# How to obtain the exact value of $\sin(3n)$?

Suppose $3n$ is an angle between $0$ and $90$ degrees and that $n$ is an integer. By considering half squares and half equilateral triangles it's easy to obtain expressions for $\sin(3n)$ if $n=10$, $n=15$ and $n=20$. (By "expressions" I mean closed expressions involving addition, subtraction, multiplication, division and square roots.)

Using the addition/subtraction formulas for sine and cosine we can easily handle the cases $n=5$, and $n=25$ as well. I suppose that if I can find $\sin3$ then I can use these formulas to find $\sin(3n)$ for any $n$. Right? So my question is really this:

How do I find $\sin3$ (where the angle is measured in degrees)?

$\sin 3^{\circ} = \sin (18^{\circ} – 15^{\circ})$, and you can use the addition formula.
$\sin (18^{\circ})$ can be found by the half-angle formula once you know $\sin (36^{\circ})$, which is related to the pentagon.
$\sin (15^{\circ})$ can be found by the half-angle formula with $\sin (30^{\circ})$, which you probably know.