Trigonometrical Functions, 201 



have in view is as follows. Setting out from the known for- 

 mulas — (See Young's Dili*. Calc, p. SO.) 



e x+Aii _ CQS x ^_ ^ __^ sm x 



e —x*/—\ __ CQS ^ _ y^ __j # sm ^ 



and taking the logarithms, we have 



x \f — 1 = log (cos x + ^ — 1 . sin x) 



—x s/ — 1 = log (cos x — s/ — 1. sin #) ; 



therefore, by subtraction, 



a / — r , COS.T+ \/ — \. smx , 1 -f s/ — 1. tan x 



2x v — 1 = log ■ == ss log = . 



cos x — s/ — 1 . sin .r 1 — \/ — 1 . tan * 



Now, log j— = 2 |« + — + T + T + ,&c.| 



hence, substituting */ — l. tan ,r for w, we have 



/ — r . ^ tan 3 jc tan 5 x tan 7 # ? ' — - 

 2x\/ — 1 = 2]tan.r 1 h,&c>\/— 1 



tan 3 # . tan 5 x " tan 7 .r , 

 .•. # = tan x - 1 = 1 = h> &c. 



o o ( 



The reason that we have obtained this defective form arises 

 from an omission in the general expression for the logarithm 



of , which omission would lead to no error if we were 



dealing with real quantities only. It was proved by Euler, 

 that every number has, besides the logarithm usually consi- 

 dered, an infinite number of others, all imaginary*: thus, if 

 we represent the usually received value of log A by Log A, 

 then will the general expression for log A be 



log A = Log A + 2 kir\/ — I. 

 When, therefore, we are dealing with imaginary quan- 

 tities, we are not warranted in omitting the imaginary values 

 of log A, as is done in the foregoing process. Restoring, there- 

 fore, what has been improperly omitted, we have 



log \£ -V{«f -! + f + f +, &c.} MW% 



* This may be proved as follows. By Young's Calculus, page 33, 

 »J —\ •=. e% •• — 1 = e .*.!=<? .*. A = Ar 



e 



.'. log A = log A + 2 h x J- 1. 



Third Series. Vol. 5. No. 27. Sept. 1834. 2 1} 



