CMClearMathAcademy

Periodic Modeling

A free Trigonometry lesson from the “Applications of Trigonometry” unit, with a worked example and practice problems including step-by-step solutions.

A sinusoidal model turns repeating real-world data into y = a*sin(b*t)+d (or cosine), where the amplitude a is half the gap between the max and min, the midline d is their average, and b = 2pi/period sets how fast it cycles. Pick sine when the value starts at the midline and rises, and cosine when it starts at a peak or trough. Once the model is built you can plug in a time t to predict the value.

What you'll learn

Why it matters: Engineers and scientists model tides, Ferris-wheel heights, daily temperature swings, and hours of daylight with sinusoids to predict the value at any future time. The same y = a*sin(b*t)+d shape describes ocean tides on a harbor chart and the seat height on a carnival ride.

Worked example

Problem. A tide ranges from a low of 2 ft to a high of 10 ft, and one full cycle takes 12 hours. Write a sine model y = a*sin(b*t) + d for the tide height.

  1. Amplitude a = (10 - 2)/2 = 4 and midline d = (10 + 2)/2 = 6.
  2. Period is 12, so b = 2pi/12 = pi/6.
  3. Combine: y = 4*sin(pi/6*t) + 6.

Answer: y = 4*sin(pi/6*t) + 6

Practice problems

1. A tide rises to a high of 10 ft and falls to a low of 2 ft. Find the amplitude of the sinusoidal model.

Show solution
  1. Warm-up: First identify exactly what the question is asking: A tide rises to a high of 10 ft and falls to a low of 2 ft. Find the amplitude of the sinusoidal model.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Amplitude = (max - min)/2.
  4. (10 - 2)/2 = 8/2 = 4.
  5. The amplitude is 4.
  6. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

2. Daylight ranges from 14 hours in summer to 10 hours in winter. Find the midline of the sinusoidal model. Write it as y = a number.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Daylight ranges from 14 hours in summer to 10 hours in winter. Find the midline of the sinusoidal model. Write it as y = a number.
  2. For range questions, identify the possible output values after the input restrictions and graph shape are considered.
  3. Midline = (max + min)/2.
  4. (14 + 10)/2 = 24/2 = 12.
  5. The midline is y = 12.
  6. Check the result by substituting or estimating: the response should match y = 12 and make sense in the original problem.

Answer: y = 12

3. A Ferris wheel completes one full turn every 60 seconds. Using b = 2pi/period, find b for the height model.

Show solution
  1. Warm-up: First identify exactly what the question is asking: A Ferris wheel completes one full turn every 60 seconds. Using b = 2pi/period, find b for the height model.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. b = 2pi/period.
  4. b = 2pi/60.
  5. b = pi/30.
  6. Check the result by substituting or estimating: the response should match pi/30 and make sense in the original problem.

Answer: pi/30

4. A tide ranges from 2 ft to 10 ft with a 12-hour period and starts at the midline rising. Write the sine model y = a*sin(b*t) + d.

Show solution
  1. Core Practice: First identify exactly what the question is asking: A tide ranges from 2 ft to 10 ft with a 12-hour period and starts at the midline rising. Write the sine model y = a*sin(b*t) + d.
  2. For range questions, identify the possible output values after the input restrictions and graph shape are considered.
  3. Amplitude = (10 - 2)/2 = 4 and midline = (10 + 2)/2 = 6.
  4. b = 2pi/12 = pi/6.
  5. Model: y = 4*sin(pi/6*t) + 6.
  6. Check the result by substituting or estimating: the response should match y = 4*sin(pi/6*t) + 6 and make sense in the original problem.

Answer: y = 4*sin(pi/6*t) + 6

5. Use the tide model y = 4*sin(pi/6*t) + 6 to find the height (in ft) at t = 3 hours.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Use the tide model y = 4*sin(pi/6*t) + 6 to find the height (in ft) at t = 3 hours.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. At t = 3, pi/6*t = pi/2, and sin(pi/2) = 1.
  4. y = 4*(1) + 6.
  5. y = 10 ft.
  6. Check the result by substituting or estimating: the response should match 10 and make sense in the original problem.

Answer: 10

Practice this interactively with instant feedback and an AI tutor.

Practice Periodic Modeling Take the free placement check

More Trigonometry lessons