Small Angle Approximations (AQA A-Level Mathematics): Flashcards

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Small Angle Approximations
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Required units for θ\theta in small angle approximations

Radians

Small angle approx. for sin⁡θ\sin \theta

sin⁡θ≈θ\sin \theta \approx \theta

Small angle approx. for cos⁡θ\cos \theta

cos⁡θ≈1−θ22\cos \theta \approx 1 - \frac{\theta^2}{2}

Small angle approx. for tan⁡θ\tan \theta

tan⁡θ≈θ\tan \theta \approx \theta

How are small angle approximations derived?

Taylor series expansions

Higher-order terms (e.g., θ3\theta^3) for small θ\theta

Become negligible

Example physics application of small angle approx.

Pendulum motion

Value of lim⁡θ→0sin⁡θθ\lim_{\theta \to 0} \frac{\sin \theta}{\theta}

11

Why tan⁡θ≈θ\tan \theta \approx \theta for small θ\theta

sin⁡θcos⁡θ≈θ1=θ\frac{\sin \theta}{\cos \theta} \approx \frac{\theta}{1} = \theta

Why cos⁡θ\cos \theta stays near 11 for small θ\theta

Cosine graph is nearly flat near θ=0\theta = 0

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