Talk:Law of cosines

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Intended meaning of Euclid's Proposition should be made clear[edit]

The text quoted by Euclid concerning the geometric interpretation of the Law of Cosines is helpful but unclear in the sense that Euclid (or possibly Heath) did not name the thing Euclid meant to "namely" identify. In the quote Euclid states 'namely "that" on which the perpendicular falls', but it is unclear if by 'that' he means namely, 'the rectangle' or namely, 'the side about the obtuse angle' which he is describing. It seems to me either Euclid was not expressing himself clearly here or the translation by Heath is at fault but the end of the Euclid quote does not seem to describe a rectangle very well. It might have been clearer if Euclid had stated something closer to the following (CAPS mark my paraphrasing):

"Proposition 12 In obtuse-angled triangles the square on the side subtending the obtuse angle is greater than the EITHER OF THE squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides about the obtuse angle, namely that RECTANGLE on WHOSE LENGTH the SIDE SUBTENDING the perpendicular falls, and WHOSE WIDTH IS EQUAL TO the straight line SEGMENT cut off outside by the perpendicular towards the obtuse angle."

I am not suggesting that the quote be changed but only that a clarification be given by somebody more in the know on this than myself.

P.S. I only place this talk section above the others to more closely align it with the portion of the main article to which it pertains.

132.45.121.6 (talk) 17:48, 27 August 2012 (UTC)

Proof "Using the distance formula" unclear[edit]

In "Using the distance formula" the article states as a fact that "We can place this triangle on the coordinate system by plotting..." and then proceeds to use some trigonometry in the coordinates, without any explanation for how those coordinates are derived / where they came from.

Just a comment about the readability- I can't actually see clearly the labels on Figure 2. 67.255.12.129 (talk) 20:52, 19 April 2011 (UTC)

Phythagorean Theorem also Proved by Law of Cosines?[edit]

Okay, in the article, it states the since cos 90 is 0 the Law of Cosines is reduced to the Phythagorean Theorem. I don't think that's the case cause if a^2=b^2+c^2-2bccosA, then wouldn't a^2=b^2+c^2? —Preceding unsigned comment added by 70.107.165.59 (talkcontribs)

Yes, and that's the Pythagorean theorem (in the case where the right angles is the angle between the sides of lengths b and c). Michael Hardy 23:20, 1 March 2007 (UTC)

Circular proof?[edit]

In the dot product article, a dot b = a b cos theta is proved using the Law of Cosines. And the Law of Cosines is proved using vector dot products. Shouldn't somebody fix this? --Orborde 07:48, 9 September 2005 (UTC)

The dot product proof should be removed. It's circular. Law of cosines comes first historically and all of vector calculus is assumed to depend on it, not vice versa. Might as well use vectors to prove the pythagorean theorem. Pfalstad 10:22, 9 September 2005 (UTC)
I have hopefully solved this problem, by stating the law of cosines is equivalent to the dot product formula from theory of vectors. --345Kai 10:45, 30 March 2006 (UTC)
Wow that's pretty darn smart. --M1ss1ontomars2k4 | T | C | @ 04:04, 21 May 2006 (UTC)
If you define the dot product (a,b)*(c,d) as ac+bd (so it is linear) and then prove (a,b)*(c,d)=|(a,b)|*|(c,d)|*cos(α) then there is nothing circular. The definition used is a matter of taste, I guess some others are possible. I like the argument with dot product because it gives a different light on the cosine law and I think it deserves a place on this article. Ricardo sandoval 23:20, 12 April 2007 (UTC)
I never saw the story of the dot product but I would guess it came from the scalar product of physics (as in the definition of work done in a particle) is that right? Ricardo sandoval 23:20, 12 April 2007 (UTC)
I guess we could also define (a,b)*(c,d) as |(a,b)|*|(c,d)|*cos(α) prove the linearity by geometrical arguments and use it to prove the cosine law. right? Ricardo sandoval 23:20, 12 April 2007 (UTC)

Section moving[edit]

I removed the following section because this is said in the first paragraph and follows from elementary algebra:

Another use for Law of Cosines[edit]

The Law of Cosines can also be used to find the measure of the three angles in a triangle if you know the length of the three sides. This is how you do it: &nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp

a^2 + b^2 - 2ab\cos C = c^2
          - 2ab\cos C = -a^2 - b^2 + c^2
               \cos C = (-a^2 - b^2 + c^2) / (-2ab)
               \cos C = (a^2 + b^2 - c^2) / (2ab)
                    C = \arccos [(a^2 + b^2 - c^2) / (2ab)]

Now you can find the measure of angle C!


