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Small angle approximationsAQA A-Level Maths: Mind map

The three results
Why they work
Replacing the angle

Small angles

radians only

sin⁡θ≈θ\sin\theta\approx\thetatan⁡θ≈θ\tan\theta\approx\thetacos⁡θ≈1−θ22\cos\theta\approx1-\frac{\theta^2}{2}
Simplifying
Equations
Exam tips

Exam questions on Small angle approximations

  1. θ\theta is a small angle measured in radians.
    Use small angle approximations to find an approximate value of 1−cos⁡2θθsin⁡θ\frac{1-\cos2\theta}{\theta\sin\theta}.2 marks
  2. An isosceles triangle ABCABC has AB=AC=10AB=AC=10 cm and BA^C=θB\hat{A}C=\theta radians, where θ\theta is small.
    Use a small angle approximation to show that the area of triangle ABCABC is approximately 50θ50\theta cm2^2.2 marks
  3. For small θ\theta (in radians), let f(θ)=1−cos⁡4θθtan⁡2θf(\theta)=\frac{1-\cos4\theta}{\theta\tan2\theta}.
    Show that f(θ)≈4f(\theta)\approx4.3 marks
See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).