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 $b = 1cm$?

Consider using Bhaksara I's sine approximation formula:
$$
\sin x^\circ \approx \frac{4x(180-x)}{40500 - x(180-x)}
$$
where $x$ is measured in degrees.  This gives a result that is accurate to within about $\pm 0.0015$ over the entire range of angles from $0$ to $180°$.  The fractional error maxes out at a little less than 2% near $0°$ and $180°$.
