What is the Connection Between \(\theta\) and Trigonometric Function Outputs?

[brendan_foo]

I believe that calculators use Taylor expansions to compute sines, cosines and tan's based upon the argument $\theta$ (in radians of course). However, my question is, aside from these expansions, is there some sort of link between $\theta$ and the output of the function itself.

I mean I know that $\cos{\theta} = \frac {adj}{hyp}$ and the other trig ratios, but was this just worked out by hand, pencil and paper and kept in a tabular form before the Taylor expansion was devised? Is there a direct link between $(\frac{adj}{hyp})$ and $\theta$.

Get me?!

 

[arildno]

Trig values were, yes, worked out by hand ()

One of Ptolemy's major contributions to Greek maths was his trig tables.
The Indian mathematicians did the same, but independently of the Greeks.

EDIT:
Hmm..now that I reread your question, it seems you were after something else..

 

[brendan_foo]

No that's a great answer.. just curious. I know the Maclaurin series for trig functions takes the parameter and manipulates it to get a solution. However I wanted to know if there was some other relation between the argument and the answer. Say I had the angle $\frac{\pi}{9}$ and I wanted to know the cosine of it, that is the ratio of the adjacent to the hypotenuse, then was there some algebraic manipulation you could do with the value $\frac{\pi}{9}$ to yield the solution.

Aside from doing it by hand, I was curious whether it could be done another way before the days of calculus.