— Sverdrup (talk) 21:00, 25 Feb 2004 (UTC)


If noone objects, I'm going to put the vector-based proof first and move the other one down since the former is more simple and universal as opposed to the latter.. or someone else could do it.. or whatever... - Evil saltine 00:33, 22 May 2005 (UTC)

As noted above, the vector-based proof is circular: the proof that the dot product of two vectors has a geometric interpretation in Euclidian space is itself based on the law of cosines. So using the geometric interpretation of a dot product to prove the law of cosines is a bit problematic... --Delirium 03:17, 13 November 2005 (UTC)
see above --345Kai 10:45, 30 March 2006 (UTC)

We?[edit]

Is the usage of "we" throughout this article proper? Shouldn't "we can easily prove" be "can be proved" (wlong with some sentence rearrangement). BrokenSegue 04:00, 30 March 2006 (UTC)

Rewrite[edit]

I have rewritten the article and expanded it considerably. I have taken a lot of material from the French article, according to the above suggestion. I hope you like it!!! Please improve further... (and sorry that in the history of this page the big change was signed "Euklid". That was me and and I know we shouldn't use these kinds of names on Wikipedia, so I've changed my login. --345Kai 10:45, 30 March 2006 (UTC)

Great work!! The French article was way better by any standards, I fell bad I can't understand it. I only see the general flow of ideas. Now it is all clear I will read in finer detail, Thanks!!! Ricardo sandoval 23:42, 12 April 2007 (UTC)

Faster demonstrations[edit]

The last demonstrations using the power of a point can be simplified, all of them can be worked out with just one application of the power point theorem, and the the first two cases can be made one by considering b<c(always can be done by flipping the sides). Someone willing to make new pictures? Ricardo sandoval 08:02, 16 April 2007 (UTC)

If you need any diagrams drawn, altered, fixed etc. Please post a request at the Graphics Lab. Please read instructions at the top of the page before posting. Thank you! XcepticZP 18:20, 15 November 2007 (UTC)


Alternative proof using Power of a Point Theorem[edit]

This proof along with that using Pythagoras theorem are of considerable historical significance since both have their origins in Euclid's Elements and both are referred to in the ground breaking work of Nicolaus Copernicus: "De Revolutionibus Orbium Coelestium". On Page 20 and Page 21 of Book 1 Copernicus describes two techniques for determining angles given all three sides of a triangle. These techniques correspond respectively to the Pythagorean and Power of Points derivations of the Law of Cosines.

Replica of diagrams in Book 1, Page 21 of "De Revolutionibus Orbium Coelestium"

Fig 1 and Fig 2 are replicas of the diagrams from Page 21 of De Revolutionibus Orbium Coelestium.In Fig 1 triangle ABC is drawn with B an acute angle. With C as centre and BC (=a) as radius semi-circle DBF is constructed with DF a diameter and D a point on AC. The circle meets side AB at E and EC is joined forming isosceles triangle ECB with EB = 2acos(B).Power of Points Theorem (intersecting secants theorem) is applied to point A outside the circle:


\begin{array}{lcl}
\quad\quad AD \times AF = AB \times AE\\
\Rightarrow (b-a)(b+a)=c(c-2a\cos\hat{B})\\
\Rightarrow \quad b^2-a^2=c^2-2ac.\cos\hat{B}\\
\Rightarrow \quad \quad b^2=a^2+c^2-2ac\cos\hat{B} 
\end{array}


It will be no different in the case of Fig 2 where angle B is obtuse:


\begin{array}{lcl}
\quad\quad AD \times AF = AB \times AE\\
\Rightarrow (b-a)(b+a)=c(c+2a\cos(180-\hat{B}))\\
\Rightarrow \quad b^2-a^2=c^2-2ac.\cos\hat{B}\\
\Rightarrow \quad \quad b^2=a^2+c^2-2ac\cos\hat{B} 
\end{array}


Proof of Law of Cosines using Power of a Point Theorem

In addition to the two cases dealt with above, we also need to consider the situation shown in the diagram where Power of Point theorem is applied about point B inside the construction circle.

