<< Chapter < Page Chapter >> Page >

Given z = 1 7 i , find | z | .

| z | = 50 = 5 2

Got questions? Get instant answers now!

Writing complex numbers in polar form

The polar form of a complex number    expresses a number in terms of an angle θ and its distance from the origin r . Given a complex number in rectangular form expressed as z = x + y i , we use the same conversion formulas as we do to write the number in trigonometric form:

x = r cos θ y = r sin θ r = x 2 + y 2

We review these relationships in [link] .

Triangle plotted in the complex plane (x axis is real, y axis is imaginary). Base is along the x/real axis, height is some y/imaginary value in Q 1, and hypotenuse r extends from origin to that point (x+yi) in Q 1. The angle at the origin is theta. There is an arc going through (x+yi).

We use the term modulus to represent the absolute value of a complex number, or the distance from the origin to the point ( x , y ) . The modulus, then, is the same as r , the radius in polar form. We use θ to indicate the angle of direction (just as with polar coordinates). Substituting, we have

z = x + y i z = r cos θ + ( r sin θ ) i z = r ( cos θ + i sin θ )

Polar form of a complex number

Writing a complex number in polar form involves the following conversion formulas:

x = r cos θ y = r sin θ r = x 2 + y 2

Making a direct substitution, we have

z = x + y i z = ( r cos θ ) + i ( r sin θ ) z = r ( cos θ + i sin θ )

where r is the modulus    and θ is the argument    . We often use the abbreviation r cis θ to represent r ( cos θ + i sin θ ) .

Expressing a complex number using polar coordinates

Express the complex number 4 i using polar coordinates.

On the complex plane, the number z = 4 i is the same as z = 0 + 4 i . Writing it in polar form, we have to calculate r first.

r = x 2 + y 2 r = 0 2 + 4 2 r = 16 r = 4

Next, we look at x . If x = r cos θ , and x = 0 , then θ = π 2 . In polar coordinates, the complex number z = 0 + 4 i can be written as z = 4 ( cos ( π 2 ) + i sin ( π 2 ) ) or 4 cis ( π 2 ) . See [link] .

Plot of z=4i in the complex plane, also shows that the in polar coordinate it would be (4,pi/2).
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Express z = 3 i as r cis θ in polar form.

z = 3 ( cos ( π 2 ) + i sin ( π 2 ) )

Got questions? Get instant answers now!

Finding the polar form of a complex number

Find the polar form of 4 + 4 i .

First, find the value of r .

r = x 2 + y 2 r = ( 4 ) 2 + ( 4 2 ) r = 32 r = 4 2

Find the angle θ using the formula:

cos θ = x r cos θ = 4 4 2 cos θ = 1 2 θ = cos 1 ( 1 2 ) = 3 π 4

Thus, the solution is 4 2 cis ( 3 π 4 ) .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Write z = 3 + i in polar form.

z = 2 ( cos ( π 6 ) + i sin ( π 6 ) )

Got questions? Get instant answers now!

Converting a complex number from polar to rectangular form

Converting a complex number from polar form to rectangular form is a matter of evaluating what is given and using the distributive property. In other words, given z = r ( cos θ + i sin θ ) , first evaluate the trigonometric functions cos θ and sin θ . Then, multiply through by r .

Converting from polar to rectangular form

Convert the polar form of the given complex number to rectangular form:

z = 12 ( cos ( π 6 ) + i sin ( π 6 ) )

We begin by evaluating the trigonometric expressions.

cos ( π 6 ) = 3 2 and sin ( π 6 ) = 1 2

After substitution, the complex number is

z = 12 ( 3 2 + 1 2 i )

We apply the distributive property:

z = 12 ( 3 2 + 1 2 i )    = ( 12 ) 3 2 + ( 12 ) 1 2 i    = 6 3 + 6 i

The rectangular form of the given point in complex form is 6 3 + 6 i .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Finding the rectangular form of a complex number

Find the rectangular form of the complex number given r = 13 and tan θ = 5 12 .

If tan θ = 5 12 , and tan θ = y x , we first determine r = x 2 + y 2 = 12 2 + 5 2 = 13 . We then find cos θ = x r and sin θ = y r .

z = 13 ( cos θ + i sin θ ) = 13 ( 12 13 + 5 13 i ) = 12 + 5 i

The rectangular form of the given number in complex form is 12 + 5 i .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Convert the complex number to rectangular form:

z = 4 ( cos 11 π 6 + i sin 11 π 6 )

z = 2 3 2 i

Got questions? Get instant answers now!

Finding products of complex numbers in polar form

Now that we can convert complex numbers to polar form we will learn how to perform operations on complex numbers in polar form. For the rest of this section, we will work with formulas developed by French mathematician Abraham de Moivre (1667-1754). These formulas have made working with products, quotients, powers, and roots of complex numbers much simpler than they appear. The rules are based on multiplying the moduli and adding the arguments.

Questions & Answers

prove that [a+b, b+c, c+a]= 2[a b c]
Ashutosh Reply
can't prove
Akugry
i can prove [a+b+b+c+c+a]=2[a+b+c]
this is simple
Akugry
hi
Stormzy
x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
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
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