Plane trigonometry, by S.L. Loney.

462

TRIGONOMETRY.

considering; whilst in measuring the distance from the
Earth to the Moon an error of one inch would be absolutely inappreciable.
394. We shall assume that the errors we have to
consider are so small that their squares (when measured
in radians if they be angles) may be neglected and we
shall give some examples of finding the errors in derived
quantities.
We shall assume that our tables and calculations are
correct, so that we have not to deal with mistakes in
calculation but only with errors in the original observation.
395. Ex. 1. MP (Fig. Art. 42) is a vertical pole; at a point 0
distant a from its foot its angular elevation is found to be 0 and its height
then calculated; if there be an error 6 in the observation of 0 find the
consequent error in the height.
The calculated height h=atan 0, clearly.
If the error 6 be in excess, the real elevation is 6- 6, and hence the
real height h' = a tan (0 - ).
Hence the error h - h' = a tan 0 - a tan (0 - 6)
sin 6
= a - a sec2 6,
cos 0 cos (0 - ) 0
if we neglect squares and higher powers of 6.
The ratio of the error to the calculated height
= sec2 0- tan 0=i20
sin 20
Except when sin 20 is small this ratio is small since 6 is small. It is
least when sin 20 is greatest, i.e. when 0 is
The ratio is large when 0 is near zero and when it is near.
Hence a small mistake in the angle makes a relatively large mistake
in the calculated result when the angle subtended is very small or when
it is very nearly -.