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

Proof. According to the parent entry, $\sin{n\varphi}$ and $\cos{n\varphi}$ can be expressed as polynomials with integer coefficients of $\sin\varphi$ or $\cos\varphi$ , respectively, when $n$ is an integer. Thus we can write $$\sin{n\varphi} \;=\; P(\sin\varphi), \quad \cos{n\varphi} \;=\; Q(\cos\varphi),$$ where $P(x),\,Q(x) \in \mathbb{Z}[x]$ . If $\displaystyle r = \frac{m}{n}$ where $m,\,n$ are integers and $n \neq 0$ , we have $$P(\sin{r\pi}) \;=\; \sin{nr\pi} \;=\; \sin{m\pi} \;=\; 0, \quad Q(\cos{r\pi}) \;=\; \cos{nr\pi} \;=\; \cos{m\pi} \;=\; \pm1,$$ i.e. both $\sin{r\pi}$ and $\cos{r\pi}$ satisfy an algebraic equation. Q.E.D.

For example, $$\cos{7\varphi} \;=\; 64\cos^7\varphi-112\cos^5\varphi+56\cos^3\varphi-7\cos\varphi,$$ whence we have the identity $$64\cos^7\frac{\pi}{7}-112\cos^5\frac{\pi}{7}+56\cos^3\frac{\pi}{7}-7\cos\frac{\pi}{7}+1 \;=\; 0,$$ and therefore $\cos\frac{\pi}{7}$ is algebraic over $\mathbb{Z}$ .