Recently, I learned that \(\cos{(\pi/7)}\) and \(\sin{(\pi/7)}\) cannot be written in terms of algebraic numbers. This is pretty shocking.
The half-angle formulas
\begin{align*}
\cos^{2}{\left( \frac{x}{2} \right)} &= \frac{1 + \cos{(x)}}{2}, & \sin^{2}{\left( \frac{x}{2} \right)} &= \frac{1 - \cos{(x)}}{2}
\end{align*}
can be used to find expressions for fractions of \(\pi\) with powers of two as denominators.
You begin with
\begin{align*}
\cos{(\pi)} &= -1, & \sin{(\pi)} &= 0
\end{align*}
Using the half-angle formulas gives
\begin{align*}
\cos{\left( \frac{\pi}{2} \right)} &= 0, & \sin{\left( \frac{\pi}{2} \right)} &= 1
\end{align*}
The idea is to repeat this procedure to get other fractions.
Using the half-angle formulas gives
\begin{align*}
\cos{\left( \frac{\pi}{4} \right)} &= \frac{\sqrt{2}}{2}, & \sin{\left( \frac{\pi}{4} \right)} &= \frac{\sqrt{2}}{2}
\end{align*}
Using the half-angle formulas gives
\begin{align*}
\cos{\left( \frac{\pi}{8} \right)} &= \frac{\sqrt{2 + \sqrt{2}}}{2}, & \sin{\left( \frac{\pi}{8} \right)} &= \frac{\sqrt{2 - \sqrt{2}}}{2}
\end{align*}
Using the half-angle formulas gives
\begin{align*}
\cos{\left( \frac{\pi}{16} \right)} &= \frac{\sqrt{2 + \sqrt{2 + \sqrt{2}}}}{2}, & \sin{\left( \frac{\pi}{16} \right)} &= \frac{\sqrt{2 - \sqrt{2 + \sqrt{2}}}}{2}
\end{align*}
Using the half-angle formulas gives
\begin{align*}
\cos{\left( \frac{\pi}{32} \right)} &= \frac{\sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2}}}}}{2}, & \sin{\left( \frac{\pi}{32} \right)} &= \frac{\sqrt{2 - \sqrt{2 + \sqrt{2 + \sqrt{2}}}}}{2}
\end{align*}
You can already see the pattern. As the denominator grows, the cosine tends to 1, and the sine tends to 0.
Let \(R_{k}\) be the following unitary matrix
\begin{equation*}
R_{k} = \begin{bmatrix}
1 & 0 \\
0 & \exp{\left( -\dfrac{2 \pi i}{2^{k}} \right)}
\end{bmatrix}
\end{equation*}
Note that
\begin{equation*}
R_{19} = \begin{bmatrix}
1 & 0 \\
0 & 0.9999999999 - 0.00001198422491i
\end{bmatrix} \approx \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
\end{equation*}
So after \(k = 19\), the matrix \(R_{k}\) can be approximated by the identity.