Triangle ABC is drawn with side AB=c,BC=a and CA=b. With A as centre construct circle DCE with radius b and diameter DE passing through B. CB produced meets the circle at F. Since triangle CAF is isosceles:


CF=2b\cos\hat{C}\quad.


Now apply the Power of a Point Theorem (intersecting chords theorem) to point B inside the circle:


\begin{array}{lcl}
\quad\quad BD \times BE = BC \times BF\\
\Rightarrow (b-c)(b+c)=a(2b\cos\hat{C}-a)\\
\Rightarrow \quad b^2-c^2=2ab.\cos\hat{C}-a^2\\
\Rightarrow \quad \quad c^2=b^2+a^2-2ab\cos\hat{C} 
\end{array} —Preceding unsigned comment added by Neil Parker (talkcontribs) 10:28, 17 December 2007 (UTC)

Distance Proof[edit]

I really like this proof, but it seems to be there are multiple cases. All of which work out to the same correct result.

I will ignore cases when any of the angles are right angles, as then the distance formula simply leads to the Pathagorian Theorem, which is a special case of the law of cosines.

The proof given works when angle at vertex C is acute and angle at vertex B is acute. Below are the other cases:

1) The angle at vertex C is acute, and the angle at vertex B is obtuse. a - b*cos(theta) becomes b*cos(theta) - a. (where as in the original theta is the angle at vertex C). Since the term is squared the result is the same since x**2 = (-x)**2

2) The angle at vertex C is obtuse. in this case the side:

 a - b*cos(theta)  becomes a + b*cos(180-theta) which is

equal to a + b*(-cos(theta)) or a - b*cos(theta). Again leading to the same result.

Does this seem reasonable to anyone? —Preceding unsigned comment added by NeilAFraser (talkcontribs) 01:30, 12 July 2008 (UTC)

Alternate forms[edit]


\begin{align}
c^2 & = a^2 + b^2  - 2ab\cos \theta \\
 & = (a + b)^2  - 4ab\cos^2 \tfrac{\theta}{2}\\
 & = (a - b)^2  + 4ab\sin^2 \tfrac{\theta}{2}\\
\end{align}

The alternate forms were found after reading about rounding errors in spherical triangles, as mentioned in the great-circle distance article.

The arithmetic and geometric mean lengths of the known sides are:

\begin{align}
m_a & = (a+b)/2\\
m_g & = \sqrt{a b}\\
\end{align}

And


\begin{align}
c^2 & = (a + b)^2  - 4ab\cos^2 \tfrac{\theta}{2}\\
 & = 4 m^2_a  - 4 m^2_g \cos^2 \tfrac{\theta}{2}\\
 & = 4(m_a + m_g \cos\tfrac{\theta}{2}) (m_a - m_g \cos\tfrac{\theta}{2}) \\
 & = (2m_a + 2m_g \cos\tfrac{\theta}{2}) (2m_a - 2m_g \cos\tfrac{\theta}{2}) \\
\end{align}

Or


\begin{align}
(\tfrac{c}{2})^2 & =  m^2_a  -  m^2_g \cos^2 \tfrac{\theta}{2}\\
 & = (m_a + m_g \cos\tfrac{\theta}{2}) (m_a - m_g \cos\tfrac{\theta}{2}) \\
\end{align}

- Ac44ck (talk) 23:40, 10 December 2008 (UTC)

Proof using distance formula moved to talk page.[edit]

