1821.] a Method of applying Maclaurin's Theorem* 103 



Lemma. — To find the sines, cosines, tangents, 8cc. of circular 

 arcs in functions of the arcs, radius being unity. 

 Given sin. z — function oi z — fz, then will 



COS. X = (1 — sin.^x)^ r= 1 — f^ z, 



sill, g _ / s __ 



COS. z 



cot. Z 



tan. z = — = , " ". — f^^Zy 



COS. Z 1 — /, Z ^ 1 > ' 



lan. z /„ %' 



11 



cosec. z •=. = -r-. 



sin. z f z 



Example 10. — To find the sine of an arc in terms of the arc, 

 radius unity. 

 u = sin. z=:0-\-fz = A+Bz + Cz^ + I>z''+&ic. = 

 A +/;s.-. A = 0. .. .>^-» 



^ = cos. x=l — /, s; = B — /i X .-.B = 1, ' 



^ = .- sin.« = - (0 -rfu z) = 2C+f,,z.'.2C=^- 0, 



^= -cos.;.= -(1 -/,,;.)=:2.3D -/„;«. -.2.3 

 D= -1, ^-. - ■■ ^-.■^•'^■---- -^ 



&c 



By substituting for A, B, C, &c. we have 



sin. z=z- j-^^ + r:^^_-j-g - See. 



Example 11. — To find the cosine of an arc in terms of the arc, 

 radius unity. 

 u = cos. z=z I —fzz= A — (Bz + C z" + D z' + &c.) 

 = A -/;.... A = 1, 

 j^ = - sin. z= - (0 +/,;?;) = B +/ z.'.B =z - 0, 



^= -cos.«= -(1 -/„;.) = 2C-/„;..-.2C=-l, 



^ = sin.a = +/,,, 21 = 2 .3D +/„i .•?;.•. 2 .3 D = 0, 



d4 u 



^,, ----- =!-/> = 2.3.4 E-/,«. -.2.3. 4E=1, 



= COS. z 



&c 



By substituting for A, B, C, &c. we have ^ * - - • 



Example 12. — To find the tangent of an arc in terms of the 

 arc, radius unity. 



