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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.

30 min Trigonometry

The big picture

A sinusoidal model has a midline, amplitude, period and phase. From maximum M and minimum m, the midline is (M+m)/2 and amplitude (M-m)/2. Convert the observed period into the angular coefficient. A repeated model assumes the pattern continues. Tides, temperatures and seasonal demand also respond to external conditions, so use an appropriate time window and interpret units. A negative physical prediction can indicate that a model is being used beyond its useful range.

Edexcel A level Maths: 7.7 Modelling with Trigonometric FunctionsCurated by Kepler, not produced by us. Watch it, then work through 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 $H = a + R\sin(t + \alpha)$ 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 + \alpha): \ \max = a + R, \ \min = a - R

Baseline plus amplitude.

When extremes occur

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

Example 1A tide height is

H = 6 + 3\sin(30t)°

. Find the max height and when it first occurs.

1Max when

\sin(30t) = 1

, i.e.

30t = 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.

Construct a model and state its useful range

A sinusoidal model has a midline, amplitude, period and phase. From maximum M and minimum m, the midline is (M+m)/2 and amplitude (M-m)/2. Convert the observed period into the angular coefficient.

A repeated model assumes the pattern continues. Tides, temperatures and seasonal demand also respond to external conditions, so use an appropriate time window and interpret units. A negative physical prediction can indicate that a model is being used beyond its useful range.

Example 1Water depth varies between 2 m and 8 m with period 12 hours. It is deepest at t = 0. Construct a model and find when it first falls to 5 m.

1Midline is

(8+2)/2=5

m; amplitude is

(8-2)/2=3

2A cosine starts at its maximum, so

h(t)=5+3\cos(\pi t/6)

with t in hours.

h=5

gives

\cos(\pi t/6)=0

. The first positive angle is

\pi/2

\pi t/6=\pi/2

gives

t=3

hours.

AnswerThe model is h(t) = 5 + 3 cos(pi t/6); the first mid-depth time is 3 hours.

Think like an examiner

Write the defining equations and show the substitution before evaluating.
Check the domain, units and interpretation of the final result.

Common misconceptions

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 results to remember

\max = a + R, \ \min = a - R

From R-form model.

\sin(t+\alpha) = 1 \text{ at the maximum}

Find the time of the peak.

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.

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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