Trigonometric Functions in Context — Edexcel A-level Mathematics (Pure)

SPEC 5.8 • TRIGONOMETRY

Trigonometry in Context

Modelling real-world periodic phenomena using trigonometric functions

Periodic Phenomena

Definition

A periodic phenomenon is any process that repeats itself at regular intervals. The time (or distance) for one complete cycle is called the period.

Trigonometric functions are ideal for modelling periodic behaviour because they:

Oscillate smoothly and continuously

Have a predictable, repeating pattern

Can be adjusted for amplitude (size), period (timing), and position

Key Insight

If something goes up and down in a regular pattern, it can probably be modelled with sin or cos!

Real-World Applications

Natural Phenomena

Tides — height of water over time

Daylight hours — seasonal variation

Temperature — daily/annual cycles

Moon phases — lunar cycle

Mechanical/Physical

Ferris wheels — height of rider

Pendulums — displacement from rest

Pistons — engine mechanics

Sound waves — air pressure variation

Exam Tip

Questions often involve tides, Ferris wheels, or temperature — learn the typical setup for each!

The General Model

Standard Forms

y = A sin (Bt + C) + D

y = A cos (Bt + C) + D

ParameterNameEffect

A

AmplitudeHeight of oscillation from midline

B

Angular frequencyControls the period:

T = \frac{2 π}{B}

C

Phase shiftHorizontal translation

D

Vertical shiftPosition of the midline

Approach Depends on the Question

Every contextual modelling question is different. The information given will vary, so you need to adapt your approach based on what the question provides.

Information You Might Be Given

Maximum and minimum values — use these to find amplitude and midline

The period directly — "repeats every 12 hours"

Frequency — "440 cycles per second" (period = 1/frequency)

When max/min occurs — helps determine phase shift

Starting value — what happens at t = 0?

A complete model — you interpret or use it

Key Skill

Read the question carefully and identify what information you have. Then decide what you need to calculate and what's already given. The worked examples will show different scenarios!

01Worked Example — Tidal Heights

The height of water at a harbour varies with the tides. At high tide, the water is 8 metres deep. At low tide, it is 2 metres deep. The time between consecutive high tides is 12 hours. High tide occurs at 3:00 AM.

Find a model for the height $h$ metres at time $t$ hours after midnight.

1

Find amplitude:

A = \frac{8 - 2}{2} = 3

2

Find midline:

D = \frac{8 + 2}{2} = 5

3

Find B from period:

B = \frac{2 π}{12} = \frac{π}{6}

4

High tide (max) at

t = 3

, so use cos with shift

Model:

h = 3 cos (\frac{π}{6} (t - 3)) + 5

Check: When t = 3, h = 3cos(0) + 5 = 8 ✓ (high tide)

02Worked Example — Ferris Wheel

A Ferris wheel has a diameter of 40 metres. The centre of the wheel is 25 metres above the ground. It takes 8 minutes for one complete revolution. A rider starts at the lowest point.

Find a model for the height $h$ metres above the ground after $t$ minutes.

1

Diameter 40m → Amplitude

A = 20

2

Centre 25m above ground → Midline

D = 25

3

Period = 8 mins →

B = \frac{2 π}{8} = \frac{π}{4}

4

Starts at minimum → use

- cos

Model:

h = - 20 cos (\frac{π t}{4}) + 25

Check: When t = 0, h = −20(1) + 25 = 5m ✓ (lowest point)

03Worked Example — Daylight Hours

In London, the longest day (21st June, day 172) has 16.5 hours of daylight. The shortest day (21st December, day 355) has 8 hours of daylight.

Find a model for the number of daylight hours $D$ on day $d$ of the year.

1

Amplitude:

A = \frac{16.5 - 8}{2} = 4.25

2

Midline:

D = \frac{16.5 + 8}{2} = 12.25

3

Period = 365 days →

B = \frac{2 π}{365}

4

Maximum at day 172 → use cos with shift

Model:

D = 4.25 cos (\frac{2 π}{365} (d - 172)) + 12.25

Model Construction Method

7-Step Process

Identify — Is the phenomenon periodic?

