Recently I am investigating some geometric construction (i.e., compass and ruler constructions) and encounter into the issue of finding sine for some angles. I remember most whole number degree angles could be expressed in surd forms, only it is less pretty than the standard angles of 30°, 45°, 60°. Mostly I would start from the standard angles and break down to the degree I am interested by half angle formula or angle sum and difference formula. But some angle could result in surd under surd, for example:
Whether we can simplify this or express it in another format is the issue.
Let us start with the example. First consider if we want to factor it out into a product. Obviously it is not possible to be a product of two surd forms, so let us assume
We use $\sqrt{b}c$ instead of $\sqrt{b}+c$ because we observed the negative sign in $\sqrt{3}$. We may also try with $\sqrt{2  \sqrt{3}} = \sqrt{a}(c\sqrt{b})$ but we will know in the following if this is actually the case. Move the bracket term to under the surd gives:
Therefore,
Expressing $c$ in terms of $a$ gives the quadratic equation $12a^28a+1=0$ and solving this gives $(a,b,c)=(\frac{1}{2},3,1)$ or $(a,b,c)=(\frac{1}{6},3,3)$. These two, indeed, corresponds to
which are the same (and explains why we assume the form $\sqrt{a}(\sqrt{b}c)$ is enough).
Another approach^{1} is not to factor it out but express into sum of two surds:
Again, we take $\sqrt{a}\sqrt{b}$ instead of $\sqrt{a}+\sqrt{b}$ because we observed the negative sign of $\sqrt{3}$. This time, we squared both side and found that $2  \sqrt{3} = a + b  2\sqrt{ab}$. By solving $a+b=2$ and $4ab=3$, we have $(a,b)=(\frac{1}{2},\frac{3}{2})$ or $(a,b)=(\frac{3}{2},\frac{1}{2})$. Because we take only positive values for the surd, comparing the expression for correct polarity we found that the solution should be
It turns out, simplifying surds directly does not seems possible but we can easily do it if we first assume the form we want to obtain. Here contains a list of sine values for all integer angles from 0° to 90°, but the surd forms are ugly. Probably we can simplify them using this approach.

This form is described in Tutorial Algebra Volume 1 ↩