AQA A Level Maths: Pure复习笔记5.4.3 Small Angle Approximations

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

5.4.3-Small-Angle-Approximations-Notes-Diagram-1

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

5.4.3-Small-Angle-Approximations-Notes-Diagram-3

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

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

tan θ ≈ θ

5.4.3-Small-Angle-Approximations-Notes-Diagram-5

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, …

5.4.3-Small-Angle-Approximations-Notes-Diagram-7

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

5.4.3-Small-Angle-Approximations-Notes-Diagram-8

Exam 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

5.4.3-Small-Angle-Approximations-Example-Diagram-1-1 5.4.3-Small-Angle-Approximations-Example-Diagram-2

 

 

 

 

 

 

 

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