Copy-pasting in comments that essentially answer the question, so that there's an answer here and the question might leave the unanswered list (CW so that it's editable—if you commented and want to turn your comment into an answer of your own, feel free to edit it out of this answer).
I doubt there is a simple method. $\sin(10^\circ)$ is a root of the irreducible cubic $8x^3-6x+1$. – Jonas Meyer
$\sin 3\alpha=3\sin\alpha−4\sin^3\alpha$ – pedja
There is no straightedge and compass construction of the $10^\circ$ angle, so there is no "geometric method" using only the classical [t]ools. – André Nicolas
If one allows for neusis (i.e. a marked straightedge), one could trisect a $30^\circ$ angle... – J. M.
In addition, it is a not easy theorem using Tools from Galois Theory that there is no expression for $\sin(10^\circ)$ in terms of real $n$-th roots. By Cardano's Formula, there is an expression using complex cube roots, but that is not a useful answer to your question. – André Nicolas