5.5 Small-angle approximations — Practise

Work through these on your own. Check that each angle is in radians and small before you replace a function with its approximation. Where a question asks for an estimate, compare it with a calculator value afterwards.

Both the questions and the answers below come from a set that is checked by computer every time the book is built.

Use a small-angle approximation to estimate $\sin 0.1$ .
Use a small-angle approximation to estimate $\cos 0.2$ .
Use a small-angle approximation to estimate $\tan 0.05$ .
A pendulum of length $0.6$ m swings through a small angle $\theta = 0.12$ radians. Estimate the horizontal displacement of the bob, in metres.
The error in $\sin\theta \approx \theta$ is about $\dfrac{\theta^3}{6}$ . Estimate the error at $\theta = 0.2$ , to three significant figures.
Light meets a surface at a small angle $i = 0.04$ radians, and $\sin i = 1.5\sin r$ . Estimate the angle $r$ in radians, to three significant figures.
Use a small-angle approximation to estimate $\cos 0.1$ .
Use a small-angle approximation to estimate $\sin 0.02$ .
Use small-angle approximations to estimate $\sin 0.1 + \tan 0.1$ .
Work out the error in the approximation $\sin\theta \approx \theta$ at $\theta = 0.5$ , to three significant figures.

Answers and concise solutions

1. The estimate

Answer: For a small angle in radians, $\sin\theta \approx \theta$ . So $\sin 0.1 \approx 0.1$ . The exact value is $0.09983$ , so the estimate is within $0.0002$ .

2. The estimate

Answer: $\cos\theta \approx 1 - \dfrac{\theta^2}{2}$ . So $\cos 0.2 \approx 1 - \dfrac{0.04}{2} = 0.98$ . The exact value is $0.98007$ , so the estimate is within $0.0001$ .

3. The estimate

Answer: For a small angle in radians, $\tan\theta \approx \theta$ . So $\tan 0.05 \approx 0.05$ . The exact value is $0.05004$ , so the estimate is within $0.0001$ .

4. The displacement

Answer: The displacement is $x = L\sin\theta \approx L\theta = 0.6 \times 0.12 = 0.072$ m, which is about $7.2$ cm.

5. The error

Answer: The error is about $\dfrac{0.2^3}{6} = \dfrac{0.008}{6} \approx 0.00133$ . The exact difference is $0.2 - \sin 0.2 = 0.00133$ , so the estimate matches.

6. The angle $r$

Answer: Replace both sines with their angles: $i \approx 1.5r$ . Then $r \approx \dfrac{0.04}{1.5} \approx 0.0267$ radians.

7. The estimate

Answer: $\cos 0.1 \approx 1 - \dfrac{0.01}{2} = 0.995$ . The exact value is $0.99500$ , so the estimate is accurate to five decimal places.

8. The estimate

Answer: $\sin 0.02 \approx 0.02$ . The exact value is $0.019999$ , so the error is about $1.3 \times 10^{-6}$ .

9. The estimate

Answer: Both $\sin\theta$ and $\tan\theta$ are approximately $\theta$ , so the sum is approximately $0.1 + 0.1 = 0.2$ . The exact value is $0.19999$ .

10. The error at $0.5$

Answer: The error is $0.5 - \sin 0.5 = 0.5 - 0.47943 \approx 0.0206$ . At $0.2$ radians the error was only $0.00133$ , so the error grows quickly with the angle.