Plane trigonometry, by S.L. Loney.

398 TRIGONOMETRY.
Comparing this with equation (1) we see that n equals
- 1 when 0 lies between
37r 07r
and
4 4
If 0 lie between 7 and it may be similarly shewn,
by putting 0 equal to 27r + a, that in this case n is equal
to - 2.
In general if 0 lie between p7r - and pr +4, the
equation (1) of Art. 340 is
I 1
0- pvr = tan 0 - tan3 0 + tan5 0 -......
o3 5
344. The series (2) of Art. 340 may be slightly
transformed by writing tan 0 = x, so that x must be not
less than -1 and not greater than 1.
It then becomes
tan-'x=x-. x+ 5x5- iX7+...... ad inf.,
'3 5 7
where tan-l x is that value which lies between
- -and + -
4 4
345. The results of the preceding articles may be more directly
connected with the preceding chapter in the following manner.
If tan 0 be numerically less than, or numerically equal to, unity we
have, by Art. 336,
1 1
Log (1 + i tan )= 2pi+ i tan0 -i2tan20+ i3 tan3...,
2 3
and Log(1 - itan )= 2q7ri - i tan0n-i tan2 0 3 tan3 0...,
2
where p and q are both integers.