# Why is pi = 16 arctan(1/5) - 4 arctan( 1/139)

1. Jan 3, 2006

### joex444

Not actually a HW question, but, is there a quick explanation that could show why:

$$\pi = 16\arctan(\frac{1}{5}) - 4\arctan(\frac{1}{239})$$

I read that somewhere, and it turned out to be true, so I was just wondering how that came about...

2. Jan 3, 2006

### Integral

Staff Emeritus
We have the series expansion:
$$Tan^{-1}(x) = \frac {\pi} 2 - \frac 1 x + \frac 1 {3x^3} - \frac 1 {5x^5} ....$$
By working with this you should be able to verify your claim. It is not clear to me whether your relationship is exact or just good to more decimal places then are being displayed.

Last edited: Jan 3, 2006
3. Jan 3, 2006

### DarkEternal

http://mcraefamily.com/MathHelp/GeometryTrigEquiv.htm [Broken] (see Machin's formula)

Last edited by a moderator: May 2, 2017