TITLE: When is $\sin x$ an algebraic number and when is it non-algebraic?
QUESTION [22 upvotes]: Show that if $x$ is rational, then $\sin x$ is algebraic number when $x$ is in degrees and $\sin x$ is non algebraic when $x$ is in radians. 

Details: so we have $\sin(p/q)$ is algebraic when $p/q$ is in degrees, that is what my book says. of course $\sin (30^{\circ})$, $\sin 45^{\circ}$, $\sin 90^{\circ}$, and halves of them is algebraic. but I'm not so sure about $\sin(1^{\circ})$.
Also is this is an existence proof or is there actually a way to show the full radical solution.
One way to get this started is change degrees to radians. x deg = pi/180 * x radian. 
So if x = p/q, then sin (p/q deg) = sin ( pi/180 * p/q rad). Therefore without loss of generality the question is show sin (pi*m/n rad) is algebraic. and then show sin (m/n rad) is non-algebraic.

REPLY [4 votes]: Lindemann-Weierstrass theorem implies that for $\alpha$ non-zero algebraic, $\sin \alpha$ is transcendental.