# Do these (physics and other) mathematics writings make sense? **URL:** \ **Category:** Factual Questions\ **Created:** [November 22, 2012, 12:40am UTC](https://boards.straightdope.com/t/do-these-physics-and-other-mathematics-writings-make-sense/641783 "2012-11-22T00:40:09Z")\ **Posts on this page:** 17\ **Page:** 3
**Author:** ![friedo](https://avatars.discourse-cdn.com/v4/letter/f/8edcca/32.png) [@friedo](https://boards.straightdope.com/u/friedo)\ **Post date:** [May 11, 2017, 9:51pm UTC](https://boards.straightdope.com/t/do-these-physics-and-other-mathematics-writings-make-sense/641783/41 "2017-05-11T21:51:33Z")
> [@Hari\_Seldon](#): > > How do you define sin and cos? By some dubious pictures or by the power series? > [@Chronos](#): > > Well, from my point of view as a physicist, the “dubious pictures” are the one and only definition of sin and cos, and if you want to use the power series, you have to first prove that they’re faithful descriptions of the pictures. This exchange is full of LOLz. It demonstrates the quintessential difference in how mathematicians and physicists view the world. > [@Bertrand Russell](#): > > Mathematics, rightly viewed, possesses not only truth, but supreme beauty. > [@Richard Feynman](#): > > Shut up and calculate. ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [May 11, 2017, 10:00pm UTC](https://boards.straightdope.com/t/do-these-physics-and-other-mathematics-writings-make-sense/641783/42 "2017-05-11T22:00:36Z")
> [@Hari\_Seldon](#): > > It’s the only way I know of. How do you define sin and cos? By some dubious pictures or by the power series? I prefer the latter. Unless you are going to get them from y’'=-y, plus initial conditions, which is legitimate but requires an existence and uniqueness proof, Picard’s little theorem. Proof of convergence of the power series is far the easiest, AFAIK. What??? “Dubious pictures”? Sine and cosine are _defined_ by the lengths of the height and and base of a triangle with unit hypotenuse for the given angle between the hypotenuse and base. That’s it. That is the _definition_. From that definition one can construct innumerable many theorems. Note that the big thing that people ignore in jumping right to the Taylor’s series is that it requires knowing the derivatives of sine and cosine. This is _not_ a trivial thing to do. Not at all. Skipping over much of the hard work to “start” with really misses a lot of the issues of the matter. ---
**Author:** ![Ruken](https://avatars.discourse-cdn.com/v4/letter/r/f475e1/32.png) [@Ruken](https://boards.straightdope.com/u/Ruken)\ **Post date:** [May 11, 2017, 10:05pm UTC](https://boards.straightdope.com/t/do-these-physics-and-other-mathematics-writings-make-sense/641783/43 "2017-05-11T22:05:28Z")
Link in the OP no longer works. Anyone have a replacement? ---
**Author:** ![markn\_1](https://avatars.discourse-cdn.com/v4/letter/m/f9ae1b/32.png) [@markn\_1](https://boards.straightdope.com/u/markn_1)\ **Post date:** [May 11, 2017, 10:12pm UTC](https://boards.straightdope.com/t/do-these-physics-and-other-mathematics-writings-make-sense/641783/44 "2017-05-11T22:12:19Z")
> [@ftg](#): > > Sine and cosine are _defined_ by the lengths of the height and and base of a triangle with unit hypotenuse for the given angle between the hypotenuse and base. > > That’s it. That is the _definition_. That’s A definition. It’s certainly not the only definition, and it’s definitely not the most convenient definition from a mathematical viewpoint, since it restricts the domain to real numbers (actually, the way you’ve expressed it, to real numbers between 0 and pi). As [wikipedia](https://en.wikipedia.org/wiki/Sine) says: > [@](#): > > The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse). > > More generally, the definition of sine (and other trigonometric functions) can be extended to any real value in terms of the length of a certain line segment in a unit circle. **More modern definitions** express the sine as an infinite series or as the solution of certain differential equations, **allowing their extension to arbitrary positive and negative values and even to complex numbers**. ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [May 11, 2017, 11:00pm UTC](https://boards.straightdope.com/t/do-these-physics-and-other-mathematics-writings-make-sense/641783/45 "2017-05-11T23:00:06Z")
> [@Ruken](#): > > Link in the OP no longer works. Anyone have a replacement? Check the cafepress link in post #33. ---
