<< Chapter < Page Chapter >> Page >

Finding the n Th root of a complex number

Evaluate the cube roots of z = 8 ( cos ( 2 π 3 ) + i sin ( 2 π 3 ) ) .

We have

z 1 3 = 8 1 3 [ cos ( 2 π 3 3 + 2 k π 3 ) + i sin ( 2 π 3 3 + 2 k π 3 ) ] z 1 3 = 2 [ cos ( 2 π 9 + 2 k π 3 ) + i sin ( 2 π 9 + 2 k π 3 ) ]

There will be three roots: k = 0 , 1 , 2. When k = 0 , we have

z 1 3 = 2 ( cos ( 2 π 9 ) + i sin ( 2 π 9 ) )

When k = 1 , we have

z 1 3 = 2 [ cos ( 2 π 9 + 6 π 9 ) + i sin ( 2 π 9 + 6 π 9 ) ]     Add  2 ( 1 ) π 3  to each angle. z 1 3 = 2 ( cos ( 8 π 9 ) + i sin ( 8 π 9 ) )

When k = 2 , we have

z 1 3 = 2 [ cos ( 2 π 9 + 12 π 9 ) + i sin ( 2 π 9 + 12 π 9 ) ] Add  2 ( 2 ) π 3  to each angle. z 1 3 = 2 ( cos ( 14 π 9 ) + i sin ( 14 π 9 ) )

Remember to find the common denominator to simplify fractions in situations like this one. For k = 1 , the angle simplification is

2 π 3 3 + 2 ( 1 ) π 3 = 2 π 3 ( 1 3 ) + 2 ( 1 ) π 3 ( 3 3 ) = 2 π 9 + 6 π 9 = 8 π 9
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the four fourth roots of 16 ( cos ( 120° ) + i sin ( 120° ) ) .

z 0 = 2 ( cos ( 30° ) + i sin ( 30° ) )

z 1 = 2 ( cos ( 120° ) + i sin ( 120° ) )

z 2 = 2 ( cos ( 210° ) + i sin ( 210° ) )

z 3 = 2 ( cos ( 300° ) + i sin ( 300° ) )

Got questions? Get instant answers now!

Access these online resources for additional instruction and practice with polar forms of complex numbers.

Key concepts

  • Complex numbers in the form a + b i are plotted in the complex plane similar to the way rectangular coordinates are plotted in the rectangular plane. Label the x- axis as the real axis and the y- axis as the imaginary axis. See [link] .
  • The absolute value of a complex number is the same as its magnitude. It is the distance from the origin to the point: | z | = a 2 + b 2 . See [link] and [link] .
  • To write complex numbers in polar form, we use the formulas x = r cos θ , y = r sin θ , and r = x 2 + y 2 . Then, z = r ( cos θ + i sin θ ) . See [link] and [link] .
  • To convert from polar form to rectangular form, first evaluate the trigonometric functions. Then, multiply through by r . See [link] and [link] .
  • To find the product of two complex numbers, multiply the two moduli and add the two angles. Evaluate the trigonometric functions, and multiply using the distributive property. See [link] .
  • To find the quotient of two complex numbers in polar form, find the quotient of the two moduli and the difference of the two angles. See [link] .
  • To find the power of a complex number z n , raise r to the power n , and multiply θ by n . See [link] .
  • Finding the roots of a complex number is the same as raising a complex number to a power, but using a rational exponent. See [link] .

Section exercises

Verbal

A complex number is a + b i . Explain each part.

a is the real part, b is the imaginary part, and i = 1

Got questions? Get instant answers now!

What does the absolute value of a complex number represent?

Got questions? Get instant answers now!

How is a complex number converted to polar form?

Polar form converts the real and imaginary part of the complex number in polar form using x = r cos θ and y = r sin θ .

Got questions? Get instant answers now!

How do we find the product of two complex numbers?

Got questions? Get instant answers now!

What is De Moivre’s Theorem and what is it used for?

z n = r n ( cos ( n θ ) + i sin ( n θ ) ) It is used to simplify polar form when a number has been raised to a power.

Got questions? Get instant answers now!

Algebraic

For the following exercises, find the absolute value of the given complex number.

For the following exercises, write the complex number in polar form.

8 4 i

4 5 cis ( 333.4° )

Got questions? Get instant answers now!

For the following exercises, convert the complex number from polar to rectangular form.

z = 7 cis ( π 6 )

7 3 2 + i 7 2

Got questions? Get instant answers now!

z = 4 cis ( 7 π 6 )

2 3 2 i

Got questions? Get instant answers now!

z = 3 cis ( 240° )

1.5 i 3 3 2

Got questions? Get instant answers now!

