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 : $x = \pi - \frac{\pi}{3} = \mathbf{\frac{2\pi}{3}}$.

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 $x = \mathbf{2\pi}$.

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 $x = \mathbf{\frac{2\pi}{3}}$.

Solve the equation $\tan^2 x - 3 = 0$ for $0 \le x \le 2\pi$ . Give your answers exactly in radians.

Rearrange: $\tan^2 x - 3 = 0 \implies \tan^2 x = 3$. Take square roots (both signs): $\tan x = \pm\sqrt{3}$. (M1)A1 The reference angle is $\frac{\pi}{3}$. Because it is $\pm$, there is a solution in all four quadrants. $x = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}$. A2 (Award only one of these two marks if just the positive-root solutions $\frac{\pi}{3}, \frac{4\pi}{3}$ are found).

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IB Maths AA SL · Unit 3: Geometry and Trigonometry

Solving Trigonometric Equations (IB Maths AA SL): notes, exam questions and worked examples

Updated October 2026 · By Pete Bromfield

Find one solution with the inverse function, use the symmetry of the graph (or the unit circle) to find the second, then add whole periods until you leave the interval. With a multiple angle such as $2 x$ , widen the interval first. Rearranging with an identity often turns the equation into a quadratic in $\sin x$ or $\cos x$ .

30 questions
Paper 1: 23
Paper 2: 7
3 worked examples

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Notes
Formulas
Worked examples
Questions
FAQ

Solving Trigonometric Equations notes

From one solution to all of them

For $\sin x = k$ : $x = \arcsin k$ or $x = π - \arcsin k$ . For $\cos x = k$ : $x = \pm \arccos k$ . For $\tan x = k$ : $x = \arctan k$ , then add $π$ . Then add $2 π$ (or $π$ for tan) until you leave the interval. Example: $\sin x = \frac{1}{2}$ on $0 \leq x \leq 2 π$ gives $x = \frac{π}{6}$ and $x = \frac{5 π}{6}$ .

Multiple angles

For $\sin 2 x = \frac{1}{2}$ on $0 \leq x \leq 2 π$ , solve for $2 x$ on $0 \leq 2 x \leq 4 π$ : $2 x = \frac{π}{6}, \frac{5 π}{6}, \frac{13 π}{6}, \frac{17 π}{6}$ . Halve at the end: $x = \frac{π}{12}, \frac{5 π}{12}, \frac{13 π}{12}, \frac{17 π}{12}$ . Four solutions, not two.

Quadratics in sin x or cos x

Factorise as you would a quadratic. $2 \sin^{2} x - \sin x - 1 = 0$ gives $(2 \sin x + 1) (\sin x - 1) = 0$ , so $\sin x = 1$ or $\sin x = - \frac{1}{2}$ . On $0 \leq x \leq 2 π$ : $x = \frac{π}{2}, \frac{7 π}{6}, \frac{11 π}{6}$ .

Using identities first

If the equation mixes $\sin$ and $\cos$ , or angles $x$ and $2 x$ , use $\sin^{2} x + \cos^{2} x = 1$ or a double-angle formula to get one function of one angle. Never divide both sides by $\sin x$ or $\cos x$ : you lose the solutions where it is zero. Move everything to one side and factorise instead.

On the calculator (Paper 2)

Graph both sides over the given interval and find every intersection. Check the angle mode: radians unless the question uses degrees.

Key formulas

Identities In the booklet: $\tan θ = \frac{\sin θ}{\cos θ}, \cos^{2} θ + \sin^{2} θ = 1$
Double angle In the booklet: $\sin 2 θ = 2 \sin θ \cos θ, \cos 2 θ = \cos^{2} θ - \sin^{2} θ = 2 \cos^{2} θ - 1 = 1 - 2 \sin^{2} θ$
Exact values at $0, \frac{π}{6}, \frac{π}{4}, \frac{π}{3}, \frac{π}{2}$ Learn it: $\sin : 0, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1; \cos : reverse order; \tan : 0, \frac{1}{\sqrt{3}}, 1, \sqrt{3}, undefined$
Symmetry Learn it: $\sin (π - θ) = \sin θ, \cos (π - θ) = - \cos θ, \cos (- θ) = \cos θ, \sin (- θ) = - \sin θ$
$\sin x = k$ : $x = \arcsin k$ or $π - \arcsin k$ , then add $2 π n$ ; keep solutions in the interval Learn it

The IB formula booklet is provided in every exam paper. In the booklet = printed there, so know where to find it; Learn it = not in the booklet, so learn it by heart. All AA SL formulas →

Explained, with a worked example: Trig identities

Solving Trigonometric Equations worked examples

Worked example 1: Simple exact trigonometric equations · easy

Solve $2 \sin x - \sqrt{3} = 0$ for $0 \leq x \leq 2 π$ . Give exact radian answers.

