TRIGONOMETRY. 



Trigonometry, 5^AmtY//, relates to the refolution and 

 calculation of the iides and angles of fpherical triangles, 

 which are formed by the interfeftion of three great circles 

 of the fphere, and which, Uke plane triangles, confiil of fix 

 parts, vi-z. three fides and three angles. See Spherical 

 Triaiig/e. 



In plane trigonometry, any three of the fix parts of a 

 triangle being given, except the three angles, the other 

 parts may be found ; but in fpherical trigonometry this 

 exception has not place, for any three of the fix parts 

 being given, the red may thence be determined, the fides 

 being meafured or eftimatcd by degrees, minutes, &c. the 

 fame as the angles. 



Spherical trigonometry is divided into right-angled and 

 obhque -angled, or the relolution of right and obhque-angled 

 fpherical triangles. When a fpherical triangle has a right 

 angle, it is called a right-angled fpherical triangle ; and 

 when one of its fides is a quadrant, or 90°, it is called a 

 quadrantal triangle. 



The folution of all the cafes of fpherical trigonometry, 

 although much more numerous than thofe in plane trigo- 

 nometry, depends upon only three fundamental equations. 



Theorem I. — In any fpherical triangle, the fines of the fides 

 have the fame ratio as the fines of their oppofite angles. 



Let O {fig 2.) be the centre of the fphere, and having 

 joined OA, O C, OB, draw AD perpendicular to the 

 plane O B C ; alfo make D E perpendicular to O B, and 

 D F to O C ; and join A E, A F. 



Then, becaufe AD is perpendicular to the plane OBC, 

 each of the planes A D E, A F D, which pafs through A D, 

 will alfo be perpendicular to that plane ; and fince E D is per- 

 pendicular to O B, and the plane A D E to the plane OBC, 

 the line AE, which lies in the plane ADE, and is drawn 

 from the fame point E, is alfo perpendicular to OB. 

 Again, in like manner, becaufe FD is perpendicular to O C, 

 and the plane A F D to the plane OBC, the fine F A, which 

 lies in the plane A F D, and is drawn from the fame point F, 

 is perpendicular to O C ; and, therefore, the angles A E D 

 and A F D, which meafure the inchnation of the planes 

 AOB, AOC, will meafure the angles CB A, BC A, of the 

 fpherical triangle ABC. Alfo A F, being perpendicular 

 to O C, is the fine of the angle A O F, or of the arc A C ; 

 and A E, which is perpendicular to O B, is the fine of the 

 angle A O B, or of the arc A B. But A D E, A F D, being 

 xight-angled plane triangles, right-angled at D, we fiiall 

 have AD = AEfin.AED, andAD = AFfin. AFD. 

 Whence, by equality, AE fin. AED = AF fin. AFD ; 

 confequently 



AE : fin. AFD :: AF : fin. AED, or 



fin. AB : fin. opp. < C :: fin. AC : fin. opp. < B ; 

 that is, the fines of the fides have the fame ratio as the fines 

 of their oppofite angles. 



Hence, if A, B, C, be fuppofed to denote the three 

 angles of any fpherical triangle, and a, b, c, correfponding 

 oppofite fides, we may from the above deduce tiie follow- 

 ing fundamental equation, wz. 



fin. A _ fin. B _ fin. C\ 

 fin. a fin. b fin. c J 



(I-) 



Theorem 2. — In any fpherical triangle. 



As the rettangle of the fines of any two fides : 

 Is to the radius :: 



So is the reftangle of radius and the cofine of the 

 other fide, minus the reftangle of the cofines of the 

 fame two fides : 

 To the cofine of the angle included by thofe fides. 

 I 



For, havmg joined OA, B, O C, [fg. 3.) draw F D 

 m the plane OBC, and DE in the plane OAB, each per- 

 pendicular to their common fedion O B, and join E F 

 Then, becaufe the angle EDF is the meafure of the in- 

 chnation of the planes OBC, OAB, it is alfo the meafure 

 ot the fpherical angle A B C or B. And becaufe 



cof. EDF = 



cof. EOF = 



and 



2OE X OF 

 EF^= OE' -f OF'- 2OE X OF X cof.EOF- 



See Form (V.) Plane Trigonometry. 



And by fubftituting this in the firft equation, cof. EDF 



_ DE' + DF'-OE'- O P + 2 OEx OF cof. EOF 

 2 D E x D F 



But OE' - EDS and 0F= - DF% are each equal tp 

 O D " ; whence cof. E D F, or its equal. 



cof. B = 



cof. B = 



OE X OF cof. EOF - OD* 

 D'E~x^DF • 



OE X OF cof. AC - OD' 



"D E X DF 

 I 



Or, fince =-^ = „ „ „ „ 

 DE fin. DOE 



I J OF 



IhTAB' '"'^ D F 



^, &c. if thefe values be fubilituted in the former 



fin. D OF' 

 equation, we fhall have 



_ cof. AC - cof. AB.cof. BC 

 *'°'' ^ ~ fin.AB.fin.BC ' 



which neceflarily involves the conditions given in the enun- 

 ciation of the theorem. 



Here again, afliiming A, B, C, to denote the angles, and 

 a, b, c, the correfponding oppofite fides, we deduce the fol- 

 lowing fet of equations, vi%. 



cof. a = cof. b cof. c + fin. b fin. c cof. A -J 



cof. b = cof. a cof. c + fin. a fin. c cof. B > (II.) 



cof. c = cof. a cof. b + fin. a fin. b cof. C J 



Thefe equations will apply equally to the fupplemental 

 triangle; thus putting for the fides a, b, c, 180° — A', 

 180° - B', 180° - C; and for the angles A, B, C, 

 i8o° — a', 180° - b', 180"^ - c', we fhall have 



- cof. A' = cof. B' cof. C - fin. B' fin. C cof. a'. 



And here, again, we have three fymmetrical equations ap- 

 plying to any fpherical triangles, -viz. 



cof. A = cof. a fin. B fin. C - cof. B cof. Cn 



cof. B = cof. * fin. A iln. C - cof. A cof. C i- (III.) 



cof. C = cof. c fin. A fin. B - cof. A cof. B J 



Another important relation may alfo thence be readily de- 

 duced ; for, fubftituting for the cof. b in the third of the 

 equations N°(.n-) 'ts '^a'"<^ '" ^^'^ focond ; fubftituting alfo 

 for cof.' a, its value i — fin.- a, and then ftriking out the 

 common foftor fin. a, we ftiall have 



cof. c fin. a = fin. c cof. a cof. B h fin. b cof. C. 



fin. B fin. c , 



But equation N° (I.) gives fin. b = — jf^^TcT"' 



hence. 



