IB Maths AA SL · Unit 3: Geometry and Trigonometry
IB Maths AA SL The Unit Circle and Exact Values of Trigonometric Ratios Questions
Exam-style IB Maths AA SL the unit circle and exact values of trigonometric ratios 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.
- 17 questions
- Paper 1: 15
- Paper 2: 2
- 4 easy
- 7 medium
- 3 hard
- 3 very hard
- 3 worked examples
Practise The Unit Circle and Exact Values of Trigonometric Ratios questions →
AA SL formula booklet
What's examined in AA SL the unit circle and exact values of trigonometric ratios
The question bank covers these the unit circle and exact values of trigonometric ratios question types (number of questions in brackets):
- Solving Trigonometric Equations (7)
- Trigonometric Identities (6)
- Unit Circle Exact Values (4)
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
- \(c^2 = a^2 + b^2 - 2ab\cos C\)
- Double-angle: sine
- \(\sin 2\theta = 2\sin\theta\cos\theta\)
- Sine rule
- \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\)
- Pythagorean identity
- \(\sin^2\theta + \cos^2\theta = 1\)
In the same notation as the IB formula booklet. All AA SL formulas →
The Unit Circle and Exact Values of Trigonometric Ratios worked examples
Worked example 1: Finding exact ratios using quadrants · easy
Given $\sin\theta = \frac{3}{5}$ and $\frac{\pi}{2} < \theta < \pi$, find the exact values of $\cos\theta$ and $\tan\theta$.
1. Quadrant 2: cosine and tangent are negative.
2. Pythagorean identity: $\cos^2\theta = 1 - \frac{9}{25} = \frac{16}{25}$.
3. Square root, negative in Q2: $\mathbf{\cos\theta = -\frac{4}{5}}$.
4. Tangent: $\tan\theta = \frac{3/5}{-4/5} = \mathbf{-\frac{3}{4}}$.
Examiner tip: A 3-4-5 reference triangle is a valid alternative, provided you remember to apply the ASTC sign rules to the final answer.
Worked example 2: Solving equations with squares · medium
Solve $3\tan^2 x - 1 = 0$ for $0^\circ \le x \le 360^\circ$.
1. Rearrange: $\tan^2 x = \frac{1}{3}$.
2. Square root: $\tan x = \pm\frac{1}{\sqrt{3}}$.
3. Reference angle: $30^\circ$.
4. Solutions in all four quadrants: $\mathbf{30^\circ, 150^\circ, 210^\circ, 330^\circ}$.
Examiner tip: Forgetting the $\pm$ symbol when square-rooting loses half of the solutions. In a full cycle, expect FOUR solutions for a squared trig equation.
Worked example 3: Proving identities algebraically · hard
Prove $\sin 2x + \cos 2x - 1 \equiv 2\sin x(\cos x - \sin x)$.
1. Identities: $\sin 2x = 2\sin x\cos x$, $\cos 2x = 1 - 2\sin^2 x$.
2. Substitute: LHS $= 2\sin x\cos x + (1-2\sin^2 x) - 1$.
3. Cancel constants: $2\sin x\cos x - 2\sin^2 x$.
4. Factorise out $2\sin x$: $2\sin x(\cos x - \sin x) = $ RHS. $\blacksquare$
Examiner tip: The cosine double angle has three versions; the presence of the $-1$ on the LHS makes $\cos 2x = 1 - 2\sin^2 x$ the strategic choice for immediate cancellation.
Try these IB Maths AA SL the unit circle and exact values of trigonometric ratios questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Question 1 · easy · 5 marks · Paper 1
Using your knowledge of the unit circle, write down the exact values of the following trigonometric ratios:
\(\sin\left(\frac{\pi}{6}\right)\)
\(\cos\left(\frac{3\pi}{4}\right)\)
\(\tan\left(\frac{4\pi}{3}\right)\)
\(\cos\left(\frac{3\pi}{2}\right)\)
\(\sin\left(\frac{11\pi}{6}\right)\)
Question 2 · medium · 5 marks · Paper 1
Given that \(\tan\theta = -\frac{4}{3}\), find the two possible exact values of \(\sin\theta\) and their corresponding exact values of \(\cos\theta\).
Question 3 · hard · 4 marks · Paper 1
Using the double angle identities, prove the trigonometric identity:
\[3\sin 2\theta + \cos 2\theta - 1 = 2\sin\theta(3\cos\theta - \sin\theta)\]
All 17 the unit circle and exact values of trigonometric ratios questions with mark schemes →
FAQ
How many IB Maths AA SL the unit circle and exact values of trigonometric ratios questions are there?
There are 17 exam-style the unit circle and exact values of trigonometric ratios questions in the AA SL question bank (Paper 1: 15 · Paper 2: 2), graded 4 easy, 7 medium, 3 hard, 3 very hard. Every question has a full IB-style mark scheme (M, A and R marks).
Is the unit circle and exact values of trigonometric ratios on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 15 · Paper 2: 2. 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