Small angle approximation for \cos\theta?

\cos\theta\approx1-\frac{\theta^2}{2}

Approximate \frac{\sin\theta\tan2\theta}{1-\cos\theta} for small \theta.

\frac{\theta\times2\theta}{\theta^2/2}=4

How do you find a percentage error?

\frac{approximate-exact}{exact}\times100

Why can a large root of an approximated equation be rejected?

The approximations only hold when \theta is small.

Why do the approximations work?

For a small angle the arc, the opposite side and the tangent length are almost equal.

Convert 3^\circ to radians before approximating.

3\times\frac{\pi}{180}=\frac{\pi}{60}\approx0.0524

Small angle approximationsAQA A-Level Maths: Flashcards

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Small angle approximation for $\sin\theta$?

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Small angle approximation for $\sin\theta$ ?: $\sin\theta\approx\theta$ ( $\theta$ in radians)
Small angle approximation for $\tan\theta$ ?: $\tan\theta\approx\theta$
Small angle approximation for $\cos\theta$ ?: $\cos\theta\approx1-\frac{\theta^2}{2}$
In what units must $\theta$ be?: Radians.
Approximate $\sin3\theta$ .: $3\theta$
Approximate $\cos4\theta$ .: $1-8\theta^2$
Approximate $1-\cos2\theta$ .: $2\theta^2$
Approximate $\tan\frac{\theta}{2}$ .: $\frac{\theta}{2}$
Approximate $\frac{\sin\theta\tan2\theta}{1-\cos\theta}$ for small $\theta$ .: $\frac{\theta\times2\theta}{\theta^2/2}=4$
How do you find a percentage error?: $\frac{\text{approximate}-\text{exact}}{\text{exact}}\times100$
Why can a large root of an approximated equation be rejected?: The approximations only hold when $\theta$ is small.
Why do the approximations work?: For a small angle the arc, the opposite side and the tangent length are almost equal.
Convert $3^\circ$ to radians before approximating.: $3\times\frac{\pi}{180}=\frac{\pi}{60}\approx0.0524$

Exam questions on Small angle approximations

$\theta$ is a small angle measured in radians.
Use small angle approximations to find an approximate value of $\frac{1-\cos2\theta}{\theta\sin\theta}$ .2 marks
An isosceles triangle $ABC$ has $AB=AC=10$ cm and $B\hat{A}C=\theta$ radians, where $\theta$ is small.
Use a small angle approximation to show that the area of triangle $ABC$ is approximately $50\theta$ cm $^2$ .2 marks
For small $\theta$ (in radians), let $f(\theta)=\frac{1-\cos4\theta}{\theta\tan2\theta}$ .
Show that $f(\theta)\approx4$ .3 marks

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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).