Find max and min — From data or description

Calculate amplitude —

A = \frac{Max - Min}{2}

Calculate midline —

D = \frac{Max + Min}{2}

Determine period — Time for one complete cycle, then

B = \frac{2 π}{T}

Determine phase shift — When does max/min occur? Choose sin/cos

Write model and verify — Check key values!

Important

Always verify your model by substituting the time when max/min occurs — the output should match!

04Worked Example — Sound Wave

A sound wave has displacement $d$ mm from equilibrium at time $t$ seconds. It oscillates between +0.5mm and −0.5mm with a frequency of 440 Hz. At $t = 0$ , the displacement is 0 and increasing.

Find a model for $d$ in terms of $t$ .

1

Amplitude:

A = 0.5

mm

2

Midline:

D = 0

(oscillates about equilibrium)

3

Frequency 440 Hz → Period =

\frac{1}{440}

s

4

B = 2 π \times 440 = 880 π

5

At

t = 0

,

d = 0

and increasing → use sin

Model:

d = 0.5 sin (880 π t)

05Worked Example — Using a Given Model

The depth of water in a tidal river is modelled by $D = 4 + 2.5 sin (\frac{π t}{6})$ where $D$ is in metres and $t$ is hours after midnight.

(a) Find the maximum and minimum depths. (b) Find times between midnight and noon when depth is 5.5m.

a)

Max = 4 + 2.5 = 6.5m, Min = 4 − 2.5 = 1.5m

b)

Solve:

4 + 2.5 sin (\frac{π t}{6}) = 5.5

•

sin (\frac{π t}{6}) = 0.6

•

\frac{π t}{6} = 0.6435...

or

\frac{π t}{6} = π - 0.6435...

•

t = \frac{6 \times 0.6435}{π} = 1.23

hours or

t = 4.77

hours

Times: 1:14 AM and 4:46 AM

06Worked Example — Interpretation

The height $h$ cm of a piston in an engine is modelled by $h = 12 + 8 sin (50 π t)$ where $t$ is in seconds.

Interpret the values 12 and 8, and find how many cycles occur per second.

•

12 = vertical shift = average height of piston = 12 cm from reference

•

8 = amplitude = distance piston travels from average = 8 cm either side

•

So piston moves between 4cm and 20cm (total travel = 16cm)

•

B = 50 π

→ Period =

\frac{2 π}{50 π} = \frac{1}{25}

second

•

Frequency =

\frac{1}{Period} = 25

cycles per second

The piston completes 25 cycles per second (= 1500 rpm)

Using the R-Formula in Context

When a model involves

a sin θ + b cos θ

, the R-formula converts it into a single function, making it easier to find maximum/minimum values.

Sine Form

a sin θ + b cos θ \equiv R sin (θ + α)

where

tan α = \frac{b}{a}

Cosine Form

a sin θ + b cos θ \equiv R cos (θ - α)

where

tan α = \frac{a}{b}

For Both Forms

R = a^{2} + b^{2}

When to Use

Use the R-formula when you need to find the maximum/minimum of a combined sin + cos expression, or solve equations like

a sin θ + b cos θ = k

.

07Worked Example — R-Formula

The temperature $T$ °C at a weather station is modelled by $T = 15 + 6 sin (\frac{π t}{12}) + 8 cos (\frac{π t}{12})$ where $t$ is hours after midnight.

(a) Express $6 sin (\frac{π t}{12}) + 8 cos (\frac{π t}{12})$ in the form $R cos (\frac{π t}{12} - α)$ .

