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.
$\sin 3^\circ=\frac{(\sqrt{3}+1) (\sqrt{5}-1)}{8 \sqrt{2}}-\frac{(\sqrt{3}-1) \sqrt{5+\sqrt{5}}}{8}$.

Solving a cubic equation you can get a huge expression for $\sin 1^\circ$ in radicals, and therefore, for any $\sin n^\circ$.
Algebraically exact value of Sine for all integer angles is possible. Please visit https://archive.org/details/ExactTrigonometryTableForAllAnglesFinal for the list of exact values for Sine of integer angles in degrees.