Triangle and periodic-model applications | Year 11

01 / LEARN

Understand the idea.

Choose the model

Convert a description into geometry before choosing a formula. For a triangle, label the actual interior angles and side lengths; a bearing is measured clockwise from north and may need conversion before it becomes an interior angle.

For a repeating quantity, identify its extreme values, period, starting value and initial direction. These determine a midline, amplitude, inside factor and phase. Keep the units of the input and output with the model.

Interpret the roots

Interpret every result in its setting. A repeated height may be reached on the upward and downward parts of a cycle, and only times in the stated physical interval count. A periodic model also assumes the conditions, such as rotational speed, continue to repeat.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

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

Worked solution

1
Substitute the starting time.
$h (0) = 20 - 18 cos 0 = 2 m$
At time zero, cosine equals 1, placing the rider at the bottom.
2
Find the highest possible height.
$h_{max} = 20 - 18 (- 1) = 38 m$
The negative cosine coefficient makes the smallest cosine give the largest height.
3
Calculate one full-cycle time.
$T = \frac{2 π}{\frac{π}{30}} = 60 s$
The cosine argument must increase by $2 π$ for the motion to repeat.

Answer

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

EXAMPLE 02

Worked example 2

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

Worked solution

1
Use the perpendicular displacements.
$d = \sqrt{4^{2} + 3^{2}} = 5 km$
East and north form the legs of a right triangle, with direct distance as hypotenuse.
2
Measure the angle from north.
$tan β = \frac{4}{3}, β = {tan}^{- 1} (\frac{4}{3}) \approx {53.1301}^{\circ}$
Relative to the north direction, the east displacement is opposite and the north displacement is adjacent.
3
Write the bearing to one decimal place.
$Bearing = {053.1}^{\circ}$
The destination is northeast, so the clockwise angle from north is the acute angle found; bearings use three degree digits.

Answer

5 km on bearing 053.1°.

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

8 time units.

The argument increases by $2 π$ when $t$ increases by 8. Equivalently, $T = \frac{2 π}{\frac{2 π}{8}} = 8$ time units.

Keep practising

6 worksheet questions · 18 marks

Open worksheet Worked solutions

Read the questions online

For tide $d = 4 + 2 cos (\frac{π t}{6})$ , $t$ in hours, state range, period and starting depth.
3 marks
For $d = 4 + 2 cos (\frac{π t}{6})$ , find $d = 4$ on $0 \leq t < 12$ .
3 marks
Paths 8 km and 11 km leave a camp $60^{\circ}$ apart. Find endpoint separation exactly.
3 marks
A triangle has $a = 20$ m opposite $A = 40^{\circ}$ , and $B = 70^{\circ}$ . 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 $d = 4 + 2 cos (\frac{π t}{6})$ , $t$ in hours, state range, period and starting depth.
3 marks
Read $D = 4$ , amplitude 2 and $B = \frac{π}{6}$ ; evaluate $cos 0$ .
Range $[2, 6]$ m; period 12 h; start 6 m.
For $d = 4 + 2 cos (\frac{π t}{6})$ , find $d = 4$ on $0 \leq t < 12$ .
3 marks
$cos (\frac{π t}{6}) = 0$ ; arguments $\frac{π}{2}$ and $\frac{3 π}{2}$ give the times.
$t = 3, 9$ h.
Paths 8 km and 11 km leave a camp $60^{\circ}$ apart. Find endpoint separation exactly.
3 marks
$d^{2} = 64 + 121 - 176 (\frac{1}{2}) = 97$ .
$\sqrt{97} km$
A triangle has $a = 20$ m opposite $A = 40^{\circ}$ , and $B = 70^{\circ}$ . Find $b$ to one decimal place.
3 marks
$b = \frac{20 sin 70^{\circ}}{sin 40^{\circ}} \approx 29.2380$ .
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.
$B = \frac{2 π}{40} = \frac{π}{20}$ .
$h (t) = 12 - 10 cos (\frac{π t}{20})$ 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 ↗

In this lesson

Understand the idea.

Choose the model

Interpret the roots

See the method in action.

Worked example 1

Substitute the starting time.

Find the highest possible height.

Calculate one full-cycle time.

Worked example 2

Use the perpendicular displacements.

Measure the angle from north.

Write the bearing to one decimal place.

Try it yourself.

Check your understanding

Keep practising