# On the behavior of transcendental functions

Given that single-variable algebraic functions take on algebraic values when the input is algebraic, and take on transcendental values when the input is transcendental, and knowing that transcendental functions are all those functions that are not algebraic,

how do I determine the nature of the output of a transcendental function in a similar fashion?

For example, in the case of $\sin : \mathbb R \to \mathbb R$, there are transcendental numbers (i.e. integer multiples of $\pi$) that map to algebraic numbers; and what can I say about the image of natural numbers or rational numbers ($\sin (\mathbb N)$, $\sin (\mathbb Q)$)?

$f(\mathbb{A})\subseteq\mathbb{A}$
where $\mathbb{A}$ is the set of algebraic numbers. If you have a function that is transidental, i.e. not algebraic, then all you know is that the converse of this holds:
$f(\mathbb{A})\nsubseteq\mathbb{A}$.