On the behavior of transcendental functions

Question

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 ($\sin (\mathbb N)$, $\sin (\mathbb Q)$)?

Answer 1

Algebraic functions have the property that for any algebraic number $\alpha$, $f( \alpha)$ is also an algebraic number. This, however, is not their defining property, and there are also transcendental functions that have this property. See https://en.wikipedia.org/wiki/Transcendental_function section titled Exceptional set.

Answer 2

An injective continuous transcendental function $f\colon D$ open $\subseteq\mathbb{C}\to\mathbb{C}$ has algebraic and transcendental values.

1.) Examples for algebraic values at algebraic points

Each elementary function can be represented as composition of $\exp$, $\ln$ and/or finitary algebraic functions. For the elementary functions, we can therefore apply Schanuel's conjecture or the theorems it includes. see e.g.:
Bauer, N.; Slobin, H. K.: Some transcendental curves and numbers. 1913
Anai, H.; Weispfenning, V.: Deciding Linear-Trigonometric Problems. ISSAC 2000
Achatz, M.; McCallum, S.; Weispfenning, V.: Deciding Polynomial-Exponential Problems. ISSAC 2008
McCallum, S.; Weispfenning, V.: Deciding polynomial-transcendental problems. 2012

For the theorem of Lin and the theorem of Chow see point 3.).

Let $E$ denote the exceptional set of the given function.

Examples:
$E=\{\}\colon$ $\cot(z)$, $\csc(z)$, $\text{arccot}(z)$, $e^{1+\pi z}$, $e^{e^z}$, $e^z+e^{1+\pi z}$
$E=\{0\}\colon$ $e^z$, $\sin(z)$, $\cos(z)$, $\tan(z)$, $\sec(z)$, $\arcsin(z)$, $\arctan(z)$, $\sinh(z)$, $\cosh(z)$, $\tanh(z)$, $\text{sech}(z)$, $\text{arcsinh(z)}$, $\text{arctanh(z)}$
$E=\{1\}\colon$ $\ln(z)$, $\arccos(z)$, $\text{arcsec}(z)$, $\text{arccosh}(z)$, $\text{arcsech}(z)$
$E=\mathbb{Z}\colon$ $\sin(\pi z)e^z$
$E=\mathbb{Q}\setminus\{0\}\colon$ $z^z$, $z^{\frac{1}{z}}$
$E=\mathbb{Q}\colon$ $e^{2i\pi z}$, $a^z$ ($a\in\mathbb{Q}\setminus\{0,1\}$)
$e^{P(z)}$ with $P\in\mathbb{C}[z]\colon$ $E$ is the set of zeroes of $P$.

Lindemann-Weiserstrass theorem implies: $\sin(a)$, $\cos(a)$, $\tan(a)$, $\sinh(a)$, $\cosh(a)$, $\tanh(a)$, $\arcsin(a)$ are transcendental for any non-zero algebraic $a$.

If $a$ is an algebraic number and $r$ is a rational number, $\sin(r\arcsin(a))$ is algebraic.
Trott, M.: Algebraic Values of Trigonometric Functions of Inverse Trigonometric Functions. 2011. Wolfram Demonstrations Project

But finding the exceptional set of a function can be a difficult mathematical problem because there are only a few transcendence theorems and conjectures known.

For the number of algebraic values of transcendental functions at algebraic points, we have Schneider-Lang theorem for meromorphic functions and Siegel–Shidlovsky theorem​ for Siegel E-functions.

Now there is an algorithm that gives the algebraic values of a given E-function and the corresponding algebraic points. Adamczewski/Rivoal 2018:
"The celebrated Siegel-Shidlovskii theorem deals with the algebraic (in)dependence of values at algebraic points of E-functions solutions of a differential system. However, somewhat paradoxically, this deep result may fail to decide whether a given E-function assumes an algebraic or a transcendental value at some given algebraic point. Building upon André’s theory of E-operators, Beukers refined in 2006 the Siegel-Shidlovskii theorem in an optimal way. In this paper, we use Beukers’ work to prove the following result: there exists an algorithm which, given a transcendental E-function f(z) as input, outputs the finite list of all exceptional algebraic points α such that f(α) is also algebraic, together with the corresponding list of values f(α). This result solves the problem of deciding whether values of E-functions at algebraic points are transcendental."

References:

Diamond, J.: The Schneider-Lang theorem for functions with essential singularities. Proc. Amer. Math. Soc. 80 (1980) (2) 223-223
Waldschmidt, M.: Algebraic values of analytic functions. J. Comput. Appl. Math. 160 (2003) (1–2) 323-333
Beukers, F.: A refined version of the Siegel-Shidlovskii theorem. Annals Math. 163 (2006) (1) 369-379
Jingjing Huang; Marques, D.; Mereb, M.: Algebraic values of transcendental functions at algebraic points. Bull. Austral. Math. Soc. 82 (2010) (2) 322-327
Herblot, M.: Algebraic points on meromorphic curves. 2012
Marques, D.; Lima, F. M. S.: Some transcendental functions with an empty exceptional set. 2012
Fischler, St.; Rivoal, T.: Arithmetic theory of E-operators. J. École polytechn. Math. 3 (2016) 31-65
Waldschmidt, M.: Irrationality and transcendence of values of special functions. 2017
Adamczewski, B.; Rivoal, T.: Exceptional values of E-functions at algebraic points. Bull. London Math. Soc. 50 (2018) (4) 697-708

2.) Examples for rational values at rational points

Reddy, R.: Irrationality of basic trigonometric and hyperbolic functions at rational values

3.) Examples for algebraic values at elementary points

If you ask for the transcendental values at algebraic/rational/integer points or for the algebraic/rational/integer values at transcendental points, you need to represent this transcendental numbers in closed form.

Wikipedia: Niven's theorem
Wikipedia: Trigonometric constants expressed in real radicals

You can also ask for representing the transcendental numbers as elementary numbers. This is the question of solving an equation of one unknown by elementary numbers. see e.g.:
[Lin 1983] Ferng-Ching Lin: Schanuel's Conjecture Implies Ritt's Conjectures. Chin. J. Math. 11 (1983) (1) 41-50
[Chow 1999] Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448
Khovanskii, A.: Topological Galois Theory. Solvability and Unsolvability of Equations in Finite Terms. Springer, 2014. and other publications of this author