Chapter 11.P, Problem 7P

(a) Show that for $xy\ne -1$,

$\text{arctan }x-\text{arctan }y=\text{arctan }\frac{x-y}{1+xy}$

if the left side lies between $-\pi /2$ and $\pi /2$.

(b) Show that $\text{arctan }\frac{120}{119}-\text{arctan}\frac{1}{239}=\pi /4$.

(c) Deduce the following formula of John Machin $\left(1680-1751\right)$:

$4\text{arctan}\frac{1}{5}-\text{arctan}\frac{1}{239}=\frac{\pi }{4}$

(d) Use the Maclaurin series for arctan to show that

$0.1973955597<\text{arctan}\frac{1}{5}<0.1973955616$

(e) Show that

$0.004184075<\text{arctan }\frac{1}{239}<0.004184077$

(f) Deduce that, correct to seven decimal places, $\pi \approx 3.1415927$.

Machin used this method in 1706 to find $\pi$ correct to 100 decimal places. Recently, with the aid of computers, the value of $\pi$ has been computed to increasingly greater accuracy. In 2013 Shigeru Kondo and Alexander Yee computed the value of $\pi$ to more than 12 trillion decimal places!