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Calculus: Help and Review13 chapters | 148 lessons

Instructor:
*Emily Cadic*

Emily has a master's degree in engineering and currently teaches middle and high school science.

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.

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.' 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.

**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).

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.

To find the theta value of our polar coordinates, we must determine the angle between the hypotenuse we have just found and a fixed direction. The standard direction for this type of calculation is east, aka the positive x-axis. To find θ, you may use one of three possible inverse trigonometric expressions.

So, our Cartesian coordinates (3,4) correspond to the polar coordinates (5,53.13 degrees).

The process of converting from polar coordinates to Cartesian coordinates is similar. The equations we have just learned need only to be reworked to isolate and solve for x and y.

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

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 θ values, which are summarized in Table 1.

We will now work through example problems that deal with the other quadrants.

**Quadrant II Example**

Convert (-4, 3) from Cartesian to polar coordinates.

Using the formulas we have learned, we solve for r and then θ.

The final step is to adjust the angle so that it falls within the θ range for Quadrant II, which can be accomplished by adding 180 degrees. This needs to be done in order to correctly reference the angle counterclockwise from the positive x-axis.

θ 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 Example**

Convert (4, -3) from Cartesian to polar coordinates.

Using the formulas we have learned, we solve from r and then θ.

The final step is to correct to adjust the angle so that it falls within the θ 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.

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.

- Plug values of θ into a given function r=f(θ)
- Convert r and θ into an x-coordinate
- Convert r and θ into a y-coordinate

**Example: Plotting a Circle**

Plot the polar function r = 4cos(θ)

To solve, pick an array of θ values to use in steps 1-3

**θ = 0**

- r = 4*cos(0) = 4*1 = 4
- x = r*cos(0) = 4*cos(0) = 4*1 = 4
- y = r*sin(0) = 4*0 = 0

**θ = 45**

- r = 4*cos(45) = 4*(√2/2) = 2√2
- x = r*cos(45) = 2√2*(√2/2) = 2
- y = r*sin(45) = 2√2*(√2/2) = 2

**θ = 90**

- r = 4*cos(90) = 4*0 = 0
- x = r*cos(90) = 0*cos(90) = 0*0 = 0
- 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.

In this lesson, we learned that **polar coordinates** are a set of values that quantify the location of a point based on distance and direction. We learned how to convert between polar coordinates and Cartesian coordinates, and how to approach different polar coordinate calculations and graphs. You are now prepared to tackle the end of lesson quiz.

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Calculus: Help and Review13 chapters | 148 lessons

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