Worksheets · Precalculus

Sinusoidal models worksheet

The middle of the motion is the midline, half the swing is the amplitude, and one full cycle is the period. A rider who starts at the bottom follows a flipped cosine: h(t) = center − radius × cos(2πt / period).

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Sinusoidal Models

Date Period

Answer each question.

  1. A Ferris wheel is 82 feet across, its center is 43 feet up, and it turns once every 10 minutes. A rider starts at the bottom. Give the amplitude and period of the height.
  2. A Ferris wheel is 82 feet across, its center is 49 feet up, and it turns once every 8 minutes. A rider starts at the bottom. How high is the rider after 2 minutes?
  3. A Ferris wheel is 60 feet across, its center is 34 feet up, and it turns once every 4 minutes. A rider starts at the bottom. What is the greatest height?
  4. A Ferris wheel is 110 feet across, its center is 59 feet up, and it turns once every 4 minutes. A rider starts at the bottom. How high is the rider after 1 minute?
  5. A Ferris wheel is 34 feet across, its center is 19 feet up, and it turns once every 6 minutes. A rider starts at the bottom. How high is the rider after 3 minutes?
  6. A Ferris wheel is 118 feet across, its center is 66 feet up, and it turns once every 10 minutes. A rider starts at the bottom. What is the greatest height?
  7. A Ferris wheel is 28 feet across, its center is 17 feet up, and it turns once every 6 minutes. A rider starts at the bottom. Give the amplitude and period of the height.
  8. A Ferris wheel is 106 feet across, its center is 60 feet up, and it turns once every 6 minutes. A rider starts at the bottom. What is the greatest height?
  9. A Ferris wheel is 46 feet across, its center is 26 feet up, and it turns once every 4 minutes. A rider starts at the bottom. Give the amplitude and period of the height.
  10. A Ferris wheel is 114 feet across, its center is 63 feet up, and it turns once every 4 minutes. A rider starts at the bottom. How high is the rider after 1 minute?
  11. A Ferris wheel is 72 feet across, its center is 45 feet up, and it turns once every 6 minutes. A rider starts at the bottom. How high is the rider after 4.5 minutes?
  12. A Ferris wheel is 86 feet across, its center is 51 feet up, and it turns once every 6 minutes. A rider starts at the bottom. How high is the rider after 1.5 minutes?
  13. A Ferris wheel is 80 feet across, its center is 44 feet up, and it turns once every 6 minutes. A rider starts at the bottom. What is the greatest height?
  14. A Ferris wheel is 76 feet across, its center is 46 feet up, and it turns once every 4 minutes. A rider starts at the bottom. How high is the rider after 1 minute?
  15. A Ferris wheel is 22 feet across, its center is 19 feet up, and it turns once every 12 minutes. A rider starts at the bottom. Give the amplitude and period of the height.
  16. A Ferris wheel is 34 feet across, its center is 25 feet up, and it turns once every 4 minutes. A rider starts at the bottom. How high is the rider after 1 minute?
  17. A Ferris wheel is 42 feet across, its center is 29 feet up, and it turns once every 10 minutes. A rider starts at the bottom. Give the amplitude and period of the height.
  18. A Ferris wheel is 112 feet across, its center is 60 feet up, and it turns once every 8 minutes. A rider starts at the bottom. How high is the rider after 2 minutes?
  19. A Ferris wheel is 86 feet across, its center is 52 feet up, and it turns once every 8 minutes. A rider starts at the bottom. What is the greatest height?
  20. A Ferris wheel is 96 feet across, its center is 56 feet up, and it turns once every 20 minutes. A rider starts at the bottom. Give the amplitude and period of the height.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Sinusoidal ModelsAnswer keyVersion 1

Date Period

Answer each question.

