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 &= 3\\
x_{n+1} &= 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 $\pi$?
Best Answer
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.