Solution

5 steps

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 $0 \leq x \leq 2 π$ . Give exact radian answers.

Solution

5 steps

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 $\sin 2 x = \sqrt{3} \cos x$ for $0 \leq x \leq 2 π$ . Give exact radian answers.

Solution

4 steps

Examiner tip: NEVER divide both sides by

\cos x

— you lose the roots where

\cos x = 0

. Bring everything to one side and factorise.

Solving Trigonometric Equations exam-style questions

3 questions from our AA SL bank, written in the style of IB papers, easiest first. One has its full worked solution open here. Open the worked answer, or check your answer in the practice area.

Question 1 · easy · 4 marks · Paper 1

Solve the equation $2 \sin x - \sqrt{3} = 0$ for the interval $0 \leq x \leq 2 π$ . Give your answers exactly in radians.

Check your answer in practice →

Question 2 · medium · 4 marks · Paper 1

Solve the equation $\tan^{2} x - 3 = 0$ for $0 \leq x \leq 2 π$ . Give your answers exactly in radians.

Full worked solution Free, no sign-in

Rearrange: $\tan^{2} x - 3 = 0 ⟹ \tan^{2} x = 3$ .
Take square roots (both signs): $\tan x = \pm \sqrt{3}$ . (M1)A1
The reference angle is $\frac{π}{3}$ . Because it is $\pm$ , there is a solution in all four quadrants.
$x = \frac{π}{3}, \frac{2 π}{3}, \frac{4 π}{3}, \frac{5 π}{3}$ . A2
(Award only one of these two marks if just the positive-root solutions $\frac{π}{3}, \frac{4 π}{3}$ are found).

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

Question 3 · hard · 6 marks · Paper 1

Solve the equation $\sin 2 x = \sqrt{3} \cos x$ for $0 \leq x \leq 2 π$ . (Hint: use the double angle identity for sine, then factorise).

Attempt it and see the mark scheme →

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Where students lose marks on solving trigonometric equations

Examiner's noteTrigonometric equations: one solution only, and exact values not known

Where marks are lost: Candidates found the first solution and stopped (missing the second angle in the interval), gave the value of sin x instead of x, did not know exact values such as tan⁻¹1, sin(5π/6), cos π and cos 60°, mistook the period of tan, and on Paper 2 spent time rearranging by hand instead of solving graphically.

Examiners flagged this in 3 exam series (May 2024, May 2025, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Solving Trigonometric Equations on your calculator

A GDC is allowed on Paper 2, not on Paper 1. Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire, with a checked worked example:

Graphs: intersections, zeros, max and min on your calculator

Solving Trigonometric Equations FAQ

How many solutions should a trig equation have?

It depends on the interval and the angle. sin x = k (with k strictly between −1 and 1) has two solutions in each 2π, and sin 2x = k has twice as many. A quick sketch of the graph over the interval tells you how many to look for.

Are the exact values of sin, cos and tan in the formula booklet?

No. Learn the exact values at 0, π/6, π/4, π/3 and π/2. The booklet does give the identities sin²x + cos²x = 1, tan x = sin x / cos x and the double-angle formulas.

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, 16 medium, 7 hard, 3 very hard. 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.

Next steps

More AA SL Unit 3: Geometry and Right Angled Trigonometry · Radians · The Unit Circle and Exact Values of Trigonometric Ratios · Trigonometric Functions · Trigonometric Identities

Teaching this topic? Free ready-to-teach lesson (slides and teacher notes): Solving trig equations.

Keep revising AA SL: AA SL revision notes · IB Maths AA SL past papers · AA SL common exam mistakes · AA SL syllabus checklist · AA SL geometry and trigonometry knowledge organiser

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Solving Trigonometric Equations (IB Maths AA SL): notes, exam questions and worked examples

Try it first

Solving Trigonometric Equations notes

From one solution to all of them

Multiple angles

Quadratics in sin x or cos x

Using identities first

On the calculator (Paper 2)

Key formulas

Solving Trigonometric Equations worked examples

Worked example 1: Simple exact trigonometric equations · easy

Solution

Worked example 2: Quadratic trigonometric equations · medium

Solution

Worked example 3: Equations requiring double angle identities · hard

Solution

Solving Trigonometric Equations exam-style questions

Question 1 · easy · 4 marks · Paper 1

Question 2 · medium · 4 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 6 marks · Paper 1

Where students lose marks on solving trigonometric equations

Solving Trigonometric Equations on your calculator

Solving Trigonometric Equations FAQ

Next steps

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