Call Now to Set Up Tutoring:

(888) 888-0446

Precalculus : Convert Rectangular Coordinates To Polar Coordinates and vice versa

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 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Convert the polar coordinates $\small (10, 150^{\circ})$ to rectangular form.

Possible Answers:

$\small (10\sqrt3,10)$

$\small (-10\sqrt3,5)$

$\small (5,-5\sqrt3)$

$\small (-5\sqrt3,5)$

$\small (-10,-10\sqrt3)$

Correct answer:

$\small (-5\sqrt3,5)$

Explanation:

We begin by recalling that polar coordinates are expressed in the form $\small (r,\theta)$ , where $\small r$ is the radius (the distance from the origin to the point) and $\small \theta$ is the angle formed between the postive x-axis and the radius.

We can find our x-coordinate and y-coordinate in rectangular form quite easily by keeping in mind two equations.

$\small cos(\theta)=\frac{x}{r}$ or $\small x=r(cos(\theta))$

$\small sin(\theta)=\frac{y}{r}$ or $\small y=r(sin(\theta))$

Substituting in both of these gives respectively

$\small x=(10)cos(150^{\circ})=10(-\frac{\sqrt3}{2})=-5\sqrt3$

$\small y=10(sin(150^\circ))=10(\frac{1}{2})=5$

Therefore, the rectangular coordinates of our point are $\small (-5\sqrt3,5)$

Report an Error

Example Question #2 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Convert the polar coordinates to rectangular coordinates:

(5, 60)

Possible Answers:

(5, 9)

(4.5, 1.22)

(2.5, 4.33)

(1.56, 1.42)

Correct answer:

(2.5, 4.33)

Explanation:

To convert polar coordinates $(r, \theta)$ to rectangular coordinates (x, y) ,

$x=r\cos\theta$

$y=r\sin\theta$

Using the information given in the question,

$x=5\cos60=2.5$

y=5sin60=4.33

The rectangular coordinates are (2.5, 4.33)

Report an Error

Example Question #3 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Convert the polar coordinates to rectangular coordinates:

(8, 45)

Possible Answers:

(9.54, 9.33)

(2.33, 2.33)

(4.54, 2.36)

(5.66, 5.66)

Correct answer:

(5.66, 5.66)

Explanation:

To convert polar coordinates $(r, \theta)$ to rectangular coordinates (x, y) ,

$x=r\cos\theta$

$y=r\sin\theta$

Using the information given in the question,

$x=8\cos 45=5.66$

$y=8 \sin 45=5.66$

The rectangular coordinates are (5.66, 5.66)

Report an Error

Example Question #4 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Convert the polar coordinates to rectangular coordinates:

(10, 35)

Possible Answers:

(6.32, 7.12)

(6.35, 8.11)

(8.19, 5.74)

(8.11, 1.25)

Correct answer:

(8.19, 5.74)

Explanation:

To convert polar coordinates $(r, \theta)$ to rectangular coordinates (x, y) ,

$x=r\cos\theta$

$y=r\sin\theta$

Using the information given in the question,

$x=10\cos 35=8.19$

$y=10\sin 35=5.74$

The rectangular coordinates are (8.19, 5.74)

Report an Error

Example Question #2 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Convert the polar coordinates to rectangular coordinates:

$(4, \pi)$

Possible Answers:

(0, -4)

(4, 0)

(-4, -4)

(-4, 0)

Correct answer:

(-4, 0)

Explanation:

To convert polar coordinates $(r, \theta)$ to rectangular coordinates (x, y) ,

$x=r\cos\theta$

$y=r\sin\theta$

Using the information given in the question,

$x=4\cos \pi=4(-1)=-4$

$y=4 \sin \pi=4(0)=0$

The rectangular coordinates are (-4, 0)

Report an Error

Example Question #3 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Convert the polar coordinates to rectangular coordinates:

$\left(5, \frac{2\pi}{3}\right)$

Possible Answers:

$\left(-\frac{5}{2}, \frac{5\sqrt3}{2}\right)$

$\left(\frac{2}{5}, \frac{\sqrt3}{2}\right)$

(2.33, 21.2)

$\left(\frac{\sqrt2}{4}, \frac{5}{\sqrt3}\right)$

Correct answer:

$\left(-\frac{5}{2}, \frac{5\sqrt3}{2}\right)$

Explanation:

To convert polar coordinates $(r, \theta)$ to rectangular coordinates (x, y) ,

$x=r\cos\theta$

$y=r\sin\theta$

Using the information given in the question,

$x=5\cos \frac{2\pi}{3}=5\left(-\frac{1}{2}\right)=-\frac{5}{2}$

$y=5 \sin \frac{2\pi}{3}=5\left(\frac{\sqrt3}{2}\right)=\frac{5\sqrt3}{2}$

The rectangular coordinates are $\left(-\frac{5}{2}, \frac{5\sqrt3}{2}\right)$

Report an Error

Example Question #7 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Convert the polar coordinates to rectangular form:

$\left(10, \frac{4\pi}{3}\right)$

Possible Answers:

$(-5, -5\sqrt3)$

$(5, 5\sqrt3)$

$(-5\sqrt3, -5)$

$(5\sqrt3, -5)$

Correct answer:

$(-5, -5\sqrt3)$

Explanation:

To convert polar coordinates $(r, \theta)$ to rectangular coordinates (x, y) ,

$x=r\cos\theta$

$y=r\sin\theta$

Using the information given in the question,

$x=10\cos \frac{4\pi}{3}=10\left(-\frac{1}{2}\right)=-5$

$y=10 \sin \frac{4\pi}{3}=10\left(-\frac{\sqrt3}{2}\right)=-5\sqrt3$

The rectangular coordinates are $(-5, -5\sqrt3)$

Report an Error

Example Question #8 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Convert the polar coordinates to rectangular coordinates:

