Ferris wheel height model: $h(t) = 20\sin\left(\frac{\pi}{30}t - \frac{\pi}{2}\right) + 25$ metres, $t$ seconds. Find the maximum height and the time for one full revolution.

1. Amplitude : $20$; principal axis : $25$. 2. Maximum : $25 + 20 = \mathbf{45}$ m. 3. Period : $\frac{2\pi}{\pi/30} = 2\pi \times \frac{30}{\pi} = \mathbf{60}$ seconds.

A patient’s blood pressure, $P$ in mmHg, at time $t$ minutes is modelled by the periodic function $P(t) = 20\sin(140\pi t) + 100$ . Write down the maximum and minimum blood pressure of the patient. Find the period of the function in minutes. Hence, determine the patient’s heart rate in beats per minute.

Maximum $=100+20=120$ mmHg. A1 Minimum $=100-20=80$ mmHg. A1 Period $=\frac{2\pi}{b}$ with $b=140\pi$: (M1) Period $=\frac{2\pi}{140\pi}=\frac{1}{70}$ minutes ($\approx 0.0143$ min). A1 One heartbeat per period, so heart rate $=\frac{1}{1/70}=70$ beats per minute. A1

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IB Maths AA SL · Unit 3: Geometry and Trigonometry

IB Maths AA SL Trigonometric Functions Questions

Q: How many IB Maths AA SL trigonometric functions questions are there?

There are 28 exam-style trigonometric functions questions in the AA SL question bank (Paper 1: 16 · Paper 2: 12), graded 6 easy, 10 medium, 9 hard, 3 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Q: Is trigonometric functions on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 16 · Paper 2: 12. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Q: Where can I get the mark schemes?

Open the AA SL Unit 3 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.

Updated October 2026

Exam-style IB Maths AA SL trigonometric functions 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.

28 questions
Paper 1: 16
Paper 2: 12
6 easy
10 medium
9 hard
3 very hard
3 worked examples

Practise Trigonometric Functions 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

sin²θ + cos²θ = 1, plus the double-angle formulas. SL AA Paper 1 asks you to prove or simplify — never to memorise the derivation, but to APPLY it fluently. Trigonometric identities — Pythagorean and double-angle overview →

The unified triangle toolkit. Sine rule when you have angle-side pairs, cosine rule when you have SSS or SAS. Area = ½ab·sin(C) closes the trilogy. Sine rule, cosine rule, and area of a triangle overview →

What's examined in AA SL trigonometric functions

The question bank covers these trigonometric functions question types (number of questions in brackets):

Trigonometric Modelling (14)
Graph Properties & Transformations (11)
Trigonometric Equations & Identities (3)

Key formulas

Double-angle: cosine: $\cos 2 θ = \cos^{2} θ - \sin^{2} θ = 2 \cos^{2} θ - 1 = 1 - 2 \sin^{2} θ$
Tangent from sine & cosine: $\tan θ = \frac{\sin θ}{\cos θ}$
Cosine rule: $c^{2} = a^{2} + b^{2} - 2 a b \cos C$
Double-angle: sine: $\sin 2 θ = 2 \sin θ \cos θ$
Sine rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
Pythagorean identity: $\sin^{2} θ + \cos^{2} θ = 1$

In the same notation as the IB formula booklet. All AA SL formulas →

Trigonometric Functions worked examples

Worked example 1: Deducing parameters from a graph · easy

A graph $y = a \cos (b x) + d$ has max $4$ , min $0$ and completes one full oscillation on $0 \leq x \leq 2 π$ . Write down $a$ , $b$ , and $d$ .

Solution

3 steps

Examiner tip: The

y

-intercept of

a \sin (b x) + d

equals

d

, but for

a \cos (b x) + d

it equals

d + a

(starts at max). Don't confuse the two.

Worked example 2: Contextual periodic modelling · medium

Ferris wheel height model: $h (t) = 20 \sin (\frac{π}{30} t - \frac{π}{2}) + 25$ metres, $t$ seconds. Find the maximum height and the time for one full revolution.

