We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Precalculus : Polar Form of Complex Numbers

Study concepts, example questions & explanations for Precalculus

Create An Account

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Express Complex Numbers In Rectangular Form

Convert the following to rectangular form:

2(cos(30)+ isin(30))

Possible Answers:

$\sqrt{3}+i$

$\sqrt{3}+2i$

$2\sqrt{3}+2i$

$\sqrt{3}-1$

$2\sqrt{3}+1$

Correct answer:

$\sqrt{3}+i$

Explanation:

Distribute the coefficient 2, and evaluate each term:

$\\2(cos(30)+ isin(30))\\ \\= 2cos(30)+i(2sin(30))\\ \\=\sqrt{3}+i$

Report an Error

Example Question #1 : Express Complex Numbers In Rectangular Form

Convert the following to rectangular form: 3(cos(60)-isin(30))

Possible Answers:

$\frac{3\sqrt3}{2}-\frac{i}{2}$

$\frac{3}{2}-\frac{i}{2}$

$\frac{3}{2}-\frac{3}{2}i$

$\frac{\sqrt3}{2}-\frac{i}{2}$

$\frac{3}{2}-\frac{\sqrt3 i}{2}$

Correct answer:

$\frac{3}{2}-\frac{3}{2}i$

Explanation:

Distribute the coefficient and simplify:

$\\ 3(cos(60)-isin(30))\\ \\=3sin(60)-i(3sin(30)))\\ \\=\frac{3}{2}-\frac{3}{2}i$

Report an Error

Example Question #2 : Express Complex Numbers In Rectangular Form

Represent the polar equation:

$z = 3\left(cos(\frac{\pi}{6} )+i\cdot sin\left(\frac{\pi}{6}\right)\right)$

in rectangular form.

Possible Answers:

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

$z=i+9\pi$

$z = 3cos\left(\frac{\pi}{6}\right) + 3sin\left(\frac{\pi}{6}\right)$

$z = cos\left(\frac{\pi}{6}\right) + sin\left(\frac{\pi}{6}\right)$

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

Correct answer:

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

Explanation:

Using the general form of a polar equation:

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

we find that the value of is and the value of $\theta$ is $\frac{\pi}{6}$ .

The rectangular form of the equation appears as a + bi , and can be found by finding the trigonometric values of the cosine and sine equations.

$cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$

$sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$

distributing the 3, we obtain the final answer of:

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

Report an Error

Example Question #1 : Polar Form Of Complex Numbers

Represent the polar equation:

$z = 4\left(cos\left(\frac{\pi}{4}\right) + i\cdot sin\left(\frac{\pi}{4}\right)\right)$

in rectangular form.

Possible Answers:

$2 + \sqrt{2}i$

$4cos\left(\frac{\pi}{4}\right) + 4sin\left(\frac{\pi}{4}\right)$

$2\sqrt{2} + 2\sqrt{2}i$

$\sqrt{2} + 2i$

$cos\left(\frac{\pi}{4}\right) + sin\left(\frac{\pi}{4}\right)i$

Correct answer:

$2\sqrt{2} + 2\sqrt{2}i$

Explanation:

Using the general form of a polar equation:

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

we find that the value of r=4 and the value of $\theta =\frac{\pi}{4}$ . The rectangular form of the equation appears as a + bi , and can be found by finding the trigonometric values of the cosine and sine equations.

$cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$

$sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt2}$

Distributing the 4, we obtain the final answer of:

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

Report an Error

Example Question #1 : Express Complex Numbers In Rectangular Form

Represent the polar equation:

$z = 5\left(cos\left(\frac{3\pi}{4}\right)+i \cdot sin\left(\frac{3\pi}{4}\right)\right)$

in rectangular form.

