Small Angle Approximations (AQA A Level Maths: Pure)

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Small Angle Approximations

Small angle approximations

  • When an angle measured in radians is very small, you can approximate the value using small angle approximations
  • These only apply when angles are measured in radians
  • They can be applied to positive and negative small angles

What's the small-angle approximation of sin θ?

sin θ ≈ θ

 Small Angle Approximations Notes Diagram 1, A Level & AS Maths: Pure revision notes 

What's the small-angle approximation of cos θ?

cos θ ≈ 1 - 1 halfθ2

Small Angle Approximations Notes Diagram 3, A Level & AS Maths: Pure revision notes

  • y = cos θ (near zero) is similar to a “negative quadratic” (parabola)

What's the small-angle approximation of tan θ?

tan θ ≈ θ

 Small Angle Approximations Notes Diagram 5, A Level & AS Maths: Pure revision notes

How do I use small angle approximations in solving problems?

  • Replace sin θ, cos θ or tan θ with the appropriate approximation
  • Given angles are often 2θ, 3θ, …
    • Replace “θ” in the approximation by 2θ, 3θ, …

Small Angle Approximations Notes Diagram 7, A Level & AS Maths: Pure revision notes

 

  • Binomial expansion (see GBE) may be involved in more awkward expressions

 Small Angle Approximations Notes Diagram 8, A Level & AS Maths: Pure revision notes

Examiner Tip

  • Small angle approximations are given in the formula booklet.
  • They can be used in proofs – particularly differentiation from first principles (see First Principles Differentiation - Trigonometry).

Worked example

Small Angle Approximations Example Diagram 1, A Level & AS Maths: Pure revision notes Small Angle Approximations Example Diagram 2, A Level & AS Maths: Pure revision notes

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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.