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Products of complex numbers in polar form

If z 1 = r 1 ( cos θ 1 + i sin θ 1 ) and z 2 = r 2 ( cos θ 2 + i sin θ 2 ) , then the product of these numbers is given as:

z 1 z 2 = r 1 r 2 [ cos ( θ 1 + θ 2 ) + i sin ( θ 1 + θ 2 ) ] z 1 z 2 = r 1 r 2 cis ( θ 1 + θ 2 )

Notice that the product calls for multiplying the moduli and adding the angles.

Finding the product of two complex numbers in polar form

Find the product of z 1 z 2 , given z 1 = 4 ( cos ( 80° ) + i sin ( 80° ) ) and z 2 = 2 ( cos ( 145° ) + i sin ( 145° ) ) .

Follow the formula

z 1 z 2 = 4 2 [ cos ( 80° + 145° ) + i sin ( 80° + 145° ) ] z 1 z 2 = 8 [ cos ( 225° ) + i sin ( 225° ) ] z 1 z 2 = 8 [ cos ( 5 π 4 ) + i sin ( 5 π 4 ) ] z 1 z 2 = 8 [ 2 2 + i ( 2 2 ) ] z 1 z 2 = 4 2 4 i 2
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Finding quotients of complex numbers in polar form

The quotient of two complex numbers in polar form is the quotient of the two moduli and the difference of the two arguments.

Quotients of complex numbers in polar form

If z 1 = r 1 ( cos θ 1 + i sin θ 1 ) and z 2 = r 2 ( cos θ 2 + i sin θ 2 ) , then the quotient of these numbers is

z 1 z 2 = r 1 r 2 [ cos ( θ 1 θ 2 ) + i sin ( θ 1 θ 2 ) ] , z 2 0 z 1 z 2 = r 1 r 2 cis ( θ 1 θ 2 ) , z 2 0

Notice that the moduli are divided, and the angles are subtracted.

Given two complex numbers in polar form, find the quotient.

  1. Divide r 1 r 2 .
  2. Find θ 1 θ 2 .
  3. Substitute the results into the formula: z = r ( cos θ + i sin θ ) . Replace r with r 1 r 2 , and replace θ with θ 1 θ 2 .
  4. Calculate the new trigonometric expressions and multiply through by r .

Finding the quotient of two complex numbers

Find the quotient of z 1 = 2 ( cos ( 213° ) + i sin ( 213° ) ) and z 2 = 4 ( cos ( 33° ) + i sin ( 33° ) ) .

Using the formula, we have

z 1 z 2 = 2 4 [ cos ( 213° 33° ) + i sin ( 213° 33° ) ] z 1 z 2 = 1 2 [ cos ( 180° ) + i sin ( 180° ) ] z 1 z 2 = 1 2 [ 1 + 0 i ] z 1 z 2 = 1 2 + 0 i z 1 z 2 = 1 2
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Find the product and the quotient of z 1 = 2 3 ( cos ( 150° ) + i sin ( 150° ) ) and z 2 = 2 ( cos ( 30° ) + i sin ( 30° ) ) .

z 1 z 2 = 4 3 ; z 1 z 2 = 3 2 + 3 2 i

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Finding powers of complex numbers in polar form

Finding powers of complex numbers is greatly simplified using De Moivre’s Theorem    . It states that, for a positive integer n , z n is found by raising the modulus to the n th power and multiplying the argument by n . It is the standard method used in modern mathematics.

De moivre’s theorem

If z = r ( cos θ + i sin θ ) is a complex number, then

z n = r n [ cos ( n θ ) + i sin ( n θ ) ] z n = r n cis ( n θ )

where n is a positive integer.

Evaluating an expression using de moivre’s theorem

Evaluate the expression ( 1 + i ) 5 using De Moivre’s Theorem.

Since De Moivre’s Theorem applies to complex numbers written in polar form, we must first write ( 1 + i ) in polar form. Let us find r .

r = x 2 + y 2 r = ( 1 ) 2 + ( 1 ) 2 r = 2

Then we find θ . Using the formula tan θ = y x gives

tan θ = 1 1 tan θ = 1 θ = π 4

Use De Moivre’s Theorem to evaluate the expression.

( a + b i ) n = r n [ cos ( n θ ) + i sin ( n θ ) ] ( 1 + i ) 5 = ( 2 ) 5 [ cos ( 5 π 4 ) + i sin ( 5 π 4 ) ] ( 1 + i ) 5 = 4 2 [ cos ( 5 π 4 ) + i sin ( 5 π 4 ) ] ( 1 + i ) 5 = 4 2 [ 2 2 + i ( 2 2 ) ] ( 1 + i ) 5 = 4 4 i
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Finding roots of complex numbers in polar form

To find the n th root of a complex number in polar form, we use the n th Root Theorem or De Moivre’s Theorem    and raise the complex number to a power with a rational exponent. There are several ways to represent a formula for finding n th roots of complex numbers in polar form.

The n Th root theorem

To find the n th root of a complex number in polar form, use the formula given as

z 1 n = r 1 n [ cos ( θ n + 2 k π n ) + i sin ( θ n + 2 k π n ) ]

where k = 0 , 1 , 2 , 3 , . . . , n 1. We add 2 k π n to θ n in order to obtain the periodic roots.

Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
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Practice Key Terms 4

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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