Possible Answers:

$z = \frac{-5}{2} + \frac{5}{2}i$

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

$z = \frac{-5\sqrt{2}}{2} + \frac{5\sqrt{2}}{2}i$

$z = cos(\frac{3\pi}{4}) + sin(\frac{3\pi}{4})i$

$z = \frac{5\sqrt{2}}{2} + \frac{-5\sqrt{2}}{2}i$

Correct answer:

$z = \frac{-5\sqrt{2}}{2} + \frac{5\sqrt{2}}{2}i$

Explanation:

Using the general form of a polar equation:

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

we find that the value of r=5 and the value of $\theta =\frac{3\pi}{4}$ . The rectangular form of the equation appears as a + bi , and can be found by finding the trigonometric values of the cosine and sine equations.

$cos\left(\frac{3\pi}{4}\right) = \frac{-1}{\sqrt{2}}$

$sin\left(\frac{3\pi}{4}\right) = \frac{1}{\sqrt{2}}$

distributing the 5, we obtain the final answer of:

$\frac{-5\sqrt{2}}{2} + \frac{5\sqrt{2}}{2}i$

Report an Error

Example Question #1 : Express Complex Numbers In Rectangular Form

Convert $6(\cos \frac{\pi}{5} + i \sin \frac{ \pi }{5} )$ in rectangular form

Possible Answers:

- 3.527 + 4.854 i

4.854 - 3.527 i

4.854 + 3.527 i

3.527 + 4.854i

Correct answer:

4.854 + 3.527 i

Explanation:

To convert, just evaluate the trig ratios and then distribute the radius.

6(0.8090 + i 0. 5878) = 4.854+ 3.527i

Report an Error

Example Question #1 : Express Complex Numbers In Rectangular Form

Convert $3(\cos 200^o + i \sin 200^o )$ to rectangular form

Possible Answers:

2.819 - 1.026i

-2.819 +1.026i

-1.026 - 2.819i

2.819 + 1.026i

-2.819 - 1.026i

Correct answer:

-2.819 - 1.026i

Explanation:

To convert to rectangular form, just evaluate the trig functions and then distribute the radius:

3(-0.9397 -0.3420 ) =-2.819 -1.026i

Report an Error

Example Question #1 : Express Complex Numbers In Rectangular Form

Convert $5(\cos \frac{ 8 \pi }{5} + i \sin \frac{ 8 \pi }{5} )$ to rectangular form

Possible Answers:

- 4.755+1.545 i

-1.545 - 4.755 i

4.755 - 1.545i

1.545 + 4.755 i

1.545 - 4.755 i

Correct answer:

1.545 - 4.755 i

Explanation:

To convert, evaluate the trig ratios and then distribute the radius:

5(0.3090 -0.9511i) =1.545 -4.755i

Report an Error

Example Question #1 : Express Complex Numbers In Polar Form

The following equation has complex roots:

x^2 - 3x + 9 = 0

Express these roots in polar form.

Possible Answers:

$3(\cos 60 \pm i\sin 60)$

$3(\cos 30 \pm i\sin(30))$

$\sqrt{3}(\cos 60 \pm i\sin 60)$

$\sqrt{3}(\cos 30 \pm i \sin 30)$

Correct answer:

$3(\cos 60 \pm i\sin 60)$

Explanation:

Every complex number can be written in the form a + bi

The polar form of a complex number takes the form r(cos $\Theta$ + isin $\Theta$ )

Now r can be found by applying the Pythagorean Theorem on a and b, or:

r = $\sqrt{a^{2} + b^{2}}$

$\Theta$ can be found using the formula:

$\Theta$ = $tan^{-1}(b/a)$

So for this particular problem, the two roots of the quadratic equation

x^2 - 3x + 9 = 0

are:

$x = \frac{3\pm 3\sqrt{3}i}{2}$

Hence, a = 3/2 and b = 3√3 / 2

Therefore r = $\sqrt{a^{2} + b^{2}}$ = 3

and $\Theta$ = tan^-1 (√3) = 60

And therefore x = r(cos $\Theta$ + isin $\Theta$ ) = 3 (cos 60 + isin 60)

