Triangle and periodic-model applications | Year 11

01 / LEARN

The idea, step by step

Choose the model

For a triangle, sketch the actual directions and select a rule from the supplied data. Bearings are clockwise from north, so work out the interior angle rather than treating every bearing as a triangle angle. For a periodic quantity identify its minimum, maximum, period, units and starting position before writing a sine or cosine rule.

Interpret the roots

A repeated height may occur while a rider is rising and again while falling. Keep only times in the stated physical interval. Periodic rules idealise smooth repeated behaviour: variable speed, weather or changing conditions can invalidate long extrapolations. State these assumptions when using a model to make predictions.

A common trap

Keeping all mathematical times, including those outside the observed cycle.

Instead: State the physical interval first and restrict every root to that interval.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

A wheel model is $h = 20 - 18 cos (π t / 30)$ metres, t in seconds. Find start height, maximum and period.

$h (0) = 20 - 18 = 2$ .
Cosine ranges from−1 to 1, so height ranges from 2 to 38.
$T = 2 π / (π / 30) = 60$ .

Start 2 m; maximum 38 m; period 60 s.

Example 2

Worked example 2

A boat travels 4 km east then 3 km north. Find direct distance and bearing to one decimal place.

$d = \sqrt{4^{2} + 3^{2}} = 5$ .
The angle from north satisfies $tan β = 4 / 3$ .
The destination is northeast; write the bearing with three digits.

5 km on bearing 053.1°.

Check your understanding

Find the period of $h = 10 + 3 sin (2 π t / 8)$ .

Show answer

8 time units.

$T = 2 π / (2 π / 8) = 8$ .

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

For tide $d = 4 + 2 cos (π t / 6)$ , t in hours, state range, period and starting depth.
3 marks
For $d = 4 + 2 cos (π t / 6)$ , find d=4 on $0 \leq t < 12$ .
3 marks
Paths 8 km and 11 km leave a camp 60° apart. Find endpoint separation exactly.
3 marks
A triangle has a=20 m opposite A=40°, and B=70°. Find b to one decimal place.
3 marks
A wheel has centre height 12 m, radius 10 m and period 40 s. It starts at its lowest point. Give a model.
3 marks
Why might today’s fitted sine temperature model fail to predict next month’s temperatures?
3 marks

Read the worked solutions

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

For tide $d = 4 + 2 cos (π t / 6)$ , t in hours, state range, period and starting depth.
3 marks
Read D=4, amplitude 2 and B=π/6; evaluate cos 0.
Range[2,6] m; period 12 h; start 6 m.
For $d = 4 + 2 cos (π t / 6)$ , find d=4 on $0 \leq t < 12$ .
3 marks
$cos (π t / 6) = 0$ ; argumentsπ/2 and 3π/2 give the times.
t=3,9 h.
Paths 8 km and 11 km leave a camp 60° apart. Find endpoint separation exactly.
3 marks
$d^{2} = 64 + 121 - 176 (1 / 2) = 97$ .
$\sqrt{97} km$
A triangle has a=20 m opposite A=40°, and B=70°. Find b to one decimal place.
3 marks
$b = 20 sin 70^{\circ} / sin 40^{\circ} \approx 29.2380$ .
29.2 m.
A wheel has centre height 12 m, radius 10 m and period 40 s. It starts at its lowest point. Give a model.
3 marks
Midline 12, amplitude 10, and negative cosine starts low.
$B = 2 π / 40 = π / 20$ .
$h (t) = 12 - 10 cos (π t / 20)$ metres.
Why might today’s fitted sine temperature model fail to predict next month’s temperatures?
3 marks
A fixed-period model assumes stable parameters; seasonal and weather changes may violate them.
The mean, amplitude and timing can change.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: triangle rules, circular measure and trigonometric functions; school sequencing may vary.

Official curriculum source ↗

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