Timeline for coordinate free foundations of trigonometry [closed]
Current License: CC BY-SA 4.0
29 events
| when toggle format | what | by | license | comment | |
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| Apr 20, 2020 at 3:04 | review | Reopen votes | |||
| Apr 20, 2020 at 13:29 | |||||
| Apr 15, 2020 at 20:18 | comment | added | Kugutsu-o | What do you mean I'm not asking about how to tech these concepts? | |
| Apr 15, 2020 at 20:16 | comment | added | Kugutsu-o | this is exactly what the question is about, new approach to circ functions and pi and lim nsin(x/n). As long as it's not usi g coordinates or ortho normal basis | |
| Apr 15, 2020 at 20:12 | comment | added | Kugutsu-o | Yea I wish to avoid adopting a basis or anything like that. | |
| Apr 14, 2020 at 22:47 | comment | added | Yemon Choi | @KonstantinosKanakoglou BTW, if the OP is somehow hoping to avoid a choice of orthonormal basis (a.k.a fixing x-axis and y-axis) then I suspect at some point one will have to say "let u and v be orthonormal vectors" since otherwise we have problems even talking about right-angled triangles, but of course there is no need to fix a particular isomorphism of our Euclidean space E with R^2 at the start, if that is what the OP is seeking to avoid | |
| Apr 14, 2020 at 22:46 | comment | added | Yemon Choi | @KonstantinosKanakoglou Then if one wishes to show that these functions model certain aspects of Euclidean geometry I guess you posit a 2-dimensional real inner product space (abstractly), you set up some axioms for what you believe an "angle" is, and then you show that the functions defined as in my previous concept do indeed have the desired properties when you use the chosen definition of "angle". But this all seems to be what Alexandre Eremenko has already pointed out in his answer. | |
| Apr 14, 2020 at 22:40 | comment | added | Yemon Choi | @KonstantinosKanakoglou I don't think those tags are relevant; the OP is not asking about how to teach these concepts. FWIW I was taught as an undergraduate an approach that I believe is quite standard, defining the power series of exp, defining cos and sin in terms of exp, showing that cos has at least one positive real root and defining pi/2 to be the smallest such, deriving the familiar addition formulas and calculus formulas for cos and sin, etc etc | |
| Apr 14, 2020 at 22:39 | history | edited | Yemon Choi |
It is not clear that the OP asks this question for pedagogical reasons, there are plenty of rigorous treatments of trig functions via power series in books on real analysis
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| Apr 14, 2020 at 19:51 | history | edited | Konstantinos Kanakoglou |
edited tags
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| Apr 14, 2020 at 19:50 | comment | added | Konstantinos Kanakoglou | I have added two more tags, which i think are relevant. | |
| Apr 14, 2020 at 19:49 | comment | added | Konstantinos Kanakoglou | It is interesting to mention, that the method described in the preceding comments is far from being the only one: A variant of the above method (based on functional eqs and taking continuity as an assumption beforehand) can be found at Proposition 1 of the following answer: math.stackexchange.com/a/834572/195021. (a proof is included by the author, following Robison's paper). | |
| Apr 14, 2020 at 19:40 | comment | added | Konstantinos Kanakoglou | And since the high school functions can be shown (through geometric arguments) that they also satisfy the preceding conditions these are also identified with the functional and the series definitions. | |
| Apr 14, 2020 at 19:39 | comment | added | Konstantinos Kanakoglou | Thus, (due to the uniqueness of the $S$, $C$ functions, shown in the paper, for any particular value of $p$) it is directly shown that the series definitions are the same with the sine and the cosine functions defined through the functional relation. | |
