In other words, you know the values of $\arcsin x$/$\arctan x$/$\arccos x$ for some specific values of $x$. That's just fine. Inverse trigonometric functions are transcendental functions and with exceptions of a few well-known values, the result is not nicely expressible with elementary functions (you can use a calculator or any number of "approximations by hand" to get a numerical value for the angle, but that's not the same).
In addition to the classical angles of multiples of $30^\circ$, which have known values of trigonometric functions, you can use the half-angle formulas and addition theorems to get other angles (inverse functions can be computed by recognizing the half-angle and angle addition expressions and reducing the calculation to a simpler expression).
In fact, the angles you can construct by adding and halving of elementary angles ($45^\circ$ and $60^\circ$), are precisely the angles you can construct with a compass and a straightedge (constructible angles). Such constructions also mean that all the angles, for which you can get the trigonometric function without a calculator (or the inverse problem), have a nice geometric representation of how to carry out such a calculation. Construction with similar triangles and circles is thus a good idea if you have a more complicated expression that you think you can get analytically.
Otherwise, there's nothing wrong about knowing a few special values. We do that all the time.