The bit in the Ptolemy's theorem section references the Pythagorean theorem, citing it as . I'm no mathematics scholar, but I'm quite sure that isn't the theorem it speaks of. I could be dead wrong, I'm not the most educated fellow, but I think it's worth some speculation. Mason0190 (talk) 13:05, 6 May 2015 (UTC)[reply]
a²+c²=b² is the Pythagorean theorem. What do you mean? Jam ai qe ju shikoni (talk) 13:41, 24 November 2019 (UTC)[reply]
Yeesh! don't be obtuse. Standardization (i.e.: Mathematics)— given a right triangle, the square of the length of the hypothenuse c, opposite to the right angle, equals the sum of the squares of the lengths of the two legs a & b, adjacent to, rendering the right angle.
I don't see how that puts my argument down, however, I do not think that it is reasonable for me to argue about this. Jam ai qe ju shikoni (talk) 21:45, 22 April 2020 (UTC)[reply]
Intended meaning of Euclid's Proposition should be made clear
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.
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)[reply]
Pythagorean Theorem also Proved by Law of Cosines?
Okay, in the article, it states the since cos 90 is 0 the Law of Cosines is reduced to the Pythagorean Theorem. I don't think that's the case cause if , then wouldn't ? —Preceding unsigned comment added by 70.107.165.59 (talk • contribs)
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)[reply]
The last time I checked, the Pythagorean theorem is . I put something about this near the top of the page. Mason0190 (talk) 13:08, 6 May 2015 (UTC)[reply]
It seems that your answer would technically be correct, Michael, but it's a bit unclear because the variables tend to be such that A and B are the legs and C is the hypotenuse. It's not a correctness issue, it's clarity. Mason0190 (talk) 13:10, 6 May 2015 (UTC)[reply]
Circular proof?
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)[reply]
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)[reply]
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)[reply]
"a dot b = a b cos theta" is the definition of a vector dot product. No proof is not required. Jmheneghan (talk) 17:16, 20 May 2016 (UTC)[reply]
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)[reply]
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)[reply]
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)[reply]
Section moving
I removed the following section because this is said in the first paragraph and follows from elementary algebra:
Another use for Law of Cosines
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:
           
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... -
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)[reply]
see above --345Kai 10:45, 30 March 2006 (UTC)[reply]
We?
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)[reply]
Rewrite
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)[reply]
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)[reply]
Faster demonstrations
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)[reply]
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)[reply
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.