Is $x = \arccos ( \frac{3}{5})$ a rational number?

Question

Wolfram tells us $x = \arccos ( \frac{3}{5})$ is an irrational number. How can we prove it? (Without using a computer obviously)

Answer 1

Suppose $x=\arccos(\frac{3}{5})$

Then, we have $\cos(x)=\frac{3}{5}$

Obviously, we have $x\ne 0$

With the Lindemann-Weierstrass-theorem, we can show that $\cos(x)$ is transcendental for every algebraic non-zero $x$.

Therefore, $x$ cannot be algebraic, since $\frac{3}{5}$ is not transcendental.

Hence $x$ is transcendental, in particular irrational.