Can the trigonometric functions be expressed, explained, or proven in terms of arithmetic?

I'm trying to wrap my head around sine, cosine, and tangent. I'm aware that they're commonly defined in high schools as ratios of the various parts of triangles set in the unit circle, but that's not particularly intuitive.

Can the trigonometric functions be expressed, explained, or proven in terms of something more intuitive, say, arithmetic? Is there a way to work out sine, cosine, and tangent on paper, without the aid of a calculator or computer?

How were they first proven or discovered? What proof might have been worked through to reach the trigonometry we have today?

Edit:

I discussed this question with a friend today. I think ultimately, what I'm looking for, is a way to work out the sine, cosine, or tangent of a given angle using paper and pencil.

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The trig functions were certainly first discovered through geometry, which many people consider to be at least as intuitive as arithmetic. – Gerry Myerson Aug 23 '12 at 3:55
You may be interested in the following Wikipedia article. Originally, a circle of fixed radius was chosen ($60$, $10000000$, more exotic choices) and people were interested in the length of the chord, given the length of the arc. So originally sine was a length, not a ratio. – André Nicolas Aug 23 '12 at 3:59
There are formulas that approximate trigonometric functions with arbitrary given precision, that have been used and perfected since the ancient times. Those include infinite summations, infinite products, and such. It sounds like you might call them arithmetic approach to trigonometric. – timur Aug 23 '12 at 4:22
$x-\frac{x^3}{6}$ is a good (and reasonably quick to compute) approximation then. Just convert all angles to their acute equivalents and go crazy. In general this has to do with what's known as the Taylor Series of $\sin(x)$, but you won't get to that till calculus. – Robert Mastragostino Aug 23 '12 at 4:30
Another easy way to approximate the sine for small arguments: $$\sin\,x\approx x\frac{60-7x^2}{60+3x^2}$$ For cosine, $$\cos\,x\approx \frac{72}{x^2+12}-5$$ For tangent: $$\tan\,x\approx x\frac{15-x^2}{15-6 x^2}$$ From that point, you can use the double-angle identities repeatedly for evaluating sines and cosines and tangents of larger angles. – J. M. Aug 23 '12 at 5:22

Here is a very precise version of your first question:

Can a trigonometric function such as $\sin x$ be defined in terms of a finite sequence of arithmetic operations (say addition, subtraction, multiplication, division, and root extraction) on $x$ (say together with a finite number of constants)?

The answer is no. Before I get to the proof, let me make a philosophical comment. The conception of $\sin x$ as a function that takes as input a real number and returns another real number is modern. In antiquity, I would imagine that the input to $\sin$ was not a real number, it was an angle, and an angle was a geometric object, not a numerical one: you draw two lines and an angle is some property of how they intersect. Trigonometric ratios describe certain relationships between angles and lengths. I am not sure what your background is, but if you want to understand trigonometry you should pick up a good book on trigonometry.

Anyway, the proof. We will show a stronger statement, namely that $\sin x$ is not an algebraic function of $x$. Suppose to the contrary that there exist nonzero polynomials $c_0(x), ... c_n(x)$, say with complex coefficients, such that

$$c_n(x) (\sin x)^n + c_{n-1}(x) (\sin x)^{n-1} + ... + c_0(x) = 0$$

for all real $x$. The LHS is a holomorphic function of $x$, so by the identity theorem the above identity must actually hold for all complex $x$. But by Euler's formula we have

$$\sin x = \frac{e^{ix} - e^{-ix}}{2i}$$

and letting $t = ix$ be real, we see that as $t \to \infty$ the first term of the above sum grows in absolute value like a polynomial times $e^{tn}$ while the remaining terms grows in absolute value like a polynomial times $e^{t(n-1)}$; in particular, the first term eventually becomes much larger than the remaining terms in absolute value, so they cannot sum to zero, which is a contradiction. Similar arguments work for the other trigonometric functions.

As for your second question, the short answer is that it depends on what kind of angles you're interested in. For angles which (in radians) are rational multiples of $\pi$ you can get pretty far by repeatedly applying the angle addition formulas and the half-angle formula and so forth, although the general problem of computing trigonometric functions exactly (in terms of algebraic numbers) is somewhat delicate and best understood in the context of Galois theory and Kummer theory.

For angles which are not a rational multiple of $\pi$, the resulting numbers are in general transcendental, and it's very unclear to me whether there's a meaningful sense in which one can compute such a thing exactly. Instead, one computes decimal approximations, for example using Taylor series.

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