5.5 Small-angle approximations — Learn

Prerequisites: radian measure; the graphs of sine, cosine and tangent; the idea of an approximation and its error.

By the end of this section you can: state the small-angle approximations for sine, cosine and tangent; use them to estimate a value; say when the approximation is too rough to trust.

1. Approximating near zero

When an angle $\theta$ is measured in radians and is close to $0$, the three trigonometric functions are almost the same as three simple expressions:

You can see why from the graphs. Near the origin the curve $y = \sin\theta$ is almost a straight line through the origin with gradient $1$, so $\sin\theta$ is almost $\theta$. The tangent curve has the same gradient there. The cosine curve is flat at its peak, so it is almost $1$, with a small quadratic correction.

The approximations are only valid in radians. In degrees the number for $\theta$ is about $57$ times larger, and the approximations fail.

Example 1. Use a small-angle approximation to estimate $\sin 0.1$ and $\cos 0.1$.

A calculator gives $\sin 0.1 \approx 0.09983$ and $\cos 0.1 \approx 0.99500$. Both estimates are close.

Example 2. A pendulum of length $L$ swings through a small angle $\theta$. The horizontal displacement of the bob is approximately

For $L = 0.8$ m and $\theta = 0.05$ radians,

2. How small is small

The size of the error grows quickly with the angle.

At $\theta = 0.2$ radians, about $11.5^\circ$, the error in $\sin\theta$ is already about $0.7\%$. At half a radian, about $28.6^\circ$, it is about $4\%$. Engineers often keep $\theta$ below about $0.2$ radians when the approximation must be accurate to three decimal places.

The cosine approximation is a better fit over the same range, because the first correction is quadratic.

Watch out. Three mistakes are common with small angles.

Using degrees. The approximation $\sin\theta \approx \theta$ is only true in radians. If $\theta = 5^\circ$ and you write $\sin 5^\circ \approx 5$, the answer is far too large. Convert first: $5^\circ \approx 0.087$ radians.

Approximating cosine by $1$. Near zero this is close, but the useful approximation keeps the quadratic term: $\cos\theta \approx 1 - \dfrac{\theta^2}{2}$. Dropping the term loses the whole reason the cosine is not exactly flat.

Forgetting the error. The approximation is not an identity. State the range of angles where it is used, and quote the approximate answer as an approximation. Re-check: substitute one angle from the table and confirm that the error is small, then try an angle near $0.5$ and watch it grow.

You should now be able to

• state $\sin\theta \approx \theta$, $\tan\theta \approx \theta$ and $\cos\theta \approx 1 - \dfrac{\theta^2}{2}$ for small angles in radians;
• use them to estimate a value without a calculator;
• explain why the approximations need radians;
• say roughly how small an angle must be for a given accuracy;
• use a small-angle approximation in a simple model.