CAtCTlATION OF TABLES.] 



MATHEMATICS. TRIGONOMETRY. 



635 



And as before the limiting value of 



Again since 



cos. 2p -\- Iw 

 2n 



in 

 The factors of x 2 " + a 2 " become 



2 * sin.*^ sin. 2 ?^. sin. 2 ^ ... sin. 2 2 "- 1;r x 

 4n 4n 4n 4n 



1+-? 2 , cotan. 2 } ( l + -^.cotan. 2 ^Vl+j cotan. 2 5 ~\. , . 

 4n 2 4y I 4n 2 4y ^ 4n 2 4y 



The limiting value of which is ,, , ., , 

 Hence the limiting value of x 2 " + a 2 " derived from the expression for its factors will be 



And the two limiting values (a) and (6) are equal. Hence 



r 1.2 T 1.2.3.4 n ^* T ) 



Write z 2 for + ** ( before) and we obtain 



1 



Z ,4 / J^N / 4,.X / 



~ 1.2 + EiTO ~ *" \ ** J V &"*) \ ~ 



A cos. . = 1- 



The series given in the preceding articles are sufficient 

 for our purpose. We now proceed to explain the method 

 of calculating trigonometrical tables. 



The Tables of the trigonometrical functions are of two 

 kinds. (1). Those which give the numerical values of 

 the sines, cosines of the angles, <kc. (2). Those which 

 give the values of the logarithms of the sines, cosines, 

 &c., of the angles. The former are called Tables of 

 natural sines ; the latter, Tables of logarithmic sines. 



We shall explain the method by which each of these 

 tables is calculated. 



(59). To show that sin. M lest than and greater than 



For sin. 







2sin. cos. -- 



/, ,0\ 



- I l-sin._J 



Now 1-sin. 2 _ must be less than 

 2 



/. sin. 2 sin. *L 



, . 

 . . sin. - Z 2 sin. -. 



.'.Sin. -^2 sin. J.. 



f\ 



Hence sin. 2" sin. . whatever n may be. 

 2'* 



sin 

 ^ - 



2- 



for all values of n, however large n 



may be ; and therefore, in the limiting case, when n 

 is infinitely large, <fc .'. 2" infinitely large, or when 



2- 



But in thiB case 











2" 

 . sin. 



or sin. < 

 In like manner we can easily prove that tan . > 



For 



tan. d = 



2 tan. T 



1-tan.'?- 

 



.'.tan. >2 tan.y 





 Hence by same reasoning as before, tan. > 2" tan. -^ 



Q 

 .", tan. n ' 2* 







Now, when n is infinitely large 



0_ 

 tan. 2- . 



0~ 



2-" 

 



and 







in this case = L 



