Section Small Angle Approximation

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Small Angle Approximation

$s$ is the arc length.

From definition of $\sin$ and $\tan$, we have: $$ \begin{eqnarray} \sin \theta &=& \frac{a}{r} \\ \tan \theta &=& \frac{a}{b} \end{eqnarray} $$ From the definition of radians, we have: $$ \begin{eqnarray} \theta &=& \frac{s}{r} \end{eqnarray} $$

When $\theta$ is small, we can see $a\approx s$ and $b \approx r$.

When $\theta$ is small, you could see from the figure $a\approx s$ and $b \approx r$, therefore: $$ \begin{eqnarray} \sin \theta &=& \frac{a}{r} \approx \frac{s}{r} = \theta \\ \tan \theta &=& \frac{a}{b} \approx \frac{s}{r} = \theta \\ \end{eqnarray} $$ Therefore we have the small angle approximation:

$$ \theta \approx \sin \theta \approx \tan \theta $$
The approximation works well in the limit $\theta \ll 1 rad$. Note that $\theta$ must be in radians, not degrees. We can see from the table below the approximation wells quite well up to around $0.5rad \approx 30^\circ$.

Proof of small angle approximation

Taylor's expansion for small $\theta$ gives: $$ \begin{eqnarray} \sin \theta &\approx& \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} + \cdots &\approx \theta \\ \cos \theta &\approx& 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} + \cdots &\approx 1 \\ \end{eqnarray} $$ Apply to $\tan \theta$: $$ \begin{eqnarray} \tan \theta &=& \frac{\sin\theta}{\cos\theta} \\ &\approx& \frac{\theta}{1} \\ &\approx& \theta \end{eqnarray} $$

$\theta$ ($rad$) $\theta$ ($ ^\circ$) $\sin \theta$ $\tan \theta$
0.01 0.57 0.01000 0.01000
0.05 2.86 0.04998 0.05004
0.1 5.73 0.09983 0.10033
0.5 28.65 0.47943 0.54630
1 57.30 0.84147 1.55741
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