A-Level — mathsStandard1 min read

Small Angle Approximations

Apply small angle approximations for sin, cos, and tan at A-Level. Simplify expressions when θ is small.

Tutor AI Team2026-02-25T01:56:12.123Z

When $\theta$ is small (in radians), trig functions can be approximated by simpler expressions.

The Approximations

For small $\theta$ (radians):

$\sin\theta \approx \theta$ $\cos\theta \approx 1 - \frac{\theta^2}{2}$ $\tan\theta \approx \theta$

Simplify $\frac{\sin 3\theta}{\theta}$ for small $\theta$ : $\approx \frac{3\theta}{\theta} = 3$ .

Simplify $\frac{1 - \cos 2\theta}{\theta^2}$ : $\approx \frac{1 - (1 - 2\theta^2)}{\theta^2} = \frac{2\theta^2}{\theta^2} = 2$ .

1. Approximate $\frac{\tan 2\theta - \sin\theta}{\theta}$ for small $\theta$ .
1. Show $\frac{\sin\theta\tan\theta}{1 - \cos\theta} \approx 2$ for small $\theta$ .

Want to check your answers and get step-by-step solutions?

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$\sin\theta \approx \theta$ , $\tan\theta \approx \theta$ , $\cos\theta \approx 1 - \frac{\theta^2}{2}$ .
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$\theta$ must be in radians and small.
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Useful for limits and simplifying expressions.

Tags:

#a-level#maths#pure#trigonometry#approximations

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