Plane trigonometry, by S.L. Loney.

ERRORS OF OBSERVATION. 463
When 0 is small, both the calculated height and the absolute error,
viz. atan 0 and a sec2 0., are small, but the latter is great compared
with the former.
When 6 is nearly 90~, both these quantities are great.
Ex. 2. The height of a tower is found as in Art. 192; if there be an
error 0 in excess in the angle a, find the corresponding correction to be
made in the height.
The real value of a is a- 0; hence the real value of the height is
found by substituting a- 0 for a in the obtained answer, and therefore
sin (a - 0) sin sin a cos 0 - cos a sin 0
a sin (3 - a + ) sin ( a - a) cos 0+ cos (3 - a) sin 0
a sin a sin ) 1- 0 cot a (Art. 280.)
sin(-a) '1 + 0 cot ( - a)
a sin a sin l3[l, ] lt( )
= asin a sip - E[1 - cot a] [1 - 0 cot (_ - a) +......
- sin (3 -ca)
a sin a sin/ [
= a sin a sin E1 - 0 {cot (p - a) + cot all
sin ( - a)
a sin a sin 03 a sin2 /
sin ( - a) sin2 (/ - a)
a sin2 f
The error in the calculated height is therefore. sin32 ( a) and is
one of excess.
Also the ratio of the error to the calculated height
0 sin j
sin a sin (/ - a)
Ex. 3. The angles of a triangle are calculated from the sides a=2,
b=3, and c =4, but it is found that the side c is overestimated by a small
quantity 8; find the consequent errors in the angles.
From the given values of the sides we easily have
7 11 1
cosA=-, cos B=, cos C= -
S 16' X n sC=
2V15 3/15 4V15
sinA=, sinB=, and sinC= -.
16 16 I6