How do calculators compute sine and cosine values?

[Loafers]

All my life I've understood Trig Functions as ratios of sides of a triangles. With this understanding I have been able to get the ratios for simple triangles like 30, 45, etc... since my teachers made me memorize the sides of it...

But now I'd like to know how would you find the ratio of a triangle with an messy angle such as 37 or 98? Most teachers would tell me to punch it in the calculator... but I'd like to have a better understanding than just what my calculator tells me... So I guess what I should be asking is, what does a calculator do to solve the sin/cos/tan of 37,94, 261, etc..? How does it determine what the measurements of a triangle are to get the ratio? I wish to understand trig functions better than just soh/cah/toa and guesstimate.

I only know a little bit of calculus so a simple explanation, if possible, would greatly be appreciated.

 

[HallsofIvy]

It's not clear what you are asking. If the "numbers" are messy, the meaning of sine, cosine, etc. is exactly what it is for a 3-4-5 right triangle.

If you are really asking how a calculator does those, check this, about the "Cordic" method:
http://www.dspguru.com/dsp/faqs/cordic

 

[Pagan Harpoon]

You could use something like the Taylor series to approximate it, it is a way to approximate functions such as sine as polynomials, the more terms of it you calculate, the more accurate your approximation will be. Obviously, a computer can do a very good approximation very fast.

http://en.wikipedia.org/wiki/Taylor_series

 

[wisvuze]

I've always found the history of trigonometry to be more empirical in the sense that they found their required values by measuring different triangles