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.

If $\mu=\lim_{n\to\infty} \frac{a_{n} }{b_{n} } $exists then 
$$
\mu=\lim_{n\to\infty} \frac{a_{n+1} }{b_{n+1} } =
\lim_{n\to\infty} \frac{b_n+2a_n + 14}{9b_n+ 2a_n+70 }\\=
\lim_{n\to\infty} \frac{\frac{b_{n} }{b_{n} }+2\frac{a_{n} }{b_{n} } + \frac{14 }{b_{n} }}{9\frac{b_{n} }{b_{n} }+ 2\frac{a_{n} }{b_{n} }+\frac{70 }{b_{n} } }\\
=\frac{1+2\lim_{n\to\infty}(\frac{a_{n} }{b_{n} }) + \lim_{n\to\infty}(\frac{14 }{b_{n} })}{9+ 2\lim_{n\to\infty}(\frac{a_{n} }{b_{n} })+\lim_{n\to\infty}(\frac{70 }{b_{n} }) }\\
=\frac{1+2\mu+ 0}{9+ 2\mu+0}
$$
Here we used the fact that ${b}_{{n}}>{n}$ and therefore $$0\leq~\lim_{{n}\to\infty}~{\frac{1}{{b}_{{n}}}}<\lim_{{n}\to\infty}~{\frac{1}{{n}}}\leq0$$
So 
$$2\mu^2+7\mu-1=0$$.
But $\mu \gt 0$, so
$$\mu={{\sqrt{57}-7}\over{4}}
  $$
