of Trigonometrical Quantities from one another. 163 



Both results, viz. the sine and cosine of 50 0' 36", may be 

 considered as correct in all the figures ; the cosine differing from 

 the true value only by an unit in the twelfth place of decimals. 



2. As an example of formulae (F) art. 12., the logarithmic 

 sines and cosines corresponding to tan (x + A ac) :r 9'5632889 

 are required. 



tan (x + A of) 9.5632889 

 Next less (HuT-ro^s Tables) tan x (20 5') = 9'5630278 



A log tan x = 2611 



. ' f 9-97276 f 9-53578 



Rfe I 9-97276 sm '* | 9-53578 



A log tan x 3-41664 Alogtana7 341664 



A log sin x = 2302 3.36216 A log cos. r = 308 2-48820 



sin x = 9.5357832 cos x = 9.727554 



sin (x + A a?) = 9-5360134 cos (x + &x) = 9'727246 



;yjqr tyinitr -.iuifnad 7t*il) t luuiia hi -i.^ rfrj.lv/ -iir.fi ,. 



15. I shall next, from the approximate values of the incre- 

 ments of the logarithmic sine and cosine of an angle, deduce 

 analytic formulae, which shall express their relations to the incre- 

 ments of the like functions of half the angle. In art. 1. it has 

 been shewn, that if we neglect the third and higher powers of 



x. then 



\.A,MM . | . -/v < '.,) 



sin (2 x + A x) m A x 



A log sm x = -. ^ ~ '- x ; Q 2) 



sin x sin (x + A x) 



sin (2 x + A x) m A x 



A log cos x = H ' -: . H 3^ 



cos x cos (x + A x 2 



From these expressions, by substituting \ x for &>, and ~ A x 

 for A an, we obtain 



A , . sin (x + 4 A x) m- A x 



A log sin A x = - ; ii rv * :- ; ; H 41 



sm \ x sin i (x + A x) 4 



sin (x + I A x) m A x 



-Alo g COSi*:= cos|igcos| ^ + ' A;r) -3- (15) 



x 2 



