of Trigonometrical Quantities from one another. 165 



may be applied in the same way. The other two, which give 

 the same things by that of log sin or, require that cos (x -\- A x), 

 or at least the logarithm of v/cos at cos (ac + A x) be known : now 



log x /cos x cos (x + A x) = log cos x + A log cos x ; (16) 



therefore, when A log cos x is determined by formula (E') art. 10. 

 these may also be applied like the others. 



17. Because A log tan x = A log sin x A log cos ae, from 

 equations (14) and (15) art. 15, we find (omitting a factor 

 cos A x 1 nearly). 



4 sin (2 x + A x) 



A log tan x = -T-TT ' 0/ , . o ; (17) 



sin 2 x sin 2 (x + A x) 2 



Hence, again, putting | a? for a?, 



A sin (a? -f A A x) m b. x 



4> log tan i * = ^ - . (18) 



sin x sin (.r + A x) 4 



This last expression, compared with (12) (13) art. 15. and (18) 

 gives 



A log tan | x ' A log sin x ; (M) 



vcos x cos (x + A x) 



\/cos x cos (a? + A x) , . , 



A log tan 1 # = - ^ , v , H^ ( A log cos #) ; (N) 



sin x sin (x + A a?) 



A log tan j a? = \/cos a,* cos (x + A a?) A log tan a- . (O) 



18. The first and last of these are very simple, and they de- 

 serve attention, because the finding the tangent of the half of 

 an angle from the sine, or from the tangent of the whole angle, 

 occurs in the resolution of quadratic equations by the Trigono- 

 metrical Tables. Their application, however, again requires, that 

 we have the logarithm of /cos x cos (ac -f- A oc) . 



When tan |- (x + A ac) is to be found from sin (ac + A x), 

 we may first find A log cos x by formula (C') ; this gives 



