Call Now to Set Up Tutoring:

(888) 888-0446

Trigonometry : Polar Form of Complex Numbers

Study concepts, example questions & explanations for Trigonometry

Create An Account Create Tests & Flashcards

All Trigonometry Resources

6 Diagnostic Tests 155 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 :Complex Numbers/Polar Form

The polar coordinates $\left ( r,\theta \right )$ of a point are $\left(2.1, \frac{7\pi }{3}\right )$ . Convert these polar coordinates to rectangular coordinates.

Possible Answers:

(1.05, 1.82)

(1.82, 1.05)

(2.1, 7.33)

(7.33, 2.1)

Correct answer:

(1.05, 1.82)

Explanation:

Given the polar coordinates $(r,\theta )$ , the-coordinate is $x= r \cos \theta$ . We can find this coordinate by substituting $r = 2.1, \theta = \frac{7\pi }{3}$ :

$x = r \cos \theta = 2.1 \cdot \cos \frac{7\pi }{3} = 2.1 \cdot 0.5 = 1.05$

Likewise, given the polar coordinates $(r,\theta )$ , the-coordinate is $y= r \sin \theta$ . We can find this coordinate by substituting $r = 2.1, \theta = \frac{7\pi }{3}$ :

$y = r \sin \theta = 2.1 \cdot \sin \frac{7\pi }{3} \approx 2.1 \cdot 0.8660 \approx 1.82$

Therefore the rectangular coordinates of the point $\left(2.1, \frac{7\pi }{3}\right )$ are (1.05, 1.82) .

Report an Error

Example Question #2 :Complex Numbers/Polar Form

Express the complex number $z=4(\cos240^\circ+i\sin240^\circ)$ in rectangular form.

Possible Answers:

$z=-4-2\sqrt{3}i$

$z=2+i\sqrt{3}$

$z=4+i\sqrt{3}$

$z=-2-2\sqrt{3}i$

Correct answer:

$z=-2-2\sqrt{3}i$

Explanation:

To convert this number to rectangular form, first think about what $\cos240^\circ$ and $\sin240^\circ$ are equal to. Because $240^\circ=180^\circ+60^\circ$ , we can use a 30-60-90^oreference triangle in the 3rd quadrant to determine these values.

D5g1377

$\cos240^\circ=-\frac{1}{2}$

$\sin240^\circ=-\frac{\sqrt{3}}{2}$

Now plug these in and continue solving:

$z=4(\cos240^\circ+i\sin240^\circ)$

$z=4(-\frac{1}{2}-i\frac{\sqrt{3}}{2})$

$z=-2-2\sqrt{3}i$

Report an Error

Example Question #3 :Complex Numbers/Polar Form

For the complex number z=7+13i , find the modulus $r=\sqrt{x^2+y^2}$ and the angle $\theta=\tan^-^1\left (\frac{y}{x} \right )$ . Then, express this number in polar form $z=r(\cos\theta+i\sin\theta)$ .

Possible Answers:

$r=2\sqrt{30}$

$\theta=61.7^\circ$

$z=2\sqrt{30}(\cos61.7^\circ+i\sin61.7^\circ)$

$r=\sqrt{218}$

$\theta=61.7^\circ$

$z=\sqrt{218}(\cos61.7^\circ+i\sin61.7^\circ)$

$r=2\sqrt{30}$

$\theta=28.3^\circ$

$z=2\sqrt{30}(\cos28.3^\circ+i\sin28.3^\circ)$

$r=\sqrt{218}$

$\theta=28.3^\circ$

$z=\sqrt{218}(\cos28.3^\circ+i\sin28.3^\circ)$

Correct answer:

$r=\sqrt{218}$

$\theta=61.7^\circ$

$z=\sqrt{218}(\cos61.7^\circ+i\sin61.7^\circ)$

Explanation:

This problem has given us formulas, so we just need to plug in x=7 and y=13 and solve.

