14 FORMULAE FOft SINES AND COSINES. 



which appeared in your Number for February last, I have been en- 

 couraged once more to trouble you with a few miscellaneous results, 

 indeed, yet curious. 



By the common trigonometrical resolutions of sines and cosines 

 we have: If w ~ 3-1415 &c. 



- (4^r0 x &c - toinfini ^ 



Sin. A 

 Hence A = - p— , t ■ 



O-7) 0-ftO 0-f?)- &c - 



Let now A z=. — tr- And .•. <rt — — . sin. ( — ^r I X 

 n ta \ n J 



n\ (2b) j . (3n)\ [An)\ &c. n . m 



. sin. — n. 



(**_ m 2 ) (2n>- m 1 ) (3^»- m a ) (4~n>- »*) . &c. m * 



n. n. 2w. 2m. _3n 3w. &c, , > 



(n— m) {n + m) (2n — m) (2 » -f m) (3w— -ro) (3b + w). &c. '( l S 



Again 



cos. a = r, _ dt\ (x - -£L^ r\ - -^-\ sa 



_(w~2w)(n + 2Mi)(3n-2wj)(3n + 2m)(5«-2m)(5n4 : ^m)^&c. <j ^ 

 "*" n '. n '. 3^T . 3 a \ 5n . 5n. &c. <? S 



In ■< l \ let - — -*-. .\ sin. — ?r — 1, and we get 

 tin n ° 



. o 4 4 fi 6.R <3 &e. 



w ~ 2. ■ 7T~- • Which is Warns s ex- 



1 . 3 . 3 . 5 . 5 . 7 • 7 • 9 • &<!. 



pression. This way of deducing it is however far shorter and more 

 direct than the usual way (see Woodhotise's Trigonometry, where, 

 however, he does not seem to have bestowed much attention on this 

 part of his subject). 



In < iX let - ~ — .*.sin. - v — — — , and the form becomes 

 * J n 4 n */2 



— a ,- *, 4. 4.8.8. 12.. 12. -16. 16.. &c. , N 



: * 3 .-5 . 7 . 9 • 11 • 13 . 15 . 17 • &c. 



