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:

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