TRIGONOMETRY. 



fin. C = 



fin. c fra. B 



fin. C fin. b 

 ^'"- ' = — finTB- 



Cafe 2. — Of the four following things ; w'z. a, b, c, and 

 an angle, any three being given, to find the fourth. 

 This cafe comprifes three problems. 

 I . When the three fides are given, to find an angle. 

 Here, from equation N° (II.) we have 



. . cof. a — cof. b cof. c 

 cof. A = — 



\A. 



II. Refolution of oblique-angled fpherual Triangles. — This to the formula: given und«r the article Sine, where we 

 may be effefted by means of four general cafes, comprehend- have 

 ing two or more problems. 



Cafe I Given three of thefe four things, iijz. two fides 



b, c, and their oppofite angles B and C, to find the fourth. 



This cafe comprehends two problems, in one of which 

 the unknown quantity is an angle, in the other a fide-; 

 which are both refolved by means of equation N° (I.) from 

 which we have 



fin. b fm. c 



The latter of thefe, divided by the former, gives 

 cof. [b — c] — cof. « 



tan. 



^A 



cof. B = 



cof. C 



fin. b fin. c 

 cof. b — cof. a cof. c 



cof. a — cof. {b + c 

 which is equivalent to 



fin. ^{a + b- 



fin. a fin. c 

 cof. c — cof. a cof. b 



, . 2 \- • - -, -i y" + <: - l) 



tan. 2 A - p,„_ i (-^ ^ ^ ^ ^) gn, L[b j^ c- a) 



fin._i_ 



See SiNE.s. 



Hence we have, for the tangents of the half angles, 



fin. a fin. b . . - . 



In this form, however, the equations are not adapted thefe three fymmetrical equations ; vi% 

 to logarithmic coraputatioo. We mult therefore refer 



tan. i A 



V 



Ifin. i (a 



+ b 



fin. i 



[a + c-b)\ 



+ b + 



tan. i B = y 



C = V 



S fin. \ [a +.,L::L-'iUlll:- 



li^n. \ [a + b + c] fin. 

 Ifi 



h{b + 

 h (b + 



- b\ 



m. 



^ {a + c - Jb)Jitu 

 i (a + i'+'O fin- 



k[b + c - a) 



The expreflions for the fines of the half angles nnight be 

 obtained with equal facility. As they are^ymmetrical, we 

 fhall put down but one ; -viz. 



/fin. i fa 4- i - c ] fin. 1 (a -^ c - b) 

 hn. i A = y/ ^ fin. b fin. c 



+ b - 



cof. c = 



(XI. 



cof. b cof. [a — a) 



I 



cof. ip 

 -The equation N° (XII. 



(XIII.) 

 obvioufly reduces to 



tan. (p 



of. C = 



tan. b 



cot. 2> cof. C = cot. b ; which 



half 



forms cof. = 



cof. 



fin. 



, cot. = -f 



tan. nn. 



When two fides, as * and c, become equal, the expreffion 



And expreflions for the cofines and cotangents of the .^ ^„^ to jsjo (yill.) So that b is the hypothe- 



If angles may be readily found from the above by the ^^^.^^ ^^^ ^ ^^^ j^^ ^^ ^ right-angled triangle. The above 



transformation, therefore, is equivalent to the divifion 

 of the propofed triangle into two, by an arc from the 

 vertical angle A falling perpendicularly upon the oppofite 

 fide a. 



3. To find the fide a, not oppofite to the given angle ; 

 b, c, and C being given 



Here find^%as before, by N°(X1I. ) then from N^ XIII. ) 



we have 



cof. c cof. $ , v-Tir I 



cof. (a - <P] = r-T— • (XIV.) 



fin. -J a 



for fin. -iA, becomes fin. -lA = -^-j • 



U a = b = c = 90', then fin. 5 A = 



* ^/2 



2 A.' 



= ■§^2 



= fin. 45° ; and A = B = C = 90°. 



Leaving other corollaries to be deduced by the reader, 

 let us proceed to the next problem in this cafe. 



2. To find the fide c oppofite to the given angle C : that is, 

 given two fides and the included angle, to find the third fide. 

 Find from the data a dependent angle (p fuch, that 

 tan. ?> = cof. C tan. i. (XII.) 



Subftitute for cof. C, in the third equation N° (II.) its 

 lalue in this, and it will become 



cof. c = cof. a cof. b + fin. a cof. * tan. (fi 

 . , /cof. a cof. <? + fin. a fin. 0^ 



= '°^- * l" ^d7^ J "' 



cof. b 



Hence a is known by adding f. 



Cafe. 3 Of the four following parts ; viz. two fides, 



a and c, and two angles, B and C, one oppofite, the other 

 adjacent ; three being given, to find a fourth. 



This cafe prefents four problems. 



I. Given a, c, B, to find C. 



Determine an arc ?' by this condition, 



, „ cot. c , 



cot. c = cot. ?' cof. B, or ^^j^ = cot. ^'. 



(XV.) 

 Subftitute 



