. Other than both being triangle identities including the tangent function, I just can't see the connection.

For something like this to have a section in an article about the "law of tangents", at the least the relevance should be made obvious, but ideally you want multiple reliable published sources directly discussing the relation between these.

There are infinitely many possible triangle identities that we can make up by doing various algebraic twiddling. The point of Wikipedia is not to describe a few randomly generated ones, but to pluck out the ones which are "notable" because they were historically important, are widely used in practical applications, were used in proofs of important other theorems, etc. A criterion like «one pseudonymous Wikipedian liked this identity» is not good enough. –jacobolus (t) 14:33, 2 May 2024 (UTC)

Thanks for your reply. If you would like to revert this (and perhaps un-revert it if the discussion warrants) I am ok with that.

Both forms aim to solve the problem of determining angles from the given angle and sides (while avoiding possible "catastrophic cancellation" when using the law of cosines to compute the missing side and then use that via law of sines to compute the angle). That's what the Application starts out by saying. Each form is trying to solve that problem; the problem is not how to write an expression using sums and differences of angles -- that was the approach taken to solve the problem. So the merits of a representation should be judged on how it solves that problem.

Does either form offer an advantage over the other? I would argue that the smaller tangent law has the advantages of (1) ease of derivation and (2) ease of use to find a missing angle. From a pedagogical perspective the smaller tangent law is immediately understood while the historical law of tangents requires additional interpretation with regards to how one is to supply sums and differences of individuals angles that are not known. I would consider this addition to be similar to showing how to calculate the second root of a quadratic by method other than the quadratic equation -- it is another (and preferable) method. In this case, however, there is not a numerical advantage: there is a conceptual advantage. Smichr (talk) 17:19, 2 May 2024 (UTC)

@Smichr – my concern is that as far as I can tell "smaller tangent law" is your own made-up name, and I have not seen this identity presented in trigonometry books as related to the law of tangents. Including it here gives the impression that it is a historically important tool that was used in practice by people trying to solve triangles. If that's an accurate impression, then fine it should stay somewhere in Wikipedia (where we could argue about). But if this is something you made up, or someone else made up recently, then it doesn't belong. Cf. Wikipedia:Original research.

"Both forms aim to solve the problem of determining angles from the given angle and sides [...] the problem is not how to write an expression using sums and differences of angles" – the law of tangents was used in a variety of ways by a variety of authors. It's overly reductive to claim that it had only one narrow purpose. (But our article here should do a better job of describing its history.)

"showing how to calculate the second root of a quadratic by method other than the [quadratic formula]" – I can find many sources directly discussing how to calculate roots of quadratic equations by various methods, which is why it's fine for Wikipedia to include a discussion of that topic in Quadratic formula § Numerical calculation. –jacobolus (t) 17:25, 2 May 2024 (UTC)

The law of cosines page shows many permutations of the law showing how it can be used to solve for a given variable. Would you be opposed to simply stating that the LoT can be re-written explicitly in terms of the given angles and sides as the equation that I gave. It need not have a different name, just be a simplification of the LoT that is more tractable in use:

>>> solve(lot.subs(B,pi-A-C).expand(trig=1).rewrite(tan),tan(A))

[2*a*tan(C/2)/(a*tan(C/2)**2 - a + b*tan(C/2)**2 + b)]

>>> _[0].rewrite(cos)

2*a*cos(C/2 - pi/2)/((-a + a*cos(C/2 - pi/2)**2/cos(C/2)**2 + b + b*cos(C/2 - pi/2)**2/cos(C/2)**2)*cos(C/2))

>> expand(_,trig=1).simplify()

a*sin(C)/(-a*cos(C) + b) <-- what I presented, given as a re-ordered result Smichr (talk) 18:15, 2 May 2024 (UTC)

I'm opposed unless you can find "reliable sources" supporting that this other identity is called something like the "law of tangents" or has been presented alongside it. Have you ever seen this identity in a book / published paper? Who came up with it? What's its history?

Law of cosines § Use in solving triangles is an entirely different situation: those are all trivial algebraic manipulations of the "law of cosines" ⁠$c^{2}=a^{2}+b^{2}-2ab\cos \gamma$⁠, just showing how you can isolate one or another variable to write it in terms of the others. Your "smaller tangent law", by contrast, seems entirely unrelated to the "law of tangents". –jacobolus (t) 18:33, 2 May 2024 (UTC)

Perhaps you did not recognize the SymPy code as demonstration that what I wrote is proof that the smaller law is just a simplified version of the law of tangents written explicitly in terms of a angles instead of being written in terms of sums and differences. Presentation of this alternate form could be given in the Application section. There is nothing new needed to arrive at this identity. Smichr (talk) 18:51, 2 May 2024 (UTC)

"My symbolic algebra package came up with it" is not a "reliable source" by Wikipedia standards. –jacobolus (t) 19:00, 2 May 2024 (UTC)

