33G 'J'keoremsforjtvd'ivg the Sums of Sims. [M a y, 



Corollary .—'^m. a + sin. 2 o + sin. 3 a &c. ad infinitum = 



\ cot. i rt. 



Per trigonometry, sin. \n a X sin. (i. a + i w «) = v cos. -i. 

 a — 4. COS. {\ a -\- no) 



But « a being infinitely great, ^ a + n a will likewise be infi- 

 nitely great, and as the cosine of an infinitely great arc may be 

 nothing, we shall have sin. a + sin. 2a+ sin, 3 a, &c. ad infi- 



i COS. ha , , , 



nitmn= ^. / = ^cot. ^ a. 



sin. ^ a 



Proh. 2. — ^To find the sum of cos. a + cos. 2 a + cos. 3 a 4- 

 .... COS. n a. 

 Per Mr. Woodhouse's Trigonometry, p. 5, 



COS. a = 4- (r + -) .-. 2 cos. a = r + - 



2 rt = X (.r"' + ^).-. 2 COS. 2 « = X' + ^ 



3 a = ^ (ar« + ^) .-.2 cos. 3 « = a ' + ^ 



COS. 



COS. 



COS 



. » « = 4- (x- + ^) ••• 2 COS. « « = .r" + — 



Then by geometrical progression, we have 

 X + x"^ + i' + o:^ + .... A" = -737-' ^"^^ + "^ + "^ + 



j:« — 1 



2 " 



x" jr" (x — 1) 



Therefore cos. a + cos. 2a + cos, 3 a + cos, 7i a = 4. 



/X" + I - J- J"- 1 \ _ , (J'+' + 1) (j" - 1) _ ^ 



\ x-\ "^ *■' (X - 1)/ ~" "^ ^"(^ - 1) 



% i 1 r ^ — i 



L .^ J 



Corollary. — Cos. a + cos. 2 a + cos. 3 fl, &c. ad infinitum 



Per trigonometry, sin. -^ n a x cos. (^ « H — «) = 4- sin. 



(-^ a +wflj — 4 sin. i a, but « a being infinitely great, 4- « + 



n fl will likewise be infinitely great ; and as the sine of an infi- 

 nitely great arc may be nothing, we shall have cos. a + cos. 2 a 



+ cos. 3 a, &c. ad infinitum = ~ "' = ■— 4-. 



sin. ^ a * 



Prob. 3. — To find the sum of sin. a + sin. {a + b) + sin. 

 (a + 2 6) + sin. (a + n b). 



