SINES. 



From which tire following particular cafes readily flow ; wiiich are the moft elegant expreflions we have for the fines 



r>\x, the 2 cof. a being denoted by y + — , we have 



2 cof. a = y H 



y 



2 cof. 2 a = y' -\- — ; 



y 



2 cof. 7a—y'-\ : 



J,! 



. I 



2 cof. 4 a = _)' ' H , 



y 



2 cof. n a = y" -{- 



2 v' — I fin- a — y 



2 ^/ — I fin. 2 a = y'' 



y' 



2 v' - I fin. 3a=j.3 . 



y 



r , I 



2^—1 lin. 4 a = y' 



2 V — I fin. n a = v" 



fin. n a ^ A cof." ' a fin. a — C cof, 

 col. n a ::=^ cof." a — B cof.""- a . fin. 



where i. A, B, C, &:c. reprefent the coefficients of ( i + i)". 

 Writing thefe under the form 



r , fin. a _, fin.^ a _ fin.^ a 



in. n a — JA — ^^ — C — p - + E - ,. — — &c 

 I col. a --J i - 



and cofines of multiple arcs. 



For a farther application of thefe formulx to the doftrine 

 of fines and cofines, fee the article Colefian THEf)REM. 



It may not be amifs to add, with res^ard to the hillory of 

 the invention of thefe formula;, that both the general and par- 

 ticular cafes ot the numbers (II) and (III) were firft, 

 given by Juhn Bernoulli, in the Leipfic Afts for 1701 ; 

 thofe of numbers (IV) and (V) were pubhilied by Vieta, 

 under the denomination angular fe Sl'ions ; hut he regarded them 

 not as properties of fines and cofines, but as thofe of chords 

 and tlieir fupplements, to which they are eafily reduced. 

 The other four formula:, TOz. (VI), (VII), (VIII), ( IX), 

 are all derived from a formula publilhed by De Moivre in his 

 " Mifcelianea Analvtica." 



la order to inveltigate formula for tlie multiple tangents 

 and cotangents, we may repeat here our formula (II) 

 and (III) ; ijiz. 



~^ a . fin.^ a + E cof."~= a fin.' a — &c. 

 a + D cof." -* a fin.' — &c. 



fin. 

 cof. n a 



B 



fin.^ a 

 cof.' a 



jf.^a 

 h D 



fin.' 



jf.'a 



af. ' a 



&c. 



f cof." a 



-A: 



and dividing the former by the latter, and obferving that — '— = tan. a ; we have 



col. a 



A tan. a — C tan.3 a + E tan.' a — &c. 



tan. a = 

 Or writing them under the form 



fin 



cof. 



B tan.- a -t- D tan.* a 



cof." ~ ' a ,, cof." ~ = a 



^' a 



cof." 



&c. 



(X.) 



f col."-' a ,, col."-= a col."-= a 1 



n a — \A -f—„'sT- — *- ?— V^-l" + ^ ?—irT~ — &t. J- fin. « a 



I fin." ' a ha." ' a fin."-* a J 



f cof." a cof."-^a , .-^ cof."-' a , 1 ^ 



cof." a 

 fi 

 and dividing the latter by the former, we have 



A cot."-' a — C cot 

 cot." a 



cot. n a ^. 



a + E cot." -' a — & c. 



B cot."-" a + D cot."-* a — &c. ^ •' 



From thefe we draw the following particular cafes. 

 I. tan. a = tan. a 



2. tan. 2 a =:: 



2 tan. 



1. cot. a = cot. a ' 



cot.' a — I 



2. cot. 2 a = 



3. tan. 3 a = -^ 



4. tan. 4 a 



1 — tan.- a 



3 tan. a — tan.' 



I — 3 tan.- a 

 4 tan. a — 4 tan. 



3. cot. 3 a = 



2 cot. a 

 cot.3 a — 3 cot. a 



3 cot.- a — I 



5. tan. 5 a = - - 



&c. 



1—6 tan.' a + tan.' a 

 5 tan. a — 10 tan.' a + tan.' a 

 I — 10 tan. a -f 5 tan.-* a 



= &c. 



cot. 4 a — 6 cot.' a + 1 



4. cot. 4 a = ~ 



4 cot. 3 a — 4 cot. a 



cot.' a — 10 cot.3 a -I- r cot. n 



5. cot. 5 a =. — - — =• 



5 cot.* a — 10 cot.- a + 1 



&c. 



&c. 



In order to obtain fimilar formulx for the fecants and cofccants, we mud avail ourfelves of the formulae (IV) and (V), 

 which may be written 



t I cof. a 1.2 cof.' a J 



fm 



cof. n a =z ^ ^ 



cof, 



■ I " (" — 3) > .7 



cof.' a ' 1.2 cof.< a J 



fin. 



af." a 