$r=\sqrt{x^2+y^2}$

$r=\sqrt{7^2+13^2}$

$r=\sqrt{218}$

$\theta=\tan^-^1\left (\frac{y}{x} \right )$

$\theta=\tan^-^1\left (\frac{13}{7} \right )$

$\theta=61.7^\circ$

$z=r(\cos\theta+i\sin\theta)$

$z=\sqrt{218}(\cos61.7^\circ+i\sin61.7^\circ)$

Report an Error

Example Question #1 :Polar Form Of Complex Numbers

Express the complex number $z=3(\cos30^\circ+i\sin30^\circ)$ in rectangular form z=a+bi .

Possible Answers:

$z=\frac{3}{2}+\frac{1}{2}i$

$z=\frac{\sqrt{3}}{2}+\frac{1}{2}i$

$z=\frac{\sqrt{3}}{2}+\frac{3}{2}i$

$z=\frac{3\sqrt{3}}{2}+\frac{3}{2}i$

Correct answer:

$z=\frac{3\sqrt{3}}{2}+\frac{3}{2}i$

Explanation:

To convert this number to rectangular form, first think about what $\cos30^\circ$ and $\sin30^\circ$ are equal to. We can use a 30-60-90^oreference triangle in the 1st quadrant to determine these values.

374137.image0

$\cos30^\circ=\frac{\sqrt{3}}{2}$

$\sin30^\circ=\frac{1}{2}$

Next, plug these values in and simplify:

$z=3(\cos30^\circ+i\sin30^\circ)$

$z=3(\frac{\sqrt{3}}{2}+\frac{1}{2}i)$

$z=\frac{3\sqrt{3}}{2}+\frac{3}{2}i$

Report an Error

Example Question #2 :Polar Form Of Complex Numbers

For the complex number z=2-5i , find the modulus $r=\sqrt{x^2+y^2}$ and the angle $\theta=\tan^-^1\left (\frac{y}{x} \right )$ . Then, express this number in polar form $z=r(\cos\theta+i\sin\theta)$ .

Possible Answers:

r=5

$\theta=-68.2^\circ$

$z=5(\cos(-68.2^\circ)+i\sin(-68.2^\circ))$

r=5

$\theta=21.8^\circ$

$z=5(\cos21.8^\circ+i\sin21.8^\circ)$

$r=\sqrt{29}$

$\theta=21.8^\circ$

$z=\sqrt{29}(\cos21.8^\circ+i\sin21.8^\circ)$

$r=\sqrt{29}$

$\theta=-68.2^\circ$

$z=\sqrt{29}(\cos(-68.2^\circ)+i\sin(-68.2^\circ))$

Correct answer:

$r=\sqrt{29}$

$\theta=-68.2^\circ$

$z=\sqrt{29}(\cos(-68.2^\circ)+i\sin(-68.2^\circ))$

Explanation:

This problem has given us formulas, so we just need to plug in x=2 and y=-5 and solve.

$r=\sqrt{x^2+y^2}$

$r=\sqrt{2^2+(-5)^2}$

$r=\sqrt{29}$

$\theta=\tan^-^1\left (\frac{y}{x} \right )$

$\theta=\tan^-^1\left (\frac{-5}{2} \right )$

$\theta=-68.2^\circ$

$z=r(\cos\theta+i\sin\theta)$

$z=\sqrt{29}(\cos(-68.2^\circ)+i\sin(-68.2^\circ))$

Report an Error

Example Question #3 :Polar Form Of Complex Numbers

Express the complex number z=-1-4i in polar form.

Possible Answers:

$r=\sqrt{17}$

$\theta=75.96^\circ$

$z=\sqrt{17}(\cos75.96^\circ+i\sin75.96^\circ)$

$r=\sqrt{15}$

$\theta=14.04^\circ$

$z=\sqrt{15}(\cos14.04^\circ+i\sin14.04^\circ)$

$r=\sqrt{15}$

$\theta=75.96^\circ$

$z=\sqrt{15}(\cos75.96^\circ+i\sin75.96^\circ)$

$r=\sqrt{17}$

$\theta=14.04^\circ$

$z=\sqrt{17}(\cos14.04^\circ+i\sin14.04^\circ)$

Correct answer:

$r=\sqrt{17}$

$\theta=14.04^\circ$

$z=\sqrt{17}(\cos14.04^\circ+i\sin14.04^\circ)$

Explanation:

In order to complete this problem, you must understand three formulas that allow you to convert from the rectangular form of a complex number to the polar form of a complex number. These formulas are $r=\sqrt{x^2+y^2}$ , $\theta=\tan^-^1\left (\frac{y}{x} \right )$ , and the polar form $z=r(\cos\theta+i\sin\theta)$ . Additionally, understand that based on the given info, x=-1 and y=-4 . Begin by finding the modulus:

$r=\sqrt{x^2+y^2}$

$r=\sqrt{(-1)^2+(-4)^2}$

$r=\sqrt{17}$

Next, let's find the angle $\theta$ , also referred to as theamplitudeof the complex number.

