This exchange is full of LOLz. It demonstrates the quintessential difference in how mathematicians and physicists view the world.
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.
Link in the OP no longer works. Anyone have a replacement?
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 says:
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.
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.
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.
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.
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.
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.
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.)
Nitpick: The Euler identity is that e[sup]iπ[/sup] + 1 = 0. The Euler formula is the one about sines and cosines.
They’re both about sines and cosines - one is just a special case of the other. 
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).
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.
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).