Talk:Law of sines/Archive 1

Triangulation
Axel, I want to write about triangualation, but I'm afraid I'll get the math wrong. In fact, I'm afraid anything I do relating to Wikipedia will go wrong under your scrutiny. Should I be bold or just wait for you to write it? --Ed Poor


 * Go right ahead, but you may want to spell it right :-) AxelBoldt 19:39 Nov 16, 2002 (UTC)

Derivation proof
The derivation proof is really incomplete because it doesn't cover what happens if angle A or B is obtuse (over 90 degrees). In that case, no perpendicular line can be drawn from line AB to point C, without extending line AB beyond A or beyond B. I'd fix it myself but I don't have experience making computer images. Art LaPella 16:23, Aug 31, 2004 (UTC)


 * I got a picture up, I hope the diagram is correct/enough. I can't remember the proof now, so if anyone wants to do it first they can... I'll probably do it in a while if I recall. Fiveless
 * http://en.wikipedia.org/wiki/Image:Sinelaw_obtuse.png

Needs Some Work
One of the main reasons that I use wikipedia for information on formulas over math world is because I do not normally get bombarded by theory and the inner workings of these formulas but more how to use them and what they can do for me. I think that it would be much better if some examples were given, like how to use it for working out a side or an angle if you have the other bits of information. —Preceding unsigned comment added by 84.92.106.157 (talk)


 * I agree. The Triangle solution(s) and Triangle equality(-ies) pages need to be made. 75.35.201.48 11:32, 15 March 2007 (UTC)

chocolateluvr88
I made radians a link, so they can easily view what a radian is if they don't know already. That is trivial to solve these problems. Also, I will try adding some examples.

--chocolateluvr88 11:08, 5 January 2006 (UTC)
 * but the law of sines is true regardless of the units used, no? Matt 12:51, 31 January 2006 (UTC)
 * oh, except for the 2R part, which I didn't see before, never mind. Matt 12:53, 31 January 2006 (UTC)
 * No proof is given for the =2R part, and the example at the bottom uses degrees for the example. Although I don't doubt that the =2R part is true, are you sure that it's part of the law of sines? my calculus text doesn't mention it. Matt 13:00, 31 January 2006 (UTC)
 * Proof is now given for the =2R part. Suck on that. (my friend did it) 195.168.237.121 17:26, 28 May 2006 (UTC)

There are 19 possible combinations of sides and anles. I would love to see the equations for each that work for acute and obtuse triangles.

Radius vs Diameter
Why not change 2R to just D? Seems like it would be more in character for wikipedia. If the 2R term is convenient for the proof of the law of sines, or some other reason then it doesn't really matter. Please let me know though. Thanks

Guardian of Light 16:02, 31 May 2006 (UTC)


 * Please see proof. D would probably make it more unclear. I have never seen it replaced with D. The reason is probably the fakt that D is not as important in triangles as the radius of the inscribed triangle.. 195.168.243.36 12:27, 6 June 2006 (UTC)

cool
oh wow, i didnt know that a/sina = b/sinb = c/sinc is ALSO equal 2r. Thats really useful. Thanks.

Erm...
I am a 6th grader and for some strange reason need to know this... *psst* I think my teachers are dillsuional *cough* so I would really love it if you could put this formula in a more 6th grader like way... Thanks! —Preceding unsigned comment added by 74.128.184.134 (talk) 01:31, 29 January 2008 (UTC)

Cyclic permutation part
Could somebody please explain about the cyclic permutation part and how it is applied? It isin't very clear...

Anyway, there is also another proof for the 2R, and it's much easier to understand. I found it on google. Link: http://planetmath.org/encyclopedia/SinesLawProof.html

—Preceding unsigned comment added by 218.186.12.11 (talk) 18:01, 15 August 2008 (UTC)

History ?
It would be nice to have a history section, identifying when the law was first stated.