$\theta=\tan^-^1\left (\frac{y}{x} \right )$

$\theta=\tan^-^1\left (\frac{-1}{-4} \right )$

$\theta=\tan^-^1\left (\frac{1}{4} \right )$

$\theta=14.04^\circ$

Finally, plug each of these into the polar form of a complex number:

$z=r(\cos\theta+i\sin\theta)$

$z=\sqrt{17}(\cos14.04^\circ+i\sin14.04^\circ)$

Report an Error

Example Question #4 :Polar Form Of Complex Numbers

Multiply the following complex numbers (in polar form), giving the result in both polar and rectangular form.

$2(\cos70^\circ+i\sin70^\circ)\cdot 3(\cos20^\circ+i\sin20^\circ)$

Possible Answers:

$6(\cos90^\circ+i\sin90^\circ)$ or

$5(\cos90^\circ+i\sin90^\circ)$ or

Correct answer:

$6(\cos90^\circ+i\sin90^\circ)$ or

Explanation:

The modulus of the product of two complex numbers is the product of their moduli, and the amplitude of the product is the sum of their amplitudes.

Therefore, the new modulus will be $2\cdot 3=6$ and the new amplitude will be $70^\circ+20^\circ=90^\circ$ . Therefore

$2(\cos70^\circ+i\sin70^\circ)\cdot 3(\cos20^\circ+i\sin20^\circ)$ $=6(\cos90^\circ+i\sin90^\circ)$

We must also express this in rectangular form, which we can do by substituting $\cos90^\circ=0$ and $\sin90^\circ=1$ . We get:

$6(\cos90^\circ+i\sin90^\circ)$

=6(0+1i)

=6i

Report an Error

Example Question #5 :Polar Form Of Complex Numbers

Find the following quotients, given that $z_1=10(\cos40^\circ+i\sin40^\circ)$ and $z_2=2(\cos220^\circ-i\sin220^\circ)$ . Give results in both polar and rectangular forms.

(a) $\frac{z_1}{z_2}$

(b) $\frac{z_2}{z_1}$

Possible Answers:

(a) $\frac{z_1}{z_2}=5(\cos180^\circ+i\sin180^\circ)$ or $\frac{z_1}{z_2}=-5$

(b) $\frac{z_2}{z_1}=\frac{1}{5}(\cos180^\circ+i\sin180^\circ)$ or $\frac{z_2}{z_1}=-\frac{1}{5}$

(a) $\frac{z_1}{z_2}=5(\cos180^\circ+i\sin180^\circ)$ or $\frac{z_1}{z_2}=-5i$

(b) $\frac{z_2}{z_1}=\frac{1}{5}(\cos180^\circ+i\sin180^\circ)$ or $\frac{z_2}{z_1}=-\frac{1}{5}i$

(a) $\frac{z_1}{z_2}=\frac{1}{5}(\cos180^\circ+i\sin180^\circ)$ or $\frac{z_1}{z_2}=-5$

(b) $\frac{z_2}{z_1}=5(\cos180^\circ+i\sin180^\circ)$ or $\frac{z_2}{z_1}=-\frac{1}{5}$

(a) $\frac{z_1}{z_2}=\frac{1}{5}(\cos180^\circ+i\sin180^\circ)$ or $\frac{z_1}{z_2}=-5$

(b) $\frac{z_2}{z_1}=5(\cos180^\circ+i\sin180^\circ)$ or $\frac{z_2}{z_1}=-\frac{1}{5}$

Correct answer:

(a) $\frac{z_1}{z_2}=5(\cos180^\circ+i\sin180^\circ)$ or $\frac{z_1}{z_2}=-5$

(b) $\frac{z_2}{z_1}=\frac{1}{5}(\cos180^\circ+i\sin180^\circ)$ or $\frac{z_2}{z_1}=-\frac{1}{5}$

Explanation:

The modulus of the quotient of two complex numbers is the modulus of the dividend divided by the modulus of the divisor. The amplitude of the quotient is the amplitude of the dividend minus the amplitude of hte divisor.

