A-level · Trigonometry

Small angle approximationsExam-style questions

The radian approximations sin θ ≈ θ, tan θ ≈ θ and cos θ ≈ 1 − ½θ², why they hold, and using them to approximate expressions and evaluate limiting ratios for small θ.

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18 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4

1
Given that $\theta$ is small and measured in radians, write down approximations for
(a) $\sin 4\theta$ ,
(b) $\cos 3\theta$ ,
(c) $\tan 2\theta$ .
Level 2 · 3 marksTry it
2
$\theta$ is small and measured in radians.

Using the small angle approximations for $\cos 2\theta$ and $\tan 5\theta$ , find an approximate value for

$\frac{\cos 2\theta - 1}{\theta\tan 5\theta}$

(3 marks)
Level 2 · 3 marksTry it
3
When $\theta$ is small, show that $2\sin\theta + 3\cos\theta \approx 3 + 2\theta - \tfrac32\theta^2$ .
Level 3 · 2 marksTry it
4
When $\theta$ is small, show that $\dfrac{\sin 3\theta}{\tan 2\theta} \approx \dfrac{3}{2}$ .
Level 3 · 2 marksTry it
5
When $\theta$ is small, show that $\dfrac{\cos\theta - 1}{\theta^2} \approx -\dfrac{1}{2}$ .
Level 3 · 2 marksTry it

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