IB Maths AA SL · Unit 3: Geometry and Trigonometry
IB Maths AA SL Solving Trigonometric Equations Questions
Exam-style IB Maths AA SL solving trigonometric equations 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.
- 30 questions
- Paper 1: 23
- Paper 2: 7
- 4 easy
- 13 medium
- 7 hard
- 3 very hard
- 3 starter
- 3 worked examples
Practise Solving Trigonometric Equations questions →
AA SL formula booklet
What's examined in AA SL solving trigonometric equations
The question bank covers these solving trigonometric equations question types (number of questions in brackets):
- Quadratic Form Trig Equations (13)
- Using Double Angle Identities (9)
- Basic Equations & GDC (8)
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 →
Solving Trigonometric Equations worked examples
Worked example 1: Simple exact trigonometric equations · easy
Solve $2\sin x - \sqrt{3} = 0$ for $0 \le x \le 2\pi$. Give exact radian answers.
1. Rearrange: $\sin x = \frac{\sqrt{3}}{2}$.
2. Reference angle: $\arcsin\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}$.
3. Sine positive in Q1 and Q2.
4. Q1: $x = \mathbf{\frac{\pi}{3}}$.
5. Q2: $x = \pi - \frac{\pi}{3} = \mathbf{\frac{2\pi}{3}}$.
Examiner tip: Sketch the sine wave or draw a CAST diagram to confirm how many solutions to expect in a given interval before you start solving.
Worked example 2: Quadratic trigonometric equations · medium
Solve $2\sin^2 x + \cos x - 1 = 0$ for $0 \le x \le 2\pi$. Give exact radian answers.
1. Substitute $\sin^2 x = 1 - \cos^2 x$: $2(1-\cos^2 x) + \cos x - 1 = 0$.
2. Simplify: $2\cos^2 x - \cos x - 1 = 0$.
3. Factorise: $(2\cos x + 1)(\cos x - 1) = 0$.
4. Case 1: $\cos x = -0.5$, Q2 and Q3 give $x = \mathbf{\frac{2\pi}{3}}$ and $x = \mathbf{\frac{4\pi}{3}}$.
5. Case 2: $\cos x = 1$ gives $x = \mathbf{0}$ and $x = \mathbf{2\pi}$.
Examiner tip: When mixing $\sin^2$ and $\cos$ (or $\cos^2$ and $\sin$), always substitute the squared term via the Pythagorean identity so the equation becomes a quadratic in ONE trig function.
Worked example 3: Equations requiring double angle identities · hard
Solve $\sin 2x = \sqrt{3}\cos x$ for $0 \le x \le 2\pi$. Give exact radian answers.
1. Substitute $\sin 2x = 2\sin x\cos x$: $2\sin x\cos x = \sqrt{3}\cos x$.
2. Factorise: $\cos x(2\sin x - \sqrt{3}) = 0$.
3. Case 1: $\cos x = 0$ gives $x = \mathbf{\frac{\pi}{2}}$ and $x = \mathbf{\frac{3\pi}{2}}$.
4. Case 2: $\sin x = \frac{\sqrt{3}}{2}$ gives $x = \mathbf{\frac{\pi}{3}}$ and $x = \mathbf{\frac{2\pi}{3}}$.
Examiner tip: NEVER divide both sides by $\cos x$ — you lose the roots where $\cos x = 0$. Bring everything to one side and factorise.
FAQ
How many IB Maths AA SL solving trigonometric equations questions are there?
There are 30 exam-style solving trigonometric equations questions in the AA SL question bank (Paper 1: 23 · Paper 2: 7), graded 4 easy, 13 medium, 7 hard, 3 very hard, 3 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is solving trigonometric equations on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 23 · Paper 2: 7. 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