SINES. 



Of the Powers tf the Sines, Coftnes, i^e. offingle Arcs. 



In order to exhibit the powers of the fine, cofine, &c. of the fingle arcs, in terms of the fines and cofinei of the multiple 

 arcs, we may refer to our formulae (VIII) and (IX), "ulz. 



2 Cof. 



2 cof. na =. y" -if 



I 



+ - 

 3 

 I 



(-■r 



(-0^ 



2 fin. 

 2 fin. na 



a = y - 



f - 



From the firft of thefe we have z " . cof." a = fy+ -^ = j-" + A ji" -^ + B >" -" + C j;" 

 A, B, C, &c. reprefenting the binomial coefficients ; or, coUefting the terms from each extreme, 



= (/ +y) + ^ (y-' +~r:) + B(y-^ + -i^,) + &c. or 



+ ace. A 



+ 



cof.' 



2'~' cof." a = cof. na + n . cof. (n — 2) a + 



n. (n - i) 



cof. (n- 4)a + &c. (XIV.) 



It is only neceffary to obferve here, that when n is even, the number of terms in the original expanfion will be odd, and 

 its middle term, into which no power of j enters, will form the laft term of the latter feries ; and having, therefore, no co- 

 fine in it, it will be only one-half of the coefficient given in the general term. 



Whence the following particular cafes. 



cof. a = cof. a 



2 cof.^ a = cof. 2 a -I- I 



4 cof.3 a = cof. 3 a -I- 3 cof. a 



8 cof.-* a = cof. 4 a -t- 4 cof. 2 a -f- 3 



16 cof.5 a = cof. 5 a -I- 5 cof. 3 a + 10 cof. a 



&c. = &c. 



For the fines, we have ( — i )"^. 2" . fin." a 



= (^-7)"=^- 



+ By"-' - C/-' 



+ &c. ± -, the 



y 



iaft fign being + when n is even, but — when n is oJJ. Collefting the terms from the two extremes, we have firft, 

 ^vhen n is even, 



(-,)-. 2-. fin." a = (y' + ^)- A(y-' +-_l_).fB(/- +y^)- &'^-°^' 



(— i)'^ . 2"-' . fin." a = cof. na — n cof. (n — 2) a ■}- "— cof. (n - 4) a — &c. (XV.) 



•bferving here the fame as above in refpeft to n, an even number ; and <when n is odd, 



(- 0^ . ^' fin.- a = (/ - ^) - A (/ -' - -i:,) + B (/- - -^) 



(-1)^' . 2" 



&c. or 



fin." a = fin. na — n fin. [n — 2) a -\ — ■ fin. (n — 4) a — &c. (XVI.) 



Obferving now the change of fign, which takes place in the moll important and ufeful transformations ; and the fol- 



... n- ■ , T , / \"-^ V lowing refults, whicli would occupy too much fpace to de- 



the imaginary coefficient ( - i )^ and ( - i ) > , according ^^j^ ^^ ^^^ ^j,;^^ ^^ ^^e fame time arc too Important to be 



as n IS of the form 4, «, 4 m -|- i, 4 m +2, at^m + 3, we n^j ^^^^ unnoticed, the reader will be able to deduce, 



draw from the above two formulas the following particular ^j^^ facihty, from the formulx drawn fmmour general ex- 



preffion, (No. I.) and many others equally curious may be 

 feen in Cagnoli's " Traitc dc Trigonometrie ;" vol. i. of 

 Eulcr's " Analyfis Infiuitorum ;" Bonnycaltlc's and Keith's 

 Treatifes of Trigonometry, Vc. &c. 



The formulx to which wc iiave above referred are the 



refults 



I. 

 2. 



3- 

 4- 

 5- 



I fin. a = -f fin. a 



a fin.^ a = — cof. 2 a — 1 



fin. 



4 fin.' a ^ — fin. 3 a -|- 



5 fin.^a = -f- cof. 4a — 4 cof. 2a -|- 3 

 16 fin.' a = -(- fin. 53—5 fin. 32 -|- 10 lin. a 



&c- = 



&c. 



We muft now come to a conclufion of our iiivcftigations 

 relative to the doftrjnc of fines ; had our limits admitted of 

 it, wc might have carried them to a much greater extent, but 

 the above will he fufficient for illullrating the principles of 



Vol. XXXIII. 



following. 



Mifcellaneous Formula of frequent /Implication in the Arith- 

 metic of Sines, 



1. fin. a . cof. A = i fin. {a + i) + ^ fin. (a - i) 



2. cof. a . fin. i = J- fin. (a -\- i) — ^ fin. (a — i) 



3. iin. a . fin. A = 5 cof. (a ^ A) — 5 cof. (a + i) 



C 4. cof. 



