TRIGONOMETRY. 



hence, by fubftitution, 

 cof. c fin. a = fin. c cof. a cof. B + 

 Dividing by fin. c, we have 



cof. ( 

 fin. ( 



fin. a = cof. a cof. B + 



fin. B cof. C fin. c 

 filTC 



fin. B cof. C 



_ cof. e 



But ;; ^ cot. C. 



fin. C. 

 See Arithmetic o/" Sines. 



Therefore, cot. c fin. a =■ cof. a cof. B -)- fin. B cot. C. 

 Thus, again, we get three fymmetrical equations ; 

 cot. a fin. b = cof. b cof C + fin. C cot. At 

 cot. 4 fin. c = cof. c cof A + fin. A cot. B V ( I V. ) 

 cot. c fin. a = cof. a cof. B + fin. B cot. C J 



The claffesot equations Nos. (I.) (II.) (III.) and (IV.) 

 comprehend the whole of fpherical trigonometry ; or, in faft, 

 N° ( II. )■, from which all the others can be derived, may be 

 regarded as comprehending the whole. They require, how- 

 ever, fome modifications to adapt them to logarithmic com- 

 putations, which we fhall now endeavour to illuftrate. 



I. Solution of right-angled fpherical Triangles. — Let us 

 fuppofe the angle A to be the right angle, then fince fin. 

 A = I, N° (I.) gives 



1 fin. B fin. C ^ , 



; eonlequently 



fin. a 

 fin.B 



fin. b 

 fin. b 



fin. 

 fin. C = 



fin. 



in. a J 



(V.) 



cot. a 



(IX. 



(X.) 



oppofite fide are always of the fame afFeAion ; and in the 

 two latter, the rules for the change of fines in the different 

 quadrants (fee Arithmetic of Sines), will determine to whic-h 

 the refult belongs. 



Cafe 2 — Given the hypothenufe a, and one of the fides, 

 to find the other parts : 



fin. giv. fide 



fin B = ' , or fin. 

 nn. a 



cof. c = — ~, or cof. fide req. = 



cof. hyp. 



fid.- 



cof. b ^ cof. giv. fide 



cof. C = tan. 6 cot. a, or cof. ang. req. = tan. giv. 

 and cot. hyp. 



Cafe 3. — Given two fides, including the right angle, 

 namely b and c, to find the other parts. 



Here, cof. a = cof. b cof. c ; 



or, cof. hyp. = reft, of cof. given fides. 



Again 



tan. B 



tan. b 



tan. 



, and tan. C 



tan. ang. req. = 



tan. c 

 c tan. b 



tan. opp. fide 



tan. adj. fide 



Cafe 4. — Given a fide c, and its oppofite angle B, to find 

 the other parts. 



fin. b . , fin. giv. fide 



f7irB'°'"^'"-^yp-~ 



Here, fin. a — 



fin. a fin. a 



fin. b = fin. B fin. a ; or fin. c = fin. c fir 



Alfo, fince cof. 90° = o, we have from N° (II- ) 



cof. a = cof. b cof. c. (VI.) 



For the fame reafon, Equation i. N° (III.) gives 



cof. a . fin. B fin. C = cof. B cof. C. ( VII. ) 



And upon the fame hypothefis, cot. A becomes = o ; 

 fo that Equation i. N° (IV.) becomes 



cot. a fin. b = cof. b cof. C, or 



fin!'*"^-^'°'^| (VIII.) 



cot. a ^ cot. b cof. C J 

 The Equations z. and 3. of N° (HI.) give alfo upon the 

 fame hypothefis, that is, angle A = 90°, 



cof. B = fin. C cof. b 1 

 cof. C = fin. B cof. c S 



And.laftly, fromN°(IV.) 



cot. B = cot. b bn. c\ 

 cot. C = cot. e fin. b^ 



From thefe equations, by a few obvious transformations, 

 the fix ufual cafes of fpherical right-angled triangles may 

 be folved as follow. 



Cafe I .. — Given the hypothenufe a, and an angle B, to 

 find the other parts. 



Here, fin i = fin. a fin B ; 



or, fin. fide req. := fin. opp. ang. x fin. hyp. 



Again, tan. c =^ tan. a cof. B ; 



or, tan. itde req. =. tan. hyp. x cof. includ. ang. 



Laftly, cot. C = cof. a tan. B ; 



or, cot. ang. req. = cof. hyp. x tan. giv. ang. 



In this cafe there can be nothing ambiguous, for in ap- 

 plying the firll form, it is known that the angle and the 



fin. opp. fide 

 Again, fin. c = tan. b cot. b ; 



or, fin. fide req. = tang. giv. fide x cot. opp. ang. 



cof. giv. ang. 



Laftly, 



iin. t, =: — ;: — -, or iin. req. aiig. 



cof. b 



fide 



cof. giv 



Cafe 5. — Given a fide c, and its adjacent angle B, to find 

 the other parts. 



Here, tang, b ■=. tang. B fin. c ; 



or, tan, fide req. = tang. opp. ang. x fin. giv. fide. 



c . tan. giv. fide 



col 



. tan. c , tan. giv. 



Again, tan. a = — ?-r=,, or tan. hyp. = — ^ 

 ° cof. B col. 



riv. ang. 

 or, cot. a =: cof. B cot. e ; that is, 



cot. hyp. = cof. giv. ang. x cot. giv. fide. 

 Laftly, cof. C =: cof. c fin. B ; 



or, cof. ang. req. = cof. opp. fide x fin. giv. ang. 



Cafe 6. — Given the two oblique angles B and C, to find 

 the reft. 



Here, cof. a = cot. B . cot. C ; 



or, cof. hyp. =: cot. of one fide x cot. of the other ; 



cof. B' 



fin 



cof. C 



of^B"! 



in. C 



? or, 



cof. c =^ 



cof3 = v ... 



r rj fcof. opp. ane- 

 coi. req. iide = -{ Cs £. 



j L fin. adj. ang. 



finTBj 



It may be proper to obferve, that the rule of the figns, 

 given under the article Arlthrttetlc of Sines, will fetve in all 

 thefe cafes to determine the kind or affeftion of the un- 

 known parts. 



In working by logarithms it muft be obferved, that 

 when the relulting logarithm is the logarithm of a quo- 

 tient, 10 muft be added to the index ; and when it is the 

 logarithm of a produft, 10 muft be fubiraSed from the 

 index. 



II. Re/olii. 



