algebraic sines and cosines

For any rational number $r$, the sine and the cosine of the number $r\pi$ are algebraic numbers.

Proof.  According to the http://planetmath.org/node/11664parent entry, $\sin n\phi$ and $\cos n\phi$ can be expressed as polynomials with integer coefficients of $\sin \phi$ or $\cos \phi$, respectively, when $n$ is an integer.  Thus we can write

$$\sin n\phi =P(\sin \phi ),\cos n\phi =Q(\cos \phi ),$$

where  $P(x),Q(x)\in \mathbb{Z}[x]$.  If  $r={\displaystyle \frac{m}{n}}$  where $m,n$ are integers and $n\ne 0$,  we have

$$P(\sin r\pi )=\sin nr\pi =\sin m\pi =\mathrm{\hspace{0.33em}0},Q(\cos r\pi )=\cos nr\pi =\cos m\pi =\pm 1,$$

i.e. both $\sin r\pi$ and $\cos r\pi$ satisfy an algebraic equation.  Q.E.D.

For example,

$$\cos 7\phi =\mathrm{\hspace{0.33em}64}{\cos }^7\phi -112{\cos }^5\phi +56{\cos }^3\phi -7\cos \phi ,$$

whence we have the identity

$$64{\cos }^7\frac{\pi }{7}-112{\cos }^5\frac{\pi }{7}+56{\cos }^3\frac{\pi }{7}-7\cos \frac{\pi }{7}+1=\mathrm{\hspace{0.33em}0},$$

and therefore $\cos \frac{\pi }{7}$ is algebraic over $\mathbb{Z}$.