Pearson Edexcel · A-Level · Mathematics
Course version: 9MA0 · Version from Mar 2026
Trigonometry
Master Edexcel A-Level Trigonometry with this comprehensive guide. We break down complex identities, radians, and reciprocal functions into manageable chunks, providing examiner insights and worked examples to help you secure top marks.
- 5 min read
- 3 worked examples
- 5 practice questions
- 3 key terms
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The start of the Trigonometry lesson, made from checked teaching notes for this specification point.
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In triangle ABC, side a = 7 cm is opposite angle A = 30°, and side b = 10 cm is opposite angle B. Find sin B.
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Sine and cosine are defined for all arguments using the unit circle, extending beyond acute angles with periodicity 360° or 2π. Tangent has periodicity 180° or π and is undefined where cos θ = 0 (e.g. 90°). The sine rule a/sin A = b/sin B = c/sin C applies to any triangle; the cosine rule a² = b² + c² − 2bc cos A applies when two sides and the included angle or all three sides are known. The area of a triangle is ½ab sin C. Radian measure uses π radians = 180°, so 60° = π/3. Arc length is l = rθ and sector area is A = ½r²θ, with θ in radians. A sector of radius 6 cm and angle π/3 has arc length 6 × π/3 = 2π cm and area ½ × 36 × π/3 = 6π cm². A strong answer earns marks for these points. Uses the unit-circle definitions to find sine, cosine and tangent of any angle, including obtuse and reflex angles, with correct signs. Applies the sine rule to find a side or angle, including the ambiguous case where two triangles may be possible. Applies the cosine rule a² = b² + c² − 2bc cos A to find a side or angle, rearranging correctly when finding an angle. Calculates the area of a triangle using ½ab sin C with the included angle. A common slip: Using degrees in l = rθ or A = ½r²θ. Instead, converting the angle to radians first. Exam tip: Decide whether the sine rule or cosine rule is needed by listing what is given and what is required.
Key words
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Area of a triangle
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Radian measure
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Arc length and sector area
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See every definition
- Area of a triangle
- Given by the formula ½ab sin C, using the included angle between the two chosen sides.
- Radian measure
- A measure in which π radians = 180°, so 60° = π/3.
- Arc length and sector area
- For θ in radians, arc length is l = rθ and sector area is A = ½r²θ.
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Study Notes
Overview
Trigonometry at A-Level is the study of the relationships between angles and side lengths of triangles, extended to waves and circles. It is a fundamental part of mathematics, forming a bridge between geometry and algebra. In your Edexcel exam, you will be expected to be fluent in manipulating complex trigonometric expressions, solving equations, and proving identities. This topic frequently links with calculus (differentiation and integration of trig functions) and mechanics, making it a cornerstone of the specification. Typical exam questions involve multi-step problem solving, requiring you to select the appropriate identity or formula and apply it accurately. Mastery of radian measure is essential, as is the ability to work with the reciprocal functions (sec, cosec, cot) and the addition and double angle formulae. This guide will equip you with the core knowledge and exam strategies to tackle this challenging but rewarding topic.
Key Concepts
Concept 1: Radians and the Unit Circle
At A-Level, angles are most commonly measured in radians. A radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius. The key relationship to remember is π radians = 180°. You must be able to convert fluently between degrees and radians. All calculus involving trigonometry requires radians, as do the small angle approximations. The unit circle is a circle with a radius of 1 centered at the origin. It is a powerful tool for understanding trigonometric functions. For any point (x, y) on the unit circle, the coordinates are given by (cos θ, sin θ), where θ is the angle measured anti-clockwise from the positive x-axis. This visual representation helps in understanding the sign of trig functions in different quadrants and deriving key identities.
Example: To convert 120° to radians, you calculate 120 * (π/180) = 2π/3 radians.
Concept 2: Reciprocal and Pythagorean Identities
The three reciprocal functions are secant (sec), cosecant (cosec), and cotangent (cot). They are defined as:
- sec θ = 1/cos θ
- cosec θ = 1/sin θ
- cot θ = 1/tan θ = cos θ/sin θ
These lead to the three Pythagorean Identities, which are essential for simplifying expressions and solving equations. The first is derived directly from the unit circle (x² + y² = 1).
- sin²θ + cos²θ ≡ 1 (Must memorise)
- 1 + tan²θ ≡ sec²θ (Given on formula sheet)
- 1 + cot²θ ≡ cosec²θ (Given on formula sheet)
To derive the second and third identities, you simply divide the first identity by cos²θ and sin²θ respectively. Examiners expect you to be able to do this.
