TRIGONOMETRY. 



an angle and its fupplement have the fame fine ; therefore, 

 the fine we have found may either anfvyerto the angle 64° 24', 

 or to its fupplement 115° 36'. This queftion, therefore, 

 involves that ambiguity to which we have above alluded ; 

 and we muft proceed to find the third fide under each of the 

 values of the angle C, and the two correfponding values of 

 the angle B. 



Since the fum of the three angles of a triangle = 1 80°, 



< A = 37° 20' 



< C = 64 24 



180" 



(loi 44) = 78° 16' = < B. 



Again, 



<A = 37° 20' 

 <C = 115 36 



,So° - (152 56) = 27° 4' = < B. 



Hence, from the firft values of the angles C and B, we 

 have. 



As fin. < A = 37''2o' 9.7827958 

 Is to fin. < B = 78^ l6' 9.9908291 

 Sois BC = 232 2.3654880 



To AC =374-56 2.5735213 



Again, affuming the fecond values of the angles B and C, 

 we have, 



As fin. < A = 37'2o' 9.7827958 

 Is to fin. < B = 27'=' 4' 9.6580371 

 Sois BC = 232 2.3654880 



To AC 



= 174.07 2.2407293 



Tlie fide and angles fought are, therefore, A C = 374-5^' 

 or 174.07 ; and the angle B = 78° 16', or 27° 4'; and 

 angle C — 64° 24', or 1 1 5° 36' ; either of which refults 

 equally anfvver .all the conditions of the original data. But 

 it is obvious, that if the given angle A had been a right 

 angle, or greater than a right angle, this ambiguity could 

 not have had place ; becaufe, in that cafe, the other two 

 angles are neceffarily acute. 



Cafe 2. When two fides and their included angle ar»the 



three given parts. 



As the fum of the fides : 



Is to the difference of the fides :: 



So is the tangent of half the fum of the reqixired angles : 



To the tangent of their half difference. 



Then to half the fum add half the difference for tht 

 greater angle, and fubtraft it for the lefs. The three angles 

 of the triangle being thus known, the required fide may be 

 found by Cafe I. 



Note The half fum of the angles is found by fubtrafting 



the given angle from 180°, and taking half the remainder ; 

 or, inftead of the tangent of the half fiim, the cotangent of 

 half the given angle may be ufed, being the fame thing. 



Let ABC [fg. :8.) be any triangle; produce A B, 

 making B E = B C ; alfo take B D = B C ; join D C and 

 C E, and draw D F perpendicular to D C. Now fince 

 D B, B C, and B E, are all equal to each other, a femi- 

 circle defcribed from the centre B, and with the radius D B, 

 would pafs through D, C, and E ; confequently D C E is a 

 right angle, or C E is perpendicular to D C, and is there- 

 4 



fore parallel to D F ; and hence D F and E C are refpec- 

 tively the tangents to the angles D C F, and C D E to the 

 fame radius DC. But C D E or C D B = half the fum 

 of the angles B A C and B C A ; for fince D B = B C, 

 the angle B D C is obvioufiy ^ to half the external angle 

 C B E, which is equal to the fum of the angles at A and C ; 

 therefore, C D B is = to half that fum, and D C F = half 

 the difference of the fame angles A C B and CAB; for 

 ACB=BCD + DCA, and confequently BAC = 

 BDC, or BCD - DCA; therefore DC A = half 

 their difference. Whence C E = tangent of half the fum, 

 and D F the tangent of half the difference of the angles A 

 and C ; and it is evident from the conft;ruftion that A E =: 

 the fum of the fides, and A D := the difference of the 

 fame ; confequently, fince D F and C E are parallel, vre 

 have 



AE : AD :: CE : DF; 

 that is. 



As the fum of the fides A B - B C : 



Is to the difference of the fides A B — B C :: 



So is the tangent of half the fum of the angles A and C : 



To the tangent of half their difference. 



Let us take as an example a triangle {Jig. 19.% in which 

 the following dimenfions are given, viz. 



A B = 75, A C = 58, A = 108° 24' ; 



then will AB = 75 AB=:75 A + B + C = 180° o' 

 AC =58 AC = 58 A = 108 24 



Sum 133 differ. 



BC 



7' 36 



^(B + C)= 3548 



Log. ofAB + AC, 2.1238516 

 Log. of A B — A C, 1.2304489 

 Log. oftan. i (B + C), 9.8580694 



Sum of log. 1 1. 0885 1 83 



Log. of tan. i(C-B) 8.9646667 The neareft 

 correfponding number to which is 5° 16'. 



i(B + C) = 35° 48' i(B + C) = 35° 48' 



i(C-B) = 5° 16' i(C-B) = 5°:6' 



C = 41° 4' 



B = 30° 32' 



If the other fide C B were required, having found the 

 angles, it may be eafily determined by the firfl cafe. 



Cafe 3. — When the three fides of ?. triangle are given, to 

 find the three angles. 



Affume any fide of the triangle as a bafe (Jig. 20. ), and let 

 fall a perpendicular upon it from the oppofite angle ; then fay, 



As tht bafe : 



Is to the fum of the other two fides :: 



So is the difference of the fame fides : 



To the difference of the fegments of the bafe. 



To the bafe or fum of the fegments add the half difference 

 for the greater fegment, and fubtraft it for the lefs. 



The fegments being thus found, the angles may be de- 

 termined by the firft cafe. The demonftration is here very 

 obvious ; for, by the 47th propofition of the firll book of 

 Euclid, 



AD' + DC = AC 

 BD^ + DC^ = BC^; 



therefore. 



