Learn how sinusoidal functions model real-world repeating patterns like Ferris wheels, tides, heart rates, and rotating objects.
Lesson 4.8 of Trigonometric Ratios & Functions in Math 30-1 — Alberta curriculum lessons.
Sinusoidal functions can model repeating real-world patterns, such as Ferris wheels, ocean tides, heart rates, and rotating machinery. The general forms are:
y = a(b(x - c)) + d or y = a(b(x - c)) + d
Step 1 — Identify Maximum and Minimum. Read the maximum and minimum values from the problem. These could be heights, temperatures, depths, etc.
Step 2 — Calculate Amplitude and Midline. a = (max - min)/(2) d = (max + min)/(2)
The amplitude a is the distance from the midline to the peak. The midline d is the average of the high and low values.
Step 3 — Find the Period and Calculate b. The period is the time for one full cycle. Then:
b = (2π)/(period)
If the problem gives cycles per unit time (frequency), convert first:
period = (1)/(frequency)
Step 4 — Choose Sine or Cosine and Determine Phase Shift. Use cosine when you know where a maximum or minimum occurs — cosine naturally starts at a peak.
Use sine when the function starts at the midline.
If the object starts at the minimum, use -. If it starts at the maximum, use +. The phase shift c is the x-value where the max or min first occurs.
Step 1 — Amplitude. a = (max - min)/(2) = (105 - 5)/(2) = (100)/(2) = 50 ft
Step 2 — Midline. d = (max + min)/(2) = (105 + 5)/(2) = (110)/(2) = 55 ft
Step 3 — Period. The wheel completes 6 rotations in 60 seconds:
period = (60 s)/(6 rotations) = 10 seconds
Step 4 — Calculate b. b = (2π)/(period) = (2π)/(10) = (π)/(5)
Step 5 — Write the Equation. The rider boards at the bottom, so at t = 0 the height is at its minimum. Since regular cosine starts at a maximum, we flip it with a negative sign:
h(t) = -50 ((π)/(5) t) + 55
A Ferris wheel has a maximum height of 105 ft and a minimum height of 5 ft. It completes 6 rotations per minute. A rider boards at the bottom.
Write an equation to model the height, h(t), of a rider in feet after t seconds.
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