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(a) Show that for xy ( - 1.
arctan x - arctan y = arctan x - y/1 + xy
If the left side lies between - (/2 and = (/4.
(b) Show that arctan 120/119 - arctan 1/239 = (/4.
(c) Deduce the following formula of John Machin (1680 - 1751):
4 arctan 1/5 - arctan 1/239 = (/4
(d) Use the Maclaurin series for arctan to show that
0.1973955597 < arctan 1/5 < 0.1973955616
(e) Show that
0.004184075 < arctan 1/239 < 0.04184077
(f) Deduce that, correct to seven decimal places, ( ( 3.1415927.
Machin used this method in 1706 to find correct to 100 decimal places. Recently, with the aid of computers, the value of has been computed to increasingly greater accuracy. In 2009 T. Daisuke and his team computed the value of to more than two trillion decimal places! |
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