For which angles we know the $\sin$ value algebraically (exact)? For example:


*

*$\sin(15^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$

*$\sin(18^\circ) = \frac{\sqrt{5}}{4} - \frac{1}{4}$

*$\sin(30^\circ) = \frac{1}{2}$

*$\sin(45^\circ) = \frac{1}{\sqrt{2}}$

*$\sin(67 \frac{1}{2}^\circ) = \sqrt{ \frac{\sqrt{2}}{4} + \frac{1}{2} }$

*$\sin(72^\circ) = \sqrt{ \frac{\sqrt{5}}{8} + \frac{5}{8} }$

*$\sin(75^\circ) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}$

*?


Is there is a list of known exact values of $\boldsymbol \sin$ somewhere?
Found a related post here.

Starting with $\tan(\pi/3)=\sqrt{3}$ and $\tan(\pi/4)=1$ and using
$$
\tan(x/2)=\frac{\sqrt{1+\tan^2(x)}-1}{\tan(x)}\tag{1}
$$
and
$$
\tan(x+y)=\frac{\tan(x)+\tan(y)}{1-\tan(x)\tan(y)}\tag{2}
$$
and
$$
(\cos(x),\sin(x))=\frac{(1,\tan(x))}{\sqrt{1+\tan^2(x)}}\tag{3}
$$
we can construct the sine and cosine of all rational multiples of $\pi$ where the denominator is a power of $2$, or $3$ times a power of $2$.
For example, $x=\pi/4$ with $(1)$ gives
$$
\tan(\pi/8)=\sqrt{2}-1\tag{4}
$$
then $(2)$ and $(4)$ yields
$$
\begin{align}
\tan(3\pi/8)
&=\tan(\pi/4+\pi/8)\\
&=\frac{1+\tan(\pi/8)}{1-\tan(\pi/8)}\\
&=\sqrt{2}+1\tag{5}
\end{align}
$$
Then $(3)$ and $(5)$ give
$$
(\cos(3\pi/8),\sin(3\pi/8))=\frac{(1,\sqrt{2}+1)}{\sqrt{4+2\sqrt{2}}}\tag{6}
$$
