Small-angle Approximation

Section 16.8 Small-angle Approximation

Most functions can be expanded into an infinite sum of polynomials for the purpose of approximating them (his is called a series expansion). The full series expansions for sine and cosine when the angle is close to \(\theta = 0\) are:

\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.

🔗

Assumption 16.8.1. Small-angle Approximation.

🔗

For sufficiently small angles, sine and cosine may be approximated as:

\begin{equation*} \sin(\theta) \approx \theta \end{equation*}

\begin{equation*} \cos(\theta) \approx 1 - \frac{\theta^2}{2!} \end{equation*}

Similarly, tangent may be approximated as:

\begin{equation*} \tan(\theta) \approx \theta \end{equation*}

🔗

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 16.8.2. Graphical representation of the small-angle approximation for the \(\sin\theta\) function.
🔗

Subsubsection Activities

Activity 16.8.1. What is the error introduced?

🔗

Calculate the value of both \(\sin\theta\) and \(\cos\theta \) at \(\theta_1 = 10^o\text{,}\) \(\theta_2 = 30^o\text{,}\) and \(\theta_3 = 60^o \text{.}\) Then, find the approximate value of each using the small-angle approximation. Last, use your answers to find the error (as a percentage of the true value) for each number.

🔗