Using Newton-Raphson to Approximate $\pi$ Is it possible to use the Newton-Raphson method to approximate $\pi$? If you use this to find the root of $\sin{x}$, could you use:

\begin{align}
x_0 &amp;= 3\\
x_{n+1} &amp;= x_n - \tan{x_n}
\end{align}

When I plug this into my calculator, it converges on $\pi$ quickly. Is this a real approximation of $\pi$ though? Does it require prior knowledge of the value of $\pi$?
Yes, it is a real approximation of $\pi$.  Whether it "requires prior knowledge of the value of $\pi$" depends on what you're using to calculate $\tan x_n$.  A typical arbitrary-precision method of calculating the $\tan$ function might use the Maclaurin series for $\sin$ and $\cos$ in a neighbourhood of $0$, and trig identities to reduce other values to those in that neighbourhood.  Those reductions do require the value of $\pi$.  However, they are not the only way.
You could use the Maclaurin series for $\sin$ and $\cos$ even for values of $x$ near $\pi$: it would just take a very large number of terms.
The approximations that you get using tangent or sine functions are real in the sense that they do give a valid sequence that converges to $\pi$. But the problem with using trig functions with the N-R method is that the approximations are just as irrational (transcendental actually) as $\pi$ itself. It is rather like the chicken and egg dilemma. The usual way of approximating $\pi$ is with infinite series like Taylor series or infinite products like Wallis's formula.