(a) The modulus for $\frac{z_1}{z_2}$ is equal to $\frac{10}{2}=5$ . The amplitude for $\frac{z_1}{z_2}$ is equal to $40^\circ-220^\circ=-180^\circ=180^\circ$ . (We have chosen to represent this as the coterminal angle $180^\circ$ rather than $-180^\circ$ as it is more conventional to represent angle measures as a positive angle between $0^\circ$ and $360^\circ$ .) Putting this together, we get $\frac{z_1}{z_2}=5(\cos180^\circ+i\sin180^\circ)$ . To represent this in rectangular form, substitute $\cos180^\circ=-1$ and $\sin180^\circ=0$ to get $\frac{z_1}{z_2}=5(-1+0i)=-5$ .

(b) The modulus for $\frac{z_2}{z_1}$ is equal to $\frac{2}{10}=\frac{1}{5}$ . The amplitude for $\frac{z_2}{z_1}$ is equal to $220^\circ-40^\circ=180^\circ$ . Putting this together, we get $\frac{z_2}{z_1}=\frac{1}{5}(\cos180^\circ+i\sin180^\circ)$ . To represent this in rectangular form, substitute $\cos180^\circ=-1$ and $\sin180^\circ=0$ to get $\frac{z_2}{z_1}=\frac{1}{5}(-1+0i)=-\frac{1}{5}$ .

Report an Error

View Trigonometry Tutors

Glenn
Certified Tutor

West Chester University of Pennsylvania, Bachelor of Science, Music Teacher Education.

View Trigonometry Tutors

Jiajun
Certified Tutor

Sam Houston State University, Bachelor of Science, Mathematics.

View Trigonometry Tutors

Messan
Certified Tutor

Indiana University-Purdue University-Indianapolis, Bachelor of Engineering, Electrical Engineering.

All Trigonometry Resources

6 Diagnostic Tests 155 Practice Tests Question of the Day Flashcards Learn by Concept

Trigonometry Tutoring in Top Cities:

Trigonometry Tutors in Top Cities:

Popular Courses & Classes

SAT Courses & Classes in New York City,SSATCourses & Classes in Dallas Fort Worth,GRE Courses & Classes in Phoenix,GRE Courses & Classes in Los Angeles,MCAT Courses & Classes in Denver,SAT Courses & Classes in Houston,SSATCourses & Classes in Denver,MCAT Courses & Classes in Philadelphia,SAT Courses & Classes in Phoenix,ISEE Courses & Classes in Boston

Popular Test Prep

GRE Test Prep in Los Angeles,SSATTest Prep in San Francisco-Bay Area,MCAT Test Prep in San Francisco-Bay Area,ISEE Test Prep in Los Angeles,LSAT Test Prep in Seattle,GMAT Test Prep in Atlanta,MCAT Test Prep in Boston,ACT Test Prep in Denver,ISEE Test Prep in Boston,MCAT Test Prep in Atlanta

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

*Full legal name

Company name

*Phone number

*Email address

Address

*Address1

Address2

*City

*State

*Zipcode

*Country

Complaint Details

*URL of copyrighted work (where your original material is located)

*Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

*Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Trigonometry : Polar Form of Complex Numbers

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #1 :Complex Numbers/Polar Form

Example Question #2 :Complex Numbers/Polar Form

Example Question #3 :Complex Numbers/Polar Form

Example Question #1 :Polar Form Of Complex Numbers

Example Question #2 :Polar Form Of Complex Numbers

Example Question #3 :Polar Form Of Complex Numbers

Example Question #4 :Polar Form Of Complex Numbers

Example Question #5 :Polar Form Of Complex Numbers

All Trigonometry Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Find the Best Tutors

Email address:
Your name:

Email address:
Your name:
Feedback: