CHAPTER 46 DE MOIVRE’S THEOREM 
                                                                        
              EXERCISE 192 Page 522 
               
              1.  Determine in polar form:  (a) [1.5∠15°]5     (b) (1 + j2)6 
                                                                                                                                                                                                            
               
              (a)             5      5           = 7.594∠75°  
                    1.5∠15° =1.5         5∠×15    °
                   [         ]
               
              (b)  1 + j2 =                  
                               5∠°63.435
               
                    Hence,  (1 + j2)6 =                   66                                           = 125∠20.61°  
                                            5∠63.435° =5                6∠63.435×      125°=380.61∠          °
                                         (               )    (    )
               
               
              2.  Determine in polar and Cartesian forms:  (a) [3∠41°]4      (b) (–2 – j) 5 
               
              (a)           4     4           = 81∠164° = 8 cos 164° + j8 sin 164° = –77.86 + j22.33     
                    3∠41° =3 ∠×4 41 °
                   [       ]
               
                            5                        5         5
              (b)                 
                   (−2−j) =5             153.435∠−     ° =5       5    153.435∠ ×−        °
                                                         (    )
                                
                                                                       = 55.90∠–767.175° = 55.90∠–47.18° 
                                                                       = 55.90 cos – 47.18° + j55.90 sin –47.18° 
                                                                       = 38 – j41 
                                     
               
              3.  Convert (3 – j) into polar form and hence evaluate (3 – j)7 , giving the answer in polar form.     
                
                                             1
                                          
                            22 −1
                             +∠ −
              (3 – j) =   3    1    tan           =   10∠−18.43° 
                                          
                                             3
                                          
                              7                       7          7
              Hence,                 = 3162∠–129° 
                       (3−j) =10             18.43∠−     ° 10=7 18.43∠ ×−                   °
                                                          (     )
                                   
               
              4.  Express in both polar and rectangular forms:   (6 + j5)3 
               
                       3                      3          3
                             
              (6+j5) =61 39.806∠                 ° 61=3 39.806∠×                   °
                                                  (     )
                           
                                                             = 476.4∠119.42° 
                                                                     779                                   © 2014, John Bird 
               
                                                                 = 476.4 cos 119.42° + j476.4 sin 119.42° 
                                                                 = –234 + j415 
                   
                                                     
                  5.   Express in both polar and rectangular forms:   (3 – j8)5   
                    
                              5                                5             5
                                     = 45530∠–347.22° = 45 530∠12.78° 
                   (3−j8) =73                    69.444∠−          ° 73=5 69.444∠ ×−                               °
                                                                    (       )
                                   
                                                                    = 45 530 cos 12.78° + j45 530 sin 12.78° 
                                                                    = 44 400 + j10 070 
                                
                                                                                                          4          
                  6.   Express in both polar and rectangular forms:   (–2 + j7)
                   
                                                                                                                           7
                                                                     22                                               −1 
                                                                                                              =tan                 74.054=                         °
                  From the diagram below,   r =                    2 +=7           53       and       α                                       
                                                                                                                           2
                                                                                                                         
                  and                                                    θ =180         74.054°−       105.945° =             °
                                                                                                              
                                              4                              4              4
                  Hence,                             
                                (−2+j7) =53 105.945∠                            ° =53           4 105.945∠×                  °
                                                                                  (       )
                                                   
                                                                                    =                      =                        
                                                                                   2809∠°423.78               2809∠°63.78
                   
                                       2809∠63.78 =2809cos63.78°+ j2809sin63.78°        
                                                           (                                                    )
                                                            = 1241+ j2520  
                                                         
                             
                  7.   Express in both polar and rectangular forms:   (–16 – j9)6                  
                   
                                                                                                                               9
                  From the diagram below,   r =                       22        and                                      −1                        
                                                                  16 +9 = 337                                 α =tan                    29.358=                         °
                                                                                                                            
                                                                                                                              16
                                                                                                                            
                  and                                                    θ =180         29.358°+       209.358° =             °
                                                6                                6                6
                  Hence,                               
                                (−16−j9) =337 209.358∠                                °337=6 209.358∠×                                  °
                                                                                      (         )
                                                     
                                                                                          780                                               © 2014, John Bird 
                   
                                                                                         =                 6                     =                    6                  
                                                                                                                                     (38.27×∠10 ) 176.15°
                                                                                            38.27×∠10          1256.148°
                                                                                                                      
                                                        66        
                                       (38.27×10 )∠176°9'=10 38.27cos176.15°+j38.27sin176.15°
                                                                                  (                                                            )
                                                                          = 106(−38.18+j2.570) 
                    
                                                                                               781                                                  © 2014, John Bird 
                    
                EXERCISE 193 Page 524 
                 
                1.   Determine the two square roots of the given complex numbers in Cartesian form and show the  
                      results on an Argand diagram:   (a) 1 + j     (b) j          
                 
                                                                  1
                                                                  2
                                    
                (a)                                                 
                      1+j =2 45∠°=2 45∠°
                                    
                                                      1    1
                      The first root is:        (  2)2 ∠ ×45°=1.1892 22.5∠(1.099°=j0.455)                             +
                                                           2
                      and the second root is:        1.1892∠(22.5°+180°) =( −1.099 −j0.455) 
                      Hence,                                             as shown in the Argand diagram below. 
                                     (1+=j)±(1.099+j0.455)
                                                                                                 
                                                                   1
                (b)                                                2  
                       jj=0+= 1∠°90 =1∠°90
                                          [         ]   [        ]
                                                   1    1
                      The first root is:         1 2 ∠ ×90°=1 ∠45 °=(0.707                +j0.707) 
                                                ( )     2
                      and the second root is:        1∠(45°+180°) =( −0.707 −j0.707) 
                      Hence,                                        as shown in the Argand diagram below. 
                                  ±+jj= (0.707          0.707)
                                                                                                 
                    
                2.   Determine the two square roots of the given complex numbers in Cartesian form and show the  
                      results on an Argand diagram:   (a) 3 – j4     (b)  –1 – j2             
                                                                             782                                        © 2014, John Bird