31151029036484.pdf

226 SPHERICAL TRIGONOMETRY.
i sin 3 = cos 4
(k positive)
k cos 3 = sin 4 cos 4
cos 3' = cos 3 cot a tan 4
(3' < 180° with the sign of cos P)
(7 = 9 -f 3'
(215)
To find c, we observe that sin 0' has the sign of sin a cos P, so that we have the
foUo-wing formulse:
/c sin 0 = sin 4 cos A (k positive)
k cos 0 = cos 4
cos 0 cos a ,„,„,
(0' < 180° with the sign of sin a cos P)
c = 0 4- 0'
Gheck. The equation (214).
128. CASE IV. Given 4, P and 4. First Solution,- when the three remaining parts
a, c and 0 are all required.
We find a by the equation
Bin A sin 4
sin P
which is determinate when the sign of cos a is given.
is by (211) and (212).
(2171
The remainder of the solution
124. CASE IV. Given 4, P and 4. Second Solution; when 0 and G are required,
without finding a.
We easily find, from (211),
A sin 3 = cos A
(k positive)
k cos 3 = sin A cos 4
And from (212),
CWeA The equation (214)
sin 9' , sin 9 cos P
cos 4 - (218)
(COB 3' and sin P cos a to have the same sign)
C =
9-}-y
A sin 0 = sin 4 cos 4
k cos 0 == cos 4
sin 0' = sin 0 tan 4 cot P
(cos 0' and cos a to have the same sign)
c = 0 4- $'
125. CASE V. Given a, 4 and c. The formula
cos a — cos 4 cos c
cos 4 =
sin 4 sin c
(A positive)
(219)
(220)
determines A when the sign of sin 4 is known. If the sign of sin P or of sin 0 if
gi fsn, that of sin A becomes known by the equation
sin 4
sin a
sin P
sin 4
sin 0
sin c