# 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, $\sin{n\varphi}$ 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

 $P(\sin{r\pi})\;=\;\sin{nr\pi}\;=\;\sin{m\pi}\;=\;0,\quad Q(\cos{r\pi})\;=\;% \cos{nr\pi}\;=\;\cos{m\pi}\;=\;\pm 1,$

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 $\mathbb{Z}$.

Title algebraic sines and cosines AlgebraicSinesAndCosines 2013-03-22 18:51:27 2013-03-22 18:51:27 pahio (2872) pahio (2872) 7 pahio (2872) Corollary msc 11R04 msc 11C08 RationalSineAndCosine MultiplesOfAnAlgebraicNumber