Small Angle Approximations (OCR A Level Maths: Pure)

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

  

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

cos θ ≈ 1 - θ²

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

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

tan θ ≈ θ

 

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, …

 

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