Expressions for the sines and cosines of multiple area. 



1. Cos. (n + 1) A rr 2 cos. n A. cos. A cos/(w 1) A. 



2. 2 Cos. mA = (2 cos. A) w ?w (2 cos. A) w " 2 -f- .*" ~ 



(3 cos. A) <w - 4 - aufi?r^J5^|L (-2 cos. A) " 6 + fte. 



3. Sin. ( 4- 1) A = 2 sin. ra A. cos. A sin. (n 1) A. 



*. Si n .m A= m sin. A- ^>_Tll> (sin . A) a + "' ( ""- 3 '-| ( f -"-' 

 (sin. A) 5 &c. (?n, odd.) 



5. Sin. m A cos. A frn. sin. A " ^*^~ ^ (si- A)* -f 



m. (wa 4) (ma 16) . . \ . 



2. 3 4. 5 (sm ' A) &c - J ( w even -) 



fl. Let 2 cos. A x -\ then 2 cos. n A x n + ^ (n any No.) 



7. (Cos. A + v'"^l sin. A) m = cos. m A + V"^T sin. m A. 



and (cos. A V 1 sin. A) m = cos. A V 1 sin. m A. 

 whence we have in another form 



8. Cos. m A = (cos. A) m S^fesU ( CO s. A) m " 2 * (sin. A) + 



g . (sin> 4 



and sin. in A m (cos. A) OT " x sin. A -- -(cos. A) m ~ 



(sin. A)3 &c. 



9. Also if e No. whose hyp. log. = 1 we have in terms of the impos- 

 feible quantity V"H1 



c nA V^^-.A vCT e nA>J-\ e -n 



Cos. n A -,&sin. n A 



2 94/17 



E.rpres* ions for the powers of the sinv and cosine of an arc. 



\. '/*- 1 (cos. A " co, n A 4. . cos. ( ) A 4-n. 

 / 4) A -f &5. 



329 T 2 



