A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Small Angle Approximations in Trigonometry — mark scheme explained
The short answer
In AQA A-Level Mathematics, understanding and using the standard small angle approximations for sine, cosine, and tangent is a crucial part of trigonometry. These approximations are particularly useful when dealing with angles that are very close to zero radians.
The question
Use the small angle approximation to estimate sin(0.1).
[Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
How the examiner actually awards the marks on this topic.
Gradora's own decode of the marking approach — not the exam board's published mark scheme.
How marks are awarded
For estimation questions, ensure you clearly state the approximation being used and show all steps of your calculation. For derivation questions, start from the basic trigonometric identity and apply the appropriate small angle approximation step-by-step.
What the command words demand
- Estimate
- Use the small angle approximation to find an approximate value for a trigonometric function.
- Derive
- Show how the small angle approximation can be used to derive a specific equation or result.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Allocate about 2-3 minutes for each question involving small angle approximations to ensure accuracy and completeness.
- The sine approximation for a very small angle θ is given by: sin(θ) ≈ θ.1 mark
- Substitute θ = 0.1 into the approximation: sin(0.1) ≈ 0.1.1 mark
Final answer: sin(0.1) ≈ 0.1
Work through every step correctly and you earn all 2 marks.
Another worked example
Use the small angle approximation to estimate cos(0.05).
- The cosine approximation for a very small angle θ is given by: cos(θ) ≈ 1 - 1 / 2 θ 2 .1 mark
- Substitute θ = 0.05 into the approximation: cos(0.05) ≈ 1 - 1 / 2 (0.05) 2 .1 mark
- Calculate (0.05) 2 : (0.05) 2 = 0.0025.0 marks
- Substitute back into the approximation: cos(0.05) ≈ 1 - 1 / 2 (0.0025) = 1 - 0.00125 = 0.99875.1 mark
Final answer: cos(0.05) ≈ 0.99875
Work through every step correctly and you earn all 3 marks.
Common mistakes
Using the sine approximation for large angles.
Why it happens: The small angle approximations are only valid for very small angles close to zero radians. Using them for larger angles can lead to significant errors.
Fix: Always check that the angle is sufficiently small before applying the small angle approximations.
Forgetting to square the angle in the cosine approximation.
Why it happens: The cosine approximation involves squaring the angle, which can be easily overlooked.
Fix: Double-check that you have squared the angle when using the cosine approximation: cos(θ) ≈ 1 - 1 / 2 θ 2 .
Using the wrong approximation for tangent.
Why it happens: Students sometimes confuse the tangent approximation with the sine or cosine approximations.
Fix: Remember that tan(θ) ≈ θ for very small angles, just like sin(θ).
Not simplifying higher-order terms in Taylor series expansions.
Why it happens: Students may include higher-order terms when they are not necessary for the approximation.
Fix: For small angle approximations, only consider the first few terms of the Taylor series that are significant for very small angles.
Using degrees instead of radians.
Why it happens: The small angle approximations are derived and valid only when the angle is in radians. Using degrees can lead to incorrect results.
Fix: Always ensure that angles are in radians when using small angle approximations.
Where the marks go
The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.
| Question type | What you’re asked to do | Marks |
|---|---|---|
| Small Angle Approximation | Estimate sin(0.1) by applying the small angle approximation for sine. | 2 |
| Small Angle Approximation | Estimate cos(0.05) by applying the small angle cosine approximation and evaluating it. | 3 |
| Small Angle Approximation | Estimate tan(0.1) by applying the small angle approximation, showing the approximation used and each step. | 2 |
| Derive Equation Of Motion | Derive the pendulum's equation of motion using the small angle approximation from basic principles. | 4 |
| Total across these question types | 11 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.