**Author:** ![Ruken](https://avatars.discourse-cdn.com/v4/letter/r/f475e1/32.png) [@Ruken](https://boards.straightdope.com/u/Ruken)\ **Post date:** [May 12, 2017, 1:26am UTC](https://boards.straightdope.com/t/do-these-physics-and-other-mathematics-writings-make-sense/641783/46 "2017-05-12T01:26:04Z")
> [@Exapno\_Mapcase](#): > > Check the cafepress link in post #33. Alas, compressed to illegibility, even when using their supposed zoomed-in view. Perhaps I need to use a real 'puter and not my phone. Or maybe that’s why the OP was asking all those years ago. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [May 12, 2017, 2:22am UTC](https://boards.straightdope.com/t/do-these-physics-and-other-mathematics-writings-make-sense/641783/47 "2017-05-12T02:22:22Z")
But if you define them by their power series, then you have to answer the question of just why we care about those particular power series. I mean, I could define thousands of different functions by specifying power series for them, but nobody would care about any of them. Why should we care about these particular ones? The reason is either because they correspond to the ratios of sides of triangles, or because they’re equal to the negative of their own second derivative. ---
**Author:** ![markn\_1](https://avatars.discourse-cdn.com/v4/letter/m/f9ae1b/32.png) [@markn\_1](https://boards.straightdope.com/u/markn_1)\ **Post date:** [May 12, 2017, 3:33am UTC](https://boards.straightdope.com/t/do-these-physics-and-other-mathematics-writings-make-sense/641783/48 "2017-05-12T03:33:03Z")
There are many reasons why the trig functions are significant mathematically. I’d say the most important is they are the functions that make the Euler formula work: e[sup]ix[/sup] = cos x + i sin x The fact that they also have a nice relation to geometric triangles is secondary, although obviously important in everyday life. ---
**Author:** ![markn\_1](https://avatars.discourse-cdn.com/v4/letter/m/f9ae1b/32.png) [@markn\_1](https://boards.straightdope.com/u/markn_1)\ **Post date:** [May 12, 2017, 3:52am UTC](https://boards.straightdope.com/t/do-these-physics-and-other-mathematics-writings-make-sense/641783/49 "2017-05-12T03:52:20Z")
But anyway, even if you don’t care about the Euler formula, the fact that they correspond to properties of triangles DOES tell you one reason why they’re important. That doesn’t imply that they must be defined in terms of triangles. There’s nothing wrong with defining them as infinite series (where they have a complex domain) and using them in geometry (where only the real domain is used). Similarly, the fact that factorials are important in combinatorics (where only the integer domain is used) doesn’t imply that there’s something wrong with defining the gamma function as an integral, since the gamma function gives the factorial for integers, but is useful in a wider range of other applications where real and complex values are used. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [May 12, 2017, 11:23am UTC](https://boards.straightdope.com/t/do-these-physics-and-other-mathematics-writings-make-sense/641783/50 "2017-05-12T11:23:18Z")
They work in Euler’s identity precisely because they correspond to triangles. Haven’t you ever seen a picture of the unit circle before? Draw a radius to the edge of the circle, drop a vertical from there to the X axis, and then take a horizontal back to the origin: There’s your triangle. ---
**Author:** ![markn\_1](https://avatars.discourse-cdn.com/v4/letter/m/f9ae1b/32.png) [@markn\_1](https://boards.straightdope.com/u/markn_1)\ **Post date:** [May 12, 2017, 12:42pm UTC](https://boards.straightdope.com/t/do-these-physics-and-other-mathematics-writings-make-sense/641783/51 "2017-05-12T12:42:46Z")
Yes of course. But I could equally validly say that they work in triangles precisely because they correspond to Euler’s identity. A mathematician would say that the Euler identity is what is fundamental; their application to geometry is a nice side effect. If you want a practical reason for defining them in terms of infinite series, it gives you a way to **calculate** their values, something that the triangle definition does not provide. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [May 12, 2017, 1:25pm UTC](https://boards.straightdope.com/t/do-these-physics-and-other-mathematics-writings-make-sense/641783/52 "2017-05-12T13:25:02Z")
Eh, I’m with **Chronos**. The history of trigonometric functions goes back centuries, so evidently they have prime importance for reasons having nothing to do with complex numbers or with Euler. It is preposterous to consider them a “side effect”. And calculating their value from the Taylor series is completely _impractical_ (unless your argument is really small). Try working out sin(123) like that, using pencil and paper. (By the way, how do you prefer to introduce the function e^x ? Nothing wrong with using power series, but you are abusing notation until you make rigorous what that has to do with exponentiating a certain real number.) ---
