How can I calculate the relative lengths of triangle sides if all angles are known? I've read about this on other occasions on here before but I think my problem isn't a duplicate.
I'm trying to find the lengths of sides of a triangle where I know all three angles.
Let's say $A = 60^\circ$, $B = 90^\circ$, $C = 30^\circ$ and $b = 1cm$.
How can I find the lengths of $A$ and $C$ without using the law of sines for any angle triangle where $B = 90^\circ$ and $b = 1cm$?

See Taylor Series for Trigonometric Functions.
You will have to convert the angles (measured in degrees) to angles (measured in radians), where $180^\circ = \pi~$ radians.
Then, you can use the linked article to express the sine function to any desired degree of accuracy that your software/hardware can handle.  In fact, I suspect that programs such as Java and C, that do have the sine
function do the exact same thing.
Once you are able to have your own estimation of the sine function, for any pertinent angle $(\theta)$, then (assuming that $b$ represents the hypotenuse), it is a simple matter to compute (for example) $b\sin(\theta).$
