The limit of $\sin(n!)$

It is known that $\lim\limits_{n\to\infty}\sin n$ does not exist.

$\lim\limits_{n\to\infty}\sin(n!)$ exists or not?

I think there is a potentially different answer if the functions use radians or degrees. I say this because trigonometric functions relate to the circle. A complete circle is a whole number of degrees, but a transcendental number of radians. Factorials, meanwhile, are whole numbers.

For the sine function in degrees, the answer is that the limit is zero. I can say this because for every $n \ge 360$, $360$ divides $n!$. And if $360$ divides the number, then the sine of that number is zero.

For the sine function that uses radians, I can't think how to prove it at the moment, but I suspect the function does not converge.