Talk:Law of cotangents

A number of problems

I see significant problems with this article. A factor making it difficult to know what to do about it is the fact that the theorem is extremely obscure. The Law of Cotangents is not referenced at all in Wikiversity, Wikibooks, Wikisource, Proofwiki, or MathWorld. Google and Bing searches turn up only a few pages, of which only one has a proof:

• pediaview
• formulasearchengine
• formulasheet
• ameriwiki (disclaimer: written by me)

There is also a book that gives the theorem: The Universal Encyclopedia of Mathematics page 530. (German edition apparently by Meyers Rechenduden, 1960; the English-language version has a forward by James Newman in 1963.) My guess is that this book may be the original source of the idea that there is a "law of cotangents".

I don't know whether I would be allowed to put in a proof without violating the various policies like WP:RS, WP:N, or WP:OR, or the general disdain for proofs given in WP:NOTTEXTBOOK. The only external page giving a proof of this, found by Google or Bing, is the 4^th item above, small open wiki, that would violate WP:LINKSTOAVOID point #12.

That said, here's the problem:

The article seems to confuse the hypotheses of a theorem with its conclusions. It says that

.... the law of cotangents states that if

r = sqrt((s-a)(s-b)(s-c)/s) (the radius of the inscribed circle of the triangle) and

s = (a+b+c)/2 (the semiperimeter of the triangle)

then blah blah blah .....

The second of these apparent hypotheses is fine, though saying something like "letting s = (a+b+c)/2" would be better.

But the parenthetical note on the first is troublesome. Is it saying

1. if r (the name we will give to the inradius) = sqrt(...) then ... ? or
2. if r = sqrt(...), and furthermore if that is the inradius, then ... ? or
3. if r is taken to be the inradius, and of course it is trivially true that the inradius is sqrt(...), then ... ?

If it's saying if the inradius is sqrt(...), that's all wrong. It is a fact, though a highly nontrivial one, that the inradius is always the given square root. You can't sensibly make that one of the theorem's hypotheses. Also, you can't assume that the reader accepts that from the beginning; it is quite nontrivial. It is one of the principal conclusions of the theorem.

The article should say something like

Letting s = (a+b+c)/2 and r = the radius of the inscribed circle of the triangle, then

cot(alpha/2) = (s-a)/r cot(beta/2) = (s-b)/r cot(gamma/2) = (s-c)/r

and furthermore

r = sqrt((s-a)(s-b)(s-c)/s)

Stated this way, even if no proof is given, it is at least plausibly stated.

Finally, the article ends with

"In words, the theorem states ..."

That is incredibly awkward. A theorem should be written out in such a lucid way that no such restatement is necessary. Mathematical notation is rich enough that people don't ever need to say "in words, ..." In those rare cases in which an informal statement in words is helpful (example: "the square of the hypotenuse is the sum ...") that statement should be first, and the more formal and precise statement, in symbols, should follow.

I note with wry amusement that the first two of the web pages cited above include that wording. The first was apparently cribbed from the from 3 Mar version here at WP, and the second from the 26 Feb version.

I suppose I could bring the Ameriwiki article over, but that would probably be a blatant violation of WP:OR, WP:RS, or WP:N.

SamHB (talk) 23:07, 20 October 2013 (UTC)

• Was it that hard to fix it by yourself? Fixing the references and linking to the Law of tangents is left as an exercise. Pldx1 (talk) 09:52, 24 October 2013 (UTC)

Rewrite

I see that the most egregious aspects of the page's hideousness have been cleaned up. However, problems remained in

• spelling ("substitued")
• usage ("reminds of", "allows to", "results into")
• irrelevance of quoting the Law of Sines
• nonsensical notion of "masking the irrationality". "Rationalized units" are used in electrical units and Maxwell's equations, for a specific purpose. But rationality of the items appearing in this theorem are irrelevant.

For this and other reasons (most importantly, the lack of an actual proof), I have rewritten this page.

SamHB (talk) 17:40, 30 October 2013 (UTC)

Pythagorean theorem

Currently the last sentence of the article says that the Pythagorean theorem can be derived from the law of cotangents. But all of trigonometry is derived from the Pythagorean theorem, so this is circular: we can prove the Pythagorean theorem by starting with the Pythagorean theorem. So I'm deleting this assertion. Loraof (talk) 13:45, 3 May 2015 (UTC)

New Heron-type area formula

I found a new formula,which is similar to Heron's formula.denoting the sum of the half-angles cotangents as S=cot ⁠A/2⁠+ cot ⁠B/2⁠ + cot ⁠C/2⁠,t=⁠2cot⁠A/2⁠/(cot²⁠A/2⁠+1)(S −cot ⁠A/2⁠)⁠
T=⁠r²/ 2√R⁠√rS(S −cot ⁠A/2⁠)(S −cot ⁠B/2⁠)(S −cot ⁠C/2⁠)= ⁠r²/ 2⁠√St(S −cot ⁠A/2⁠)(S −cot ⁠B/2⁠)(S −cot ⁠C/2⁠)
where r is the radius of incircle.I show proof.
According to this article's statement,s-a=rcot⁠A/2⁠,s=⁠a + b + c/2⁠=s-a+s-b+s-c=rcot⁠A/2⁠+rcot⁠B/2⁠+rcot⁠C/2⁠=rS,a=s-(s-a)=rS-rcot⁠A/2⁠=r(S-cot⁠A/2⁠),sin x=⁠2tan⁠x/2⁠/1+tan²⁠x/2⁠⁠^[1]=⁠2/tan⁠x/2⁠/(1+tan²⁠x/2⁠)/tan²⁠x/2⁠⁠=⁠2cot⁠x/2⁠/cot²⁠x/2⁠+1 ⁠,1=⁠2RsinA/r(cot⁠B/2⁠+cot⁠C/2⁠)⁠,⁠r/R⁠=⁠2sinA/cot⁠B/2⁠+cot⁠C/2⁠⁠=⁠2cot⁠A/2⁠/(cot²⁠A/2⁠+1)(S −cot ⁠A/2⁠)⁠,T=rs=r√⁠sabc/4Rr⁠=⁠1/2⁠√⁠r/R⁠sabc=⁠1/2⁠√rStr(S −cot ⁠A/2⁠)r(S −cot ⁠B/2⁠)r(S −cot ⁠C/2⁠)[Special:Contributions/240D:1E:309:5F00:4990:3AE4:1986:E35B]] (talk) 12:53, 16 October 2021 (UTC)(Ankert)

• Don't break chronology. Pldx1 (talk) 19:02, 18 October 2021 (UTC)