710 PHILOSOPHICAL TRANSACTIONS. [aNNO I796. 



I- ' a^ + '- z* — 1 x'' + &c. 



, Sin. z, 1.2.3 I . 2 . . 1; J . 2 . . 7 

 Tang. 2( = ) = z X ~ ; : 



1.2 I. 2. .4 I. 2. .6 



^ 15. After having expressed the sine, cosine, and tangent of an arc, in 

 terms of the arc, we can reciprocally, by reversion of series, express the arc by 

 these functions of itself. The following is the most elementary method, and 

 most conformable to the foregoing processes. 



,^^ , ^, , (cos. z + sin. z -/ — 1)" + (cos. z — sin. z ^ — 1)" 



We know that cos. J2z = ;3 ^^ — 



, . (cos. z + sin. z aJ — l)' — (COS. z — sin. z ^ — 1)" 

 and sin. nz = — ,, ^ _ i 



Hence cos. vz + sin. nz / — 1 = (cos. z + sin. z </ - 1)"; 



I 



and (cos. 7iz + sin. 7iz V — -l)" = cos. ,:; + sin. s v' — 1 ; 



also cos. nz — sin. nz v/ — 1 =(cos. z — sin. z ^ — If; 



I 



and (cos. nz — sin. nz s/ — l)~ = cos. z — sin. s -/ — 1. 



(COS. nz + sin. nz ^ — 1)" + (c os, nz — sin, nz ^ — i)~ 



Therefore cos. % = i ■ :; -. 



I 



(cos. m + sin, nz ^/ — 1)" — (cos, nz - sin, nz ^/ — i) -r 



and sm. z = — - — o ^ _ i ~ • 



JL 1 - 



Hence n sin. z = cos. nz " X (tang. nz. Therefore also n sm. -z = cos. z " (tang.z 



' ' I — J. 2 — 



_ Ll . Ill tang. %z - ~ . -7^ tang. \ 



.-'I 4-i- '-- 4--^ 



+ ,— ,--^ ta»g- '«2 + T.i""'"~ *^"^- '^ 



■ 8ic. &c. 



Hence also the limits of the '2 members of this equation are equal to each 

 other. But, by augmenting n, the limits are x, and tang, z — I tang.^ x + i 



tang.' z — i tang.' z + &c. 



Therefore ;. = ^ - -i- i^ + M' - i ^^ + i <' - ^^c - (making < = tang, z.) 



§ 16. From the preceding formulas we easily deduce the differential formulas 

 of the trigonometric functions of circular arcs. 



Since sin. z = z — j-^— z' + TTo" *' ~ 1 . 2 ... 7 *' "^ ^^' 



rr\ C ,^ f'sin-- _ , 1 ,21 i „■« \ -,/+&C. = COS.5;. 



Therefore — ,— = ' f;~ * ^1.2. .4 i.->..6^ 



. , (/ cos. z , 1 3 i ,5 I \ 7 _ gjc. = — sin. z. 



^"■^ '^IhT = ~ '^ + T'JTl} ^ 1 . 2 . . 5 ~ ^ 1 . 2 . . . 7 

 Since tang. 



d . sin. z . J ■ C OS. z 

 1 . tos. J; . — sin. a . ■ ^ z + sin. Z 1 « 



Sin. z (/ . tang, z ,1^ '^^ __ cos, ^ -r »u ■ __ __ gg^^Sj. 



* — cos. z' dz ~ cos.^ z. cos.« z cos.^z 