Is it in Euclid? Jheald (talk) 10:30, 14 September 2008 (UTC)

Applying cyclic permutation:
This is the final step in this article's proof section. Can anybody link me to an explanation of this step or help me out? Maybe the step needs to be a little more explicit. Thanks 98.199.206.122 (talk) 02:22, 20 September 2008 (UTC)


 * I've commented out the whole section. It said one should make an "axis" through b.  I couldn't figure out what that meant.  The illustration should a line through the vertex B but there was no indication of how one could know WHICH line.  Later it is implied that the line goes through the center of the circumscribed circle, but somehow one was to find that center by using that line, so it's as if one doesn't know where the center is at the time when one draws that line.  So how does one know where to draw that line?  No clue.  One could have drawn perpendicular bisectors of TWO sides rather than one, and they would have intersected at the center of the circumscribed circle, and then maybe the rest of the argument works.  But that's not what it said and not what the illustration showed. Michael Hardy (talk) 15:23, 20 September 2008 (UTC)


 * OK, I've found the edit where the proof was added and at that time it made sense, but was expressed in imperfect English. I'll see if I can restore it to what it should have been. Michael Hardy (talk) 15:30, 20 September 2008 (UTC)

Mansur citation problem
The article cites Jacques Sesiano, "Islamic mathematics", p. 157, in for the proposition The spherical law of sines was discovered in the 10th century by the Persian mathematician, Abu Nasr Mansur.  Unfortunately that book does not mention Abu Nasr Mansur at all. On page 157 it does say: ... theorem 111.1) and gave rise through Abū׳l-Wafā׳ al-Būzjānī (940—997/8) to the law of sines for spherical triangles. I am afraid that we must delete the reference as inapplicable.  Does anyone have a replacement citation to a reliable source that actually verifies Abu Nasr Mansur as the discoverer? --Bejnar (talk) 16:58, 15 April 2010 (UTC)

[In the following, please forgive my uneasy and likely-inconsistent transliterations of Arabic names.] The Biographical Dictionary of Mathematics (New York: Scribner's, 1991) entry on "Abu Nasr Mansur ibn 'ali ibn 'Iraq" contains the following on p.1656: "[Abu Nasr] is one of the three authors (the others being Abu'l Wafa' and Abu Mahmud al-Khujandi) to whom al-Tusi attributed the discovery of the sine law...." (The continuation of the quote covers both the spherical and plane cases.) Continuing in the article the author (Julio Samso) argues for the likely primacy of Abu Nasr, though he admits the impossibility of definitively resolving the question. I'm not an experienced editor, so am hesitant to edit the main article. Please let me know if you think this would be a better citation than exists in the article. Nitsua60 (talk) 01:46, 6 May 2013 (UTC)

Merge proposal for Hinge theorem
This is a very short article and the theorem is an obvious corollary of the Law of sines, which can be used to prove it. Since it appears to be very poorly-sourced in its own right, I'm proposing that article should be merged into this. Rodhull andemu  20:11, 27 May 2010 (UTC)
 * How do you propose to derive the hinge theorem from the law of sines? I see an obvious way to derive it from the law of cosines: The two sides remain constant while the angle between them increases, and the cosine of the angle decreases.  In the law of cosines, the one term in which the angle appears has a factors equal to the lengths of the sides that don't change, and has a minus sign in front of it, so the whole thing increases as the angle increases.  But now think of the law of sines: the angle between the two constant sides increases, so the sine of the angle goes up to 1 and after the angle passes 90&deg;, then the sine decreases from 1 down to 0 while the angle continues to increase.  But in the mean time, the other two angles are also changing, and along with them their sines, which appear in the other terms in the law of sines.  It doesn't seem immediate.  Hasty comments&mdash;I'll look at it again later. Michael Hardy (talk) 23:37, 28 May 2010 (UTC)
 * Well, I am not a mathematician any more, and took my lead from here. If the Hinge theorem is more closely related to the Law of cosines, fine. Merge it there instead. Rodhull  andemu  23:42, 28 May 2010 (UTC)


 * I disagree. These two articles are only loosely related, and merging them would only cause confusion. SquallBL (talk) 20:16, 27 May 2010 (UTC)
 * really? How loosely? Rodhull  andemu  20:19, 27 May 2010 (UTC)


 * I can see how that hinge theorem is very closely related, but I was talking about the other one, which can be found at spp15.emagc.com. This hinge theorem is not at all related. The only suitable solution would be to create a new and separate page for this hinge theorem.
 * Sorry, hinge theorem does not appear in the search box on that website, so a more precise reference would be welcome. In any event, it looks like some schoolkid's spamsite, so we wouldn't regard it as a reliable source. Rodhull  andemu  20:29, 27 May 2010 (UTC)


