SINES. 



In the formula of (II) and (III), the fin. a and cof. a both enter ; but as fin.- a= l - cof.' a, we may, by fubfti- 

 lon in the latter, exprefs the cof. n a in terms of the cof. a only ; but in the former, in confequence of the odd powers 



In 



tution m the latter, expr , , ^ , „ „ r , r -, , r i. l 



of the fine, we cannot exterminate either the fm. a or cof. a entirely ; the fimplell itate of thefe formula, after the above 



fubititution, is as follows ; viz. 



fm. n a 



cof. 



{ (a cof. .)-' - '^-^- (2 cof. .)"- + (!LzJU1^A<^ cof. a)" 



_ (r^_-^^(n-_s)^n-_6l (^cof.a)""' + &c.j fin. <z (IV.) 



1.2.3 J 



a= if (2 cof. a)" - « (2 cof. «)"-' + —"-^ (z cof- ")"* 



_ njn_-4)_^L_-Z_ll (2 cof. fl)--^ + &c.| (V.) 



1.2.3 J 



1.2.3 



From which the following particular cafes are readily 

 drawn, 173. 



1. fin. a = I fin « 



2. fin. 2 a =. 2 cof. a . fin. n 



3. fin. 3 a ^ ( 4 cof." n — i) fin. a 



4. fin. 4 ^ = ( 8 cof. 'a — 4 cof. a) fin. a 



5. fin. 5 a =z (16 cof.^ rt — 12 cof ' rt + l) fin. /i 

 &c. = &c. 



1. cof. ff = I cof. a 



2. cof. 2 n =r 2 cof.' a — I 



3. cof, 3 a = 4 cof.' a — 3 cof. a 



4. cof. 4 a = 8 cof.'' a — 8 cof.' a + l 



5. cof. 5 a = 16 cof' a — 20 cof.' a + 3 cof. a 

 &c. = 8cc. 



If we refer again to our formula (I), viz. 



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

 cof. {a + b) = cof. a . cof. i — fin. a . fin. A 



■we readily draw from thefe the following, viz. 



fin. (n + l) a = fin. « a . cof. a 4 cof. n a fin. a 

 cof. (n 4- I ) a = col. n a . cof. a — fin. n a fin. a 



Multiplying now the firft by any indefinite quantity 11, and 

 adding it to the fecond, we obtain 



cof. (n 4- l) a + Ti fin. (n ^- i) a 



(cofv a + V fin. a) cof. n a 



4- (cof. a fin. a) -y fin. « <« 



If, inftead of addinjr the two formulx, after multiplyin]^ 

 the firit by v, we fubtraft the firft from the fecond, we fhall 

 get 



cof (n 4- I ) a — -u fin. ^n -f i ) a 



(cof. a — V fin. a) cof. n a 

 (cof. a H fin. a) v fin. n a 



v/ — I, to II, we 



{- 



-^{ 



Where, by taking for the indefinite quantity ti, the particular 



value V — 1' f° t^3' ■" = ' thefe will become 



cof. (n + I) a 4- -u fin. (n 4- l) a = (cof. a -f i; fin. a) 

 (cof. n a 4- ■" fin- " ") 



whence, making n fucceffively = i, 2, 3, &c. we obtain 



cof. 2 a 4- I* fin. 2 a = (cof. a -\- v fin. a)^ 

 cof, 3 a 4- t) fin. 3 a = (cof. a 4- t fin. a)^ 

 cof. 4 a 4- t* fin. 4 a = (cof. a -|- ^| fin. a)* 

 cof. f a + V fin. 5 a = (cof. a + v fin. a)'' 



cof, n a + ^ fin, n a = (cof. a 4- ai fin. a)" (VI.) 



Where, by giving the fame value, vi 

 have 



cof. (n 4 I ) a — v fin. (n + I ) a 

 (cof. a — "u fin. a) (cof. n a — "u fin. n a) 

 and making again n = i, 2, 3, &c. we obtain 



cof. 2 a — V fin. 2 a — (cof. a — v fin. a)^ 

 cof. 3 a — V fin. 3 a := (cof. a ~ v fin. a)' 

 cof. 4a — ■ufin.4a=: (cof. a — v fin. a)* 

 cof, ^ a — V fin. 5 a = (cof, a — -u fin, a)' 



cof. n a — 11 fin. n a = (cof. a — v fin. a)" (VII.) 



Whence by addition and fubtraftion, and eftablifhing the 

 value of V, 



cof, n a = ^ i (cof. a 4- ^/ — I . fin. a)''4- 

 (cof. a — ^/ — I , fin. a)' {■ 



fin, n a = J (cof. a + ^^ — I fin. a)' — 



2 x/ - I 1 



(cof. a — v' — 1 • fin- ")" f 

 If in thefe two formula we fubftitute cof. a 3= *, fin. a =1 

 •v/ I — «- ; then ^/ — I (fin. a) zzz V x^ — I, and we have 



= i I (* + aV— I)" 4- (*• — A''.r'- — 1)» 



cof. 



f 



fin. n a z= — ) (* 4- aV — l)" + (j — V.r^ — l)" 



And afluming .r s= jr 4. — , fo that x' — i may be a 



complete fquare, or .\/a:' — i = jj ; thefe expreflions 



reduce to 



coL«a= i (/+-J;) (VIII.) 



rin.„a= -^—[r-r) (IX.) 



From 



