Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with 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. Need help with PreCalculus? Explore our rich collection of videos, games, activities and worksheets that are suitable for the study, practice or review of PreCalculus.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. Parallel Lines, Skew Lines and Planes. Proving Parallel Lines. Difference Quotient of Function I. Difference Quotient of Function II. Domain of a Function. Greatest Integer Function I. Greatest Integer Function II.

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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. You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics.

Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.In mathematicsthe polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

The reference point analogous to the origin of a Cartesian coordinate system is called the poleand the ray from the pole in the reference direction is the polar axis.

The distance from the pole is called the radial coordinateradial distance or simply radiusand the angle is called the angular coordinatepolar angleor azimuth.

Polar/Rectangular Coordinates Calculator

The initial motivation for the introduction of the polar system was the study of circular and orbital motion. Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point. The Archimedean spiral for example is much simpler to express using polar forms. Moreover, many physical systems—such as those concerned with bodies moving around a central point or with phenomena originating from a central point—are simpler and more intuitive to model using polar coordinates.

The polar coordinate system is extended into three dimensions with two different coordinate systems, the cylindrical and spherical coordinate system. Alexis Clairaut was the first to think of polar coordinates in three dimensions, and Leonhard Euler was the first to actually develop them. The concepts of angle and radius were already used by ancient peoples of the first millennium BC. The Greek astronomer and astrologer Hipparchus — BC created a table of chord functions giving the length of the chord for each angle, and there are references to his using polar coordinates in establishing stellar positions.

The Greek work, however, did not extend to a full coordinate system. From the 8th century AD onward, astronomers developed methods for approximating and calculating the direction to Mecca qibla —and its distance—from any location on the Earth. The calculation is essentially the conversion of the equatorial polar coordinates of Mecca i. There are various accounts of the introduction of polar coordinates as part of a formal coordinate system.

Saint-Vincent wrote about them privately in and published his work inwhile Cavalieri published his in with a corrected version appearing in Cavalieri first used polar coordinates to solve a problem relating to the area within an Archimedean spiral. Blaise Pascal subsequently used polar coordinates to calculate the length of parabolic arcs.

In Method of Fluxions writtenpublishedSir Isaac Newton examined the transformations between polar coordinates, which he referred to as the "Seventh Manner; For Spirals", and nine other coordinate systems. Coordinates were specified by the distance from the pole and the angle from the polar axis. Bernoulli's work extended to finding the radius of curvature of curves expressed in these coordinates.

The actual term polar coordinates has been attributed to Gregorio Fontana and was used by 18th-century Italian writers. Degrees are traditionally used in navigationsurveyingand many applied disciplines, while radians are more common in mathematics and mathematical physics.

In mathematics, e. The polar angles decrease towards negative values for rotations in the respectively opposite orientations. The equation defining an algebraic curve expressed in polar coordinates is known as a polar equation. Different forms of symmetry can be deduced from the equation of a polar function r. Because of the circular nature of the polar coordinate system, many curves can be described by a rather simple polar equation, whereas their Cartesian form is much more intricate.

For the circle, line, and polar rose below, it is understood that there are no restrictions on the domain and range of the curve. In the general case, the equation can be solved for rgiving. A polar rose is a mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation.

polar coordinates word problems

If k is an integer, these equations will produce a k -petaled rose if k is oddor a 2 k -petaled rose if k is even. If k is rational but not an integer, a rose-like shape may form but with overlapping petals. Note that these equations never define a rose with 2, 6, 10, 14, etc. The variable a directly represents the length or amplitude of the petals of the rose, while k relates to their spatial frequency.

The Archimedean spiral is a spiral that was discovered by Archimedeswhich can also be expressed as a simple polar equation. It is represented by the equation.Teachers Pay Teachers is an online marketplace where teachers buy and sell original educational materials. Are you getting the free resources, updates, and special offers we send out every week in our teacher newsletter?

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For All Subject Areas. See All Resource Types. Polar Coordinates PreCalculus - Unit 6. Your PreCalculus Honors students will have foldables, guided notes, homework, and a content quiz in the Polar Coordinates lesson of a nine-lesson unit on Applications of Trigonometry that cover the concepts of advanced trigonometry in depth.

PreCalculusTrigonometryMathematics. Interactive WhiteboardHomeworkGraphic Organizers. Add to cart. Wish List. Polar Coordinates This file contains one interactive page that your students could use to practice plotting polar coordinates.Here a polar graph with some points on it. You will probably be asked to convert coordinates between polar form and rectangular form.

Converting from rectangular coordinates to polar coordinates can be a little trickier since we need to check the quadrant of the rectangular point to get the correct angle; the quadrants must match. Here are the formulas:. Thus is is typically easier to convert from polar to rectangular. Here are some examples of conversions both way; note you may be asked to convert back to polar into degrees or radians.

And again, note that when we convert back to polar coordinates, we may not always get the same representation as the polar point we started out with. I find that drawing polar graphs is a combination of part memorizing and part knowing how to create polar t -charts. First, here is a table of some of the more common polar graphs.

I included t -charts in both degrees and radians. These graphs always go through the pole center. Try it!

Cardioid or Heart. Positive would be on the right side. Limacon Bean. Here are a couple more polar graphs Spirals and Lemniscates that you might see:. You might be asked to obtain the equation of a polar function from a graph:. We can do this with the following equations:. We can do this with the following equations, depending on what we have in the polar equation:. Here are some examples.

Note that sometimes we may be asked to Complete the Square to get the equation in a circle form; we learned how to do this in the Factoring Quadratics and Completing the Square section here. To get all these elusive points, you put in the r value in both curves to see what additional points you get. Find the intersection points for the following sets of polar curves algebraically and also draw a sketch.

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polar coordinates word problems

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Convert between Polar and Rectangular Coordinates

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polar coordinates word problems

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Polar/Rectangular Coordinates Calculator

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