 * If you click on the "FMA" tab, then click on the image gallery in the bottom left corner, it is there. I have several other people who can vouch for this (as I said in the discussion page for Hinge Theorem). SquallBL (talk) 20:37, 27 May 2010 (UTC)


 * As stated on Talk:Hinge theorem, there is nothing to indicate that this site is anything more than yet another blog. Favonian (talk) 20:42, 27 May 2010 (UTC)


 * A very reliable source is vouching for the site at this moment. SquallBL (talk) 20:47, 27 May 2010 (UTC)

Comment. The hinge theorem remains true in a nonpositively curved space form, and an appropriate version is true for sufficiently small geodesic triangles in any nonpositively curved Riemannian manifold. So there is certainly room for expansion in the existing article having nothing to do with the law of sines. Sławomir Biały (talk) 01:18, 28 May 2010 (UTC)
 * I won't argue with that, but I get the impression (without generalisation) that we are talking about plane geometry here. It's a little confusing because there are two competing "hinge theorems"; one related to congruence of triangles, mentioned in passing on some educational websites, and the other a novel theorem related to linear transformations in the plane which seem on the face of it to ignore all previous mathematics, including Euclid. This is simply not rigorous enough to be acceptable here, and that's ignoring that it appears to be original research. There may be an isomorphism between the two, but even that would have to be rigorously argued. Rodhull  andemu  01:33, 28 May 2010 (UTC)
 * There is certainly no original research in noting that the theorem is true as stated in a nonpositively curved space form (which includes the case of plane geometry). See, for instance, Chavel "Riemannian geometry".   Sławomir Biały  (talk) 01:45, 28 May 2010 (UTC)
 * Confusion exists. There are two "hinge theorems"; the first, referred to in some online sources, is a corollary of the Law of Sines, which can be used as a proof thereof, generalised to some arbitrary angle. The second is a novel and somewhat trite proposition about linear transformations in the plane, and is set out [here], in the left hand column, with neither argument nor proof. The latter is original research, pure and simple. The former is arguably worthy of inclusion in this article, as far as I can see, even if generalised to N dimensions. Rodhull  andemu  02:38, 28 May 2010 (UTC)

Saying it's an immediate corollary of the law of sines does not justify a merger. It is possible that the "hinge theorem" can be proved in some very simple axiomatic systems without anything anywhere near as sophisticated as a sine function; indeed, possibly in some models in which the law of sines does not hold. Michael Hardy (talk) 04:35, 28 May 2010 (UTC)


 * And indeed, as I pointed out above, this last possibility in fact prevails. Sławomir Biały  (talk) 09:28, 28 May 2010 (UTC)


 * It's obvious that something needs to be done with it. If Law of sines isn't a geed fit then find a better on and merge it there or expand it and add refs. It's AfD bait as it stands now.--RDBury (talk) 18:50, 28 May 2010 (UTC)


 * We don't delete articles simply because they need improvement or are incomplete stubs. (And yes, we do have lots of stubs&mdash;what's the rush?)  At any rate, the AfD would not be successful, and I would sternly remind the hypothetical nominator about WP:BEFORE.  There are thousands of hits for this theorem on google, google books, and google scholar.  Deletion is totally inappropriate.  Sławomir Biały  (talk) 19:07, 28 May 2010 (UTC)

Why is law of sines, rather than law of cosines, the target of the proposed merger? This has to do with two sides and the angle between them, i.e. "SAS". That is normally handled using the law of cosines. Michael Hardy (talk) 19:43, 28 May 2010 (UTC)


 * I agree that the law of cosines would be a better page for a merger. The hinge theorem and the law of sines are only loosely related. The law of cosines is more closely related. Squall B L 21:55, 28 May 2010 (UTC)  —Preceding unsigned comment added by SquallBL (talk • contribs)

One of the problems with merging this into law of cosines is that the hinge theorem holds in other spaces than Euclidean spaces where the law of cosines might not be valid. Someone pointed out negatively curved manifolds. But I'm wondering: what if we look at things like models of subsets of Hilbert's axioms for Euclidean geometry. For which of those subsets is the hinge theorem true? Is it possible that looking at that question sheds some light on foundations of geometry? In that way, some non-trivial content may be found in what would seem like a trivial proposition in a more elementary context. Michael Hardy (talk) 03:35, 29 May 2010 (UTC)