The interesting thing is that, like routes up Everest which end at the peak (some more difficult than others), getting to the simple result one way is more difficult than getting there another. I showed in a previous edit the *few* steps that are needed to arrive at the compact and memorable (because of symmetry) expression `tan(A)=sin(C)/(b/a-cos(C)`. The peak attained (so to speak) is an expression that can be used to calculate a missing angle given two sides and an angle. I suspect it can be no more simple that this. What I showed (for your sake) with the CAS is that basic trig identities can take you from the traditional form to this alternate form. But the route there is much longer than simply starting from the law of sines and deriving as I showed. Since you are moderating this trigonometry related page, you may be familiar with Fu's trigonometric simplification approach. The CAS that I am using has those primitives defined and, step by step, you can see the circuitous route from the traditional law of tangents to the form I gave:

>>> eq

(a - b)/(a + b) - tan(A/2 - B/2)/tan(A/2 + B/2)

>>> _.subs(B,pi-A-C) # use C as gamma

(a - b)/(a + b) + cot(A + C/2)/cot(C/2)

>>> TR2(_)

(a - b)/(a + b) + sin(C/2)*cos(A + C/2)/(sin(A + C/2)*cos(C/2))

>>> TR10(_)

(a - b)/(a + b) + (-sin(A)*sin(C/2) + cos(A)*cos(C/2))*sin(C/2)/((sin(A)*cos(C/2) + sin(C/2)*cos(A))*cos(C/2))

>>> isolate(sin(A),_) # so sin(A) = the following

-2*a*sin(C/2)*cos(A)*cos(C/2)/(-a*sin(C/2)**2 + a*cos(C/2)**2 - b*sin(C/2)**2 - b*cos(C/2)**2)

>>> _/cos(A) # so lhs is sin(A)/cos(A) = tan(A)

-2*a*sin(C/2)*cos(C/2)/(-a*sin(C/2)**2 + a*cos(C/2)**2 - b*sin(C/2)**2 - b*cos(C/2)**2)

>>> TR5(_)

-2*a*sin(C/2)*cos(C/2)/(-a*(1 - cos(C/2)**2) + a*cos(C/2)**2 - b*(1 - cos(C/2)**2) - b*cos(C/2)**2)

>>> TR7(_)

-2*a*sin(C/2)*cos(C/2)/(-a*(1/2 - cos(C)/2) + a*(cos(C)/2 + 1/2) - b*(1/2 - cos(C)/2) - b*(cos(C)/2 + 1/2))

>>> TR8(_)

-a*sin(C)/(a*cos(C) - b)

>>> numer(_)/-a/expand(denom(_)/-a) # divide top & bottom by -a

sin(C)/(-cos(C) + b/a)

So we see that alternate form can be derived by additional step from the historical law of tangents. But the same can be derived from the same starting point as which led to the traditional form (the law of sines) with the application of two rules: sin(pi-x)=sin(x) and the rule for converting sine of a sum to a sum of products):

>>> sin(A)/a-sin(B)/b # same starting point as for traditional law of tangents

-sin(B)/b + sin(A)/a

>>> _.subs(B,pi-A-C)

-sin(A + C)/b + sin(A)/a

>>> TR10(_)

-(sin(A)*cos(C) + sin(C)*cos(A))/b + sin(A)/a

>>> isolate(sin(A),_)

a*sin(C)*cos(A)/(-a*cos(C) + b)

>>> _/cos(A)

a*sin(C)/(-a*cos(C) + b)

>>> numer(_)/-a/expand(denom(_)/-a) # divide top & bottom by -a

sin(C)/(-cos(C) + b/a)

I think it is worth showing the simple form as an alternate route to the goal for which law of tangents was derived. While it is true that many forms could be derived, the simplicity of derivation and economy of presentation makes the alternate form a particularly nice resting place. Whether it is more beautiful when taking an additional step and inverting the alternate form is another matter:

cot(A) = (b/a)*sin(C) - cot(C)

But it seems worth mentioning an alternate form. Already, the presentation shows simplification of the rhs by replacing tan((A+B)/2) = cot(C/2). But if the pupose of wikipedia is to only show what is shown elsewhere, then I agree that this alternate (and equivalent) form should not be shown on this page. If published, it might be worth mentioning.

/c Smichr (talk) 15:51, 3 May 2024 (UTC)

if the pupose of wikipedia is to only show what is shown elsewhere – yes this is correct. It is fundamental Wikipedia policy that "[Wikipedia's] content is determined by previously published information rather than editors' beliefs, opinions, experiences, or previously unpublished ideas or information." See Wikipedia:Verifiability, Wikipedia:Reliable sources, and Wikipedia:No original research. Novel trigonometric identities can be published in e.g. a blog post, arXiv preprint, or (if you are feeling ambitious) journal article. They shouldn't be published here, because it is misleading for our readers. –jacobolus (t) 16:19, 3 May 2024 (UTC)

I appreciate the time that you took to discuss this. I do not often make edits, but will keep this in mind. Smichr (talk) 16:24, 3 May 2024 (UTC)

correction: cot(A)=(b/a)*csc(C) - cot(C) Smichr (talk) 16:21, 3 May 2024 (UTC)

By the way, it is entirely fine to mention that ⁠$\textstyle \tan {\tfrac {1}{2}}(\alpha +\beta )=\cot {\tfrac {1}{2}}\gamma$⁠ somewhere if it seems helpful (though also not necessary). –jacobolus (t) 16:25, 3 May 2024 (UTC)