Is $\mathrm{arccos}$ of an irrational algebraic number necessarily transcendental?

Question

I know the sine of a non-zero algebraic number is necessarily transcendental; but what about the inverse cosine of an irrational algebraic number?

Answer 1

Isn't the second statement a consequence of the first?

Let $\alpha=\arccos x$, where $x$ is algebraic. However, then $y=\sqrt{1-x^2}$ is also algebraic, as a solution of the quadratic $y^2+x^2-1=0$.

Now, suppose $\alpha$ is algebraic. It follows that either $\sin\alpha=\pm\sqrt{1-x^2}=\pm y$ is transcendental - (contradiction), or that $\alpha=0$, which actually gives you one case ($\arccos 1=0$) where $\arccos$ of an algebraic number is algebraic. (However, in this case $1$ is not irrational.)