Report an Error

Example Question #2 : Express Complex Numbers In Polar Form

Express the roots of the following equation in polar form.

x^2+3x+3

Possible Answers:

$\sqrt{3}(\cos(30)+i\sin(30))$

$\sqrt{3}(\cos(150)+i\sin(150))$

$\sqrt{3}(\cos(-30)+i\sin(-30))$

$(\cos(150)+i\sin(150))$

$3(\cos(150)+i\sin(150))$

Correct answer:

$\sqrt{3}(\cos(150)+i\sin(150))$

Explanation:

First, we must use the quadratic formula to calculate the roots in rectangular form.

$\frac{-3\pm \sqrt{9-36}}{2}=\frac{-3\pm i\sqrt{3}}{2}$

Remembering that the complex roots of the equation take on the form a+bi,

we can extract the a and b values.

$a=-\frac{3}{2}$

$b=\frac{\sqrt{3}}{2}$

We can now calculate r and theta.

$r=\sqrt{a^2 + b^2}$

$\theta=\tan^{-1}\frac{b}{a}$

Using these two relations, we get

$r=\sqrt{3}$

$\theta=-30$ . However, we need to adjust this theta to reflect the real location of the vector, which is in the 2nd quadrant (a is negative, b is positive); a represents the x-axis in the real-imaginary plane, b represents the y-axis.

The angle theta now becomes 150.

$\theta=150$ .

You can now plug in r and theta into the standard polar form for a number:

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

Report an Error

View Pre-Calculus Tutors

Muhammad
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Mechanical Engineering.

View Pre-Calculus Tutors

Joseph
Certified Tutor

Rutgers University-New Brunswick, Bachelor in Arts, Chemistry. Rutgers Graduate School of Biomedical Sciences, Current Grad S...

View Pre-Calculus Tutors

Zara
Certified Tutor

University of Manitoba, Bachelor in Arts, Mathematics.

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Algebra Tutors in Washington DC, English Tutors in Boston, MCAT Tutors in Chicago, French Tutors in Miami, GMAT Tutors in Los Angeles, LSAT Tutors in Atlanta, Biology Tutors in Los Angeles, LSAT Tutors in San Diego, Statistics Tutors in Atlanta, SAT Tutors in Phoenix

Popular Courses & Classes

Spanish Courses & Classes in Washington DC, MCAT Courses & Classes in San Francisco-Bay Area, LSAT Courses & Classes in Houston, GRE Courses & Classes in New York City, SAT Courses & Classes in Dallas Fort Worth, GMAT Courses & Classes in Chicago, LSAT Courses & Classes in New York City, GRE Courses & Classes in Philadelphia, SSAT Courses & Classes in Phoenix, Spanish Courses & Classes in Boston

Popular Test Prep

GRE Test Prep in Washington DC, LSAT Test Prep in Denver, LSAT Test Prep in San Francisco-Bay Area, SSAT Test Prep in San Francisco-Bay Area, ACT Test Prep in Philadelphia, GRE Test Prep in Phoenix, LSAT Test Prep in Atlanta, GRE Test Prep in Philadelphia, MCAT Test Prep in Washington DC, SSAT Test Prep in Phoenix

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

Precalculus : Polar Form of Complex Numbers

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Express Complex Numbers In Rectangular Form

Example Question #1 : Express Complex Numbers In Rectangular Form

Example Question #2 : Express Complex Numbers In Rectangular Form

Example Question #1 : Polar Form Of Complex Numbers

Example Question #1 : Express Complex Numbers In Rectangular Form

Example Question #1 : Express Complex Numbers In Rectangular Form

Example Question #1 : Express Complex Numbers In Rectangular Form

Example Question #1 : Express Complex Numbers In Rectangular Form

Example Question #1 : Express Complex Numbers In Polar Form

Example Question #2 : Express Complex Numbers In Polar Form

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: