LOGARITHMIC SERIES.] 



MATHEMATICS. TRIGONOMETRY. 



629 



1-1 



_ , 0\* 



/ 0V * mf 0V- 1 f im ' m 



,= I cos. Z- 1 ( cos. -- I 3- 0s 



\ m/ 12 V / * IV 



1.2.3.4 



/ 0Y~* 



I cos. -i- ] 

 V m/ 



/. ,v/l. sin. = 0^ 1 



1.2.3 



v V X 



1. -. o. 4: .;_> 



.*. cos. + Y/ 1. sin. = 1 + e .vA^Tf 



0V^I_ _0*_ 0^/^i 



+ 1.2.3 1.2. 3. 4 + 1.2. 3. 4. 5 



_ 

 1.2 



&cos. </ 1. sin. = 1 ^7 1 + - 



1.2 



This formula is true for every value of m, however 

 large TO may Be ; but if we suppose m to become very 

 large, then in the limiting or extreme case when m is 



infinity is zero, and hence in this limiting case 



Bin. 

 m 



Alsol = 



(V i 

 ros. ( ^ J 



And therefore 

 cos.0 = 1 ^- 



L -TO = 1 - 



0> 



By reasoning in precisely the same way from the 

 formula for sin. m we shall obtain 



COR. If be so small that we can omit s , we shall 

 clearly have 



gin. = and cos. = 1 g" 



(51). To obtain sin. 0. cos. 0. and tan. in terms of 

 Exponentials. 



Since 



0* 



sin. = j 23 + j.2.3.4.5 ' 



cos. 



0* 

 1 + 







__ 



72 1.2.3.4 



<fec. 



1.2.3 



, __ _ 



""" 



__ _ 



1.2.3.4 1.2.3.4.^ 



Kow 



_ 



- 1)* (,/_ 



.'. cos. + V^l sin. = 1 + v/^ 



and 



1.2 



1.2.3 



/. cos. + v/ 1 sin. = e . . .. (4C). 

 And cos. ,v/^7sin.0=e--/^"r. . . . (47). 

 by the exponential theorem (Art. on Series)* 

 .'. dividing the former by the latter 



cos. + v/ 1 sin. 



cos. / 1 sin. 



1+ J 1 tan. 2 , , rrp 



,". = ~ = 6 ... (48). 



1 -v 1 tan. 



Also adding (46) and (47) we obtain 



2 cos. = e -/^i -f e V"^"i . . . (49). 

 and subtracting (47) from (46) we obtain 



2\/ Isin. = e * V~^i e. * V^T. . (50) 



(52). To obtain an expression for in terms of tan. and its powers. 

 Since from equation (48) 



1 V/ 1 tan. 



Taking logarithms on both sides. 



2 y^Tl = log. (1 -f- V 1 tan. 0) log. (1 V 1 tan. 0) 



"^! tan. 



l tan. 0)' (y/ 1 tan. 6) 3 . <feo. 



2 





8 



j V Itan. (V ltan.0 (/ 



~T~ 1.2 ~3 



'^TI tan. 9 (V Itan. Of (^ 1 tan. 6) 8 



~1T~ 3 5 



tan. '0 , tan. 

 2 v/^-'T < tan - 6 1 s &0 - 



.'.0 = tan.0 



tan.'0 , tan.0 



Sec ante, p. 516. 