  1. A Ferris wheel is 82 feet across, its center is 43 feet up, and it turns once every 10 minutes. A rider starts at the bottom. Give the amplitude and period of the height.amplitude 41 feet, period 10 minutes
  2. A Ferris wheel is 82 feet across, its center is 49 feet up, and it turns once every 8 minutes. A rider starts at the bottom. How high is the rider after 2 minutes?49 feet
  3. A Ferris wheel is 60 feet across, its center is 34 feet up, and it turns once every 4 minutes. A rider starts at the bottom. What is the greatest height?64 feet
  4. A Ferris wheel is 110 feet across, its center is 59 feet up, and it turns once every 4 minutes. A rider starts at the bottom. How high is the rider after 1 minute?59 feet
  5. A Ferris wheel is 34 feet across, its center is 19 feet up, and it turns once every 6 minutes. A rider starts at the bottom. How high is the rider after 3 minutes?36 feet
  6. A Ferris wheel is 118 feet across, its center is 66 feet up, and it turns once every 10 minutes. A rider starts at the bottom. What is the greatest height?125 feet
  7. A Ferris wheel is 28 feet across, its center is 17 feet up, and it turns once every 6 minutes. A rider starts at the bottom. Give the amplitude and period of the height.amplitude 14 feet, period 6 minutes
  8. A Ferris wheel is 106 feet across, its center is 60 feet up, and it turns once every 6 minutes. A rider starts at the bottom. What is the greatest height?113 feet
  9. A Ferris wheel is 46 feet across, its center is 26 feet up, and it turns once every 4 minutes. A rider starts at the bottom. Give the amplitude and period of the height.amplitude 23 feet, period 4 minutes
  10. A Ferris wheel is 114 feet across, its center is 63 feet up, and it turns once every 4 minutes. A rider starts at the bottom. How high is the rider after 1 minute?63 feet
  11. A Ferris wheel is 72 feet across, its center is 45 feet up, and it turns once every 6 minutes. A rider starts at the bottom. How high is the rider after 4.5 minutes?45 feet
  12. A Ferris wheel is 86 feet across, its center is 51 feet up, and it turns once every 6 minutes. A rider starts at the bottom. How high is the rider after 1.5 minutes?51 feet
  13. A Ferris wheel is 80 feet across, its center is 44 feet up, and it turns once every 6 minutes. A rider starts at the bottom. What is the greatest height?84 feet
  14. A Ferris wheel is 76 feet across, its center is 46 feet up, and it turns once every 4 minutes. A rider starts at the bottom. How high is the rider after 1 minute?46 feet
  15. A Ferris wheel is 22 feet across, its center is 19 feet up, and it turns once every 12 minutes. A rider starts at the bottom. Give the amplitude and period of the height.amplitude 11 feet, period 12 minutes
  16. A Ferris wheel is 34 feet across, its center is 25 feet up, and it turns once every 4 minutes. A rider starts at the bottom. How high is the rider after 1 minute?25 feet
  17. A Ferris wheel is 42 feet across, its center is 29 feet up, and it turns once every 10 minutes. A rider starts at the bottom. Give the amplitude and period of the height.amplitude 21 feet, period 10 minutes
  18. A Ferris wheel is 112 feet across, its center is 60 feet up, and it turns once every 8 minutes. A rider starts at the bottom. How high is the rider after 2 minutes?60 feet
  19. A Ferris wheel is 86 feet across, its center is 52 feet up, and it turns once every 8 minutes. A rider starts at the bottom. What is the greatest height?95 feet
  20. A Ferris wheel is 96 feet across, its center is 56 feet up, and it turns once every 20 minutes. A rider starts at the bottom. Give the amplitude and period of the height.amplitude 48 feet, period 20 minutes

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

A wheel 40 feet across, center 25 feet up, one turn every 8 minutes. Height after 2 minutes?

  1. 2 minutes is a quarter turn.
  2. A quarter turn from the bottom is level with the center.

The answer is 25 feet.

Where students go wrong

Using the diameter as the amplitude. The swing either side of the center is the radius, half the diameter.

Also called Ferris wheel problem, modeling with trig functions or periodic models.

Questions about these worksheets

Yes. Every sinusoidal models sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the sinusoidal models questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

Using the diameter as the amplitude. The swing either side of the center is the radius, half the diameter.