Bob,

I am not quite sure what the intent of this question is.

*How do you find the value of (eg) sin(x) without geometric data?
Practically this is usually done with a computer program, calculator, set of tables, or slide rule. The last two are rare these days.
It can also be done using various approximations or infinite series. For instance,

sin(x) = x - x³/6 + x⁵/120 - x⁷/5040 + ...

where x is in radians (divide degree measure by 57.3). This eventually converges (surprisingly) for any x whatsoever, converges quite fast for x < 1, and very fast for small x.
In particular, for angles less than about 0.2 radian (say ten degrees) it is often useful for practical work to assume sin(x) = tan(x) = x, cos(x) = 1. Once again, the angle must be in radians .

There is also a formula using complex numbers that is very useful in advanced applications:

sin(x) = [e^ix - e^-ix]/2i

*How do you find the value of (eg) sin(x) given data other than edge lengths?
This can be done from other trig functions using identities like the "sum of angles", "double angle", "half angle", and "Pythagorean" identities: eg, sin(A+B) = sin(A)cos(B) + cos(A) sin(B)

*How do you find the value of (eg) sin(x) if the triangle is not right?
Use the sine law [sin(A)/a = sin(B)/b = sin(C)/c] ,
the cosine law [a² = b² + c² - 2bc cos(A)],
and the sum-of-angles rule.

Good Hunting!
RD