31151029036484.pdf

DIFFERENCES OF PLANE TRUNQLES. 107
This equation gives by (135)
_ J A5 _ a
sin 1 AC cos 1 AC " siir(CTXC)
and dividing (380) by this
Aa _ cos(C4-iAC)
A6 cos i AC
(383)
It is to be observed that the increments (or half increments) of
the angles must be deduced from their sines or tangents, since it is
only by these functions that a small angle can be accurately determined.
Moreover, a small arc being nearly equal to its sine or tangent,
the equations (380), (381) and (382) express very nearly the
ratios of the increments of the sides to the increments of the angles,
or rather to those increments reduced to arc by Art. 9, or Art. 54.
198. CASE II. A and a constant. We have as in the preceding
case AB = — A C; and in the two triangles •
b ain A = a sin B
(5 4- *A b) sin A = a sin {B 4- AB)
the difference and sum of which give
I Ab ain A = a cos (-S 4- J A.B) sin J AB
(5 4- i Ab) sin A = a sin {B -\- I AB) cos J AB
whence by di-vision
lAb ^ _ i-A5 ^ b-^^Ab
tan 1 AB tan J AC ~ tan (-B 4- i AB)
{p)
(384)
In the same way
hAe I'-C _ e4-iAc
taniAC tanjA^ tan (C 4-i AC)
(385)
From the equations
c sin A = a sin C
{c 4- Ac) sinA =• a sin (C 4- AC)
we find I Ac sin A = a cos (C 4- J AC) sin | AC {q)