<s>[INST] 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.
 [/INST] $\sin 3^\circ=\frac{(\sqrt{3}+1) (\sqrt{5}-1)}{8 \sqrt{2}}-\frac{(\sqrt{3}-1) \sqrt{5+\sqrt{5}}}{8}$.
Solving a cubic equation you can get a huge expression for $\sin 1^\circ$ in radicals, and therefore, for any $\sin n^\circ$.
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