Finding $\sin^{-1}(x)$ without using a calculator I don't understand how to compute $\sin^{-1} (0.6293)$, to figure out the angle without using a calculator. I understand how to find the answer if I use a calculator but I don't understand the necessary steps to solve the problem without a calculator. 
Am I wrong to assume you have to know at least the value of the opposite or hypotenuse length of the given triangle? Can you find the angle if you only have the sine and nothing else and you don't have a calculator?  

Probably your best bet would be a Taylor/Maclaurin series:
$$
\sin^{-1}(x)= x + 1/6\cdot x^3 + 3/40\cdot x^5 + 5/112\cdot x^7 + O(x^9)
$$