Concept 3: Addition and Double Angle Formulae
These formulae allow you to find the trigonometric ratio of a sum or difference of angles, or of a multiple of an angle. They are crucial for solving more complex equations and are provided on the formula sheet.
Addition Formulae (Given on formula sheet):
- sin(A ± B) ≡ sinAcosB ± cosAsinB
- cos(A ± B) ≡ cosAcosB ∓ sinAsinB
- tan(A ± B) ≡ (tanA ± tanB) / (1 ∓ tanA tanB)
Double Angle Formulae (Derived from addition formulae, given on formula sheet):
- sin(2A) ≡ 2sinAcosA
- cos(2A) ≡ cos²A - sin²A ≡ 2cos²A - 1 ≡ 1 - 2sin²A
- tan(2A) ≡ 2tanA / (1 - tan²A)
Candidates must be adept at selecting the correct form of the cos(2A) identity to simplify an equation. For example, if the equation also contains a cosA term, using the 2cos²A - 1 form is often the best approach.
Mathematical/Scientific Relationships
- Radians to Degrees: Radians = Degrees × (π / 180)
- Degrees to Radians: Degrees = Radians × (180 / π)
- Arc Length: s = rθ (θ in radians) - (Must memorise)
- Area of a Sector: A = ½r²θ (θ in radians) - (Must memorise)
- Small Angle Approximations (for small θ in radians):
- sin θ ≈ θ
- cos θ ≈ 1 - θ²/2
- tan θ ≈ θ
(All given on formula sheet)
Practical Applications
Trigonometry is not just an abstract topic; it has numerous real-world applications. It is used in:
- Navigation: GPS systems use trigonometric principles (triangulation) to pinpoint locations.
- Engineering: Civil engineers use trigonometry to calculate forces in structures like bridges and to design curved roads.
- Physics: It describes the motion of waves, including sound, light, and water waves, as well as oscillations like pendulums.
- Computer Graphics: Trigonometric functions are used to rotate and transform objects in 3D space, essential for video games and animated films.
Visual Resources
2 diagrams and illustrations
Interactive Diagrams
2 interactive diagrams to visualise key concepts
Conceptual Flow Outline
A flowchart outlining the general process for solving trigonometric equations.
Conceptual Flow Outline
A process diagram for applying the R-formula (harmonic form).
Worked Examples
3 worked examples — open one to explore the question and available guidance.
Practice Questions
Try the question, then reveal an answer plan or example response
Show that the equation 2cos²x = 4 - 5sinx can be written as 2sin²x - 5sinx + 2 = 0.
Hint: Use a Pythagorean identity to replace the cos²x term.
Solve, for -180° ≤ θ < 180°, the equation 3tanθ = 2cosθ.
Hint: Express tanθ in terms of sinθ and cosθ, then form a quadratic in sinθ.
Given that sinA = 3/5 and cosB = 5/13, where A is obtuse and B is acute, find the exact value of cos(A - B).
Hint: You will need to find cosA and sinB first. Use Pythagorean identities and consider the quadrants to determine the correct signs.
Find the smallest positive solution to the equation tan(2x - π/4) = 1.
Hint: Let y = 2x - π/4 and solve for y first. Then solve for x.
Using the small angle approximation, find an approximate value for cos(0.1) - sin(0.2).
Hint: Use the small angle approximations for cos(x) and sin(x).
Exam technique
How to turn what you know into marks
How to approach the question
1. Read the question carefully, noting the required range for solutions and whether the answer must be exact. 2. Identify the type of question (e.g., solving, proving, simplifying). 3. Select the most appropriate identity or formula from the formula sheet or memory. 4. Show all steps of your working clearly. 5. Check your final answer(s) against the range and conditions.
How to structure your answer
For proofs, start with one side and manipulate it logically until it matches the other. For solving equations, show the identity used, the simplified equation, the factorisation (if any), the principal values, and all final solutions.
Timing
Approximately 1.2 minutes per mark. A 5-mark question should take around 6 minutes.
Common pitfalls
- Working in degrees when radians are required (e.g., in calculus). Always check your calculator mode.
- Dividing by a trig function when solving an equation, which leads to lost solutions. Always factorise instead.
- Forgetting to adjust the interval when solving for a multiple angle (e.g., 2x or x/2).
- Assuming the identity is true at the start of a proof and moving terms across the equals sign.
What the command words want
- Prove
- Start with one side of the statement and manipulate it logically to show it is equivalent to the other side.
- Solve
- Find the value or set of values that satisfy the given equation or inequality.
- Explain
- Provide reasons, often using 'because'.
- Describe
- Provide a written account of the key features.
- Calculate
- Work out a numerical answer, showing your working.
- State/Give/Write down
- A short, direct answer with minimal or no working needed.
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