Find the ratio $\frac{R}{r}$ rounded to four decimals 
A large circle of radius R is surrounded by 12 smaller circles of radius r. Find the ratio $\frac{R}{r}$ rounded to four decimals.
Hint: Draw an isosceles triangle with one vertex at the centre of the R-circle
and the other vertices at the centres of two adjacent r-circles.


I drew the triangle, and the two equal sides (let's call them A and B) have a length of $R + r$, and the different side (let's call it C) has a length of $2r$.
I understand that the angle AB is 30 degrees (I divided 360 by 12), and as it's an isosceles triangle the other two angles are equal. They both measure 75 degrees.
I don't understand how to calculate the ratio though.

You can use the law of sines.
\begin{align}
\frac{2r}{\sin{30^\circ}}&=\frac{R+r}{\sin75^\circ}\\
2r\sin75^\circ&=(R+r)\sin30^\circ\\
2r\left(\frac{\sqrt2+\sqrt6}4\right)&=(R+r)\left(\frac12\right)\\
(\sqrt2+\sqrt6)r&=R+r\\
(\sqrt2+\sqrt6-1)r&=R\\
\frac{R}r&=\sqrt2+\sqrt6-1\approx2.8637
\end{align}
