Modelling with Trigonometric Functions
Tides rise and fall. Temperatures peak in summer and dip in winter. Sound waves travel through air. All of these phenomena can be modelled by sinusoidal functions. In this lesson, you will learn how to extract real-world data, build trigonometric models, and use them to make predictions.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
Quick warm-up — the height of the tide at a particular beach varies between a low of 1.2 m and a high of 3.6 m. What is the average tide height? What is the maximum deviation from this average?
There are only two moves to building any trigonometric model from real-world data: extract the amplitude and midline from max and min, then find $b$ from the period. Everything else is choosing the right starting function.
Every periodic model follows the same general form: $y = a\sin(b(x-c)) + d$. The parameter $a = \dfrac{\text{max}-\text{min}}{2}$ is the amplitude, and $d = \dfrac{\text{max}+\text{min}}{2}$ is the midline. The coefficient $b = \dfrac{2\pi}{P}$ comes from the period $P$.
general form
Key facts
- How to calculate amplitude and midline from maximum and minimum values
- How to determine the period from real-world cycles
- How to write a trigonometric model from given data
Concepts
- Why periodic phenomena are naturally modelled by trig functions
- How the parameters $a$, $b$, $c$, $d$ correspond to physical quantities
- The limitations of simple sinusoidal models
Skills
- Build a trig model from maximum, minimum, and period data
- Use the model to predict future values
- Interpret the meaning of each parameter in context
Many real-world phenomena repeat in regular cycles. If you can identify the maximum value, minimum value, and period of the cycle, you can build a sinusoidal model.
Step 1: Find the amplitude and midline
Given the maximum and minimum values of the phenomenon:
$$a = \frac{\text{max} - \text{min}}{2}, \quad d = \frac{\text{max} + \text{min}}{2}$$
Step 2: Find $b$ from the period
If the phenomenon repeats every $P$ units of time (or angle), then:
$$b = \frac{2\pi}{P}$$
Step 3: Determine the phase shift
The phase shift $c$ depends on when the cycle starts. If the model uses sine, the cycle normally starts at the midline going up. If it starts at a maximum, cosine might be more natural. You can always convert between sine and cosine using phase shifts.
Tide height modelled by a sinusoidal function with period 12 hours, amplitude 1.2 m, and midline 2.4 m
General form: $y = a\sin(b(x-c)) + d$; Amplitude: $a = \dfrac{\text{max} - \text{min}}{2}$ — half the total range
Pause — copy the four-step modelling process ($a$, $b$, $c$, $d$ from max/min data) and the amplitude formula $a = \frac{\text{max} - \text{min}}{2}$ into your book.
Did you get this? True or false: the amplitude of a sinusoidal model is calculated as $\text{max} - \text{min}$ (the full range).
Worked examples · 3 in a row, reveal as you go
The tide height at a harbour varies between a low of 1.2 m and a high of 3.6 m, with a period of 12 hours. Write a model for the tide height $h$ (in metres) $t$ hours after low tide, using a sine function.
The average monthly temperature in a city varies from a minimum of $8^\circ$C in July to a maximum of $24^\circ$C in January. The cycle repeats annually. Model the temperature $T$ as a function of time $t$ in months, with $t = 0$ representing January, using a cosine function.
Using the temperature model $T = 8\cos\!\left(\dfrac{\pi}{6}t\right) + 16$, find the temperature in April ($t = 3$).
Quick check: A ferris wheel has a minimum height of 2 m and a maximum height of 18 m. What is the amplitude of its height model?
Common errors · the 3 traps that cost marks
Fill in the blank: If a sinusoidal phenomenon has a period of 24 hours, then $b = \dfrac{2\pi}{\square}$ = (leave in terms of $\pi$).
Quick-fire practice · 5 reps
A ferris wheel has a minimum height of 5 m and a maximum height of 25 m. It completes one revolution every 4 minutes. Let $h$ be the height $t$ minutes after reaching the minimum height. Write a model.
