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
What each mark is really for — in plain English — and the wording trap that loses it.
- S1
The sine approximation for a very small angle θ is given by: sin(θ) ≈ θ.
- S2
Substitute θ = 0.1 into the approximation: sin(0.1) ≈ 0.1.
Model answer
Worked through, with each step tagged to the mark it earns.
- S1
The sine approximation for a very small angle θ is given by: sin(θ) ≈ θ.
- S2
Substitute θ = 0.1 into the approximation: sin(0.1) ≈ 0.1.
Final answer: sin(0.1) ≈ 0.1
Common mistakes
- Using the sine approximation for large angles. — Always check that the angle is sufficiently small before applying the small angle approximations.
- Forgetting to square the angle in the cosine approximation. — 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. — Remember that tan(θ) ≈ θ for very small angles, just like sin(θ).
- Not simplifying higher-order terms in Taylor series expansions. — 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. — Always ensure that angles are in radians when using small angle approximations.
Where the marks go
- Full worked solution (all marking points)2 marks