If $x$ is given in degrees, are $\sin(x)$ and $\arcsin(x)$ algebric for $x$ rational?

Question

If $x$ is given in degrees, are $\sin(x)$ and $\arcsin(x)$ algebric for $x$ rational?

For the sin I think the answer is yes. We know that $\sin(1°)=\frac{\sqrt[90]{i}-\sqrt[90]{-i}}{2i}$, and we can construct $\sin(p x)$ using Chebyshev polynomials, and $\sin(x/q)$ by solving the root of a Chebyshev polynomial. Is this sound?

For arcsin I don't think this is true, but I can't find any counter-example, nor a proof.