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.

In January 2008 I posted several references published in the 1800s of tables that give exact values for the sine and cosine of $3$, $6$, $9$, …, $90$ degree angles. (Among the integer degree angles, only those that are multiples of $3$ can be expressed in real-radical form.) See google-groups archive for 1st post and google-groups archive for 2nd post.
The best table I know of was prepared by the Belgium mathematician E. Gelin in the 1880s. His table gives a list of values, with rationalized denominators, for all six trig. functions evaluated at $3$, $6$, $9$, …, $90$ degree angles. I know of three places where his table has been published:
Mathesis Recueil Mathematique (1) 8 (1888), Supplement 3. [See pp. 327-333 of the downloaded .pdf file.]
Mathesis Recueil Mathematique (3) 6 (1906), Supplement 3. [See pp. 338-348 of the downloaded .pdf file.]
E. Gelin, Éléments de Trigonométrie Plane et Sphérique (1888). [See pp. 59-62, which is equivalent to pp. 66-69 of the downloaded .pdf file.]
I believe Johann Heinrich Lambert was the first person who published exact radical values for the sine of $3$, $6$, $9$, etc. degree angles. A table of values is in Volume 1 of his Collected Works. The table is in an item that was published in 1770. Lambert’s table was reprinted two or three times in the first half of the 1800s (e.g. one was in Crelle’s Journal [= Journal für die reine und angewandte Mathematik]), but I don’t have the exact references with me now.
