IB Maths AA SL · Unit 3: Geometry and Trigonometry
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 the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
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 →
The question bank covers these trigonometric functions question types (number of questions in brackets):
A graph $y = a\cos(bx) + d$ has max $4$, min $0$ and completes one full oscillation on $0 \le x \le 2\pi$. Write down $a$, $b$, and $d$.
1. Amplitude: $a = \frac{4-0}{2} = \mathbf{2}$.
2. Principal axis: $d = \frac{4+0}{2} = \mathbf{2}$.
3. Period: $2\pi = \frac{2\pi}{b} \implies \mathbf{b = 1}$.
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.
Daylight model $S(t) = 3.5\sin\left(\frac{2\pi}{365}(t-80)\right) + 12$ hours, $1 \le t \le 365$. A crop needs strictly more than 14 hours of continuous daylight. Use your GDC to find the number of consecutive full days.
1. Inequality: $3.5\sin\left(\frac{2\pi}{365}(t-80)\right) + 12 > 14$.
2. Graph Y1 and Y2 $= 14$; ensure Radian mode.
3. Intersection points: $t_1 = 115.34\ldots$, $t_2 = 227.15\ldots$
4. Duration: $227.15\ldots - 115.34\ldots = 111.81\ldots$ days. Consecutive full days = $\mathbf{111}$.
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Consider the function $f(x) = \cos x$ with domain $-2\pi \le x \le 2\pi$.
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.
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 (\(0 \le t \le 24\)). The minimum temperature of \(14^\circ\text{C}\) occurs at \(04:00\) (\(t = 4\)) and the maximum temperature of \(28^\circ\text{C}\) occurs at \(16:00\) (\(t = 16\)).
Find the values of \(a, b, c\), and \(d\), assuming \(a > 0\) and \(0 \le c < 24\).
Use your graphic display calculator to find the amount of time during the day when the temperature is strictly above \(25^\circ\text{C}\). Give your answer in hours to two decimal places.
All 21 trigonometric functions questions with mark schemes →
There are 21 exam-style trigonometric functions questions in the AA SL question bank (Paper 1: 10 · Paper 2: 11), graded 4 easy, 6 medium, 7 hard, 3 very hard, 1 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Both. In the bank, Paper 1: 10 · Paper 2: 11. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.
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.