A very curious rational fraction that converges. What is the value? Is there any closed form for the following limit?

Define the sequence
  $$ \begin{cases} 
  a_{n+1} = b_n+2a_n + 14\\
  b_{n+1} = 9b_n+ 2a_n+70
\end{cases}$$
  with initial values $a_0 = b_0 = 1$. Then $\lim_{n\to\infty} \frac{a_n}{b_n} = ? $

The limit is approximately $0.1376$. My math teacher Carlos Ivorra says that this limit have a closed form involving the sine of an angle. What is the closed form for is limit?
NOTE: I have found this (and another series of converging sequences) by the use of an ancient method for calculating sines recently rediscovered. I'll give the details soon as a more general question.

One idea to get the limit in closed form :
Let $A=\begin{pmatrix}
2 & 1 \\
2 & 9
\end{pmatrix}$, $X_n = \begin{pmatrix} a_n \\ b_n \end{pmatrix}$, and $b=\begin{pmatrix} 14 \\ 10 \end{pmatrix}$.
You can write $X_{n+1} = AX_n + b$. The idea is to solve the equation $X=AX+b$ ($X$ being a two-dimensional vector) - this equation has a unique solution as $(A-I)$ is non-singular. Let's call $X$ this solution; you can write $(X_{n+1}-X)=A(X_n - X)$ then, for all $n$, you have :
\begin{equation}
(X_{n} - X ) =A^n (X_{0} - X)
\end{equation}
$A^n$ can be evaluated by diagonalizing $A$. This will give you $X_n$ (and then $a_n$ and $b_n$ in closed form).
