The question says it all.
Given a triangle, find its angles without a calculator. Is this even possible without tables or making tables?
Summary:
Is it possible to determine the inverse sin, cos of a triangle with an equation not involving calculators and tables?
Use the Law of Cosines. If $C$ is the measure of the angle opposite the side of length $c$, and $a,b$ the other two side lengths, then $$c^2=a^2+b^2-2ab\cos C,$$ so $$\cos C=\frac{a^2+b^2-c^2}{2ab},$$ and so since $C$ is an angle of a triangle, then $$C=\cos^{-1}\left(\frac{a^2+b^2-c^2}{2ab}\right).$$
Similarly, if $A$ and $B$ are the respective measures of the angles opposite the sides of length $a$ and $b$, then $$A=\cos^{-1}\left(\frac{b^2+c^2-a^2}{2bc}\right)$$ and $$B=\cos^{-1}\left(\frac{a^2+c^2-b^2}{2ac}\right).$$
If you want a numerical value, you're probably out of luck, but the above work does give us exact and correct values for $A,B,C$.