How to convert polar coordinates into rectangular form

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page. Related Topics: More Lessons for Pre-Calculus Math Worksheets Examples, videos, worksheets, solutions, and activities to help PreCalculus students learn how to convert between polar coordinates and rectangular coordinates or Cartesian coordinates The following diagrams show how to convert between Polar coordinates and Rectangular or Cartesian coordinates.

Scroll down the page for more examples and solutions. How to convert the rectangular coordinates of a point into polar coordinates? We will often be asked to convert rectangular to polar coordinates, and this conversion will be very important to understand in Calculus. In order to convert rectangular to polar coordinates, we use the distance formula to find the radius, and the inverse tangent function to find the angle. We may also sometimes be asked to convert from polar coordinates to rectangular coordinates.

This video provides two examples of converting a point given in rectangular coordinates to polar coordinates using degrees. Show Step-by-step Solutions.He still trains and competes occasionally, despite his busy schedule. To unlock all 5, videos, start your free trial. We will often be asked to convert rectangular to polar coordinatesand this conversion will be very important to understand in Calculus. In order to convert rectangular to polar coordinateswe use the distance formula to find the radius, and the inverse tangent function to find the angle.

We may also sometimes be asked to convert from polar coordinates to rectangular coordinates. Converting from rectangular coordinates to polar coordinates.

Convert between Polar and Rectangular Coordinates

And that can be kind of tricky because remember that the polar coordinates for a point are not unique. So let's make a rule here that we're going to get r to be greater than or equal to 0 and theta between 0 and 2 pi. This will allow us to get a unique set of polar coordinates for a point and there will be a lot less confusion.

Your teacher may actually have a requirement like that as well. So all of these are rectangular coordinates. I want to convert them to polar. So let's start with this one. Now the first thing I can do is use this formula here to find the r value, right?

The distance of the point from the pole. So r squared is root 3 squared plus 3 squared. Remember you should the positive value for r so there's no plus or minus here. We want the positive value. And then we can use these formulas to find theta. Now x is root 3, r is 2 root 3.

And so that's one half. And I also use this formula, right? So sine theta is y over r. The y value is 3, the r value is 2 root 3 and I can rationalize this denominator by multiplying by root 3 over root 3 and I get 3 root 3 over 2 times 3.

The threes cancel. Root 3 over 2. So the question is what angle has a cosine of one half and a sine of root 3 over 2? Well, that's pi over 3. Theta's pi over 3. And remeber we want theta to be between 0 and 2 pi. I mean there is another angle of 7 pi over 3 that works but it's not in the interval that we want.

So this is the theta that we want and the polar coordinates are going to be, remember r comes first, so 2 root 3 theta. Pi over 3. Okay, let's try another example. The point -5 5. First, r squared equals x squared plus y squared. So -5 squared plus 5 squared.By Yang Kuang, Elleyne Kase.

You can use both polar coordinates and Cartesian x, y coordinates also known as rectangular coordinates at any time to describe the same location on the coordinate plane. Cartesian coordinates are much better suited for graphs of straight lines or simple curves. When changing to and from polar coordinates, your work is often easier if you have all your angle measures in radians.

You can make the change by using the conversion factor.

How do you convert rectangular coordinates to polar coordinates?

You may choose, however, to leave your angle measures in degrees, which is fine as long as your calculator is in the right mode. These equations simplify into two very important expressions for x and y in terms of r and. Furthermore, you can use the Pythagorean theorem in the right triangle to find the radius of the triangle if given x and y:. With respect to the final equation, keep in mind that your calculator always returns a value of inverse tangent that puts.

You need to look at your x- and y- coordinates and decide whether that placement is actually correct for the problem at hand. Time for an example in reverse. Given the point —4, —4find the equivalent polar coordinate:. Mary Jane Sterling aught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois for more than 30 years.

How to Change between Polar and Cartesian Coordinates. About the Book Author Mary Jane Sterling aught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois for more than 30 years.Sign in to comment. Sign in to answer this question. Unable to complete the action because of changes made to the page. Reload the page to see its updated state. Choose a web site to get translated content where available and see local events and offers.

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Answers Support MathWorks. Search Support Clear Filters. Support Answers MathWorks. Search MathWorks. MathWorks Answers Support. Open Mobile Search. Trial software. You are now following this question You will see updates in your activity feed. You may receive emails, depending on your notification preferences. Converting polar form complex numbers into rectangular form.

Dalton Houghton-Schaffer on 4 Sep Vote 1. Alex Mcaulley on 4 Sep Cancel Copy to Clipboard. Some references which will help you. Thank you for your help!

Accepted Answer. Vote 3.Below is an interactive calculator that allows you to easily convert complex numbers in polar form to rectangular form, and vice-versa.

