﻿44 Exponential and Logarithmic Theorems. 



(m). Now (as is well known) we have t&n.pv>pv>sin.pv } or 



v 3 



substituting the values of tan.pt?, sin.pv, we get v+ "g" + &c- 



t?3 ^2 V 3 



>pv>v—Y22+&c-> or l + y+&c.>^>l — j~23 + &c, (the 



sign > not excluding equality,) which requires that j9= 1, for 

 the inequality must obtain when v is diminished in infinitum, 

 .\ by the first condition p is not >1, and by the second p is not 

 <1, r. p = \\ the same result is evident from the circumstance 



v 2 



that ^-fo^+^c is the development of sin.pv, .*. since p and 



v are equally involved in sin.pv, they must be equally involved 

 in the development, that is, v must have p as a factor in the de- 

 velopment, so thatj?*; = i;, andp = l. 



v 3 v 5 



Hence we get sin.t;=t; - fol+TYoT^ — & c -* cos.v = 1 — 



v 2 v* tan.^v tan. 5 v ,_ % "' f 



O+L^I'^ ' t? = tan.t;— — 3— +— g &c, (A) and (*) 



become e ~= =sm.v. e - — = cos.v, (A') ; we 



can easily deduce the known formulae for the sines, cosines, tan- 

 gents, &c. of the sum and difference of any two arcs (rad. 

 from (A'), but as the process is sufficiently simple and obvious, 

 we shall not stop to give it. If we denote the semi-circumfer- 

 ence of a circle whose radius equals unity by *, then since 

 sin.v=0. when v=0, v=n, v=—*, v~2n, v=—2rr, t>=3», 



3-r, and so on ; we shall have sin.r = v ( 1 --] ( l+~ 



v 



1 



v 



2/ 



1 +|i)-( 1 -| I )-( l +^) ><&C - =r ( 1 -^)'( 1 -^ 



1% 



3 



1-Q^I X vtc, (B); also cos.u = 0, when v = ±%, v=±g*, 



9jt 

 5* 



2v\ I 2v 



v=±~2, and so on, .-. we shall have cos.v= ^1— — j'[l+— 



x( 1 -|)-( 1 +l)x^- = (l-^ ! )-( 1 -^)-( 1 -£) >< 



&c, (B'). By comparing the values of sin.r, as given by (A) 



j> ' -a 



and (B), we get *~ i^*^^-^^-^)'^-^ 



