These days, I'd watched some youtube movies regarding manual computation of $\pi$ and find out about 239, 1/5 and their inverse tangent functions ($arctan$/$arctg$/$\tan^{-1}$) and so on.
Today, I've done a search on these things, being curious about how you end up with these numbers ... and I found this (summarized bellow):
- firstly, you consider an angle $x$ whose tangent is $1/5$
- then you compute $\tan(2x)$
- compute $\tan(4x)$. At the end of this calculation, you substitute 1 with $\tan(\frac{\pi}{4})$
- compute $\tan(4x-\frac{\pi}{4})$ which gives 1/239
- you do arctan on both sides, ending up with something like $\tan^{-1}(1/239) = 4x - \frac{\pi}{4}$, the right side being equal to $4\tan^{-1}(1/5) - \frac{\pi}{4}$ (using step 1)
- you can get a formula for $\pi$ like this one: $\frac{\pi}{4} = 4\tan^{-1}\frac{1}{5} - \tan^{-1}\frac{1}{239}$
Now, my questions are these ones:
A. can one start with another initial angles? (I'd tried with $\tan(x)=1/2$)
B. can one try to force the $1 = \tan(\frac{\pi}{4})$ substitution earlier, i.e. don't wait till the step (3)? (I'd tried to skip step 3, and forced the $\tan(2x) = \tan(\frac{\pi}{4}) + 1/3$; then I computed $\tan(2x-\frac{\pi}{4})$ and get 1/7 and then I'd performed the arctan on both sides, ending with something like: $\frac{\pi}{4} = 2\tan^{-1}\frac{1}{2} - \tan^{-1}\frac{1}{7}$)
C. Am I'm loosing something (precision / anything else)/Am I doing something wrong by skipping the step (3)? I get the above equation, which to me, looks much more easier to use to manually compute $\pi$ than the initial one (from step 6)?
Thank you!
R
There's nothing wrong with your formula; see for example this Wikipedia page.
The thing with 1/5 is that it's easy to handle when calculating in base 10, and the thing with 1/239 is that it's a smallish number; the closer to zero $x$ is, the faster the series for $\arctan x$ converges, which means that you need fewer terms in order to get a good approximation.