Plane trigonometry, by S.L. Loney.

CALCULATION OF 7r.

399

Hence, by subtraction,
Lo1 +itanO
Log t = 2 (p q) r+ 2i [tan 0 - tan3 0+......(1).
I - 1 tan 0 a GS^+ l ~
But Log l+ i tan0 [cos 0 + i sin O
1 - i tan 0 [os 0 - i sill
= Log (cos 0 + i sin 0)2= Log [cos 20 + i sin 20]
= 2r ri + i 28................................................... (2),
where r is an integer.
Some one of the values of the right hand of (1) must therefore be
equivalent to some one of the values on the right hand of (2).
Hence, by equating and putting r-p+q=n, we must have, for some
integral value of n, the relation
1 1
8 + n7r = tan 0 - 3 tan3 0.+ tan5 -....
3 5
If we consider principal values only of the logarithms then in (1) both
p and q are zero and tan 0 is numerically less than unity.
Also, by Art. 333, the value of r in (2) is zero and 0 lies between
7r 7r
-- and +
2 22
Combining these two statements we see that'p, q, and r are zero, and
therefore n is zero, when 0 lies between - and + -
4 4*
346. Value of 7r. One of the chief uses of Gregory's
series is its application to find the value of 7r.
In Art. 344 put x= 1, and we have
rr '1. 1 1' 1
4 3 5 7 9
24 1:3+^5_7+9......
=1 - 3 5 (7 9+ 111+ 13...A
= 1 -2 [3 i+779 +11.13 +. 1 J
This series may be used to calculate vr; its defect
however is that the successive terms do not rapidly