SINES. 



Or they may be reduced to two, by means of the double 

 fign +, thus : 



1. fin. {a + i) = fin. a . cof. b + fin. b . cof. a. 



2. cof. (a + 3) = cof. a . cof. b f fin. a . fin. i. 



In order to find fimilar formulx for the tangents, co- 

 tangents, fecants, eofecants, &c. we may put our fecond 

 expreffion under the form, 



r f . i\ /' _ fin. a . fin. 3\ ^ ^ , 



cof. (a + b) =1 + —= jT-r cof. a . cof. b. 



V col. a . col. b/ 



And if now we divide formula i by this, we have, by 

 obferving that — ^ = tan. a. 



cof. a 



3. tan. {a ±b) = 



4. cot. (a + b) = 



tan. a ± tan. 5 

 I '^ tan. a . tan. b' 



cot. a . cot. i + I 

 cot. b -)- cot. a 



the latter being obtained by dividing the fecond formula 

 by tlie firft. 



Again, taking the reciprocal of formula 2, we have 



cof. (a + 3) J + tan. a . tan. b ' cof. a . cof. b '' 



fee. a . fee. 3 



5. fee. (a + 3) :=: 



6. cof. {a±b) = 



I 4- tan. a . tan. i' 

 cofec. a . cofec. b 



cot. b + cot. a 

 In like manner may be found, 



7. cho. {a ±b) = cho. a . fup. cho. i + clio. I . fup. 



cho. a. 



8. verf. (a + 3) = ^' (verf, a . fup. verf. I) + 



^ (verf. b . fup. verf. a). 



The reader will find a different invefttgation of thefe pro- 

 perties under the article Geometry of Position. 



Let us now fee how far thefe general formulse will indi- 

 cate the changes of figns in the trigonometrical lines, as 

 connefted with the feveral different magnitudes of their 

 refpeftive arcs ; a fubjecT; that we have already confidered 

 geometrically. 



L,et T denote the iemicircle, and let the arc a = i t ; 

 then becaufe fin. ^ ir ~ I, and cof. i ^ = o, we have 

 from 



Formula (i) fin. (i:r + i) = + cof. 3. 



(2) cof. (iff +3) = _ fin. b. 



(3) fin. (iff - (5) =1 -f cof. b. 



(4) cof. (iff ~ b) = + fin. b. 



Here we obtai" the fame refu'.tp as before deduced ; that 

 is, it is obviou; n thefe, that if we conlider the fine and 

 cofine of an arr i ; than a qHadr.int as pofitive quantities, 

 the cofine of an .i:c greater than a quadrant, but lefs than 

 a femicii-cle, will be negative, but its fine pofitive. 



Again, let the arc a — -x ; then becaufe fin. v — o, 

 cof. TT =: — I, we have 



fin. {■;r + b) =z - fin. 3 ; cof. (- 4 3) = - cof 3 ; 

 fin. [v — b) — + fin. b; cof. [-r — b) — — cof. b. 



Whence it follows, that the fines of angles greater than a 

 femicircle, but kfs than a circle, have negative figns ; while 

 12 



thofe that are lefs than a femicircle are pofitive : whereas 

 the cofines of angles equally above and below the femicircle 

 have the fame figns. 



If now we aifume the arc a = i ir ; then, becaufe fin. 

 -J T = — I, and cof. i tt = o, we have 



fin. (-3- n- 4- i) = - cof. i ; cof. (4^4- b) = -f fin. b ; 

 fin. (4 ff — i) = — cof. b ; cof. (i 7: — b) = — fm. b. 



Therefore, when the arc is comprehended betvreen 4 'r, 2 ir, 

 its cofine is pofitive, and its fine negative. Thefe refults 

 have all been previoufly eilablifhed geometrically ; but they 

 are here very readily extended to any arcs expreffed by 



— ~ + 3 ; « being any number whatever. Thefe are all 



contained in the following general formulae, viz. 



fin. 



4« + I 



±') = 



li.L 



n.. (ilJ-J = ± j) = If r,„. t 

 r,n. (l^--3 . ± «) = - 



co(. b. 



+ fin. b. 





fin. b. 



P 



4« + 3 



+ i) = - cof. b. 



fin. 3. 



^ + b^ =z ± 



cof. 



fil±A ^ + i^ = + cof. b. 



Hence it follows, that there is an indefinite number of 



arcs that have the fame fine and cofine, as alfo the fame 



fecant and tangent ; the conditions of which latter are 



readily deduced from the above, by means of the formula 



fin. a 1 



fee. a = ■■ , &c. 



cof. a 

 tan. a =1 , cot. a 



fin. 



^a' 



_ . fin. a , ^ 



ror imce tan. a = —-. — , and cotan. a 



cof. a 

 cof. a 



, it fol- 



cof. a fin. a 



lows, that the tangent will be pofitive, while fin. a and 

 cof. a have the fame figns ; and negative, when the figns are 

 different ; and the fame alfo with the cotangents ; thefe are, 

 therefore, both pofitive in the firll and third quadrants, and 

 negative in the fecond and fourth. Alfo the tangent is in- 

 finite when the cof. :;:; o, and cotangent infinite when 



fin. =: o. Again, fince fee. a= — ^— , and cofec. a = ;; , 



col. a nn. a 



it follows that thefe lines have the fame figns as the cof. and 

 fin. rcfpeftively ; and that the fecaiit is infinite when cof. 

 = o, and the cofec. infinite when fin. = o. 



Having thus fhewn the relations; between the feveral tri- 

 gonometrical lines, tlieir mutations and determinate values, 

 under certain magnitudes of the arcs to which they apper- 

 tain, we (hall proceed in the next place to inveftigate fome 

 of the principal formuls for the fines, cofines, &c. of mul- 

 tiple arcs. 



0/ 



