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1	
Operations	with	Complex	Numbers	in	Polar	Form		
	
	
Let	𝑧! = 𝑟!(𝑐𝑜𝑠𝜃! + 𝑖 sin 𝜃!)	and	𝑧! = 𝑟!(𝑐𝑜𝑠𝜃! + 𝑖 sin 𝜃!)	
	
a)	Find	𝑧! 𝑧!		
(this	can	also	be	expressed	as	𝑟! 𝑐𝑖𝑠𝜃!×𝑟! 𝑐𝑖𝑠𝜃!)	
	
	
	
	
	
b)	Find	
!!
!!
	
	
	
	
	
What	can	you	conclude	about:		
𝑧! 𝑧! 	
	
arg	(𝑧! 𝑧!)	
	
In	Euler	Form:	
𝑟! 𝑒!!!
𝑟! 𝑒!!!
= 	
	
!!!!!!
!!!!!! =	 	
Hint	remember	your	addition	formulae:		
cos(𝜃! + 𝜃!) = 𝑐𝑜𝑠𝜃! 𝑐𝑜𝑠𝜃! − 𝑠𝑖𝑛𝜃! 𝑠𝑖𝑛𝜃!	
sin (𝜃! + 𝜃!) = 𝑠𝑖𝑛𝜃! 𝑐𝑜𝑠𝜃! + 𝑐𝑜𝑠 𝜃! 𝑠𝑖𝑛𝜃!	
	
Hint:
!!!"#!!
!!!"#!!
×
!"#(!!!)
!"#(!!!)
2	
Practice		
1)	Find	𝑧! 𝑧!		
	When	𝑧! = 3𝑐𝑖𝑠
!!
!
𝑎𝑛𝑑 𝑧! = 4𝑐𝑖𝑠
!
!
	
	
	
	
2)

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Lesson 4 operations with complex numbers in polar form p1 2

  • 1. 1 Operations with Complex Numbers in Polar Form Let 𝑧! = 𝑟!(𝑐𝑜𝑠𝜃! + 𝑖 sin 𝜃!) and 𝑧! = 𝑟!(𝑐𝑜𝑠𝜃! + 𝑖 sin 𝜃!) a) Find 𝑧! 𝑧! (this can also be expressed as 𝑟! 𝑐𝑖𝑠𝜃!×𝑟! 𝑐𝑖𝑠𝜃!) b) Find !! !! What can you conclude about: 𝑧! 𝑧! arg (𝑧! 𝑧!) In Euler Form: 𝑟! 𝑒!!! 𝑟! 𝑒!!! = !!!!!! !!!!!! = Hint remember your addition formulae: cos(𝜃! + 𝜃!) = 𝑐𝑜𝑠𝜃! 𝑐𝑜𝑠𝜃! − 𝑠𝑖𝑛𝜃! 𝑠𝑖𝑛𝜃! sin (𝜃! + 𝜃!) = 𝑠𝑖𝑛𝜃! 𝑐𝑜𝑠𝜃! + 𝑐𝑜𝑠 𝜃! 𝑠𝑖𝑛𝜃! Hint: !!!"#!! !!!"#!! × !"#(!!!) !"#(!!!)
  • 2. 2 Practice 1) Find 𝑧! 𝑧! When 𝑧! = 3𝑐𝑖𝑠 !! ! 𝑎𝑛𝑑 𝑧! = 4𝑐𝑖𝑠 ! ! 2)