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A very useful approximation for many physical applications, especially for simple harmonic motion and pendulums in particular. It states that

This follows because the sine function may be expressed as the infinite Taylor series:

\begin{large} \[ sin( \theta ) = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + ... \]\end{large}

If the angle is small enough, then we can ignore all but the first term, giving

\begin{large} \[ sin( \theta ) \approx \theta \]\end{large}

This gives us our criterion for "small enough", because it is clear that this will be the case when x is much greater than x³/3!, or in other words 6 >> x².

Similarly, the Taylor series for cos(θ) is

\begin{large} \[ cos( \theta ) = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + ... \]\end{large}

so that in the small angle approximation we have

\begin{large} \[ cos( \theta ) \approx 1 \]\end{large}

Finally, the expansion for the tangent is

\begin{large} \[ tan( \theta ) = \theta + \frac{x^{3}}{3} + \frac{2x^{5}}{15} + \frac{17x^{7}}{217} + ... \]\end{large}

So that for small angles

\begin{large} \[ tan( \theta ) \approx \theta \]\end{large}