# Introduction to Circular Functions

### The First Ferris Wheel

The original Ferris Wheel was built by George Washington Gale Ferris, Jr. in 1893 for the World's Columbian Exposition in Chicago. The wheel, 80 meters high and equipped with 36 passenger cars with 40 seats in each car, was the highlight of the Exposition.

In this lesson, you'll use an interactive model to investigate the mathematics of how you move when you're riding a Ferris Wheel. In the figure on the right, you can tap the button to stop or start the wheel, you can change the slider to change the wheel's speed, and you can even drag any of the passenger cars.

As the wheel turns, watch one of the cars, paying attention to the height of the car above the ground. How does the height depend on the rotation of the wheel?

Imagine that you are riding on a Ferris Wheel. The figure below shows only a single passenger car—the one you are riding in. Use the figure to investigate exactly how your height above the ground changes as the wheel turns.

### Check Yourself

With your group, decide which five of the questions below are most interesting. Press the button at the end of any question to type your answer. [Note: We've not yet decided on the technology to use for such answers, so only the first two questions have buttons at this time.]

1. How far does the car go in one revolution? How far does it go in three revolutions?
2. What are the minimum and maximum heights the car reaches as the wheel revolves?
3. How can you tell from the graph when the car is moving up and when it is moving down?
4. Is there a place on the graph where the car is not moving either up or down? Explain your answer.
5. How can you adjust the model so the graph shows three full revolutions of the wheel? How can you make it show five full revolutions?
6. Adjust the speed to the fastest setting, restart the wheel, and observe the spacing of the dots. What does the spacing tell you about the motion of the car?
7. If the graph goes beyond the right edge of the window, how can you still tell the number of revolutions the car has completed?

### The Unit Circle

To analyze a mathematical problem, it's often helpful to create a simplified model, eliminating as many details as you can while keeping the mathematical situation intact. Use the Unit Circle figure below to construct a simplified version of the Ferris wheel, eliminating details to make the mathematics as simple as possible.

### Check Yourself

1. What details of the original problem have been dropped? What important elements have been retained?
2. What are the minimum and maximum heights that θ reaches as it rotates?
3. How can you tell from the graph when θ is moving up and when it is moving down?
4. Where on the graph is the height changing most quickly? Where is it changing most slowly?
5. What happens when θ does more than a single revolution?
6. How can you tell the number of revolutions θ has completed?

### Problems

(Each problem button displays a websketch and tools you can use to work on the problem.)

Graph the horizontal location (x-value) of θ as a function of the rotation distance.

Graph the ratio of the y and x values (y/x) as a function of the rotation distance.

### Projects

1. Graph From Scratch: Using stand-along dynamic software such as The Geometer's Sketchpad, start with a blank screen and create the same unit circle graph that you did in this lesson: the graph of the height of point θ as a function of the length of the arc. (You'll likely need to consult the software's documentation to determine how to accomplish various steps along the way.)
2. Multiple Rotations: Once you've completed project 1, figure out a way to make the graph extend beyond a single rotation about the unit circle. (In other words, the graph should extend beyond 2π.)
3. Length of Seasons: Research the efforts made by the ancient astronomers to figure out why summer is longer than winter. How did their mathematical work relate to the functions that you graphed in this lesson?