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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 Video by Zeeshan Zamurred Trigonometry and Modelling

Edexcel A level Maths: 7.7 Modelling with Trigonometric FunctionsWatch the full walkthrough 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

Reading a model

A model like $H = a + R sin (t + α)$ has a baseline $a$ , amplitude $R$ , and a phase shift. The maximum is $a + R$ and the minimum is $a - R$ .

H = a + R sin (t + α) : max = a + R, min = a - R

Baseline plus amplitude.

When extremes occur

The maximum of $R sin (t + α)$ occurs when $sin (t + α) = 1$ , i.e. $t + α = \frac{π}{2}$ . Solve for $t$ to find the time.

1Max when

sin (30 t) = 1

, i.e.

30 t = 90°

, so

t = 3

2Max height

= 6 + 3 = 9

AnswerMax

9

m at

t = 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

From R-form model.

sin (t + α) = 1 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

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