TITLE: Are there any "nonstandard" special angles for which trig functions yield radical expressions?
QUESTION [45 upvotes]: Everyone learns about the two "special" right triangles at some point in their math education—the $45-45-90$ and $30-60-90$ triangles—for which we can calculate exact trig function outputs. But are there others?
To be specific, are there any values of $y$ and $x$ such that: 

$y=\sin(x)$; 
$x$ (in degrees) is not an integer multiple of $30$ or $45$; 
$x$ and $y$ can both be written as radical expressions? By radical expression, I mean any finite formula involving only integers, addition/subtraction, multiplication/division, and $n$th roots. [Note that I require $x$ also be a radical expression so that we can't simply say "$\arcsin(1/3)$" or something like that as a possible value of $x$, which would make the question trivial.]

If yes, are they all known and is there a straightforward way to generate them?
If no, what's the proof?

REPLY [5 votes]: A $72^{\circ}-36^{\circ}-72^{\circ}$ isosceles triangle gives $$\sin 18^{\circ}=\frac{\sqrt 5-1}{4}$$

$\Delta ABC$ is an isosceles triangle. 
$AB=AC=a, \angle A=32^{\circ}, AD\bot BC, CE$ is bisector of $\angle C$ and $EF\bot AC$. We observe that $BC=EC=EA=x$.
Now, $\Delta ABC\sim CEB.$ Hence,
$$\frac{AB}{CE}=\frac{BC}{EB}\\
\frac ax=\frac{x}{a-x}\implies x^2+ax-a^2=0\\
\frac xa=\frac{\sqrt 5-1}{2}\\
\sin \angle BAD=\frac{BD}{AB}\\
\sin 18^{\circ}=\frac{\frac x2}{a}=\frac{\sqrt 5-1}{4}$$
Similarly $\cos 36^{\circ}$ can be found from this triangle.