Simply because of all this debating, maybe the best thing to do would be to leave the article on its own. Squall BL (talk) 13:25, 29 May 2010 (UTC)  —Preceding unsigned comment added by SquallBL (talk • contribs)


 * Is it alright if we leave these articles alone? If it is, then I will go ahead and take down the merger tag on the hinge theorem article. Squall BL  (talk) 22:55, 29 May 2010 (UTC)


 * Is this proposal over? Can I take the tag on hinge theorem down? Squall BL  (talk) 01:53, 31 May 2010 (UTC)  —Preceding unsigned comment added by SquallBL (talk • contribs)

wrong page
The term "hinge" is common among Riemannian geometers in the context of Toponogov's theorem (some call it Alexandrov's theorem). Generalisations of the law of cosines to spaces other than R^n should be discussed there. Tkuvho (talk) 11:18, 30 May 2010 (UTC)

questions about a proof
Someone wrote a proposed proof that the usual identity for the sine of a sum follows from the law of sines:
 * To prove this, make an arbitrary triangle with sides a, b, and c with corresponding arbitrary angles A, B and C. Draw a perpendicular to c from angle C. This will split the angle into two different angles, α and β, that are less than 90 degrees.  Let α be on the left and β be on the right.
 * To prove this, make an arbitrary triangle with sides a, b, and c with corresponding arbitrary angles A, B and C. Draw a perpendicular to c from angle C. This will split the angle into two different angles, α and β, that are less than 90 degrees.  Let α be on the left and β be on the right.

The part about left and right makes no sense, since nothing before that had been specified as being on the left or on the right. To people's diagrams could be mirror images of each other until they get to this line about α being on the left, and then they both put α on the left, and then one has angle α adjacent to side a and the other has α adjacent to b.
 * Apply the sine law to side c and side a.
 * Apply the sine law to side c and side a.

This is unclear at best. What does it mean? Here's a guess: it means write the identity
 * $$ {\sin A \over a} = {\sin C \over c}. $$
 * $$ {\sin A \over a} = {\sin C \over c}. $$

Just a guess.
 * Solve this equation for the sine of the multiple angles.
 * Solve this equation for the sine of the multiple angles.

Again unclear. What are the "multiple angles"? Does it mean solve for the sines (plural) if the two angles A and C? That would give
 * $$ \sin A = {a\sin C \over c}. $$

and similarly for sin C, but we'd only be solving for each in terms of the other.
 * Notice that the perpendicular makes two right angles triangles, also note that sin(A) = cos(α),
 * Notice that the perpendicular makes two right angles triangles, also note that sin(A) = cos(α),

But sin(A) = cos(α) only if α is on the same side that angle A is on. If that is what was meant by "Let α be on the left, then it works only if A is on the left, although that was never stated. Perhaps "Let α be on the left" really meant "Let α be on the same side that A is on."  We shouldn't be made to guess about these things.
 * sin(B) = cos(β) and that c = a sin(β) + b sin(α). After making these substitutions you should have sin(α + β) = sin(β)cos(α) + (b/a)sin(α)cos(α).  Now apply the sine law to sides b and a and make the substitutions noted before. Now substitute this expression for (b/a) into the original equation for sin(α + β) and you will have the angle sum identity for α and β in terms of sine.
 * sin(B) = cos(β) and that c = a sin(β) + b sin(α). After making these substitutions you should have sin(α + β) = sin(β)cos(α) + (b/a)sin(α)cos(α).  Now apply the sine law to sides b and a and make the substitutions noted before. Now substitute this expression for (b/a) into the original equation for sin(α + β) and you will have the angle sum identity for α and β in terms of sine.

OK, from this I should be able to reconstruct it. I think these things should have been written clearly in the first place. Michael Hardy (talk) 22:14, 13 August 2010 (UTC)

I apologize for the lack of clarity in my addition to this article. "Multiple angles" was terminology used in this proof to mean angle C which is now equivalent to the sum of α and β. Also by "apply sine law to side c and side a" I did mean the identity you proposed but I meant to solve it in term of the sine of angle C. I did not mean to solve the sine of each angle in terms of each other. Finally your suggestion to replace "left" and "right" with "the same side as "A"" and "the same side as "B"" respectively is also what I intended. Are there any more assumptions that I made that you feel should be addressed before I start modifying the article? User:Mickeyjetson428 (talk) 02:22, 28 August 2010 (UTC)

A alpha, B beta, C gamma
I have been editing the trigonometry articles (Law of cosines, Law of tangents and Law of cotangents) to match each other by adding a standard triangle like the one on the right.

