56 PRACTICAL MATHEMATICS 



If the fraction - is less than 1, all terms on the right-hand side, 

 r 



except the first, may be neglected, and approximately 



8h- c 



i ' 



35. If we put x = cos a + i sin a 



then - = cos a i sin a 



x 



and x + - = 2 cos a (1) 



00 



u - - = 2i sin a (2) 



x 



Also x n = (cos a + i sin a)" = cos na + i sin na 



11 



- = : : = cos na i sin na 



x n cos na + i sin na 



and x n H - n = 2 cos na (3) 



\ 



x n = 2i sin na (4) 



x n 



These results are very useful for expressing powers of sines and 

 cosines, and also products of powers of sines and cosines, in terms 

 of sines and cosines of multiple angles. 



(a) Working with sin 5 a 



( 1\ 6 



(2i sin a) 5 = ( x -- ) 

 \ x/ 



32i 5 sin 5 a = x 5 - 5x 3 + Wx - 10 - + 5-^ -- ^ 



x x s # 5 



32i sin 5 a = (x 5 - i) - 5 (x* - 1) + 10 (x - -) 



x & / or/ \ x/ 



= 2i sin 5a 10i sin 3a + 20i sin a 



155 



and sin 5 a = sin Sa-^rr sin 3a + - sin a 

 16 ID o 



(&) Working with cos 4 a 



/ 1\ 4 



(2 cos a) 4 = lfl?+-J 

 \ xj 



16 cos 4 a= 4 +4 2 +6+4 + -, 



2 4 



+ 6 



= 2 cos 4a + 8 cos 2a + 6 



113 



and cos 4 a = - cos 4a + - cos 2a + - 



o ^ o 



