Solutions To Complex Numbers In Polar Form (Part 2) Problem

 We want to divide: z₁  = -2 + 2i by z₂  = -2 - 3i

We graph each of these first on an Argand diagram (graph) to obtain the respective angles q₁  and   q₂.

Then: let A = z₁    and B =  z₂       

arg(z₁ ) = arctan (y1/x1)  =

arctan (y1/x1) = arctan (-2/2) = arctan (-1) = - 45 deg

 So q₁ = - 45 deg

arg(z₂) = arctan(y₂/x₂) =  arctan(-3/ - 2)  

=  arctan(3/ 2) = 56.3 deg

 so q₂ = (56.3 deg) 

And (by Pythagoras) r2  =  Ö (x2 ²  +  y2 2 )

= Ö (-2) ²  +  (-3) 2   = Ö(4) + (9) = Ö 13  = 3.6

 Now  z₁  and z₂  may be written respectively:

z₁ = 1.41 cos (-45) + i sin (-45) = 1.41 cis (-45)

z₂ = 3.60 cos (56.3) + i sin (56.3) = 3.60 cis (56.3)

We may now proceed with complex division, i.e.:

(z₁/z₂) =  (r₁ cis(q₁)/ r₂ cis(q₂)) = (r₁/ r₂) cis (q₁ –  q₂)

Where:  (r ₁/r ₂) = 1.41/ 3.60  =   0.39

And: (q₁ – q₂)   = arg(z₁) – arg(z₂) = (-45) – (56.3) = -101.3

Finally: (z₁/z₂) =  0.39 cis (q₁ –  q₂) =

 0.39 cos (-101.3) + i sin (-101.3) 

= 0.39 [(- 0.195) + i (-0.98)]  =  - 0.07 + 0.38 i