DE MOIVRE'S THEOREM 45 



be proved in the same way as (cos + t sin 0)~ n = cos ( - n0) 

 + i sin ( n0) has been proved. 



Thus in general (cos + i sin 0) n = cos n0 + i sin n0, whatever form 

 n may take : n may be positive or negative, integral or fractional 



. 4 4 



/5 4t)-(3t 2) 



Example 1. Reduce * -*- to the form a + bi. 



( - 8 - 3i)* 



Now 5 - 4t = 41* (cos + i sin 0) 



where = 360 - A and tan A = - = 0-8 



5 



A = 38 40' 

 = 321 20' 



Then (5 - 4*)* = 41*(cos 321 20' + i sin 321 20')* 



= 41^(cos 160 40' + i sin 160 40') 

 Next - 2 + Si = 13*(cos + i sin 0) 



where = 180 - A and tan A = = 1-5 



A = 56 19' 

 = 123 41' 



Then (3i - 2)' = 13*(cos 123 41' + i sin 123 41') ^ 



= 13*(cos 82 27' + i sin 82 27') 

 Next - 8 - 3i = 73*(cos + i sin 0) 



where = 180 + A and tan A = | = 0-375 



8 



A = 20 32' 

 = 200 32' 



Then ( - 8 - 3*) f = 73 T (cos 200 32' + i sin 200 32') T 

 = 73^(cos 80 13' + i sin 80 13') 



Hence ^~ 



( - 8 - 3i)* 



41^ x 13* f (cos 160 40' + Isinl60 40')(cos 82 27 r + t sin 8227h 



ro 4 \ (cos 80 13' + i sin 80 13') / 



73 



fcos 243 7' + i sin 243 7'} 

 ~ \cos 80 13' + i sin 80 13'f 



= 2-523(cos 162 54' + i sin 162 54') 

 = 2-523( - cos 17 6' + i sin 17 60 

 = 2-523( - 0-9558 + 0-2940J) 

 = - 2-411 + 0-7415i 



