Is the sine of a transcendental angle transcendental or algebraic?

Question

Let $x$ be a transcendental number Algebraically Independent from $\pi$. It is known if $ \sin x $ is also transcendental or algebraic?

For example, is $\sin \sqrt{2}^\sqrt{2}\pi$ algebraic or transcendental?

NOTE: Then the sine of an transcendental number is not necessary transcendental. Are there any known example in which $\sin x$ is algebraic for another transcendental number, different of $\pi$ and that is not defined with the use of inverse trigonometric functions?

Answer 1

$\arcsin(1/3)$ is known to be transcendental, so it is a transcendental number whose sine is algebraic, indeed, rational.

More detail: Let $\arcsin(1/3)=x$, so $\sin x=1/3$ is algebraic (indeed, rational). By the Lindemann-Weierstrass Theorem, if $x$ is a nonzero algebraic number, then $\sin x$ is transcendental (says Wikipedia, https://en.wikipedia.org/wiki/Transcendental_number). Therefore, $x$ is transcendental.