The temperature, $T$ in $^{\circ}\text{C}$, of a greenhouse is modelled by $T(t) = 4\cos(15t)^{\circ} + 22$, where $t$ is the time in hours after midnight. Calculate the period of the model and explain what it represents in context. Calculate the temperature of the greenhouse at 06:00 (when $t = 6$).

Period $= \frac{360}{15} = 24$ hours A1 The temperature pattern repeats every 24 hours, i.e. one full daily cycle. R1 Substitute $t = 6$: $T(6) = 4\cos(15 \times 6)^{\circ} + 22$ M1 $= 4\cos(90^{\circ}) + 22 = 0 + 22 = 22^{\circ}\text{C}$ A1

Home › IB Maths AI SL › Questions by topic › Sinusoidal Modelling

IB Maths AI SL · Unit 2: Functions

IB Maths AI SL Sinusoidal Modelling Questions

Updated October 2026

Exam-style IB Maths AI SL sinusoidal modelling questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

13 questions
Paper 1: 3
Paper 2: 10
2 easy
5 medium
3 hard
3 very hard
3 worked examples

Practise Sinusoidal Modelling questions →

🖩 How do I do this on my calculator? Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire.

What you need to know

Modelling tides, daylight hours, or Ferris-wheel height with y = a·sin(bx + c) + d. Amplitude, period, and phase shift each carry an A-mark. Sinusoidal models for periodic data overview →

Slope, intercept, and interpreting the gradient in context. SL AI dresses linear models as taxi fares, phone plans, and rental costs — always ask what the gradient MEANS in the real-world context. Linear functions and modelling overview →

What's examined in AI SL sinusoidal modelling

The question bank covers these sinusoidal modelling question types (number of questions in brackets):

Analysing sinusoidal models (8)
Constructing sinusoidal models (4)
Determining model parameters (1)

Sinusoidal Modelling worked examples

Worked example 1: Reading amplitude and principal axis · easy

Water depth in a harbour: $D (t) = 4 \cos (30 t^{\circ}) + 12$ ( $t$ in hours after midnight). Find the exact amplitude and principal axis.

Solution

4 steps

Examiner tip: Amplitude is a POSITIVE value: the distance from principal axis to max (or min). It is NOT the peak-to-trough distance (which is

2 a

Worked example 2: Max and min of a sinusoidal temperature model · medium

Greenhouse temperature: $T (t) = 15 - 5 \sin (15 t^{\circ})$ ( $t$ in hours). Find the exact maximum and minimum temperatures.

Solution

5 steps · 1 to fill in

Examiner tip: Regardless of the sign of the sine/cosine coefficient, max = principal axis + amplitude and min = principal axis − amplitude.

Worked example 3: Time above a threshold — sinusoidal inequality · hard

Using $D (t) = 4 \cos (30 t^{\circ}) + 12$ , use your GDC to find the total time between midnight ( $t = 0$ ) and noon ( $t = 12$ ) when depth is strictly greater than $14 m$ .

Solution

6 steps · 1 to fill in

Examiner tip: Sketch the sine/cos curve from your GDC screen onto paper when solving inequalities — visualising the peaks and troughs prevents you picking the wrong interval.

Play with it Amplitude, period and principal axis: drag the sliders

Change a, b and d so that y = a sin(bx) + d passes through all the points (x in radians). Then write down the amplitude, the period and the equation of the principal axis.

Want more? Open the full Grapher.

Try these IB Maths AI SL sinusoidal modelling questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 3 marks · Paper 1

The height of a point on a rotating wheel, $h$ in metres above the ground, is modelled by the function $h (t) = 5 \sin (30 t)^{\circ} + 12$ , where $t$ is the time in seconds.

Write down the amplitude of the function.
Write down the equation of the principal axis.
State the maximum height reached by the point.

Attempt it and see the mark scheme →

Question 2 · medium · 4 marks · Paper 2

The temperature, $T$ in $^{\circ} C$ , of a greenhouse is modelled by $T (t) = 4 \cos (15 t)^{\circ} + 22$ , where $t$ is the time in hours after midnight.

Calculate the period of the model and explain what it represents in context.
Calculate the temperature of the greenhouse at 06:00 (when $t = 6$ ).

Full worked solution Free, no sign-in

Period $= \frac{360}{15} = 24$ hours A1
The temperature pattern repeats every 24 hours, i.e. one full daily cycle. R1
Substitute $t = 6$ : $T (6) = 4 \cos (15 \times 6)^{\circ} + 22$ M1
$= 4 \cos (90^{\circ}) + 22 = 0 + 22 = 22^{\circ} C$ A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 6 marks · Paper 2

The depth of water, $D$ in metres, in a harbour is modelled by the function $D (t) = a \cos (b t)^{\circ} + d$ , where $t$ is the time in hours after high tide. High tide occurs at $t = 0$ with a maximum depth of $14 m$ . The next low tide occurs at $t = 6$ hours with a minimum depth of $6 m$ .

Find the exact values of $a$ and $d$ .
Find the value of $b$ .
Using your Graphic Display Calculator, find the amount of time during the first 12 hours when the depth of the water is strictly greater than $12 m$ .

Attempt it and see the mark scheme →

Free account — sign in to download Topic PDF: 7 questions with worked answers · 4 pages · 214 KB

All 13 sinusoidal modelling questions with mark schemes →

Save your progress and get 5 more questions on every topic, checked instantly. Free account, no card. Join free

Revise sinusoidal modelling for the exam

FAQ

How many IB Maths AI SL sinusoidal modelling questions are there?

There are 13 exam-style sinusoidal modelling questions in the AI SL question bank (Paper 1: 3 · Paper 2: 10), graded 2 easy, 5 medium, 3 hard, 3 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is sinusoidal modelling on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 3 · Paper 2: 10. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI SL Unit 2 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

What else is free here

Save your answers and track your progressA free account remembers the questions you have done on every topic and brings you back to this page.

Join free

Already have an account? Sign in

Teaching this topic?

Free ready-to-teach lesson (slides and teacher notes): Sinusoidal models.

IB Maths AI SL Sinusoidal Modelling Questions

Try it first

What you need to know

What's examined in AI SL sinusoidal modelling

Sinusoidal Modelling worked examples

Worked example 1: Reading amplitude and principal axis · easy

Solution

Worked example 2: Max and min of a sinusoidal temperature model · medium

Solution

Worked example 3: Time above a threshold — sinusoidal inequality · hard

Solution

Try these IB Maths AI SL sinusoidal modelling questions

Question 1 · easy · 3 marks · Paper 1

Question 2 · medium · 4 marks · Paper 2

Full worked solution Free, no sign-in

Question 3 · hard · 6 marks · Paper 2

Revise sinusoidal modelling for the exam

FAQ

What else is free here

Teaching this topic?

More AI SL Unit 2 topics