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The elementary "opposite over hypotenuse" definition of the sine function defines the sine of an angle, not a real number. As discussed in the article "A Circular Argument" [Fred Richman, The College Mathematics Journal Vol. 24, No. 2 (Mar., 1993), pp. 160-162. Free version here. Thanks to Aaron Meyerowitz's answer to question 72792 for the reference.], angles might be measured either by the area of a sector of unit radius having the angle or by the arc length of such a sector. If the former convention is adopted then it can be proven using a completely unexceptionable Euclidean argument that $\lim_{x\to 0} \sin(x)/x = 1$. Also, whichever convention is adopted (or so it seems to me), using completely unexceptionable Euclidean arguments, it is possible to prove the angle addition formulas for sine and cosine. Using these two ideas, it is straightforward to find the derivatives of sine and cosine, and from there one can derive an algorithm for computing digits of sine and cosine (and for computing $\pi$) using the relatively sophisticated mean-value version of Taylor's theorem.

The equivalence of the two definitions of sine (or of angle measurement) apparently depends on something like Archimedes' postulate: "If two plane curves C and D with the same endpoints are concave in the same direction, and C is included between D and the straight line joining the endpoints, then the length of C is less than the length D." (Again, thanks to Aaron Meyerowitz.) Of course, it is just this postulate that Archimedes needed to prove that the area of a circle is equal to the area of a triangle with base the circumference of the circle and height the radius. And something like it is surely necessary to derive any algorithm for computing digits of $\pi$. (Except, and this confuses me a bit, it seems that if we used the area definition of angle, we could derive an algorithm for computing sine without depending on this postulate, and from there we could get an algorithm for computing digits of $\pi$ since $\sin(\pi)=0$.)

I am looking in general for elucidation of the conceptual connections between the ideas I have so far discussed and of their background. But here are two more specific questions. First, in what sense is a postulate like Archimedes' needed in the foundations of geometry? (I wonder, in particular, if in a purely formal development we might get by without it, but we would somehow be left without assurance that what we had axiomatized was really geometry.) Also, are more intuitive alternatives to Archimedes' postulate? Second, what is really needed to get an algorithm for computing digits of sine? Does it really require such complicated technology as Taylor series? It seems like if one uses the area definition of angle, one might be able to give an algorithm using unexceptionable Euclidean techniques and without so much as invoking the notion of limit.

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I don't understand your claim that something like Archimedes' postulate is necessary to derive an algorithm for computing the digits of $\pi$. I can define $\sin(x)$ as the unique solution of the differential equation $y'' = -y$ satisfying $y(0) = 0, y'(0) = 1$ and derive everything else (angle addition, the geometric interpretation) from this, then define $\pi$ as the smallest positive real number such that $\sin(\pi) = 1$ and show that the area of the unit circle is $\pi$. As far as I can tell none of this requires anything like Archimedes' postulate. –  Qiaochu Yuan Jul 14 '12 at 21:05
    
Sorry, that should say $\sin (\pi) = 0$. –  Qiaochu Yuan Jul 14 '12 at 21:06
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Well, you could also "define" $\pi$ by as the result of some algorithm that outputs its digits. I am supposing that $\pi$ is defined as the ration of the circumference of a circle to its diameter. However, as suggested by my parenthetical comment at the ends of the second paragraph of my question, things might turn out differently if you defined $\pi$ as the area of a circle of unit radius. –  Marian Jul 14 '12 at 21:20
    
By the way, the definition in terms of area seems conceptually more fraught since it presupposes commensurability between areas and linear distances instead of just arc lengths and linear distances. It is only relatively recently in the history of mathematics that people thought it made sense to compare areas to distances. –  Marian Jul 14 '12 at 21:26
    
It presupposes nothing of the sort. All I am claiming is that the circle takes up a proportion $\frac{\pi}{4}$ of the area of the square with vertices $(\pm 1, \pm 1)$. –  Qiaochu Yuan Jul 15 '12 at 0:19
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