TRIGONOMETRY. 



therefore, A D« - B D" = A C ' - BC ; 



confequently, 



AD + DB:AC + BC::AC-BC:AD-DB; 



which is the fame as the theorem in words. 



As an example, let a triangle be propofed {Jig. 2i.), in 

 which the three fides are as follow, vi%. 



A B = 36, A C = 45, B C = 40 

 A C = 45, A C = 45 

 ABr= 36, AB = 36 



AC + AB : 

 Log. of B C 

 Log. of A C 4- 

 Log. of F C 



81 



AB = 



FC = 9 

 1.6020600 

 1.9084850 

 0.9542425 



Sum of log. 



= 2.8627275 



Log. of C G = 1.2606675 Tlic neareil 



correfponding number to which, in the tables, is 18.22 



B C = 40.00 

 CG= 18.22 



EG = 10.89 

 CG = 18.22 



BG= 21.78 CE = 29.H 



BE = 10.89 

 Log. of A B 

 Log. of whole fine 

 Log. of E B 



= 1-5563025 



:= 10.0000000 

 = 1-0370279 



Log. of fin. of E A B = 9.4807254 The corre- 

 fponding number to which, in the tables, is 17* 36'; 

 confequently, the angle ABE 72° 24'. 



Log. of AC = 1. 6532 1 25 



Log. of the whole fine r:^ ic.ooooooo 

 Log. of C E = 1.4640422 



Whence B C ; 



AC 



ABxDF ABx tan 



AD 



AB X AF 



AD 



rad. 



When the hypothcnufc is given, each of the legs will 

 reprefent, or have the ratio of, the fines of their oppofitc 

 angles, the hypothcnufe ilfelf being alTumed for the radius. 



In tills cafe, therefore, it will be 



As radius : 



Is to ths hypothcnufc :: 



So is the fine of cither acute angle ; 



To the oppofite fide. 



AE : AC :: AG 



That 



rad. : AC ::fm.A 



rad. : A C :: fin. B 



rad. 



= BC 



:AB 



Log. of fin. of E AC = 9.8108297 To which 

 the neareft correfpondent number, in the tables, is 40° 18' ; 

 therefore A C E 49° 42', and C A B 57° 54'. 



The three preceding cafes include all the poflible varieties 

 that can arifc in the folution of plane triangles ; but, 

 under certain relations of the data, more fimple operations 

 may frequently be employed. Some of thefe folutions we 

 fhall inveftigate analytically at the conclufion of this article, 

 and it will therefore be fufficient to point out in this place a 

 few particulars relative to the folution of right-angled plane 

 triangles. 



In any right-angled triangle, any of the unknown parts 

 may be found by the following proportions. 



As radius : 



Is to either leg of the triangle :: 



So is the tangent of the adjacent angle : 



To the oppofite leg ; and : : 



So is the fecant of the fame angle : 



To the hypothenufe. 



For AB [Jig. 22.) being fuppofed the given leg, let 

 A D reprefent the tabular radius, defcribe the arc D E, 

 and draw D F perpendicular to A D ; fo (hall D F re- 

 prefent the tabular tangent, and A F the tabular fecant of the 

 angle A ; and becaufe of the parallels, as A D : A B : : D F 

 : BC :: A F : AC, which is the fame as the theorem in 

 words. 



Note The radius is equal to the fine of 90', or to the 



tangent of 45°. 



The preceding theorems have been deduced from the 

 geometrical properties of triangles and of their feveral 

 parts ; and they exhibit the fimplcft and mod dirett mode 

 of folution that can be obtained generally for each cafe ; 

 but there are certain other forms of folution which are 

 much more readily applied under particular relations of the 

 data, which it will be proper now to confider, and in which 

 we (hall adopt the analytical mode of inveftigation inftead of 

 the geometrical one hitherto purfued. 



Let A B C (Plate lU.Jig. I.) be any plane triangle ; C 

 the vertical angle ; C D a perpendicular let fall upon the bafe 

 A B ; and let a, b, c, denote the fides refpcftively that arc 

 oppofite to the angles A, B, C. 



Then becaufe A C = *, A D is the eofine of A to that 

 radius ; confequently, when radius — i, A D = i cof A. 

 In like manner, B D = a cof. B ; therefore A D -1- B D = 

 A B = a cof. B + ^ cof. A. If one of the angles, as A, 

 were obtufe, the refult would ftlll be the fame ; becaufe, 

 while on the one hand cof. A would be negative, A D, 

 lying on the contrary fide of A to what it does in the figure 

 referred to, it muft be dedufted from B D to leave A B ; ar.d 

 a negative quantity fiibtrafted, is equivalent to a pofitive 

 quantity added : and by letting fall perpendiculars from 

 the angles A and B upon the oppofite fides, or upon their 

 continuations, precifely analogous refults will be obtained ; 

 and hence we derive immediately the following fundamental 

 equations. 



a = * cof. C + c cof. B -> 



4 = d cof. C + c cof. A S- (I.) 



c — a cof. B + * cof. A J 



Again, it is obvious, that in the lame manner as we found 

 A D = * cof. A, and D B = a cof. B, we might alio obtain 

 D C = 4 fin. A, and D C = a fin. B ; therefore a lin. 



fin. A 



B =: i fin. A.; whence alfo - = ^r-—. 



fin. A 



B' 

 fin. 



and 



B 



Uke 



manner we 



ing the denominators, the relations of all the fix quantities 

 may be thus exprcfTed : 



fin. A _ fiii.Ji _ (in^l ,jj J 



~^~ I12 ' ■ Thefe 