For the following exercises, find z 1 z 2 in polar form.

z 1 = 2 3 cis ( 116° ) ;   z 2 = 2 cis ( 82° )

4 3 cis ( 198° )

Got questions? Get instant answers now!

z 1 = 2 cis ( 205° ) ;   z 2 = 2 2 cis ( 118° )

Got questions? Get instant answers now!

z 1 = 3 cis ( 120° ) ;   z 2 = 1 4 cis ( 60° )

3 4 cis ( 180° )

Got questions? Get instant answers now!

z 1 = 3 cis ( π 4 ) ;   z 2 = 5 cis ( π 6 )

Got questions? Get instant answers now!

z 1 = 5 cis ( 5 π 8 ) ;   z 2 = 15 cis ( π 12 )

5 3 cis ( 17 π 24 )

Got questions? Get instant answers now!

z 1 = 4 cis ( π 2 ) ;   z 2 = 2 cis ( π 4 )

Got questions? Get instant answers now!

For the following exercises, find z 1 z 2 in polar form.

z 1 = 21 cis ( 135° ) ;   z 2 = 3 cis ( 65° )

7 cis ( 70° )

Got questions? Get instant answers now!

z 1 = 2 cis ( 90° ) ;   z 2 = 2 cis ( 60° )

Got questions? Get instant answers now!

z 1 = 15 cis ( 120° ) ;   z 2 = 3 cis ( 40° )

5 cis ( 80° )

Got questions? Get instant answers now!

z 1 = 6 cis ( π 3 ) ;   z 2 = 2 cis ( π 4 )

Got questions? Get instant answers now!

z 1 = 5 2 cis ( π ) ;   z 2 = 2 cis ( 2 π 3 )

5 cis ( π 3 )

Got questions? Get instant answers now!

z 1 = 2 cis ( 3 π 5 ) ;   z 2 = 3 cis ( π 4 )

Got questions? Get instant answers now!

For the following exercises, find the powers of each complex number in polar form.

Find z 3 when z = 5 cis ( 45° ) .

125 cis ( 135° )

Got questions? Get instant answers now!

Find z 4 when z = 2 cis ( 70° ) .

Got questions? Get instant answers now!

Find z 2 when z = 3 cis ( 120° ) .

9 cis ( 240° )

Got questions? Get instant answers now!

Find z 2 when z = 4 cis ( π 4 ) .

Got questions? Get instant answers now!

Find z 4 when z = cis ( 3 π 16 ) .

cis ( 3 π 4 )

Got questions? Get instant answers now!

Find z 3 when z = 3 cis ( 5 π 3 ) .

Got questions? Get instant answers now!

For the following exercises, evaluate each root.

Evaluate the cube root of z when z = 27 cis ( 240° ) .

3 cis ( 80° ) , 3 cis ( 200° ) , 3 cis ( 320° )

Got questions? Get instant answers now!

Evaluate the square root of z when z = 16 cis ( 100° ) .

Got questions? Get instant answers now!

Evaluate the cube root of z when z = 32 cis ( 2 π 3 ) .

2 4 3 cis ( 2 π 9 ) , 2 4 3 cis ( 8 π 9 ) , 2 4 3 cis ( 14 π 9 )

Got questions? Get instant answers now!

Evaluate the square root of z when z = 32 cis ( π ) .

Got questions? Get instant answers now!

Evaluate the cube root of z when z = 8 cis ( 7 π 4 ) .

2 2 cis ( 7 π 8 ) , 2 2 cis ( 15 π 8 )

Got questions? Get instant answers now!

Graphical

For the following exercises, plot the complex number in the complex plane.

Technology

For the following exercises, find all answers rounded to the nearest hundredth.

Use the rectangular to polar feature on the graphing calculator to change 5 + 5 i to polar form.

Got questions? Get instant answers now!

Use the rectangular to polar feature on the graphing calculator to change 3 2 i to polar form.

3.61 e 0.59 i

Got questions? Get instant answers now!

Use the rectangular to polar feature on the graphing calculator to change 3 8 i to polar form.

Got questions? Get instant answers now!

Use the polar to rectangular feature on the graphing calculator to change 4 cis ( 120° ) to rectangular form.

2 + 3.46 i

Got questions? Get instant answers now!

Use the polar to rectangular feature on the graphing calculator to change 2 cis ( 45° ) to rectangular form.

Got questions? Get instant answers now!

Use the polar to rectangular feature on the graphing calculator to change 5 cis ( 210° ) to rectangular form.

4.33 2.50 i

Got questions? Get instant answers now!

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.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
Practice Key Terms 4

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask