46 PRACTICAL MATHEMATICS 



Example 2. Reduce (3 7*) ~ 3 to the form a + hi. 

 Now 3 - 7i = 58 2 (cos + i sin 0) 



where = 360 - A and tan A = - = 2-3333 



o 



A = 66 48' 

 = 293 12' 



Then (3 - 7*)~ 3 - 58~^(cos 293 12' + * sin 293 12') ~ 3 



= 0-002264 (cos ( -879 36') +i sin ( - 879 36') } 

 = 0-002264(cos 200 24' + i sin 200 24') 

 = 0-002264( - cos 20 24' - * sin 20 24') 

 = 0-002264J - 0-9373 - 0-3486*') 

 = - 10- 4 (21-21 + 7-894*) 



EXAMPLES III 



Simplify the following expressions, giving each in the form 

 a + bi, the values of a and b given correct to four significant 

 figures. 



(1) (5 + 4t) (3 + 7*) (2 - 3*) (2) (7 - 2*) (5* - 3) (8 + 3*) 

 (-)(+) (4) 5- 



(4-3*) (2 +3*) (3 -5*) 



/tf (7*-5)(5-2) (2* - 7) (3 +10*) 



( ' (8 + 5*) (3 - 7*) ( ' (8 - *)(4 + 3*) 



Extract the square roots of : 



(7) 15 + 7* (8) 9 + 13* 



(9) 12 - 19* (10) 15 - 8* 



(11) -8+13* (12) - 18 + Hi 



(13) - 14 - 19* (14) - 21 - I6i 



(15) * (16) 1 



v 



Express the following complex quantities in the form r(cos 

 + * sin 0), always keeping r positive : 

 (17) 8 + 3* (18) 18+11* 



(19) 11 - 15* (20) 9-8* 



(21) -7+5* (22) - 10 + 17* 



(23) - 14 - 9* (24) - 12 - 17* 



1 * 



(25) Express j= -\ j= in the form r(cos + i sin 0), giving the 



A/2 A/2 



(1 * \ 4 /I * 

 r=-\ 7= ) > ( 7=H 7= 

 A/2 V2/ \V2 V2 



/I * \ 3 , / 1 i A 



( ?= + ~7= . and ~7= H 7= ) 



\V3 A/2/ \A/2 A/2/ 



