214 So77te Observations on Dr. Taylor's Theorem 



these values in the preceding formula we shall find 



Sin. a + A' = Sin. a + -• Cos. a 4- x A • Sin. a A- x — 



1 1" 2 



Cos. a -I- ;« : — • Sin. a + .v -1 • p„_ 



1- 2- 3 1-2-3-4 1-2-3-4-5 



a -\- X + Sec; and arranging the terms of ihis equation ac- 

 cording to the sine and cosine of a + x we have. Sin. a + x. 



\ 1 + + — &c. I — Cos. a + X. ( 



V 1-2 ^ 1-2-3-4 / \1 1-2-3 



+ 8ic. ) = Sin. a. Now if we develop Sin x 



l'2-3-4'5 / ^ 



and Cos. x by Taylor's theorem, we shall find 



-r ^^ ■■>■' 



Sin. X = J- — — Sec. 



1 r2-3 1-2-3-4J 



-'■* ■'■■' 



Cos. .r = 1 1 See; 



1- 2 1*2- 3- 4 ' 



and putting these values into the former result, we obtain 



the following property of the functions, viz. Sin. a + x. 



Cos. X — Cos. a + X. Sin. x = Sin. a. 



We now proceed to consider the development of the 



Cos. a + x; and having in this case <p {a + x) = Cos. 



d (p (a + t) ^. d.'^ <P (a + a) 



« 4- u; ^^j ' - =3 - Sin. a + x; — ^) J -' = - 

 du du- 



d*. (p (a -\- x) ^. „ , 



Cos. a -\- X ; —^r-^ = Sm. a + x; &c. we calcu- 



late Cos. a ->r X — Cos. a — - Sin. a + x + — Cos. 



a ^ X A Sin. a -{■ x Cos, a + x 



1-2-3 1-2-3-4 l-2'3-4-5 



Sin a ■\- X -\- &.c.; and putting this result under the form 

 Cos. a + X. (1 — 37^ + i.2-i-\ ~ ^^-^ + ^'"' * + "• 



(X X'^ X^ \ 

 --— -\ r— r^ — &.C. ) = Cos. a, we obtain tiiis 

 i l'2-3 1-2-3-4-5 J ' 



Other property of these functions : Cos. a + x. Cos. oe + 



6in. a + X. Sin. x = Cos. a. 



From 



