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.

We use radian notation. Every rational multiple of $\pi$ has trigonometric functions that can be expressed using the ordinary arithmetic operations, plus $n$-th roots for suitable $n$.  
This is almost immediate if we use complex numbers, since $(\cos(2\pi/n)+i\sin(2\pi/n)^n=1$. 
But it is known, for example, that there is no expression for $\sin(\pi/9)$ that starts from the integers, and uses  only the ordinary operations of arithmetic and roots in which every component is real.
The following more restricted problem has a long history because of its close connection with the problem of which angles are constructible by straightedge and compass.  
Let $\theta=\frac{m}{n}\pi$, where $m$ and $n$ are relatively prime.  Restrict our algebraic operations to the ordinary operations of arithmetic, plus square roots only,  The trigonometric functions of $\theta$ are so expressible iff $n$ has the form
$$n=2^k p_1p_2\cdots p_s,$$
where the $p_i$ are distinct Fermat primes.
A Fermat prime is a prime of the form $2^{\left(2^t\right)}+1$. There are only five Fermat primes known: $3$, $5$, $17$, $257$, and $65537$. It is not known whether or not there are more than five.