| Apr 14, 2020 at 19:38 | comment | added | Konstantinos Kanakoglou | Furthermore, the above presentation provides an easy way of identifying the above definition of the trig functions with the power series definitions: Starting from the series defs we can show that conditions (a), (b), (c), laid above, are satisfied. (This is a little technical, it uses the binomial theorem and the Cauchy product; see for example: math.stackexchange.com/a/626793/195021). | |
| Apr 14, 2020 at 19:25 | comment | added | Konstantinos Kanakoglou | Then, the usual sine function is defined through: $\sin(x) = S(\frac{2x}{\pi})$ and the same for the cosine. (this is equivalent to saying that the usual sin, cos are the unique solutions of the above functional condition for $p=\frac{\pi}{2}$ but the way the author works through this, enables him to define $\pi$ from skratch). | |
| Apr 14, 2020 at 19:25 | comment | added | Konstantinos Kanakoglou | Actually the author lays the foundations carefully: in the same paper, it is shown that $S(x)$ and $C(x)$ defined by the three conditions above are continuous functions.Then, he defines $\pi$ as follows: Taking the (unique) solutions $S(x)$, $C(x)$ resulting from the above conditions for $p=1$, he shows that $\lim_{x\to 0}\frac{S(x)}{x}$ exists (for $p=1$) he proposes the definition: $\pi = 2\lim_{x\to 0} \frac{S(x)}{x}$. | |
| Apr 14, 2020 at 19:12 | comment | added | Konstantinos Kanakoglou | This result is shown in detai in: G. B. Robison, "A New Approach to Circular Functions, π and lim(sinx)/x", Math. Mag. 41 (1968), 66–70. | |
| Apr 14, 2020 at 19:11 | comment | added | Konstantinos Kanakoglou | There are nice characterizations/definitions of the trig functions in terms of functional relations; these do not make any direct reference to any kind of coordinates. I believe the most well known is the following one: Let $p$ be a real number and $C(x)$, $S(x)$ be real valued functions of a real variable satisfying the following conditions: (a). $C(x-y)=C(x)C(y)+S(x)S(y)$ for all real $x,y$, (b). $S(p)=1$ and (c). $S(x)\geq 0$ for all $x\in [0,p]$. Then $S$ and $C$ are uniquely determined. | |
| Apr 12, 2020 at 9:08 | comment | added | Kugutsu-o | It doest matter what I consider coordinate free(thaugh I explained it clearly in all my posts those lines if for some reason that is of interest here.. I staded it perfectly unambiguously here too look at the description:"without in any way introducing coordinates",... | |
| Apr 12, 2020 at 4:55 | comment | added | Yemon Choi | @KonstantinosKanakoglou I did not vote to close, but I found the wording of the question unclear, and I couldn't get a good idea of what would satisfy the OP as being "co-ordinate free" -- see mathoverflow.net/questions/352362/… and math.stackexchange.com/questions/3484627/… for context | |
| Apr 11, 2020 at 21:10 | comment | added | Konstantinos Kanakoglou | So i have voted to reopen. | |
| Apr 11, 2020 at 21:10 | comment | added | Konstantinos Kanakoglou | I am not really sure why this question has been closed. It might have been more clearly formulated but it is still interesting and -imo- there are still more and interesting things to be said. Alexandre Eremenko's answer is certainly valuable but i think there are various ways to define trigonometric functions in "axiomatic" or "coordinate-free" ways, coming from different areas (from rep theory to differential equations and from complex analysis to the study of functional equations). | |
| Apr 11, 2020 at 20:10 | review | Reopen votes | |||
| Apr 12, 2020 at 4:55 | |||||
| Apr 11, 2020 at 17:44 | history | closed |
YCor user44191 Bugs Bunny ARG Alex M. |
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| Apr 10, 2020 at 0:00 | review | Close votes | |||
| Apr 11, 2020 at 17:44 | |||||
| Apr 9, 2020 at 12:08 | answer | added | Alexandre Eremenko | timeline score: 7 | |
| Apr 9, 2020 at 8:29 | comment | added | Kugutsu-o | I'm looking something akin to a synthetic approach, but augmented with formulas that can be used without adopting coordinate systems | |
| Apr 9, 2020 at 8:22 | comment | added | Carlo Beenakker | see en.wikipedia.org/wiki/Synthetic_geometry ; isn't that what you are looking for? | |
| Apr 9, 2020 at 8:18 | history | asked | Kugutsu-o | CC BY-SA 4.0 |