FORMULA FOR SINES AND COSINES. 15 



Let ~ = -■ .% sin. rr » being; = - form ^ 1 1 becomes 



6 6 



• — 3 6 • 6 - 12 . 12 . 18 . 18 . 24 . 24 . &C „ 



* ~~ * 5 . 7 • 1 1 • 13 . 17 . 19 • 23 . 25 . &c. ' 



Let - — - * and form \ 1 f becomes 

 n 3 (.3 



— 3 '/S 3.3 .6.6.9-9. 12 . 12. &c. 

 2 * 2 . 4. 5. 7 • 8. 10. 11 . 13 . &c. 



In the same way by making - zz ■ — , or — , we find 

 n 10 20 



_ 5 [Vb — 1) v 10 . 10 . 20 . 20 . 30 . 30 . 40 . 40 . &C. .^ 

 * — i 9.11.19. 21 . 29 . 31 . 39 . 41 . &c. * ' 



And 



(c) 



-7? 



CvT + 1 _ •) 20 . 20 . 40 . 4 . 60 . 60 . &c. 

 ~~ 5 /" W V ' 5 ~ ^ J 19 • 21 . 39 . 41 . 59 • Cl . &c. ^' 



to 



and so on, whenever sin. -it can be found in algebraic terms, as if 

 n 



* L ±: ; -L,&c. . 



■a ■ 17 17 



Let us now take form | 2 > » an< * for — write - . Now 



cos. — <sr ~ — — . .*. V'if = » and the form becomes 



4 \^2 cos. ir 



_ 4 . 4 . 12 ♦ 12 . 20 . 20 . 28 . 28 . &C. 



^* ~ 2.6. 10 . 14 . 18 . 22 . 26 . 30 . &c. 



2.2 . 6.6. 10 . 30 . 14. 14 . &c. . 

 - 1.3.5.7. 9 .11.13. 15.&C Which,san 

 expression due (if I remember rightly) to Euler. 



In form ) 2 > for nt write - m, and it becomes 



2. 



m [n — to) (« -f- to) (3 n — to) (3 n -\- nt) &e. 



cosin. tr ~ — — — — — - — 1 — 



2m n n . 3a . 3ra . &c. 



1 's/jr 



Let to — 1 , n = 3 . .*. cos. - » being = — — - we get 



- . 2 . 4 . 8 . 10 . 14 . 16 . 20 , 22 . &c. 

 V* ~ 2 * 3 . 3 . 9 . 9 . 15 . 15 . 21 . 21 . &'c. '**' ' V} 



