# Talk:Law of tangents

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Field:  Geometry

Can someone improve this page, it is too short and confusing

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

## Expansion

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.

## error in example

${\displaystyle {-\tan {\alpha +60 \over 2}}={5}{\tan {\alpha -60 \over 2}}.}$
Take the inverse tangent of both sides:
${\displaystyle {-\left({\alpha +60 \over 2}\right)}={{5}\left({\alpha -60 \over 2}\right)}.}$

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

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

Somehow this got changed to

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

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

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

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 (talkcontribs)

• The exemple was: Find ${\displaystyle \gamma }$, given that Triangle ABC has sides b = 3 and a = 2, as well as angle ${\displaystyle \beta }$ = 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 ${\displaystyle \gamma =\pi /3}$". 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
${\displaystyle ax^{2}+bx+c=0{\text{ and }}a\neq 0\,}$
then
${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac\,{}}}}{2a}}.}$
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.
${\displaystyle \tan \left({\frac {\alpha +\beta }{2}}\right)={\frac {\sin \alpha +\sin \beta }{\cos \alpha +\cos \beta }}}$
and
${\displaystyle \tan \left({\frac {\alpha -\beta }{2}}\right)={\frac {\sin \alpha -\sin \beta }{\cos \alpha +\cos \beta }}.}$
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:
{\displaystyle {\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)