Timeline for Taking a theorem as a definition and proving the original definition as a theorem
Current License: CC BY-SA 4.0
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| Sep 28, 2021 at 1:29 | comment | added | KConrad | @Stef textbooks on real analysis often define trigonometric functions by their power series, so indeed it is reasonable from the perspective of mathematicians to have the textbook approaches by elementary geometry and by analysis treated on an equal footing. There are many textbooks on analysis, after all. | |
| Sep 27, 2021 at 10:33 | comment | added | Robin Saunders | For me, the most natural way to conceptualize trigonometric functions is in terms of the linear relationships between them, rotation maps, and the differential equations they satisfy. This approach can be introduced intuitively before making it more rigorous, and is particularly natural if exponentials are taken as part of the background. The relationship with circles, complex versions, power series etc. then follow directly as soon as the relevant concepts are present. What's more, exponentials and rotations lead naturally to Euler's formula and Lie theory. | |
| Sep 27, 2021 at 9:42 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| Sep 27, 2021 at 9:25 | comment | added | Stef | "In some textbooks"?! Trigonometric functions are often introduced in middle school, at least 5 years before power series. Defining trig functions as their power series would require either teaching power series much earlier, or trigonometric functions much later. Several identities involving trig functions appear to have been known in Ancient Greece already; on the other hand, power series were not introduced before the 17th century, as far as I can tell. So, it's probably a little more than "some textbooks" which define trig functions without power series. | |
| Sep 27, 2021 at 7:51 | history | answered | user21820 | CC BY-SA 4.0 |