Read a cosine model
Identify a midline, amplitude and period in a supplied periodic function.
Supplied materials
- Fictional sensor model: h(t) = 5 + 3 cos(πt/6), with t in hours and h in units. Angles inside cosine are in radians.
- Use these supplied values: cos(0) = 1, cos(π/2) = 0, cos(π) = −1, cos(2π) = 1. The task needs substitution and function notation; no real sensor is involved.
Learn the idea
Cosine repeats every 2π radians and has values from −1 to 1. Multiplying by 3 gives a deviation from −3 to 3; adding 5 sets the midline at 5. The model’s range is therefore 2 through 8.
The input angle πt/6 increases by 2π when t increases by 12, so the period is 12 hours. A modelled repeating pattern describes these fictional values; it does not establish that real data follow it.
- Substitute t = 0 and use cos(0).
- Substitute t = 3 and simplify the angle to π/2.
- Substitute t = 6 and use cos(π).
- Confirm that t = 12 repeats the t = 0 value.
See the method in action
h(0) = 5 + 3 = 8. h(3) = 5 + 3 × 0 = 5. h(6) = 5 + 3 × (−1) = 2. h(12) = 8 again, consistent with a 12-hour period.
Check your reasoning
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See the answer and explanation
3
At t = 6 the angle is π and its cosine is −1, so k(6) = 5 − 2 = 3. Reducing the amplitude changes the range to 3 through 7 while the period stays 12 hours.
Take it further
Replace πt/6 with πt/3 while keeping the other terms. Explain why the period becomes 6 hours and check that t = 3 is now the minimum.
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