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One learns trigonometry in high school/secondary school and either forgets it if one continues onto a career less mathematical or, possibly, uses it extensively in their work, as do engineers and physicists.

As a field of study in mathematics however, it seems that trigonometry is mostly "solved", at least it seems so for the familiar trigonometry in $\mathbb{R}^2$. Is this true, or are there still interesting questions that deal with trigonometry or, perhaps, generalizations of it?

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  • $\begingroup$ it seems that long ago trigonometry is at precalculus level, long enough to became folklore $\endgroup$
    – janmarqz
    Feb 28, 2014 at 2:40
  • $\begingroup$ Unfortunately, there are no unsolved topics or new topics to discover. $\endgroup$
    – NasuSama
    Feb 28, 2014 at 2:40
  • $\begingroup$ @NasuSama You sure about that? $\endgroup$
    – Jack M
    May 17, 2015 at 11:54
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    $\begingroup$ I'd say the issue is that none has found a way to generalize it, it is inherenlty bound to $R^2$ which limit it severely, if one could generalize it somehow it would probably pick up speed again. $\endgroup$ May 17, 2015 at 11:58
  • $\begingroup$ To older numerous loci ( circum-center,in-center, 9 point circle and center, Fermat point and lines of Euler, Soddy etc.) new one are being continuously added. $\endgroup$
    – Narasimham
    Sep 19, 2015 at 18:41

2 Answers 2

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Trigonometry on its own is for the most part no longer a very active area of research; though trigonometry remains a key tools in applied research today (for instance studying the rigidity of origami patterns). And the trig functions as eigenfunctions of the Laplacian have countless generalizations and applications.

One case I've seen of modern fundamental research on trigonometry is Wildberger's work on rational trigonometry, which seeks to reformulate trigonometry as purely algebraic relations of positions (see his web site). But NB that while as far as I can tell this work is sound, it is at the fringes of mainstream mathematical research.

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In Āryabhaṭa's sine table the jya values or 'modern values' still have to be fully computed. So the Āryabhaṭa's computational method is still being researched.

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    $\begingroup$ What do you mean by "modern values"? Furthermore, the fact that a problem is unsolved doesn't mean it's being actively researched. $\endgroup$
    – Jack M
    May 17, 2015 at 11:59

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