**Author:** ![CurtC](https://avatars.discourse-cdn.com/v4/letter/c/ce73a5/32.png) [@CurtC](https://boards.straightdope.com/u/CurtC)\ **Post date:** [May 12, 2017, 1:46pm UTC](https://boards.straightdope.com/t/do-these-physics-and-other-mathematics-writings-make-sense/641783/53 "2017-05-12T13:46:07Z")
> [@Chronos](#): > > They work in Euler’s identity precisely because they correspond to triangles. Haven’t you ever seen a picture of the unit circle before? Draw a radius to the edge of the circle, drop a vertical from there to the X axis, and then take a horizontal back to the origin: There’s your triangle. Nitpick: The Euler identity is that e[sup]iπ[/sup] + 1 = 0. The Euler formula is the one about sines and cosines. ---
**Author:** ![friedo](https://avatars.discourse-cdn.com/v4/letter/f/8edcca/32.png) [@friedo](https://boards.straightdope.com/u/friedo)\ **Post date:** [May 12, 2017, 2:01pm UTC](https://boards.straightdope.com/t/do-these-physics-and-other-mathematics-writings-make-sense/641783/54 "2017-05-12T14:01:45Z")
They’re both about sines and cosines - one is just a special case of the other. 🙂 ---
**Author:** ![Quercus](https://avatars.discourse-cdn.com/v4/letter/q/7ab992/32.png) [@Quercus](https://boards.straightdope.com/u/Quercus)\ **Post date:** [May 12, 2017, 2:14pm UTC](https://boards.straightdope.com/t/do-these-physics-and-other-mathematics-writings-make-sense/641783/55 "2017-05-12T14:14:00Z")
> [@markn\_1](#): > > Yes of course. But I could equally validly say that they work in triangles precisely because they correspond to Euler’s identity. A mathematician would say that the Euler identity is what is fundamental; their application to geometry is a nice side effect. > > If you want a practical reason for defining them in terms of infinite series, it gives you a way to **calculate** their values, something that the triangle definition does not provide. I’m curious how you view definition of the sin/cos family as the functions that are the negative of their own second derivatives. Is that a separate, third, definition, or trivially implied by the Euler identity (and vice versa)? (I’m a physicist/engineer, so, of course, I’m with **Chronos** : the trig definition is fundamental, while the negative second derivative property is perhaps more useful/important. I can understand that for formal proofs, the Taylor series might be an easier/more useful definition, since as an engineer/physicist I’m all about usefulness, of course). ---
**Author:** ![markn\_1](https://avatars.discourse-cdn.com/v4/letter/m/f9ae1b/32.png) [@markn\_1](https://boards.straightdope.com/u/markn_1)\ **Post date:** [May 12, 2017, 3:15pm UTC](https://boards.straightdope.com/t/do-these-physics-and-other-mathematics-writings-make-sense/641783/56 "2017-05-12T15:15:42Z")
This is really kind of a pointless argument. The triangle property is derivable from the series definition and vice versa, so the two are exactly mathematically equivalent and it’s just a matter of taste which one to use as the definition and which is a derived property. If I were writing a mathematical paper and wanted to define the functions unambiguously from first principles, it seems clear to me that writing a concise expression using simple mathematical operators is going to be clearer than a paragraph about triangles, what a right angle is, what “opposite” and “adjacent” means, etc. But use whichever definition suits your taste. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [May 12, 2017, 7:48pm UTC](https://boards.straightdope.com/t/do-these-physics-and-other-mathematics-writings-make-sense/641783/57 "2017-05-12T19:48:41Z")
> [@](#): > > Quoth **DPRK** : > > (By the way, how do you prefer to introduce the function e^x ? Nothing wrong with using power series, but you are abusing notation until you make rigorous what that has to do with exponentiating a certain real number.) My preferred pedagogical approach is to start by defining exponentiation to an integer power, then to use the properties of exponents to extend the definition to rational powers, then defining irrational powers as the limit of rational powers. Having done that, you then note that any function f(x) = b^x has a derivative that’s proportional to the function itself, and define e as the base for which e^x is equal to its own derivative. Having done that, you can then derive the power series for e^x, and thus have a convenient way to calculate it, and then use properties of exponents to derive the formula for any other power in terms of e^x, and thus have a convenient way to calculate them, too. Once you’ve done all of that, you can define ln(x) to be the inverse of e^x, and then prove that the integral of 1/x is ln(x). If desired, you can also calculate the power series for ln(x), but that one isn’t nearly as useful as for e^x (it converges slower, and only on an inconveniently-small region of convergence). [Previous page](https://boards.straightdope.com/t/do-these-physics-and-other-mathematics-writings-make-sense/641783.md?page=2)