54 PRACTICAL MATHEMATICS 



Then t L* = log e fc + -. (log e fc) 3 + . . . 



When a is made infinitely small - = 1, and the terms in- 



volving the powers of a become negligibly small 



and i = log e k, or k = e 1 

 Hence cos a + i sin a = k a = (*) a = e^ 



and cos a i sin a = fc~ a = e~ ia 

 It should be noted that since k = e 1 



i 2 i 3 i 2 

 A;=1 + ; + _. + _ 



Ill 



~ ~|ar "~]j~" TIT" v ' "r ijr [s 



= 0-5403 + 0-8415i, which agrees with the value given above. 



34. The Series for sin a and cos a. 



e = cos a + * sin a e~ ia = cos a sin a 

 Subtracting 2* sin a = e w e~ 



1 

 and sin a = . (e w e"" 1 ) 



Adding 2 cos a = e + e~ ia 



and cos a = - (e + e') 



m 



Fromtheserelationswecanreadilyfindtheseriesfor sin a and cosa. 

 For ^=i + ia+ ^VJ + g + ... 



Subtracting e ia e' = 2 Ua + -rr- + -rr- + . . . j 



and sin a --(,'-,-")- a- -- + 



Adding ^+^ = 



"* 4 



and cos a = (e io + - to ) = 1 - -r 



