IB Maths AA SL · Unit 3: Geometry and Trigonometry
IB Maths AA SL Trigonometric Identities Questions
Exam-style IB Maths AA SL trigonometric identities 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.
- 20 questions
- Paper 1: 15
- Paper 2: 5
- 4 easy
- 7 medium
- 7 hard
- 2 starter
- 3 worked examples
Practise Trigonometric Identities questions →
AA SL formula booklet
What's examined in AA SL trigonometric identities
The question bank covers these trigonometric identities question types (number of questions in brackets):
- Fundamental Identities (8)
- Trigonometric Equations & Proofs (6)
- Double and Compound Angles (6)
Key formulas
- Double-angle: cosine
- \(\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta\)
- Tangent from sine & cosine
- \(\tan\theta = \dfrac{\sin\theta}{\cos\theta}\)
- Cosine rule
- \(c^2 = a^2 + b^2 - 2ab\cos C\)
- Pythagorean identity
- \(\sin^2\theta + \cos^2\theta = 1\)
- Double-angle: sine
- \(\sin 2\theta = 2\sin\theta\cos\theta\)
- Sine rule
- \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\)
In the same notation as the IB formula booklet. All AA SL formulas →
Trigonometric Identities worked examples
Worked example 1: Double angles from right triangles · easy
A right triangle contains an acute angle $\theta$ such that $\cos\theta = \frac{3}{5}$. Find the exact values of $\sin 2\theta$ and $\cos 2\theta$.
1. Deduce third side: $\sqrt{25-9} = 4$, so $\sin\theta = \frac{4}{5}$.
2. Sine double angle: $\sin 2\theta = 2\sin\theta\cos\theta = 2\cdot\frac{4}{5}\cdot\frac{3}{5} = \mathbf{\frac{24}{25}}$.
3. Cosine double angle: $\cos 2\theta = \cos^2\theta - \sin^2\theta = \frac{9}{25} - \frac{16}{25} = \mathbf{-\frac{7}{25}}$.
Examiner tip: If the original angle is acute, its double $2\theta$ can become obtuse, making $\cos 2\theta$ negative. Don't panic if the sign flips.
Worked example 2: Geometric proof of a double angle identity · medium
An isosceles triangle has two sides of length $a$ and included angle $2\theta$. Prove $\sin 2\theta = 2\sin\theta\cos\theta$ by calculating the area in two different ways.
1. Area (SAS): $\frac{1}{2}(a)(a)\sin 2\theta = \frac{1}{2}a^2\sin 2\theta$.
2. Half-triangle: base half $= a\sin\theta$, so total base $= 2a\sin\theta$; height $= a\cos\theta$.
3. Area ($\frac{1}{2}bh$): $\frac{1}{2}(2a\sin\theta)(a\cos\theta) = a^2\sin\theta\cos\theta$.
4. Equate: $\frac{1}{2}a^2\sin 2\theta = a^2\sin\theta\cos\theta$, giving $\mathbf{\sin 2\theta = 2\sin\theta\cos\theta}$. $\blacksquare$
Examiner tip: Geometric proofs require an explicit statement of the formula being used (like Area $= \frac{1}{2}ab\sin C$) BEFORE substituting variables.
Worked example 3: Algebraic proof involving tangent · hard
Given $f(x) = \tan x\sin x + \cos x$, prove that $f(x) = \frac{1}{\cos x}$.
1. Substitute $\tan x = \frac{\sin x}{\cos x}$: $f(x) = \frac{\sin^2 x}{\cos x} + \cos x$.
2. Common denominator: $\frac{\sin^2 x}{\cos x} + \frac{\cos^2 x}{\cos x} = \frac{\sin^2 x + \cos^2 x}{\cos x}$.
3. Pythagorean identity: numerator $= 1$.
4. Conclude: $\mathbf{f(x) = \frac{1}{\cos x}}$. $\blacksquare$
Examiner tip: When proving identities mixing $\tan x$ with $\sin x$ and $\cos x$, immediately substitute $\tan x = \frac{\sin x}{\cos x}$ and find a common denominator.
Try these IB Maths AA SL trigonometric identities questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Question 1 · medium · 4 marks · Paper 2
The function \(f(x) = 4\sin x \cos x\) is plotted below.
Use a double angle identity to write \(f(x)\) in the form \(A\sin(Bx)\).
Use your GDC to find the exact coordinates of the first positive maximum point on the graph.
Question 2 · medium · 5 marks · Paper 1
Given that $\sin \theta = \frac{4}{5}$ and $\theta$ is an obtuse angle ($\frac{\pi}{2} < \theta < \pi$), find the exact value of $\sin(2\theta)$.
Question 3 · hard · 5 marks · Paper 1
Given that $\cos x = -\frac{2}{3}$ and that $\pi < x < \frac{3\pi}{2}$, find the exact value of $\sin(2x)$.
All 20 trigonometric identities questions with mark schemes →
FAQ
How many IB Maths AA SL trigonometric identities questions are there?
There are 20 exam-style trigonometric identities questions in the AA SL question bank (Paper 1: 15 · Paper 2: 5), graded 4 easy, 7 medium, 7 hard, 2 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is trigonometric identities on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 15 · Paper 2: 5. 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.
← All IB Maths AA SL topics