SINES. 



4. cof. a . cof. b = I cof. {a + l') + i cof. (a w I) 



5. fin. a + fin. i = 2 fm. i (a + A) . cof. i (a .c i) 



6. cof. a + cof. b = 2 cof. ^ (1 + A) . cof. i (a >jo i) 

 fin. (a + 3) 



tan. a + tan. ^ = -f:^^"^^ 



, r 2 fin. i (j. _ z) , cof. l{x + x) 

 therefore ^-^ ' = 



A (x- - 2^ + B (,: ' - S + C (»' - Z') 



7- 



8. cot. a + cot. b = 



9 



fin. {a + 6) 



fin. a . fin. b 

 fin. a - fin. i = 2 fin. ^ {a - I) . cof. i (« f b) 

 10. cof. a ,.r cof. A = 2 fin. i (a - A) . fin. ^ {a + b) 

 fin, (a - 3 ) _ 

 cof. a . cof. i 

 fin. (a-b) 



1 1, tan. a — tan. i = 



12. cot. a i/^ cot. b ^ 



•3- 

 14. 



16. 



fin. (a + 3) 

 fin. (a — b) 

 cof. (a + b) 

 cof. (a c/5 i) 

 fin. a + fin. i 

 fin. a — fin. 3 

 cof. a + cof. i 



fin. a . fin. i 

 tan. a + tan. 3 



tan. a 

 cot. 3 



tan. b 

 tan. a 



cot. A + tan. a 

 tan. ■§ {it + b) 

 tan. i (a — i) 

 co^t. i {a -\-b ) 

 tan. i (a — i) 



cot. a + cot. i 



cot. a — cot. A 



cot. a — tan. b 



~ cot. a + tan. b 



cof. a w col. i 



1 7. fin. (a — i) fin. (a + i) 



18. cof. (a U5 i) cof. (a + A) = cof. ' 

 fin, (a-i^)fin. (a + b) 



cof.' a . cof.' b 

 fin, (a -A) fin, (a +3) 



19 



f fin.'a - fin.'*, or 

 I cof.' a >/> cof.^ i ' 



= tan.* a 



20 



= cot.' a 



fin.' * 



— tan.' A 



— cot.' b 



fin.' a . fin.'i 



If a + i + c = 180°, then 



21. tan. a . tan. b . tan. c = tan. a + tan. 3 + tan. c 



22. fin. a. fin. A . fin. c = :^ (fin. 2a + fin. 2* + fin.2c) 



It only remains now to exhibit the method of expreffing 

 the fine, cofine, &c. of an arc, in terms of the arc itfelf ; and 

 reverfedly, to exprefs the arc in terms of its fine, cofine, &c. 

 This is performed with great facihty, if we allow ourfekes 

 to employ the principles of the doftrine of fluxions ; but our 

 objeft, in this place, is to perform the fame by means of the 

 arithmetic of Jints only. 



For this purpofe, let x reprefent any arc, then it is obvious 

 that its fine will be fome funftion of this arc, and fuch a 

 funftion that it will change its fign, without changing its 

 magnitude, when the arc, without changing its magni- 

 tude changes its fign, becaufe fin. — .t; = — fin. x ; confe- 

 quently the developement of the fine in funftions of the arc, 

 will contain only the odd powers of that arc ; if, therefore, wc 

 reprefent the coefficients of the odd powers of x by A, B, C, 

 &c. we (hall have, or may ailume. 



+ D.»', &c. 



(«) 



fin. * = Ax + Bx' + C 

 And if z is another arc, 



fin, z = Az + Bz' + Cz^ + Da' + &c. 



by fiibtrafticm, 



fin. a: - fin. z = A (x - 2) + B (x' - z') + C 

 (x» — z') + &c. 

 but fin, jf — fin. « = 2 fin. § (j — z) . cof. \{x -\- z). 



■I- &c. 



Subftituting now for 2 fin. \ (.r — z), its expanded value, 

 by changine .v, in the firft feries, into \ (x — z), "uiz. 2 fin. 

 i(..-z) = A(.r- z) +iB(«-z)' + TVC (.»•-=)* 

 -|- &c. and dividing both fides by {x — z), we have, by 

 malting, after the divifion, jr = z, 



Acof..'t= A + 3B.V' + iQ.x' + yD.v'^ f &c. 03) 

 and confequently, 



A cof. z = A + 3B2' + sCz" + 7 Dz* + &c. 

 AVhence, by fubtraftion, and obferving that 



cof. .X — cof. z = — 2 fin. i (.r + z) fin. \ (.v — 2) 



= - fin. i (« + ^) {a (.V - z) + B (.X - z) 

 + C (a- - 2)^ + &c. I 



we fhall have, after dividing both members by .r — 2, and 

 then making x =2, . 



— A' fin. .V = 2 . 3 B .V + 4 . 5 C .v3 + 6 . 7 D .T^ + iScc. 



But fin. ^ = A .V + B ;r3 + C X ' + &c. 



therefore 



- A'fin. .v= - A'x- A'Bx'- A'Cx*-|- &c. 



Whence, by equating the coefiicients of the homologous 

 terms, we have 



2.3 B = - A3; 4.5C = - A'B; 

 6. 7 D = - A'C, &c. 



B=::i^;c = 



A* 



fo that 

 fin. .V 



2-3 

 = A .V 



2-3-4-5 



; D = 



A' 



+ 



A'; 



2-3- 



A' J 



&c. 



+ &c. 



2.3 2.3.4.5 2.3.-7 



and it remains only to find the value of A ; which is readily 

 drawn from the following cor.fideration ; vix. if we divide 

 both fides of this equation by x, it becomes 



fin. X 



X 



A - 



h?x ' 

 2-3 



+ &c. 



which ought to anfwer to every value of .r ; but when x is 

 indefinitely fmall, fin. .v = x, and therefore the firtt number 

 of the equation = i ; therefore the fecond alfo equal I, 

 confequently A = i ; and we have therefore 



fin. ."« = *• -1 



2.3 2.3.4.5 



^ +&c.(XVII.) 



2. 3. ..7 



and by fubftituting, in equation (/S), the above values of 

 A, B, C, &c. we obtain 



cof, 



X' a" 



— + 



2 2.3.4 



-—:- — , + &c. (XVIII.) 

 2. 3. ..6 



In order to exprefs the arc in terms of the fine and co- 

 fine, let .V and 2 be two arcs ; let y be the fine of the former, 

 and u the fine of the latter ; and fince, for the fame reafon as 

 above, the expanded funftions oan contain only odd powers 

 of jr and u, we will fuppofe that 



x= Ajr + Bjr' + C>> + D^' + &c. (7) 



2 = A«+B«'-i-C«^ + D«'+ fcc. 



and, 