(b) Find the maximum temperature and the time it occurs.

a)

R = 6^{2} + 8^{2} = 36 + 64 = 100 = 10

•

tan α = \frac{a}{b} = \frac{6}{8} = 0.75

→

α = 0.6435

rad (or 36.87°)

•

$6 sin (\frac{π t}{12}) + 8 cos (\frac{π t}{12}) \equiv 10 cos (\frac{π t}{12} - 0.6435)$

b)

So

T = 15 + 10 cos (\frac{π t}{12} - 0.6435)

•

Maximum when

cos (\frac{π t}{12} - 0.6435) = 1

, giving T = 25°C

•

This occurs when

\frac{π t}{12} - 0.6435 = 0

→

t = \frac{12 \times 0.6435}{π} = 2.46

hours

Maximum is 25°C at approximately 2:28 AM

Model Limitations

When evaluating trigonometric models, consider these limitations:

Perfect periodicity

Real phenomena may have irregular variations (weather affecting tides, clouds affecting temperature)

Constant amplitude

Real oscillations may decay or grow over time (pendulum slowing, sound dissipating)

Sinusoidal shape

Not all oscillations are smooth sine curves (heartbeat, square waves)

Single frequency

Many phenomena combine multiple frequencies (musical harmonics, complex tides)

Exam Tip

When asked to "comment on limitations," always relate your answer to the specific context given in the question!

P1

Practice

A buoy bobs up and down in the sea. Its height above the sea bed varies between 2m and 6m with a period of 4 seconds. At $t = 0$ , the buoy is at its maximum height.

(a) Find a model for the height

h

metres at time

t

seconds.

(b) Find the height when

t = 2.5

seconds.

a)

Amplitude:

A = \frac{6 - 2}{2} = 2

, Midline:

D = \frac{6 + 2}{2} = 4

•

Period = 4s →

B = \frac{2 π}{4} = \frac{π}{2}

•

Max at

t = 0

→ use cos

•

Model: $h = 2 cos (\frac{π t}{2}) + 4$

b)

h = 2 cos (\frac{π ( 2.5 )}{2}) + 4 = 2 cos (1.25 π) + 4

•

= 2 (- 0.707...) + 4 =

2.59m

P2

Practice

The population $P$ (in thousands) of a species varies seasonally according to $P = 50 + 20 sin (\frac{π t}{6})$ where $t$ is the month number (January = 1).

(a) Find the maximum and minimum populations.

(b) In which months is the population above 60,000?

a)

Maximum:

50 + 20 = 70

thousand = 70,000

•

Minimum:

50 - 20 = 30

thousand = 30,000

b)

Solve:

50 + 20 sin (\frac{π t}{6}) > 60

•

sin (\frac{π t}{6}) > 0.5

•

\frac{π}{6} < \frac{π t}{6} < \frac{5 π}{6}

→

1 < t < 5

•

Months: February, March, April (t = 2, 3, 4)

P3

Practice

A clock pendulum swings with displacement $x$ cm from vertical given by $x = 15 cos (2 π t)$ where $t$ is in seconds.

(a) What is the amplitude and period of the swing?

(b) Find the times in the first 2 seconds when the pendulum is 10cm from vertical.

a)

Amplitude = 15 cm

•

Period =

\frac{2 π}{2 π} =

1 second

b)

Solve:

15 cos (2 π t) = 10

→

cos (2 π t) = \frac{2}{3}

•

2 π t = 0.8411, 2 π - 0.8411, 2 π + 0.8411, 4 π - 0.8411

•

t = 0.134, 0.866, 1.134, 1.866

seconds

•

Times: 0.13s, 0.87s, 1.13s, 1.87s

Summary

Key Formulas

ParameterFormula

Amplitude

A = \frac{Max - Min}{2}

Midline

D = \frac{Max + Min}{2}

Angular frequency

B = \frac{2 π}{Period}

Model form

y = A sin (Bt + C) + D

or

y = A cos (Bt + C) + D

Exam Skills

Extract parameters from contextual information

Construct models and verify with key values

Solve equations arising from models

Interpret parameters in context

Discuss model limitations appropriately