For what values of $x$ is $\cos x$ transcendental?

Question

For what values of $x$ is $\cos x$ transcendental? Is there any way I can figure out the values of $x$ where $\cos x$ is transcendental or do I have to check individually for every $x$ whether it is or not?

Answer 1

Let $T \subset (-1,1)$ be the set of transcendental numbers in the interval $(-1,1)$. Then $\cos(x)$ is transcendental for all numbers $x$ belonging to the set $\arccos(T) = \{\arccos(t): t \in T\}$. To get all the elements in the set, we can add multiple of $\pi$ to all the elements in $\arccos(T)$.

Answer 2

I'm not an expert on this, but I'm pretty sure this is not known in general. Here are some partial results:

• There are only countably many values of $x$ such that $\cos x$ is algebraic (because there are only countably many algebraic numbers and $\cos$ takes every value only countably many times). So $\cos x$ is "almost always" transcendental in a rather strong sense.
• If $x\neq 0$ is algebraic, then $\cos x$ is transcendental (this follows from the Lindemann-Weierstrass theorem).
• If $x$ is a rational multiple of $\pi$, then $\cos x$ is algebraic (this is elementary and follows from the fact that $e^{ix}$ is a root of unity). More generally, if $y$ is such that $\cos y$ is algebraic and $x/y$ is rational, then $\cos x$ is algebraic.
• If $x$ is an algebraic irrational multiple of $\pi$, then $\cos x$ is transcendental (this follows from the Gelfond-Schneider theorem). More generally, if $y$ is such that $\cos y$ is algebraic and $x/y$ is algebraic and irrational, then $\cos x$ is transcendental.

(Note that the general transcendence theorems tend to be stated in terms of exponentials; to translate them into results about cosines, you can use the fact that $\cos x$ is algebraic iff $e^{ix}$ is algebraic.)