There's also a graph which shows you the meaning of what you've found. You can change the precision of all the calculations by changing the "Decimal places" option.

Go back to the examples on the Polar Form page and try them here in the calculator, and compare the results. Enter radius: Enter angle: o. Enter real: Enter imaginary: j.

Go back to this earlier page to see how to convert polar to rectangular form the old-fashioned way:. All numbers from the sum of complex numbers? Complex conjugates by phinah [Solved! Index problem by Rapelang [Solved! Name optional. Basic Definitions of Complex Numbers 2. Basic Operations in Complex Numbers 3. Graphical Representation of Complex Numbers 4. Products and Quotients of Complex Numbers Graphical explanation of multiplying and dividing complex numbers 7. Powers and Roots of Complex Numbers 8.

AC Circuit Definitions 9. AC Circuit Exercises Exponential Form of Complex Numbers. Click to search:. Online Algebra Solver This algebra solver can solve a wide range of math problems. Go to: Online algebra solver.Rectangular coordinates, or cartesian coordinates, come in the form??? I create online courses to help you rock your math class. Read more. Polar coordinates, on the other hand, come in the form???

Instead of moving out from the origin using horizontal and vertical lines, we instead pick the angle??? The value of??? All we have to do is take the values of??? Note: There are multiple ways to indicate the same polar point. Even though??? This will always be true, so you can always get away with only using the positive solution for??? Instead of moving out from the origin using horizontal and vertical lines, we instead pick the angle, which is the direction, and then move out from the origin a certain distance. The equivalent rectangular coordinate point is??? Polar coordinates vs.

Polar coordinates start with rectangular coordinates Rectangular coordinates, or cartesian coordinates, come in the form??? I'm krista. How to convert back and forth between polar and rectangular coordinates.

How do you convert the rectangular point #(-sqrt3,1)# into polar form?

Take the course Want to learn more about Calculus 2? I have a step-by-step course for that. Learn More. Converting the rectangular point to a polar point Example Convert the rectangular point to a polar point.Over 12 kilometers from port, a sailboat encounters rough weather and is blown off course by a knot wind. How can the sailor indicate his location to the Coast Guard?

In this section, we will investigate a method of representing location that is different from a standard coordinate grid. However, there are other ways of writing a coordinate pair and other types of grid systems. The polar grid is represented as a series of concentric circles radiating out from the poleor the origin of the coordinate plane. The polar grid is scaled as the unit circle with the positive x- axis now viewed as the polar axis and the origin as the pole.

When given a set of polar coordinateswe may need to convert them to rectangular coordinates. Dropping a perpendicular from the point in the plane to the x- axis forms a right triangle, as illustrated in Figure 5. To convert rectangular coordinates to polar coordinateswe will use two other familiar relationships. With this conversion, however, we need to be aware that a set of rectangular coordinates will yield more than one polar point.

Converting from rectangular coordinates to polar coordinates requires the use of one or more of the relationships illustrated in Figure 8. This gives. There are other sets of polar coordinates that will be the same as our first solution.

We can now convert coordinates between polar and rectangular form. Converting equations can be more difficult, but it can be beneficial to be able to convert between the two forms.

Since there are a number of polar equations that cannot be expressed clearly in Cartesian form, and vice versa, we can use the same procedures we used to convert points between the coordinate systems.

We can then use a graphing calculator to graph either the rectangular form or the polar form of the equation. Figure Note that this is two separate functions, since a circle fails the vertical line test.

This equation appears similar to the previous example, but it requires different steps to convert the equation. We can still follow the same procedures we have already learned and make the following substitutions:.

Polar Equations to Rectangular Equations, Precalculus, Examples and Practice Problems

The Cartesian or rectangular equation is plotted on the rectangular grid, and the polar equation is plotted on the polar grid. Clearly, the graphs are identical. We have learned how to convert rectangular coordinates to polar coordinates, and we have seen that the points are indeed the same. We have also transformed polar equations to rectangular equations and vice versa. Now we will demonstrate that their graphs, while drawn on different grids, are identical.

We clear the fraction, and then use substitution. Hyperbolas have many interesting geometric features and applications, which we will investigate further in Analytic Geometry. In this example, the right side of the equation can be expanded and the equation simplified further, as shown above. However, the equation cannot be written as a single function in Cartesian form. To do this, we can start with the initial equation. Skip to main content.

Module Further Applications of Trigonometry. Search for:. Polar Coordinates Learning Outcomes Plot points using polar coordinates. Convert from polar coordinates to rectangular coordinates. Convert from rectangular coordinates to polar coordinates. Transform equations between polar and rectangular forms. Identify and graph polar equations by converting to rectangular equations. 