Skip to main content

Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 15.8 Small-angle Approximation

The small-angle approximation is used to approximate the values of the main trigonometric functions when the angles involved are restricted to a certain domain. Any function can be expanded into an infinite sum of polynomials. This is called a series expansion. Consider a series expansion (Taylor Series) of the sine and cosine functions:
\begin{equation*} \sin(\theta) = \sum^\infty_{n=0} \frac{(-1)^n}{2n+1}\theta^{2n+1} = \theta - \frac{\theta^3}{3!}+\frac{\theta^5}{5!} -\frac{\theta^7}{7!}-.... \end{equation*}
\begin{equation*} \cos(\theta) = \sum^\infty_{n=0} \frac{(-1)^n}{2n}\theta^{2n} = 1 - \frac{\theta^2}{2!}+\frac{\theta^4}{4!} -\frac{\theta^6}{6!}-.... \end{equation*}
Both expressions above are valid only when \(\theta\) is measured in radians. The small-angle approximation considers only the first nontrivial term in the sum.
The figure below shows a graphical representation of the small-angle approximation for the \(\sin(\theta)\) function. You can see the linear function \(\theta\) and the trigonometric function \(\sin(\theta)\) closely match each other when the angle is small.
Figure 15.8.2. Graphical representation of the small-angle approximation for the \(\sin\theta\) function.

Exercises Activities

1. What is the error introduced?

Determine the error introduced by the small-angle approximation for \(\sin\theta\) and \(\cos\theta \) when \(\theta = 0.2, 0.4, 0.8 \) radian.

References References

PrevTopNext