THE EXPONENTIAL FORM OF (cos 9 + i sin 8) 53 



Is + iku 

 Current - i> /W ^ operating on sin qt 



f / a _ R\ / a _ ft\i 



- Ar |cos( C I + sin ( r )| operating on sin qt 



- Ai> { cosf - ^-J- ) sin <# + sin ( - - E J . cos qt \ 



\ \ JP / \ 4M ' 



= At; sin{-(a- 



-[ 

 -[ 



9 x IP" 12 + 25 x IP" 18 36 x 10 6 ]* 

 36 + 9 x 10- 6 36 x 10 6 J 



9 x 10~ 12 (1 + 100)1* 



- 10- 3 ( 



86(1 + 9) 

 ) = 10- 3 (2-525)* = 1-261 x 10" 3 



a - p = 12 44' - (a - (3) = 6 22' = 0-1111 radians 



Then current = 1-261 x 10- 3 v sin (6000* + 0-1111) 

 the angle being expressed in radians. 



33. The Exponential form of (cos + i sin 0) 



(cos + i sin 0) n = cos w0 + i sin w0 



Taking to be 1 radian 



Then (cos 1 + i sin l) n = cos n + i sin n 



Putting k = cos 1 + i sin 1 



Then k n = cos n + i sin n, or k = cos a + i sin a, where n or a 

 represents any angle taken in radians. 



It must be remembered that A; is a complex quantity of the form a 

 + bi, for cos 1 = 0-5403, and sin 1 = 0-8415, and k = 0-5403 + 0-8415i. 

 If k n = cos a + i sin a 



k~ n = r : = cos a i sin a 



cos a + * sin a 



and 2i sin a = fc" k~ a by subtraction 



a 2 

 = 1 + a logjc + -j-jr- (log (S } 2 + . . . 



>...) 



(a 



