130 Mr WALLACE, ON GEOMETRICAL THEOREMS, AND FORMULAE. 



By assuming that cot 6 = cot a — cot (p, we find that 



. „ ^ ^ sin a sin ^, ^ , sin a sin 9 



if tan 9 = -7 — ; ^ ; then tan d) = —. — ^ . 



sin {<p -a) ^ sin (9 — a) 



Now, make a = (p + \l^ in the first of these formulse, and a = <p — ^ly in tlie 

 second, and we get these other formulte. 



Eighth. If tan 9 =■ ^^^ . sin (0 + >/') ; 



then tan = 



sin 9 



sin (9 + + v//) 



, sin (0 + \|/) 



8. No. 1. 



If tan(?= *4^.sin(0-x/.); 

 sin \// r 1 / 



sin • / , , \ 



then tan = -. — rr — r- pry . sin {<p - f)} 



^ sm {9-{<p-^)} 



8. No. 2. 



30. We may apply the eighth formula to the case of Trigono- 

 metry, in which there are given two sides a and i of a triangle, and C 

 the angle they contain, to find A and B, the two remaining angles, 

 of which A is opposite to A, and B to b. 



Find 9 so that tan = ,- sin C. 



b 



Then tan A = 



sin 9 



sin {C-9) 



sin C 



This formula is particularly applicable when an angle and the loga- 

 rithms of the sides containing it are given. 



To verify the solution, try if ^? „ = A ^^ ^^^^^ logarithnis. 



31. I shall now apply to the example (Art. 20.) one of the formulas 

 for determining the angles which the lines x, y, x (Fig. 5.) make with 

 the lines joining the stations: and for this purpose select No. 1. of 

 Formulae viii. 