I moved the following here to the talk page. This proof combines trigonometry and the Pythagorean theorem and quite a bit of algebra of real numbers. It is not nearly as elementary as some of the other proofs presented. I don't see what is gained by including it in the article.--345Kai (talk) 05:40, 8 September 2009 (UTC)

Consider a triangle with sides of length a, b, c, where \theta is the measurement of the angle opposite the side of length c. We can place this triangle on the coordinate system by plotting
A = (b\cos \theta ,\ b\sin \theta ),\ B = (a,0),\ \text{and}\ C = (0,0).
By the distance formula, we have
c = \sqrt{(b\cos \theta - a)^2+(b\sin \theta - 0)^2}.
Now, we just work with that equation:

\begin{align}
c^2 & {} = (b\cos \theta - a)^2+(b\sin \theta)^2 \\
c^2 & {} = b^2 \cos ^2 \theta - 2ab\cos \theta + a^2 + b^2\sin ^2 \theta \\
c^2 & {} = a^2 + b^2 (\sin ^2 \theta + \cos ^2 \theta ) - 2ab\cos \theta \\
c^2 & {} = a^2 + b^2  - 2ab\cos \theta.
\end{align}
An advantage of this proof is that it does not require the consideration of different cases for when the triangle is acute vs. obtuse.
Why banish this one? It sets up one equation and grinds it out. The trig proof that is left requires looking at the triangle in three different ways to come up with three different equations to mash together. It, too, seems to involve "quite a bit of algebra of real numbers."
I don't see what is gained by moving it. - Ac44ck (talk) 01:18, 9 September 2009 (UTC)

Law of Cosines is plural, NOT singular[edit]

11-03-09: Apparently NorweigianBlue and HambergerRadio don't know their Trigonometry. There are three Cosine formulas, NOT one formula.

a^2 = b^2 + c^2 - 2bc\cos(\alpha)\,
b^2 = a^2 + c^2 - 2ac\cos(\beta)\,
c^2 = a^2 + b^2 - 2ab\cos(\gamma)\,

If you Google "Law of Cosines", you will find the above three formulas, which corresponts to the article's triangle figure.

All three are the same formula. Therefore there is just one Cosine Law. Martin451 (talk) 23:27, 3 November 2009 (UTC)

All triangles have three Sines, three Cosines, and three Tangents. The same goes for their inverses too. The article's triangular drawing displays three angles, therefore three Cosine identies exists, as I have shown above ... Mike —Preceding unsigned comment added by 169.230.110.161 (talk) 00:54, 4 November 2009 (UTC)

In the French Wikipedia, the Law (singular) of cosines, is a featured article. The law is stated there only once, not three times as is it is here after 169.230.110.161 changed the article. I suggest that someone reverts the change, unless there is consensus to do otherwise. I have already reverted a couple of times, but don't want to engage in an edit war, so I'm taking the article off my watchlist. --NorwegianBlue talk 10:49, 4 November 2009 (UTC)

All three say the same thing. Michael Hardy (talk) 17:22, 4 November 2009 (UTC)

The identity

a^2 = b^2 + c^2 - 2bc\cos(\alpha)\,

applies if a is any of the three sides, and α the angle opposite it, and b and c the other two sides. Thus it can be applied first with one side labeled a, then with a second side labeled a, then with the third side labeled a, so stating the formula just once covers all three sides of the triangle (and also all other triangles). Michael Hardy (talk) 17:34, 4 November 2009 (UTC)

Adjusted line "CH = a cos(π – γ) = −a cos(γ)" in history[edit]

I changed the line "CH = a cos(π – γ) = −a cos(γ)" in the history subsection to "CH = (CB) cos(π – γ) = −(CB) cos(γ)". In the equations used in this section and the diagram there is no "a" side. Danshil (talk) 21:39, 19 December 2009 (UTC)

Make consistent with "Law of sines"[edit]

The diagram and symbols used for angles on the "Law of cosines" page are not the same as those on the "Law of sines" page. Would it be reasonable to change this page to match those on the "Law of sines" page? — Preceding unsigned comment added by 209.98.145.235 (talk) 14:36, 30 January 2013 (UTC)