1820.] Theorems for Jinding the Sums of Sines. 335 



Article VI. 



DemonsfratioHS of Theorems forjiiidlug I he Sums of Siues. 

 By Mr. James Adams. 



(To Dr. Thomson.) 



SIR Stonehouse, near Plymouth, Feb. 8, 1820. 



Never having seen pny demonstrations of the theorems for 

 finding the sums of the sines sin. a + sin. (« + h) + sin. {a -J- 

 2 b) + .... sin. (« + n b) and cos. a + cos. {a + b) + cos. 

 (a 4- 2 6) + .... cos. (a + « A), I, therefore, beg leave to send 

 you the following demonstrations thereof, the insertion of which 

 in the Annals of VMlosopliy will oblige, 



Your humble servant, 



James Adams. 



Prob. 1. — To find the sum of sin. a + sin. 2 a + sin. 3 rt -f- 

 .... sin. rt a. 

 Per Mr. Woodhouse's Trigonometry, p. 54, 



sin 



a = I ^ — - 1 



2 V _ 1 \ -r / 



sin. 2 a = —^ (x' - \\ 

 sin. 3 rt = Ar' — ^ > 



sin. w o = — ( cT" ) 



Then per geometrical progression, 

 X + i- + j:^ ^. ^,4 ^ ^n _ ^"^'-^ and - + -V + -V + 



X — I X X' x^ 



.... — = -^ — , we then have sin. a + sin. 2 a + sin. 3 a 



x" x"{x — 1) 



1 /x"+' — X Xt, — \ \ 1 



+ .... sm. n a — ( ; p, ) = „ . 



2t/3T \ ^-1 x^'{x-\)) 2V_i 



•Tn (-r - I ) 4 _ • 2 ^ - \ 



n + 1 



(2 ^ - 1 . sin. i;nn) (2 \/ — \ . sin. — ^^ — n) 



2 -/ — I X sill, ^n 2 V - I 



. . .1+1 



Bin. J n rt X sin. — -- — a 



sin. i a 



