Plane trigonometry, by S.L. Loney.

458

TRIGONOMETRY.

386. Natural Cosecant. Just as in the case of the
secant it may be shewn that the differences are insensible
and irregular when 0 is nearly 90~, and irregular when 0
is nearly zero. Otherwise the principle holds.
387. Tabular Logarithmic Sine. We have
L10 sin (0 + k) - L1o sin 0 = logo sin (0 k)
sin 0
=logio [cos k + cot 0 sin k] = logo 1 i+ k cot 0 -2
(Arts. 279 and 280)
k2 1
= cot 0- k2 cot2+... (Arts. 256 and 260)
= /k cot 0- — cosec2 0....
The numerical ratio of the second term to the first
1 1 _ k
2 *sin 0 cos - sin 20'
This is small except when 0 is near zero or a right angle.
Hence, with the exception of these two cases, we have
L sin (0 + k) - L sin 0 = J cot 0 x k,
so that the rule holds in general.
If 0 be small the term,k cot 0 is large, so that the
differences are large as well as irregular. We cannot
therefore apply the principle to small angles in the case
of tables constructed with difference of 1'.
Even if the tables were constructed for differences of
10" we are not sure of being free from error in the 7th
place of decimals unless 0 be > 5~.