Small angle approximations

In a nutshell

Small angle approximations involve approximating the values of $\sin \theta$ , $\cos \theta$ or $\tan \theta$ to a linear or quadratic expression when the value of $\theta$ is small. Small angle approximations are useful as they remove the need for any trigonometric functions and can therefore simplify complex calculations. The approximations only apply if the angle $\theta$ is in radians.

Small angle approximations

When $\theta$ is small and in radians, the following approximations can be used. The graphs illustrate how close the trigonometric function is to the approximation.


$\boxed{\sin \theta \approx \theta}$	$\boxed{\cos \theta \approx 1-\dfrac{\theta^2}{2}}$	$\boxed{\tan \theta \approx \theta}$

Example 1

Find an approximation for $f(x)$ when the angle $x$ is small, but non-zero.

$f(x) = \dfrac {\sin(x) \tan (x)}{1 - \cos (3x)}$ 

Substitute in the approximations and simplify:

$\begin{aligned}f(x) &= \dfrac {\sin(x) \tan (x)}{1 - \cos (3x)} \\\\&\approx \dfrac{x \times x}{1 - (1-\frac{(3x)^2}{2})} \\\\&\approx \dfrac{x ^2}{1 - 1+\frac{9x^2}{2}} \\\\&\approx \dfrac{x ^2}{\frac{9x^2}{2}} \\\\&\approx \underline{ \dfrac{2}{9}} \\\\\end{aligned}$