Plane trigonometry, by S.L. Loney.

PRINCIPLE OF PROPORTIONAL PARTS.

455

This ratio is small, except when 0 is nearly equal to o.
Hence, except when the angle is nearly a right angle, the
second term in (1) may be neglected, and we have
sin(O + k) - sin = k cos 0.
So sin (O + h) - sin 0 = h cos 0,
ad h e sin (0 + k) - sin 0 k
and hence sin ( + )-sin h.................. (3).
sin (0 + h) - sin 0 h. v /
When 0 is very nearly a right angle we cannot say
that
sin (0 + k) - sin 0 = k cos 0,
and hence in this case the relation (3) does not hold and
the difference in the sine is not proportional to the
difference in the angle. In this case then the differences
are irregular. At the same time the differences are
insensible; for, when 0 is nearly, k cos 0 is very small.
In fact k cos0 has nothing but ciphers as far as the
seventh place of decimals, so long as 0 is within a few
minutes of a right angle. Also
k2 (00
- sin 0 is always < 0003) i.e. < 00000005...
Hence when the angle is nearly a right angle a comparatively small change in the sine will correspond to a
comparatively large change in the angle; also at the same
time these changes are irregular.
382. Natural Cosines. Since the cosine of an angle
is equal to the sine of its complement this case reduces to