Existence of a $\varphi \in \mathbb{R}$ such that $\cos(\varphi)$ is transcendental

Question

Does anybody know an elementary proof that shows that there is a $\varphi \in \mathbb{R}$ such that $\cos(\varphi)$ is a transcendental number? I have read about the Lindemann-Weierstrass Theorem but would prefer not to make use of it if at all possible. Your help is appreciated.

Answer 1

Pick any trascendental number $a$. Then all the numbers $\frac{a}{n}$, $n \in \mathbb{N}$ are trascendental, so there exists a trascendental number $ b \in (\frac{1}{3}, \frac{1}{2})$.

Then, call $\varphi = \arccos b$.

Answer 2

Hm, $1/e$ is transcendental and $\cos(0) = 1$, $\cos(\pi/2) = 0$. By the intermediate value theorem, there must be an $x \in (0,\pi/2)$ such that $\cos(x) = 1/e$.

Answer 3

Note that $\cos \varphi$ is continuous for all $\varphi\in\mathbb{R}$, and the interval $\left[-1,1\right]$ is uncountable while there are only countably many algebraic numbers. Hence, by the Intermediate Value Theorem, there is some $\varphi_{0}\in\mathbb{R}$ such that $\cos\varphi_{0}$ is transcendental.