$\left(8, \frac{7\pi}{6}\right)$

Possible Answers:

$(-4\sqrt2, 4\sqrt2)$

$(4\sqrt2, 4\sqrt2)$

$\left(-\frac{8\sqrt3}{2}, -4\right)$

$(\sqrt2, \sqrt2)$

Correct answer:

$\left(-\frac{8\sqrt3}{2}, -4\right)$

Explanation:

To convert polar coordinates $(r, \theta)$ to rectangular coordinates (x, y) ,

$x=r\cos\theta$

$y=r\sin\theta$

Using the information given in the question,

$x=8\cos \frac{7\pi}{6}=8\left(-\frac{\sqrt3}{2}\right)=-\frac{8\sqrt3}{2}$

$y=8 \sin \frac{7\pi}{6}=8\left(-\frac{1}{2}\right)=-4$

The rectangular coordinates are $\left(-\frac{8\sqrt3}{2}, -4\right)$

Report an Error

Example Question #1 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

How could you express $\small \small \left(-5, \frac{\pi}{3}\right)$ in rectangular coordinates?

Possible Answers:

$\left (\frac{5\sqrt3}{2}, \frac{5}{2}\right)$

$\left(-\frac{5}{2}, -\frac{5\sqrt3}{2}\right)$

$\left (-\frac{2\sqrt3 }{2}, -\frac{2}{5}\right)$

$\left (-\frac{2}{5},-\frac{2\sqrt3}{5} \right)$

$\left(\frac{-5}{2}, \frac{5\sqrt3}{2}\right)$

Correct answer:

$\left(-\frac{5}{2}, -\frac{5\sqrt3}{2}\right)$

Explanation:

The polar coordinates given have an angle of $\small \frac{\pi}{3}$ but a negative radius, so our coordinates are located in quadrant III.

This means x and y are both negative. You can figure out these x and y coordinates using trigonometric ratios, or since the angle is $\small \frac{\pi}{3}$ , special right triangles.

The hypotenuse of this triangle is 5, but in the special right triangle it's 2, so we know we're multiplying each side by $\small \frac{5}{2}$ .

That makes the x-coordinate or adjacent side be

$\small \frac{5}{2}*1 = \frac{5}{2}$

and the y-coordinate or opposite side be

$\small \frac{5}{2}*\sqrt{3} = \frac{5\sqrt3}{2}$ .

In this case, once again, both are negative, so our answer is

$\left (-\frac{5}{2}, -\frac{5\sqrt3}{2}\right)$ .

Report an Error

Example Question #6 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

How could you express $\left(10, \frac{\pi}{10}\right)$ in rectangular coordinates?

Round to the nearest hundredth.

Possible Answers:

(3.09, 9.51)

(9.51, 3.09)

(10, 0.05)

(3.25, 3.25)

(0.05, 10)

Correct answer:

(9.51, 3.09)

Explanation:

In order to determine the rectangular coordinates, look at the triangle representing the polar coordinates:

Polar to rectangular a

We can see that both x and y are positive. We can figure out the x-coordinate by using the cosine:

$cos\left(\frac{\pi}{10}\right) = \frac{x}{10}$

$0.951 \approx \frac{x}{10}$ multiply both sides by 10.

9.51 = x

We can figure out the y-coordinate by using the sine:

$sin\left(\frac{\pi}{10}\right) = \frac{y}{10}$

$0.309 \approx \frac{y}{10}$

3.09 = y

Report an Error

View Pre-Calculus Tutors

Jonathan
Certified Tutor

University of Maryland-Baltimore County, Bachelor of Science, Mathematics.

View Pre-Calculus Tutors

Sydney
Certified Tutor

University of Vermont, Bachelor of Science, Nursing (RN). University of Connecticut, Bachelor of Science, Health Sciences, Ge...

View Pre-Calculus Tutors

James
Certified Tutor

St. Mary of the Plains College, Bachelors, Mathematics. Baker University, Masters, Business Administration and Management.

All Precalculus Resources

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

Popular Subjects

Statistics Tutors in Chicago, Reading Tutors in Phoenix, Physics Tutors in New York City, ACT Tutors in San Diego, MCAT Tutors in Houston, ISEE Tutors in Seattle, English Tutors in Denver, Chemistry Tutors in Philadelphia, ACT Tutors in Miami, SSAT Tutors in Philadelphia

Popular Courses & Classes

SAT Courses & Classes in Houston, MCAT Courses & Classes in Phoenix, SAT Courses & Classes in Miami, LSAT Courses & Classes in San Diego, LSAT Courses & Classes in Los Angeles, GRE Courses & Classes in Denver, ISEE Courses & Classes in Miami, Spanish Courses & Classes in Dallas Fort Worth, LSAT Courses & Classes in Chicago, SAT Courses & Classes in New York City

Popular Test Prep

ACT Test Prep in Washington DC, ACT Test Prep in Seattle, SSAT Test Prep in Seattle, ACT Test Prep in Boston, SSAT Test Prep in Washington DC, GMAT Test Prep in Houston, ISEE Test Prep in Miami, GRE Test Prep in New York City, GRE Test Prep in Miami, ISEE Test Prep in Dallas Fort Worth

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 : Convert Rectangular Coordinates To Polar Coordinates and vice versa

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Example Question #2 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Example Question #3 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Example Question #4 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Example Question #2 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Example Question #3 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Example Question #7 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Example Question #8 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Example Question #1 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Example Question #6 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: