Exploring the Link between \theta and Trig Functions

[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 (:yuck:)

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.