84 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 



3.70 Finite Products. 



i. sin nx 



i 



n/ sin 2 x \ 

 li- r- \ n even. 

 . 2 KIT I 



*= i \ sm 2 / 

 \ / 



M 



2 y 



sin 2 x 



2. cos w# = 1 I / i ; \ w even. 



. 2k i 

 sm 2 - 



2H 

 ni 

 2 



m 2 







k=i\ sm 2 



ni sm x 

 li -- : I n odd. 

 . kTT I 



n 



n i 

 2 / \ 



n/ sm 2 x \ , , 

 /i : 1 n odd. 

 . 2k - I 

 k = i \ Sin 2 7T / 



iTT ( / 2kTT\\ 



5. cos nx cos ny = 2 n ~ < cos x cos I y -\ ) > 



^=o 

 n i 



6. a? n - 2a n b n cos nx + b 2n = TT I a 2 - 2ab cos ( x + ] + ft 2 1 



fA 1 V n 1 j 



ROOTS OF TRANSCENDENTAL EQUATIONS 



3.800 tan x = x. 



The first 17 roots, and the corresponding maxima and minima of - 



x 



are given in the following table (Lommel, Abh. Munch. Akad. (2) 15, 123, 1886): 

 n x n Max sin x 