This required all the angles to be labeled $${\alpha}$$, $${\beta}$$ or $${\gamma}$$ as opposed to A, B or C. Should the Law of sines article be changed from

$$ \frac{a}{\sin A} \,=\, \frac{b}{\sin B} \,=\, \frac{c}{\sin C} $$

to

$$ \frac{a}{\sin {\alpha}} \,=\, \frac{b}{\sin {\beta}} \,=\, \frac{c}{\sin {\gamma}} $$

in order to match the other articles? CJ Drop me a line! • Contribs 16:33, 15 October 2011 (UTC)
 * Very late reply. Yes, please, let us do this. A, B, C clearly denote points not angles, the angles are denoted α, β, γ. The current notation is confusing and not really consistent with the image currently used in the lead section:
 * Law_of_sines_in_plane_trigonometry-20201220.svg
 * – Tea2min (talk) 08:54, 11 June 2021 (UTC)

FALSE: ambiguous case?
this says there are two ways to solve the problem.

im making a spreadsheet to do my trigo calculations and found i was getting errors when the angle i was calculating was over 90 degrees! same on a casio math calculatior! i did the same calculation in the spreadsheet and calculator as on the main page - which worked, i got the same answer - until i changed 20 to 70 (or >37.33 so a is the hypotenuse), then error! maybe its supposed to work but todays computers happen to have a limit. i have tried to use it in different ways and always have the same problem (aside from the note below). the problem is in arcsin. this calculation:

Given: side a = 20, side c = 24, and angle C = 40°
 * $$ A = \arcsin\left( \frac{20\sin 40^\circ}{24} \right) \cong 32.39^\circ. $$

quick info from ambiguous case:


 * $$B = \arcsin {b \sin A \over a}\!$$

or
 * $$B= 180^\circ - \arcsin {b \sin A \over a}$$

as a side note, i have used the law of cosines in conjunction with law of sines and it does appear faulty contineously. with a, c and B to find b (by cosines - seems fine) and then C (by sines) and A by inference from B and C. swapping a and c should swap A and C. Charlieb000 (talk) 05:53, 18 April 2012 (UTC)

I would like to thank the author of the site http://www.1728.org/trig4.htm for pointing out that it should be using an IF (both above and the one he uses). he also provided an alternative method to calculating the law of cosines.

law of sines:


 * $$A = \arctan {a \sin C \over b-a\cos C}\!$$

if A is < 0 then add pi in radians (180*)

if A > pi then subtract pi in radians (though its not needed be me)

law of cosines: appears to be pythagoras's theorm.
 * $$c = \sqrt {\left( a \sin C \right)^2 + \left( b-a\cos C \right)^2 }\!$$

Charlieb000 (talk) 23:53, 18 April 2012 (UTC)

Extention to tetrahedra?
The section on the extension of tetrahedra claims to be about the Law of Sines, but there are only angles involved, no edge lenghths or triangle areas. Other than that the presented formula has a bunch of sine functions in it, what makes it correspond to the ordinary two-dimensional case that relates sines to edge lengths? I would remove this section of the article. 192.35.44.24 (talk) 21:08, 17 January 2013 (UTC)

Simpler Criterion for Ambiguous Case?
If you are given A, a, c in ∆ABC, then you can use the Sine Law to calculate at least one solution C_1 for C where C_1 = sin^-1 (c.sin A / a). If C_1 is not 90° degrees and a < c, then there exists another solution C_2 = 180° - C_1.

Danchristensen (talk) 19:55, 28 November 2014 (UTC)

Proposal to remove long proof, please vote
Support: I don't see that the long proof, in three cases, adds any relevant information or intuition to why the Law of sines is true. I move that we eliminate the long proof, retaining only the succinct proof based upon computing the area of a triangle with three different base edges. Please indicate your opinion. 𝕃eegrc (talk) 11:59, 16 June 2016 (UTC)