Sunday 4 July 2010

A Pythagorean Generalization

Looking through the recent, and excellent, 67th Carnival of MathematicsI came across a link to a theorem at "Cut the Knot" that I had never known.

It begins with the simple fact that for a triangle ABC, Cos2(A) + Cos2(B) + Cos2(C) = 1 IFF (if and only if) the triangle is a right triangle..that is, one of Cos(A), Cos(B) or Cos(C) = 0.
But the part I loved is the more general extension... that for ANY triangle, Cos2(A) + Cos2(B) + Cos2(C)+2 Cos(A)Cos(B)Cos(C) = 1.

Since only one of the angles can be obtuse (and hence the quantity 2 Cos(A)Cos(B)Cos(C) would be negative only in the obtuse case), we can use Cos2(A) + Cos2(B) + Cos2(C) as a determinant for triangles. When the sum is > 1 the triangle is obtuse. If it is equal to one, the triangle is a right triangle; and if the sum is< 1, the triangle is acute. Not sure how I got so old without knowing that.

Can anyone tell me who/when this general identity was first discovered?


Dave L. Renfro said...

I don't know the origin of this identity, but it can be found in most older advanced trigonometry texts. For example, see Sections 114-115 (pp. 75-76) in

Isaac Todhunter, "Plane Trigonometry",
8th edition, 1880.

Note that besides the exercises that begin on p. 77, there are at least a couple of other chapters in this book that contain a lot of triangle identities (e.g. Chapters 13 & 16, as well as the 300 supplementary problems on pp. 290-324).

I knew to look in this book because I happen to own a hard copy of this book, as well as Hobson and Durell/Robson below.

Durell/Robson's "Advanced Trigonometry"

Hobson's "A Treatise on Plane and Advanced Trigonometry"

Dave L. Renfro said...

Before I leave for the day I thought I'd give you another great advanced trigonometry reference that also happens to have the triangle cosine identity.

See pp. 40-41 Serret's "Traite De Trigonometrie" (1850).

In a later edition (1900) at least part of this becomes an exercise for the reader.

See pp. 48-53 (especially VI on p. 53) of Serret's "Traite De Trigonometrie"

Dave L. Renfro said...

I looked through some of my things at home early this morning ...

In Durell/Robson's "Advanced Trigonometry", see Chapter XIV, "Conditional Equalities" on pp. 265-268. The cosine formula is #1 in Exercise XIVb on p. 266.

In Hobson's "A Treatise on Plane and Advanced Trigonometry", see "Examples" following Section 45, specifically (6) on p. 44. See also "Examples" following Section 54, specifically (6) on p. 57. Note that (6) on p. 57 asks the reader to prove the cosine identity by geometrical methods.

Also, the cosine identity appears in:

Charles R. Diminnie, "Solution to Problem 990", Pi Mu Epsilon Journal 11 #4 (Spring 2001), p. 224.

Specifically, the problem posed (by R. S. Luthar) states: "Identify all triangles ABC such that (cos A)^2 + (cos B)^2 + (cos C)^2 = 1."

Diminnie's solution involves establishing the cosine identity by algebraic methods (I'll leave the details as an exercise for your interested blog readers), and then observing that the required identity holds if and only if (cos A)(cos B)(cos C) = 0, or if and only if triangle ABC is a right triangle.

Other references that may be of interest are:

In the following excerpt from Gelin's 1888 trigonometry book, see p. 2 (= p. 330 in the google page numbering), 3rd formula in [4].

E. Gelin, "Questions diverses de trigonometrie", Mathesis Recueil Mathematique 8, Supplement 4 (1888), 15 pages.

I tried very hard to get a copy of Gelin's trig. book about 3 years ago with no luck anywhere, and I haven't tried searching for it in the past year or more, but this morning I found several copies digitized by google. By the way, there is a 1906 2nd edition to Gelin's book, but I couldn't find a digital copy of it.

Gelin's "Elements de trigonometrie plane et spherique" (1888)

As for the history of the cosine identity, I suspect this is entangled so deeply in mathematical literature that it could be debatable how to assign credit, since it's a fairly trivial (relatively speaking) consequence of the addition angle formula and some algebraic manipulations. For instance, mathematically the identity is not very far from the expansion of cos(A + B + C), and I suspect it's all but impossible to assign credit for for the expansion of cos(A + B + C).

Dave L. Renfro said...

For some reason I seem to be the only person commenting on this post.

Anyway, I had a few minutes free a short while ago and glanced through Gelin's book (freely available on the internet):

Gelin's "Elements de trigonometrie plane et spherique" (1888)

Regarding the triangle cosine identity, see Article 74 on p. 91.

For the identities also published in the Mathesis journal article I cited previously, see Article 96 on pp. 103-106.

Finally, connoisseurs of exotic exact radical expressions for certain trigonometric values will find themselves salivating over the tables on pp. 59-62.

Pat's Blog said...

Thanks for all the links... I'm still trying to look through all of them...
You are probably right, there is not going to be a clear first time for something this general.

Thanks again


Dave L. Renfro said...

In case you (or anyone else) is interested, I just came across another "nontrivial type of reference" for this identity. The identity appears as Problem #2, proposed in (Two-Year) College Mathematics Journal 4 #1 (Winter 1973), p. 73 and the solution appears in (T-Y) CMJ 4 #4 (Fall 1973), p. 83.