SINES. 



may, therefore, from what has been faid, draw the following general conclufion. 

 circumference, and making rad. =; i ; then. 



Let V be taken to reprefent the fetni- 



In addition to the precedinjT deductions, it may not be definitions themfelves, it will, we prefume, be deemed quite 



amifs to refer, in this place, to the relations fubfiftinir be- fiifficient, if we exhibit them only, without going over the 



tween the different trigonometrical lines ; but as thefe are operations by which they are obtained. The moft im- 



mere fimple analogies, which follow immediately from the portant of thefe are as follow, -viz. 



cof. a 



fin. 



3f. a = 



We may now eftablifh our fundamental propofition, 

 which, ill words at length, may be ftated thus. 



Theorem. — The fine of the fum of two arcs is equal to 

 the fum of the produfls of the fine of each, multiplied by 

 the cofine of the other, divided by radius ; or, according 

 to our notation, in which we denote one arc by a, and the 

 other by b, and make the radius = i, it becomes 



fin. {a + b) — fin. a . cof. b + fin. b . cof. a. 



Let A B (Jig. 9. ) reprefent the arc a, and A C the 

 arc i ; B D and O D being the fine and cofine of the former, 

 apd A F and O F the fine and cofine of the latter ; alfo 

 B G the fine of the fum ; we have to prove that 



OA.BG = BD.OF + AF.OD. 



Draw D H parallel to O C, or perpendicular to B G ; 

 thcnthe three right-angled triangles, HBD, ODE, OAF, 

 are fimilar, and we have 



AO : OF : 

 AO : OD : 



BD 

 AF 



BH 



D E or H G. 



BD.OF + AF.OD, 

 . OF + AF . OD. 



Alfo, 



Confequently, 



OA.BH + AO.HG=3 

 orOA.BG^BD 



And confequently, when the radius O A = 1, we have 



fin. [a + b) = fin. a . cof. b -f- fin. b . cof. a. 



This one general formula would be fufficient for all our 

 purpofe, in eftablifhing the doctrine of fines ; but we prefer 

 drawing alfo a fecond, as it follows immediately from the 

 fame principles as the above ; viz.. 



cof. (a + b) =. cof. a . cof. b — fin. a . fin. b. 



For as O.^l : AF :: BD 

 OA : OF .: OD 



therefore, 



OA . OE - OA. GE = 

 orOA.OG = OF. 



And when O A = radius = 



cof. (a + b) = cof. a 



HD or 

 OE; 



GE 



OF.OD-AF.BD, 

 OD - AF . BD. 



1, this becomes 

 cof. b — fin. u . 



fin. b. 



^{1 -hcot.'a) 

 I 



tan. a 

 ^/ ( I 4- tan. 

 cot. a 



<') 



cofec. a 

 I 



^/ ( I -f tan.' a) ^/ ( • + cot.' a) fee. a 



If now we repeat thefe two, w's. 



fin. (a -^ b) ^ fin. 3 . cof. 3 + fin. i . cof. <i 

 cof. (a -\- b) = cof. a . cof. b — fin. a . fin. i ; 



and multiply the firft by cof. b, and the fecond by fin. b, 

 and fubtraft the refults, we fhall have 



fin. [a + b) cof. b — cof. (a -(- b) fin. b = 

 fin. a (cof.'* + fin." 3); 



or, fincecof.'^ + fin.' i = i, this is 



fin. a = fin. (a + b) . cof. b — cof. (a -f b) fin. i. 



If now we make a -|- i = a', then a =: a' — b, and we 

 have fin. (a' — i) = fin. a' cof. b — fin. b . cof. a', which is 

 a third general formula. 



Again, multiplying the firft of the above formula by 

 fin. b, and the latter by cof. b, and adding the refults, we 

 obtain 



fin. (a + b) fin. b + cof. (a + /) cof. b ~ 

 cof. a (cof.' b + fin.-" b) ) 



or, fince cof.* b + fin.' i = 1, 



cof. a = cof. (a + b) cof. b 4- fin. (a 4- b) fin. b. 



Or making a -f /^ = a', as above, 



cof. (a' — b) + cof. a' cof. b = fin. a' fin. b ; 



which is a fourth general formula. 



In the two latter exprcflions we have ufed a' for a ; but 

 it is obvious that this was merely for the fake of diilinc- 

 tion, and that it is not ncccllary to retain it in our future 

 operations. 



Thefe four formulae bring the foundation of the whole 

 calculus of fines, we (liall repeat tliem here in a connefted 

 order, whereby their an.ilogy and relation will be more 

 readily difcovercd ; it will alio be convenient for the fake 

 of reference. 



of. a] 



£ 2 



Or 