Solution

3 steps

Examiner tip: When the period coefficient is a fractional

π

, multiply by its reciprocal rather than typing fraction-over-fraction into your GDC to avoid syntax errors.

Worked example 3: Intersections of models using GDC · hard

Daylight model $S (t) = 3.5 \sin (\frac{2 π}{365} (t - 80)) + 12$ hours, $1 \leq t \leq 365$ . A crop needs strictly more than 14 hours of continuous daylight. Use your GDC to find the number of consecutive full days.

Solution

4 steps

Examiner tip: "Days of the year" (

365

) models almost always use

\frac{2 π}{365}

, which strongly implies Radian mode is required.

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 AA SL trigonometric functions 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 · 5 marks · Paper 1

Consider the function $f (x) = \cos x$ with domain $- 2 π \leq x \leq 2 π$ .

Write down the amplitude and the period of $f$ . [2]
The graph of $f$ is transformed to give the graph of $g (x) = 2 \cos (\frac{x}{2})$ . Describe two transformations that map the graph of $f$ onto the graph of $g$ . [3]

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 2

A patient’s blood pressure, $P$ in mmHg, at time $t$ minutes is modelled by the periodic function $P (t) = 20 \sin (140 π t) + 100$ .

Write down the maximum and minimum blood pressure of the patient.
Find the period of the function in minutes.
Hence, determine the patient’s heart rate in beats per minute.

Full worked solution Free, no sign-in

Maximum $= 100 + 20 = 120$ mmHg. A1
Minimum $= 100 - 20 = 80$ mmHg. A1
Period $= \frac{2 π}{b}$ with $b = 140 π$ : (M1)
Period $= \frac{2 π}{140 π} = \frac{1}{70}$ minutes ( $\approx 0.0143$ min). A1
One heartbeat per period, so heart rate $= \frac{1}{1 / 70} = 70$ beats per minute. A1

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

Question 3 · hard · 6 marks · Paper 2

The ambient temperature $T$ , in degrees Celsius, in a desert on a given day can be modelled by a cosine function in the form $T (t) = a \cos (b (t - c)) + d$ , where $t$ is the time in hours past midnight ( $0 \leq t \leq 24$ ). The minimum temperature of $14^{\circ} C$ occurs at $04 : 00$ ( $t = 4$ ) and the maximum temperature of $28^{\circ} C$ occurs at $16 : 00$ ( $t = 16$ ).

Find the values of $a, b, c$ , and $d$ , assuming $a > 0$ and $0 \leq c < 24$ .
Use your graphic display calculator to find the amount of time during the day when the temperature is strictly above $25^{\circ} C$ . Give your answer in hours to two decimal places.

Attempt it and see the mark scheme →

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FAQ

How many IB Maths AA SL trigonometric functions questions are there?

There are 28 exam-style trigonometric functions questions in the AA SL question bank (Paper 1: 16 · Paper 2: 12), graded 6 easy, 10 medium, 9 hard, 3 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is trigonometric functions on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 16 · Paper 2: 12. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Where can I get the mark schemes?

Open the AA SL Unit 3 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.

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Teaching this topic?

Free ready-to-teach lesson (slides and teacher notes): Modelling with sinusoidal functions.

IB Maths AA SL Trigonometric Functions Questions

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What you need to know

What's examined in AA SL trigonometric functions

Key formulas

Trigonometric Functions worked examples

Worked example 1: Deducing parameters from a graph · easy

Solution

Worked example 2: Contextual periodic modelling · medium

Solution

Worked example 3: Intersections of models using GDC · hard

Solution

Try these IB Maths AA SL trigonometric functions questions

Question 1 · easy · 5 marks · Paper 1

Question 2 · medium · 5 marks · Paper 2

Full worked solution Free, no sign-in

Question 3 · hard · 6 marks · Paper 2

Revise trigonometric functions for the exam

FAQ

What else is free here

Teaching this topic?

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