I'm aware of some transcendence proofs of certain special numbers like $\pi$ and $e$, and I'm aware that finding certain transcendence proofs can be quite difficult and involved. I also know that most numbers, we are not certain of their transcendence, or lack thereof.
However, I just put into Wolfram Alpha for an unrelated question, arcsin$(\frac{1}{2\sqrt{2}})$, and Wolfram Alpha told me it was transcendental. What kinds of numbers do we know are transcendental, how was Wolfram Alpha able to pick this up so fast with an arbitrary input for arcsin (barring obvious ones like zero), and how would we be able to prove a statement like "arcsin $(x)$ is transcendental for all inputs $x \neq 0, \frac{\pi}{2}$"? (maybe I am missing some exceptions, but you get the idea.)
Cheers.
This all follows from the Lindemann–Weierstrass theorem: if $x$ is a nonzero algebraic real or complex number then $e^x$ is transcendental.
The arcsine can be written in terms of logarithms as $$\sin^{-1}z=-i\ln(\sqrt{1-z^2}+iz)$$ Now suppose $\sin^{-1}\frac1{2\sqrt2}$ is algebraic. Then so is $i\sin^{-1}\frac1{2\sqrt2}=\ln\left(\sqrt{\frac78}+\frac1{2\sqrt2}i\right)$, and by the above theorem $e^{i\sin^{-1}1/(2\sqrt2)}=\sqrt{\frac78}+\frac1{2\sqrt2}i$ is transcendental. But clearly $\sqrt{\frac78}+\frac1{2\sqrt2}i$ is algebraic. So $\sin^{-1}\frac1{2\sqrt2}$ is transcendental.