$\cos(x)=\frac{1}{n}, n=2k+1, k\in \mathbb Z_+$

It is a corollary of a result by Ivan Niven: We have $$\cos(x)\in\mathbb R\setminus\mathbb Q$$ whenever $x\in\mathbb Q\setminus\{0\}$. Now you can simply take the contraposition.

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Also, I am not sure if this classifies for a duplicate of Why $\arccos(\frac{1}{3})$ is an irrational number?.

Perhaps the Lindemann-Weierstrass theorem is more widely known than Niven's paper.

(Special case) If $a$ is algebraic and nonzero, then $e^a$ is transcendental.

Suppose $\cos x = \frac{1}{3}$. Then $$ \frac{e^{ix}+e^{-ix}}{2} = \frac{1}{3} $$ shows that $e^{ix}$ is algebraic (solving a quadratic equation). Therefore, by L-W we conclude that $ix$ is transcendental. And $i$ is algebraic so we have $x$ is transcendental.