Actual formulas for trigonometric functions? (no calculator)

Hi,

I need to know the actual formulas for the trigonometric functions: sine, cosine, tangent, arcsine, arccosine, arctangent, etc.

What I need is a function, like that of a calculator, for example cosine, which only takes the radian value of angle and returns a value, just like the calculator, but I need to manually calculate it. Those are formulas/procedures I need but I have never been able to find.

From what I can see, for example Cosine, would take the angle in Radians, and would asume a Radius of 1. Since Cosine of an angle is equals to Rx/R, and since Radius is assumed to be 1 and the angle is known, only Rx would need to be determined (presumably would be the direct value Cosine would return), maybe with Pythagoras, but I can't see how if Pythagoras formulas are supposed to use Sine, Cosine, etc., as well. Then I need that you tell me the actual procedure for manually finding Sine, Cosine, and the rest of functions.

If you can also, provide a rationale as to what the different functions are doing. If you can't provide a very extensive explanation, the formulas alone would suffice and I would try to figure them out myself.

If you can, please tell me also where I can find a full, in-depth analysis of the unit circle, like Pythagoras would relate and develop the different components, with the formulas I asked for above, and why they are being developed a particular way, so I can study further by myself, and any other web resource and/or book I should refer to make sense of this normally canned-only functions.

Any information will be welcome, as I see the need of being able to calculate these trigonometric functions manually, without using the calculator or tools, to greatly and really improve math understanding.