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7.7PureStretch

Modelling with Trigonometric Functions

Periodic phenomena — tides, temperatures, oscillations — are modelled with sine and cosine. The R-form is the key tool for finding maximum and minimum values and the times they occur.

28 min Trigonometry and Modelling
Edexcel A level Maths: 7.7 Modelling with Trigonometric FunctionsCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Interpret trig models in context
  • Use the R-form to find max/min
  • Find when extremes occur
  • Solve modelling equations
1

Reading a model

A model like H=a+Rsin⁡(t+α)H = a + R\sin(t + \alpha) has a baseline aa, amplitude RR, and a phase shift. The maximum is a+Ra + R and the minimum is a−Ra - R.

H=a+Rsin⁡(t+α): max⁡=a+R, min⁡=a−RH = a + R\sin(t + \alpha): \ \max = a + R, \ \min = a - R
Baseline plus amplitude.
2

When extremes occur

The maximum of Rsin⁡(t+α)R\sin(t+\alpha) occurs when sin⁡(t+α)=1\sin(t+\alpha) = 1, i.e. t+α=π2t + \alpha = \tfrac{\pi}{2}. Solve for tt to find the time.

Example 1A tide height is H=6+3sin⁡(30t)°H = 6 + 3\sin(30t)°. Find the max height and when it first occurs.
1Max when sin⁡(30t)=1\sin(30t) = 1, i.e. 30t=90°30t = 90°, so t=3t = 3.
2Max height =6+3=9= 6 + 3 = 9.
Answer
Max 99 m at t=3t = 3.

Tip — Maximum of a sin model: set the sine equal to 1 and solve for the variable.

Formula recap

max⁡=a+R, min⁡=a−R\max = a + R, \ \min = a - R
From R-form model.
sin⁡(t+α)=1 at the maximum\sin(t+\alpha) = 1 \text{ at the maximum}
Find the time of the peak.

Common mistakes to avoid

Forgetting the baseline a when stating max/min.
Max = a + R, min = a − R.
Solving in the wrong angle mode.
Match degrees/radians to how the model is written.

Key takeaways

  • Trig models capture periodic behaviour.
  • Max = baseline + R, min = baseline − R.
  • Extremes occur where the sine/cosine equals ±1.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. For H=6+3sin⁡(30t)H = 6 + 3\sin(30t), the maximum height is:

    • 99
    • 66
    • 33
    • 1818
    Show answer and explanation

    Answer: 99

    6 + 3.

  2. The minimum of H=6+3sin⁡(30t)H = 6 + 3\sin(30t) is:

    • 33
    • 66
    • 00
    • −3-3
    Show answer and explanation

    Answer: 33

    6 − 3.

  3. The maximum occurs when sin⁡(30t)\sin(30t) equals:

    • 11
    • 00
    • −1-1
    • 12\tfrac12
    Show answer and explanation

    Answer: 11

    Peak of sine.

  4. In H=6+3sin⁡(30t)°H = 6 + 3\sin(30t)°, max first occurs at t=t =

    • 33
    • 3030
    • 9090
    • 11
    Show answer and explanation

    Answer: 33

    30t = 90 ⇒ t = 3.

  5. In a model a+Rsin⁡(...)a + R\sin(...), the amplitude is:

    • RR
    • aa
    • a+Ra + R
    • α\alpha
    Show answer and explanation

    Answer: RR

    R scales the wave.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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