766
TRIGONOMETRY.
Case
Given
Sought
Solution
7
One leg AB and theadjacent angle A
The other leg
BC
As radius : sine AB :: tang. A : tang. BC (by theo-rem 4.)
8
One leg AB and theadjacent angle A
The opposite angle
C
As radius: sine A :: co-sine of AB : co-sine of C (bytheorem 3.)
9
One leg AB and theadjacent angle A
The hyp.
AC
As co-sine of A : radius:: tang. AB: tang. AC (bytheorem 1.)
10
One leg BC and theopposite angle A
The other leg
AB
As tang. A : tang. BC :: radius ; sine AB (by the-orem 4.)
11
One leg BC and theopposite angle A
The adjacent angle
C
As co-sine BC ; radius :: co-sine of A : sine C (bytheorem 3.)
•
12
One leg BC and theopposite angle A
The hyp.
AC
As sine A ; sine BC :: radius : sine AC (by the-orem 1.)
13
Both legs
AB and BC
The hyp.
AC
As radius : co-sine AB :: cosine BC :: co-sine AC (bytheorem 2.)
14
Both legs
AB and BC
An angle, suppose
A
As sine AB : radius :: larig. BC : tang. A (by the-orem 4.)
IS
Both angles
A and C
A leg, suppose
AB
As sine A •. co-sine C :: radius : co-sine AB (by the-orem 3.)
16
Both angles
A and C
The hyp.
AC
As tang. A : co-tang. C :: radius : co-sine AC (bytheorem 4.)
The Solution of the Cases of Oblique Spherical Triangles, (fig. 15 ) and 20. J
Case
Given
Sought
Solution
1
Two sides AC, BC, andan angle A opposite toone of them
The angle B oppo-site to the other
As sine BC : sine A : ; sine AC : sine B (by theorem 1.) Note, thiscase is ambiguous when BC is les than AC ; since it cannot be deter-mined from the data whether B be acute or obtuse.
2
Two sides AC, BC, andan angle A opposite toone of them
The included angleACB
Upon AB produced (if need be) let fall the perpendicular CD ; then(by theorem 4,) rad. : cosine AC : : tang. A : co-tang. ACD ; but(by theorem 1,) as tang. BC : tang. AC :: co-sine ACD ; co-sine BCD.Whence ACB = ACD = BCD is known.
3
Two sides AC, BC, andan angle opposite toone of them
The other side
AB
As rad. : co-sine A : : tang. AC : (ang. AD (by theorem 1 ;) and by(theorem 2,) as co-sine AC : co-sine BC : : AD ; cosine BD. Note,this and the last case are both ambiguous when the first is so.
4
Two sides, AC, AB,and the includedangle A
The olher side
BC
As rad. : co-sine A : : tang. AC : tang. AB (by theorem 1,) whenceAD is also known ; then (by theorem 2,) as co-sine AD : co sine BD: : co-sine AC : co-sine BC.
5
Two sides AC, AB,and the includedangle ^
Either of the otherangles suppose
B
As rad. : co-sine A : : tang. AC : tang. AD (by theorem 1,) whenceBD is known; then (by theorem 4,) as sine BD ; sine AD : : (ang. A:tang. B.
6
Two angles A, ACB,and the side ACbetwixt them
The other angle
B
As rad. : co-sine AB : ; tang. A : co-tang. ACD (by theorem 4,)whence BCD is also known ; then (by theorem 3,) as sine ACD : sineBCD : co-sine A : co-sine B.