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.

Define
$$ L = \lim_{n \to \infty} \frac{a_n}{b_n} $$
$$ r = \lim_{n \to \infty} \frac{a_{n+1}}{b_n} $$
$$ s = \lim_{n \to \infty} \frac{b_{n+1}}{b_n} $$
then it's not hard to see that $L = r/s$. Also, by substituting in the recursion, since we have $b_n \to \infty$ we can compute
$$ r = \lim_{n \to \infty} \left(1 + 2 \frac{a_n}{b_n} + \frac{14}{b_n}\right) = 1 + 2L $$
$$ s = \ldots = 9 + 2L $$
(the reason to define $r$ and $s$ is precisely because I wanted to simplify the recursions in this fashion)
Solving the system of equations, along with $L>0$,  gives
$$ L = \frac{-7 + \sqrt{57}}{4} $$
