4
$\begingroup$

I'm new to this stack community, please bear with me as I try to explain my question properly. Recently I came across with these trigonometric identities (where $ \omega + \phi + \psi = 180^\circ $):

$$\tan \omega + \tan \phi + \tan \psi = \tan \omega \tan \phi \tan \psi$$

and,

$$\sin (2\psi ) + \sin (2 \phi) + \sin (2 \omega) = 4 \sin \omega \sin \phi \sin \psi$$

and a user who answered the first one mentioned that such identity belonged to the 18th century scientist Antonio Cagnoli. I tried to look for sources regarding this but I only could find the wiki entry and this other entry in World cat. The OP of the answer had referenced his proof by this book and it is consistent with what it appears on World cat under:

Trigonométrie rectiligne et sphérique by Antonio Cagnoli( Book )
18 editions published between 1808 and 2017 in French 
and Undetermined and held by 117 WorldCat member libraries worldwide

Although it does mention in the book about the identity with the tangent function it doesn't specifically say about the one with the double angle sine function, which is the one that has triggered my attention the most.

Therefore my question arises that other than this source (the book mentioned) are any others out there which can reassure or tell if these two belong to Cagnoli?.

Initially I wanted to post this question in the Mathstack site but I felt this place would be more proper. I'm looking forward for your answers.

$\endgroup$
  • $\begingroup$ Many such identities don’t seem especially attributed to Cagnoli... e.g. (1884, nº 6, 7) or (1885, nº 7, 12). $\endgroup$ – Francois Ziegler Jul 15 '18 at 4:21
  • $\begingroup$ If you can read German (I can't), you might be able to find what you want in Vorlesungen über Geschichte der Trigonometrie by Anton Braunmühl (1903). $\endgroup$ – Dave L Renfro Jul 15 '18 at 16:10
  • $\begingroup$ @FrancoisZiegler Do you mean that he had popularized a method which was well known before him a-la Tartagia, Simpson or Newton and that eventually everyone just named that way because of him?. I don't understand very well what you imply. The documents you referenced aren't accessible from my computer hence I cannot read them, perhaps can you upload a screenshot of the reference what you meant with posting them?. $\endgroup$ – Chris Steinbeck Bell Jul 16 '18 at 8:11
  • $\begingroup$ @DaveLRenfro It just happens that I studied german long time ago but forgot most of it, therefore I'm just relying on google translate and it looks to be a book about the history of trigonometry. Can you perhaps help me to point a page on where to look exactly?. I'm still in doubt regarding the sines function identity. Maybe is there any other book on english?. Thanks in advance. $\endgroup$ – Chris Steinbeck Bell Jul 16 '18 at 8:14
  • $\begingroup$ I can't read German at all, but possibly @Francois Ziegler can help, based on this answer. I don't have time to look around anytime soon because of some work [= day job] issues, but if I did, I would look through the various books you can find cited in my comments here. $\endgroup$ – Dave L Renfro Jul 16 '18 at 10:34
3
$\begingroup$

Your first identity is indeed attributed to Cagnoli by e.g. Franchini (1805), Encyclopédie du dix-neuvième siècle (1847), or Le Cointe (1858, p. 59). It is in the second edition of his Trigonometria (1804, Chap. IV, nº 173) but apparently not in the first (1786) where Chap. IV ends at nº 128. Woodhouse (1819) writes:

$$\tan.(A+B+C)=\frac{t+t'+t''-tt't''}{1-(tt'+tt''+t't'')}.$$ If $A+B+C=\pi$, (which is the case when $A$, $B$, $C$, are the three angles of a triangle), since $\tan.\pi=0$, \begin{align} t+t'+t''&-tt't''=0, \text{ or}\\ t+t'+t''&=tt't'', \end{align} which is the theorem given in the Phil. Trans. 1808, p. 122.

But the theorem has an origin much more remote; for, the above formula for $\tan.(A+B+C)$ and similar formulas for the tangents of $A+B+C+D$, &c. were given as far back as the year 1722, by John Bernoulli, and are inserted in the Leipsic Acts for that year, p. 361, and in the second volume of his works, at p. 526 [see Lemma II].

Your second identity appears among many similar ones in e.g. Crelle (1826), Grunert (1836), Serret (1850; 1862), Colenso (1851), Müller (1852), Le Cointe (loc. cit., p. 58), Christensen (1884), Mathesis (1885), Gelin (1888, pp. 89, 104-106), etc. None of them attribute it to Cagnoli.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.