The number of daylight hours in a town varies from a minimum of 9 hours in June to a maximum of 15 hours in December, repeating annually. Let $D$ be the number of daylight hours $t$ months after December. Write a model.
A pendulum swings so that its angle from the vertical varies from $-15^\circ$ to $+15^\circ$, completing one full swing every 2 seconds. Let $\theta$ be the angle $t$ seconds after passing through the vertical in the positive direction. Write a model.
Using your ferris wheel model, find the height at $t = 2$ minutes.
Using your daylight model, find the number of hours at $t = 6$ months (June).
Odd one out: Three of these are correct steps in building a trig model. Which one is incorrect?
Match each parameter to its meaning in $y = a\sin(b(x-c)) + d$:
Earlier you were asked about tide heights of 1.2 m and 3.6 m.
The average tide height (midline) is $\dfrac{1.2 + 3.6}{2} = 2.4$ metres. The maximum deviation from this average (amplitude) is $\dfrac{3.6 - 1.2}{2} = 1.2$ metres. These two numbers, $a = 1.2$ and $d = 2.4$, are the keys to building the tide model.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. A population of rabbits varies sinusoidally between a minimum of 800 in March and a maximum of 2400 in September, repeating annually. Let $P$ be the population $t$ months after January. Write a model for $P$ in terms of $t$. (4 marks)
Q2. The depth of water in a harbour is modelled by $D = 3\sin\!\left(\dfrac{\pi}{6}t\right) + 5$, where $D$ is in metres and $t$ is hours after midnight. (a) Find the maximum and minimum depths. (b) Find the depth at 9:00 am. (4 marks)
Q3. A student claims that the tide model $h = 2 + \sin(30t)$ (where $h$ is in metres and $t$ is in hours) has a period of 12 hours. Verify this claim and find the maximum and minimum tide heights predicted by the model. (3 marks)
📖 Comprehensive answers (click to reveal)
Drill 1: $a = 10$, $d = 15$, $b = \dfrac{\pi}{2}$. Minimum at $t = 0$, so $h = 15 - 10\sin\!\left(\dfrac{\pi}{2}t\right)$.
Drill 2: $a = 3$, $d = 12$, $b = \dfrac{\pi}{6}$. December ($t = 0$) is maximum, so $D = 3\cos\!\left(\dfrac{\pi}{6}t\right) + 12$.
Drill 3: $a = 15$, $d = 0$, $b = \pi$. Passes through vertical at $t = 0$ going positive, so $\theta = 15\sin(\pi t)$.
Drill 4: $h = 15 - 10\sin\!\left(\dfrac{\pi}{2} \times 2\right) = 15 - 10\sin(\pi) = 15$ m.
Drill 5: $D = 3\cos\!\left(\dfrac{\pi}{6} \times 6\right) + 12 = 3\cos(\pi) + 12 = 9$ hours.
Q1 (4 marks): $a = \dfrac{2400 - 800}{2} = 800$ [0.5], $d = \dfrac{2400 + 800}{2} = 1600$ [0.5]. Period = 12 months, so $b = \dfrac{\pi}{6}$ [1]. March ($t = 2$) is minimum; $P = 1600 - 800\cos\!\left(\dfrac{\pi}{6}(t - 2)\right)$ or equivalent [2].
Q2 (4 marks): (a) Max = $3 + 5 = 8$ m, Min = $-3 + 5 = 2$ m [2]. (b) At $t = 9$: $D = 3\sin\!\left(\dfrac{3\pi}{2}\right) + 5 = -3 + 5 = 2$ m [2].
Q3 (3 marks): Period = $\dfrac{2\pi}{30} = \dfrac{\pi}{15}$ hours $\approx 12.57$ min [1]. The student is incorrect — the period is $\dfrac{\pi}{15}$ hours, not 12 hours [1]. Max = 3 m, Min = 1 m [1].
Five timed questions. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms by answering trigonometric modelling questions. Lighter alternative to the boss.
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