What exactly are Sin, Cos and Tan? How are they calculated

I take physics and calculus, so I use trig all the time with no problems. I am just wondering what exactly are they, and how are they calculated? I know that Sin0=o/h, Cos0=a/h and Tan0=o/a, so please don't tell me what I already know. How are they related to all right-angled triangles?

Thanks.

I'll try to explain this without pictures, but it might not help as much as it should.

a = adjacent (side length)
o = opposite (side length)
h = hypotenuse (side length)

sin theta = o/h, theta is the angle, where what sin equals is the ratio of : opposite over hypotenuse

90