44 Exponential and Logarithmic Theorems. 



(m). Now (as is well known) we have ta.n.pv'^pvysin.pv, or 



substituting the vahies of tan.^pv, sin.pv, we get Z7+ ^ + &c. 



^3 ^2 -yS 



>pv>v-Y2S+^<^; or l + -^+&c.>p>l-j-^ + &c., (the 



sign > not excluding equality,) which requires that p = l, 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, .•.^ = 1; the same result is evident from the circumstance 



that V - -, n '3 +&C. is the development of sin.pv, .*. since j? 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 that^^7 = ^?, andj9=l. 



Hence we get sin.i7=z? - ..-^-0 + 1 0345 ~ ^^-j cos.v = 1 — 



v^ V* tan.''z; tan. ^v 



l^o+ i 2 3 4 ~^^-' v = ta.n.v — —^ — + — ^ &.C., (A) and [k) 



become —= =sin.z;, — = cos.v, ik')' we 



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

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

 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 ^r^ then since 

 sin.v = 0, when «;=0, v=-'^, v=—^, v — 2n, v=—2Tr, v~3n, 



v=—3n, and so on; we shall have sin.i; = v ( 1 --) ( 1+- j 



V \ I v-\ I v' 



f v^ \ 7t 3 



•11 — 5^1 X »fcc., (B) ; also cos.?; = 0, when v = =b2)^ = ±2^j 



5n f 2v\ I 2v\ 



?; = =b^, and so on, .'. we shall have cos.i;= I 1 — — j-l 1-f ~j 



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

 and(B), vvegeti;-j-2;3 + Y;2r3A5-*^^-=^l^-;7^JV"4^/ 



