of Trigonometrical Quantities from one another. 157 



The operations have been all put down at length ; but in 

 practice, the trouble of writing down so many cyphers, and the 

 repetition of the same figures, may be spared, just as in the com- 

 mon operations of multiplication and division. 



5. Before leaving the consideration of formula (B'), (C'), 

 (art. 2), I may just observe, that they may otherwise be elegantly 

 expressed thus : 



sin(*+A;r) tan x sec x cos (x + A x) 



sinQr + Aa?)) sUTT~ (cos (# + A.r)| . (JJ"^ 



~~ sin x \ ~\ cos a? ) 



cos (* + A *) cot x cosec x sin (x + A *) 



cos(.r ~ 



cosx 



Here we see immediately, that the determination of the sine 

 from the cosine, or the cosine from the sine, by the method we 

 have followed, requires the resolution of the exponential equa- 

 tions, 



y y -p, z z = q, 



p and q being known, and y and z unknown quantities, and each 

 having the value which it denotes in art. 3. The facility with 

 which they have been found is the consequence of their being 

 nearly 1. The general solution of such an equation, how- 

 ever, is attended with more difficulty. 



6. The determination of the sine from the cosine, or the 

 cosine from the sine, enables us to determine the tangent from 



either the sine or cosine, (because tan x ^-^ ) also the sine of 



COS X J 



twice the arc, which is equal 2sin<r cos x, and hence, again, the 

 cosine and tangent of double the arc. 



7. I now proceed to investigate formulas by which the loga- 

 rithmic sine and cosine may be deduced from the tangent. By 

 the calculus of sines, 



