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?
- 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
- 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).