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Grade 12 · lesson 2 of 4

Build a Cosine Rule from Features

Construct a cosine model from extrema, period, and starting phase.

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Supplied materials

  • Paper or a note app.
  • Fictional repeating signal: maximum 10 units, minimum 4 units, period 8 hours, with a maximum at t = 0. Use cos(0) = 1 and cos(π) = −1; angles are radians.

Learn the idea

The midline is halfway between the minimum and maximum: (10 + 4)/2 = 7. The amplitude is half their separation: (10 − 4)/2 = 3. A positive cosine amplitude reaches a maximum when its angle is zero, matching the supplied start.

For period T, the angle coefficient is 2π/T because cosine repeats when its angle increases by 2π. With T = 8, use π/4. Extrema and period alone do not fix a phase, so the maximum-at-zero information is necessary for this selected rule.

  1. Calculate midline 7 and amplitude 3 from the supplied extrema.
  2. Calculate 2π/8 = π/4 and write h(t) = 7 + 3cos(πt/4).
  3. Check h(0) = 10 and h(4) = 4, then construct the changed signal below.
Worked example

See the method in action

h(t) = 7 + 3cos(πt/4). At t = 0, h = 7 + 3 = 10; at t = 4, h = 7 + 3(−1) = 4. Increasing t by 8 adds 2π to the angle.

Your turn

Check your reasoning

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A new signal has maximum 9, minimum 1, period 12 hours, and a maximum at zero. Which rule fits?

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See the answer and explanation

5 + 4cos(πt/6)

The midline is (9 + 1)/2 = 5, amplitude (9 − 1)/2 = 4, and angle coefficient 2π/12 = π/6. The first rule has all three features and the required phase.

Take it further

Keep maximum 9 and minimum 1 but shorten the period to 6. The rule becomes 5 + 4cos(πt/3). At t = 3 its angle is π and its value is 1.

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