Plane trigonometry, by S.L. Loney.

400

TRIGONOMETRY.

become small, so that a very large number of terms would
have to be taken to obtain the value of 7r correct to any
great degree of accuracy.
For this reason other series have been sought for.
347. Euler's Series. We can easily prove that
tan-l + tan-l =.
In Art. 344 put in succession x equal to
1 1
- and -
2 3'
and we have
vr tan_1 1
tan-l + tan-l
4 2 3
1 1 1 11 11
2 3'23+ 5 7227
1 1 1 1 1
3 3 53 35 7 37
This series converges more quickly than the preceding
series; but more than eleven terms of the series for
tan-l would have to be taken to give 7r correct to 7
places of decimals.
348. Machin's Series. A more convergent series
than the preceding is Machin's, which is derived from the
expression
1 1 (Artr 240
4 tan-' — tan-' - 4 (Art. 240, Ex. 4).
5 239 4