TRIGONOMETRV'. 



The tangent of an arc, is a line toucliing the circle in one 

 extremity of that arc, and continued thence to meet a line 

 drawn from the centre through the other extremity of it, 

 which laft line is called x}\e fecant oi the fame arc : thus, A H 

 is the tangent, and C H the fecant of the arc A B ; alfo 

 E I is the tangent, and C I the fecant of tlie fupplemental 

 arc B D E ; and this latter tangent and fecant are equal to 

 the former, but are accounted negative, as being drawn in 

 an oppofite or contrary direftion to the former. 



The cojlne, cotangent, and cofecant, of an arc, are the fine, 

 tangent, and fecant of the complement of that arc ; the 

 letters co. being only a contraction of the word complement. 

 Thus, the arcs A B, B D, being the complement of each 

 other, the fine, tangent, and fecant of the one of thefe, is 

 refpeftively the cofine, cotangent, and cofecant of the 

 other : thus, 



B F, the fin. of A B, is the cof. of B D. 



B K, the fin. of B D, is the cof. of A B. 



A H, the tan. of A B, is the cotan. of B D. 



D L, the tan. of B D, is the cotan. of A B. 



C H, the fee. of A B, is the cofec. of B D. 



C L, the fee. of B D, is the cofec. of A B. 



FA, the verf. of A B, is the coverf. of B D. 

 D K, the verf. of B D, is the coverf. of A B. 



The above are the principal definitions relating to plane 

 trigonometry, as far as regards the folution of plane triangles, 

 which is that part of the fubjeft to which we muft more 

 particularly confine our remarks in this article ; what con- 

 cerns the mutations or changes in the quantities above 

 defined in pafling fuccellively round the circumference, and 

 their particular values at certain points, have been already 

 explained under the article Arithmetic of SlviES, to which the 

 reader is referred. In every triangle there are fix parts ; 

 ■c/z. three fides and three angles, any three of which being 

 given (except the three angles), the other three may be 

 found ; and that either by geometrical conilruftion, arith- 

 metical or logarithmic computation, or by inftrumental 

 operation : that is, either by conftriifting the figure with the 

 fompaffcs and a fcale of chords, or other inftrument for 

 meafuring angles ; or by means of tables of natural or 

 logarithmic fines, tangents, &c. in which the computation 

 depends upon the proportionality of the fides of fimilar 

 triangles ; and laftly, by means of a Gunter or other fcale 

 conftrufted for this particular purpofe, by which the refults 

 are obtained by the proper application of a pair of compaffes 

 to certain lines on the fcale : in the prefent article, however, 

 we fhall only attempt an illuftration of the fecond method. 



There are only three diftinft cafes in trigonometry ; viz. 



1. When a fide and its oppofite angle are two of the 

 given parts. 



2. When two fides and the included angle are given. 

 ^. When the three fides are given. 



Cafe I. — When a fide and its oppofite angle are two of 

 ihe given parts. 



As any one fide : 



Is to any other fide :: 



So is the fine of the angle oppofite to the former : 



To the fine of the angle oppofite to the latter. 



This analogy fuppofes two fides and one angle to be 

 given : if two angles and one fide be given, the order of the 

 terms becomes, 



As the fine of any angle : 

 Is to the fine of any other angle :; 

 So is the fide oppofite to the former : 

 To the fide oppofite to the latter. 

 Vol. XXXVI. 



For let ABC (/^. 15.) reprefent any triangle; 

 take Ac, Be, equal to each other, and let them re- 

 prefent the tabular radius; then ca, cb, will alfo be the 

 tabular fines of the angles A and B. Now by fimilar 

 triangles, 



AC : Ac :: CTi : ca 

 BC : Be :: CD : cb. 



Confequently, fince A c = B f, and the third terms being 

 equal, we have 



AC : BC :: cb : ca, or 

 AC : BC :: fin. B : fin. A; 



which is the fame as the theorem in words. 



Hence in the triangle A B C {fg. 16.), let A B, BC, 

 and the angle A be given ; then it will be 



BC : BA :: fin. A : fin. C, 

 B A X fin. A 



or fin. C = 



BC 



or log. fin. C = log. B A -J- log. fin. A - log. B C 



Again, if the angles A and B were given, and the fide 

 BC; then^ 



fin. A : fin. B :: BC : AC, 



fin. B X B C 



or AC 



fin. A 



or log. A C = log. fin. B -I- log. B C — log. fin. A. 



It (hould be obferved here, that the angle found by the 

 firft of thefe analogies is ambiguous, or uncertain, viz. 

 whether it be acute or obtufe, unlefs its magnitude be fuch 

 as to prevent the ambiguity ; for when this is not the cafe, 

 there will be two different triangles, which have the fame 

 three parts, but the other three different in each ; and there 

 is nothing in the abflraft folution to determine which of the 

 two is the reqmred one : but in any praftical cafe, there 

 will be always found fome circumftance or other to de- 

 cide the queflion. This will be underftood from_/ff. 17, 

 where there are two triangles A B C, ABC; in each of 

 which the fides A B, B C, or B C, and the angle A, are 

 the fame ; and, therefore, the angle C, determined by the 

 analogy, maybe either BCA or BC'A, which are the 

 fupplements of each other, and which two angles, we have 

 (hewn in the definitions, have always the fame fine. The 

 tabular fine, however, is always that belonging to the acute 

 augle ; and, therefore, when the obtuie angle is required, 

 the acute angle muft be fubtrafted from 180'', which will be 

 the obtufe angle. But if the angle be a rigiit angle, or 

 greater than a right angle, the acute angle found by the 

 table muft be the required angle ; and, therefore, in this 

 cafe, there is no ambiguity. 



Let us propofe, as an example, a triangle ABC, in 

 which the fide A B = 345, B C = 232, and angle A = 



Sf 20'. 



Firft, to find the angle at C. 



As the fide B C = 232 2.3654880 



Is to fide A B =345 2-5375>9J 



So is fine < A =37° 20' 9.7827958 



Tofin. <C =64" 24' 9.9551269 



That is, by adding togetlier the logarithms of the fecond 



and third terms, and fubtradting the firft, we obtain the 



logarithmic value of the fine of < C, whicli is found ni the 



table to anfwer to the angle 64" 24'. But we have fcen tliat 



LI an 



