Periodic modelling | Year 11

01 / LEARN

The idea, step by step

Turn the context into graph features

A height or daylight model needs a midline, amplitude, period and starting phase. Use the same time units in the coefficient and input. A negative cosine suits a starting minimum; a positive cosine suits a starting maximum.

Find amplitude from half the maximum–minimum difference, not from the maximum height. The midline is the average. State the meaning and units of the variables and restrict the model to a sensible time interval.

ω = \frac{2 π}{T}

Solve, interpret and evaluate

Isolate the trigonometric term to find when a threshold is reached. Use every branch in one cycle, then distinguish the equality times from the whole interval above/below the threshold. A sketch helps determine the correct inequality interval.

A real tide, daylight pattern or machine may only approximately follow one sinusoid. Weather can change tide height and timing; machinery may vary speed. State these assumptions and check a prediction against another observation rather than assuming a fitted model is exact.

A common trap

Using the maximum height as amplitude.

Instead: Amplitude is half the difference between maximum and minimum; the midline is their average.

02 / WORKED EXAMPLES

See the thinking

Example 1

An idealised tide is 0.5 m at $t = 0$ and 3.5 m at $t = 6$ hours. The period is 12 hours. Model its height.

Midline is 2 m and amplitude 1.5 m.
The angular frequency is $2 π / 12 = π / 6$ per hour.
It starts at the minimum, so use negative cosine.

h (t) = 2 - 1.5 cos (π t / 6)

Example 2

For this tide, when is $h (t) \geq 2.75$ m in the first 12 hours?

The inequality becomes $cos (π t / 6) \leq - 1 / 2$ .
In one cycle the argument lies between $2 π / 3$ and $4 π / 3$ .
Multiply both endpoints by $6 / π$ .

From 4 to 8 hours, inclusive.

Check your understanding

A wheel turns once every 40 seconds. Find its angular frequency for a model using seconds.

Show answer

$π / 20$ radians per second.

Divide $2 π$ radians by the 40-second period.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A model has maximum $28^{\circ}$ C and minimum $16^{\circ}$ C. Find amplitude and midline.
2 marks
A wheel has radius 8 m, axle 10 m above ground and period 40 s. It starts at the bottom. Give a height rule.
2 marks
For that wheel, find the height after 10 s.
2 marks
For that wheel, find every time at height 14 m on $[0, 40)$ .
2 marks
Why is $cos (2 π t / 12)$ inappropriate for a 365-day daylight cycle when $t$ is in days?
2 marks
Give two limitations of modelling a tide with a fixed sine wave.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A model has maximum $28^{\circ}$ C and minimum $16^{\circ}$ C. Find amplitude and midline.
2 marks
Half-difference is 6; average is 22.
Amplitude $6^{\circ}$ C; midline $22^{\circ}$ C.
A wheel has radius 8 m, axle 10 m above ground and period 40 s. It starts at the bottom. Give a height rule.
2 marks
The midline is 10, amplitude 8 and angular frequency $π / 20$ ; negative cosine begins at height 2.
$h (t) = 10 - 8 cos (π t / 20)$ m.
For that wheel, find the height after 10 s.
2 marks
The argument is $π / 2$ , whose cosine is 0.
10 m.
For that wheel, find every time at height 14 m on $[0, 40)$ .
2 marks
Solve cosine $- 1 / 2$ , giving arguments $2 π / 3, 4 π / 3$ .
Multiply by $20 / π$ .
$t = 40 / 3, 80 / 3$ s.
Why is $cos (2 π t / 12)$ inappropriate for a 365-day daylight cycle when $t$ is in days?
2 marks
A 365-day cycle needs angular frequency $2 π / 365$ per day.
It repeats after 12 days.
Give two limitations of modelling a tide with a fixed sine wave.
2 marks
The fixed amplitude/period assumptions ignore those effects, so predictions must be checked.
Weather can change heights; real successive tidal cycles can have unequal highs or timing.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics, Further Trigonometry: periodic functions and models, trigonometric identities and reciprocal functions; the Library sequence follows the approved school-topic structure.

Official curriculum source ↗

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