IS FORMULA FOR SINES AND COSINES; 



Let m zz 4 , n zz 5. .*. cos. — it zz cos. ~ it zz > &c. j" 



10 5 4 



.-_! , 4 * • 9 • 1 1 * 19 . 21 ■ 29 . 3 1 . 39 . 41 . 4.9 . &c. 



~* ' 5 . 5 . 15 . 15 . 25 . 25 . 35 . 35 . 45 . 45 . &c. '^' 



A 1 



In the same way as Euler's theorem, - — sin. A — sin. 



2 2 



2 A 4- — sin. 3 A — &c. is deduced, we may obtain the following 



cos. A + cos. 3 A 4- cos. 5 A 4- &c. to infinity, always ~ "\ 



cos. 2 A 4- cos. 4 A 4- cos. 6 A 4- &c. always zz j ' 



and .*. cos. A — cos. 2 A 4- cos. 3 A — &c. = — as may also be 



2 J 



had by differencing Euler'ts series. 

 Again, if e zz 2*7182818 &c. we find 



A cos. A— ^cos. 2 A 4- -J- cos. 3 A ~|cos.4A4- &c.\ 

 2. cos. - zz e I 



2 (0 



A —(cos. A 4-1 cos. 2 A 4-} cos. 3 A 4- \ cos. 4 A4- &c.U 



and 2. sin.— zze } 



2 , * 



Again ?— — = sin. A+ - sin. 2 A 4- - sin. -3 A 4- &c. (k) 



° 2 >2 3 x * 



4 j a A a i V i 



And -g- r- 4- - = cos. A 4- ~^r • cos - 2 A + -j,, 



cos. 3 A 4- - a . cos. 4A+&C (/) 



A, ,. (sin. A)* (sin. 2 A)* , (sin. 3 A)* 



And again - (it — A) zz ^— 4- 



2* + 3* 



(sin, 4 A)* , . 



4- — £j 4- &c. (»i) 



These last theorems are so easy of deduction, that I have omitted 

 their demonstrations for; the sake of keeping within the compass of 

 a letter. 



I am, Sir, 



Your most obedient humble servant, 

 March the 23<J, 1812 ' ANALYTICUS. 



