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Discover a new way of graphing with polar coordinates. In this lesson, you will learn the definition of polar coordinates, how they can be calculated, and in what types of problems they will be useful. Practice what you have learned with example problems and a quiz after the lesson.

Visualizing Polar Coordinates

Look at the face of an analog clock or watch. There should be one on your smartphone if you don’t actually own one of these. Now let’s imagine it’s 3:30, so the hour hand is on the 3 and the minute hand is on the 6. If I asked you to describe the location of the hour hand with respect to the minute hand, what would you say? You might spend some time making exact measurements between each number, but the most concise way of answering this question would be to say ‘the 3 and the 6 are 90 degrees apart.

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‘ This, in a nutshell, is how polar coordinates can be used to simplify locating points on a graph. Now, we’re going to explore how to find and use polar coordinates.


Definition of Polar Coordinates

Polar coordinates are a set of values that quantify the location of a point based on 1) the distance between the point and a fixed origin and 2) the angle between the point and a fixed direction.Polar coordinates are a complementary system to Cartesian coordinates, which are located by moving across an x-axis and up and down the y-axis in a rectangular fashion.

While Cartesian coordinates are written as (x,y), polar coordinates are written as (r,θ).

Converting from Cartesian to Polar Coordinates

Because polar coordinates and Cartesian coordinates are both commonly used systems, it is helpful to learn how to convert between the two.We will start first with a set of Cartesian coordinates and learn how to convert to polar coordinates.

Our coordinates are (x,y) = (3,4) as we see below. To convert to polar coordinates, we want to create a triangle that has a base along the x-axis and an vertex at (3,4).


The shortest distance between the origin and (3,4) is now the hypotenuse (the longest side) of the triangle we have drawn. That is the first point of our polar coordinates: the r in (r,θ). To find the value of r, we must use the Pythagorean Theorem.


Now that we know how to convert both ways, let’s move onto some examples.

Examples: Working in Different Quadrants

In our first example, we were working in Quadrant I of the Cartesian coordinate plane. You may also encounter problems in Quadrants II, III or IV. Each quadrant encompasses a different range of ; values, which are summarized in Table 1.

Table 1

; now matches the range given in Table 1 for Quadrant II. In our next example, we will skip ahead to Quadrant IV, as Quadrant III requires the same adjustment that we have just seen in this example.

Quadrant IV ExampleConvert (4, -3) from Cartesian to polar coordinates.Using the formulas we have learned, we solve from r and then ;.


The final step is to correct to adjust the angle so that it falls within the ; range for Quadrant IV, which can be accomplished this time by adding 360 degrees. This needs to be done in order to correctly reference the angle counterclockwise from the positive x-axis.


Applications: Graphing Polar Functions

There are various tools available for graphing polar functions.

If you do not own a calculator that creates such a plot, you can plot by hand following the steps below.

  1. Plug values of ; into a given function r=f(;)
  2. Convert r and ; into an x-coordinate
  3. Convert r and ; into a y-coordinate

Example: Plotting a CirclePlot the polar function r = 4cos(;)To solve, pick an array of ; values to use in steps 1-3; = 0

  1. r = 4*cos(0) = 4*1 = 4
  2. x = r*cos(0) = 4*cos(0) = 4*1 = 4
  3. y = r*sin(0) = 4*0 = 0

; = 45

  1. r = 4*cos(45) = 4*(;2/2) = 2;2
  2. x = r*cos(45) = 2;2*(;2/2) = 2
  3. y = r*sin(45) = 2;2*(;2/2) = 2

; = 90

  1. r = 4*cos(90) = 4*0 = 0
  2. x = r*cos(90) = 0*cos(90) = 0*0 = 0
  3. y = r*sin(90) = 0*1 = 0

Some of the possible results are listed in the table below.


When the graph is plotted, we see the basic outline of a circle. As you enter more points, it will begin to look like a more complete circle.

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