SINES. 



or, fin. „ a = I2-' - ^!— -? . 2— fee' a + (« -3) (" - 4) , ,„-. f^^ , ^ 7^ 1 



J^ I '1.2 j fee." ^ ' a . cofec. a 



cof. n a = X2"-' — n , 2"-' fee." a + ~ ^' .2"-^ fee.^a — SlC.Ij^-^ — 



I 1.2 J fee." a 



The reciproeal of the latter gives, 



fee. n a = -j — — r (XII.) 



2"-' - n . 2"-' fee' a + — ^ " ~ ^^ . 2"-' fee.-* a - See. 



I . 2 



And the reeiprocal of the former, 



fee " — ' a . cofec. a 



cofec. n a = 7 tw 7^ (XIII.) 



2"-. - :^-:^ 2»- fee' a + ^" - 3) (" - 4) , ^„_, f^^_, ^ _ ^^_ 

 I 1.2 



From thefe we draw the following particular cafes ; 



1. fee. a = fee. a '■ cofec. a = cofec. a 



fee ' a r fsc, a . cofec. a 



2. fee. 2 a = — il£vV *• ''°'^"^- ^ " = ; 



2 — fee.' a 2 



fec.^ a r fee-' a . cofec. a 



3. fee. 3 a = . . - 3- cofec. 3 a = — ^ ,- , 



■' •' 4—3 fee' a 4 — fee' a 



fee* a fec.^ a cofec. a 



A., fee 4 a =E = — =— ; — ; ; — 7 — : 4- coiee. ,< a = ^ ^ — ; — - 



T- ^ 8 — 8 iee' a + fee a ^ 8—4 fee' a 



fee' a r f'^c.'' a cofec. a 



- A._ - ^ — I — C. COleC. C a = -;^ ;= ; f ■ 



5. lee. 5 a - ^g _ ^^ ^^^., ^ + ^ fee* a ^ ^ 16-12 fee" a + fee* a 



&c. = &c. &e = &e. 



Thefe latter formulse have the fame defeft as thofe ef the fines from which they are derived ; viz. that both the fecant 

 and cofecant of the fiiip;le arc enter in the geueral expreffion for the eofecant of the multiple .ire 



We (hall not inveftigate the formula for the multiple verfed fine, but (hall content ourlelves with merely exhi- 

 biting the refult, which is as follows : 



verf, na-re verf. a - — ~ - ( A ) verf.' a ( B ) verf.3 a - -^ ( C ) verf.*a - &e 



where A, B, C, &e are the coefficients of the preceding terms. 



A variety of other formuli for multiple arcs might be here inveIHgated, but our limits forbid a farther extenfion ; 

 thofe which we have given above are chiefly exhibited by means of the powers of the fingle arc j a few others, in the 

 form of continued produfts, may be here enumerated, but we muft not attempt their inveftigation, 



Let % denote the femicireumference, and we (hall have 



fin. na = 2"-' . fin. a . fin. T- — aj . fin. (— -^r a\ . fin. r— - a\ . fin. T— + aj . fin. {^ — a^ 



"'•"■='•"■"'■ ('T;r'+'") ■'°'- (tt'-') ■"'■■ (tv'+'") ■'»'■ (^''-') 



. cof. r-^^ v ■\- a\ . cof. f ^^^ ■n — a\ &c. 

 . na — tan. a . tan. {- a\ tan. f - -f a j tan. ( aj tan. { y a\ tan. f — — a j 



■Tn"^") ""' *^"- (2^ + ") 



fin. 



tan 



tan 



To thefe we add the following, which are very fimple ; viz. 

 fin. a + fin. {a + 6) + fin. (a + 2^) + fin. (a + 3 i) + &c. aJ iiifinitum = '— — j^' ; and 



fin. a + fin. (a + 4) + fin. ( a + 2 i) + &c. fin. (a + « i) = 



fin, (a + i*)fin.i (« + 1 ) * 



fin. i b 



cof. a + cof. (a + i) + coL (a + 2^) + cof. (a + 3 ^) + &e. ad infinitum = '-^ , ^ ■■ 



r . r / , i\ . r I . i\ » r/ ,\ cof- (a + i n^) fin. 5 (n + l) B 

 eof. a + cof. {a + i) + cof. (a + 2 i) + &c. cof. {a -^ n b) = i — ' i , — 



Of 



