Small-angle approximation

The small-angle approximations can be used to approximate the values of the main trigonometric functions, provided that the angle in question is small and is measured in radians:

${\displaystyle {\begin{aligned}\sin \theta &\approx \tan \theta \approx \theta ,\\[5mu]\cos \theta &\approx 1-{\tfrac {1}{2}}\theta ^{2}\approx 1.\end{aligned}}}$

Angles measured in degrees must first be converted to radians by multiplying them by ⁠${\displaystyle \pi /180}$⁠.

These approximations have a wide range of uses in branches of physics and engineering, including mechanics, electromagnetism, optics, cartography, astronomy, and computer science.^[1][2] One reason for this is that they can greatly simplify differential equations that do not need to be answered with absolute precision.

There are a number of ways to demonstrate the validity of the small-angle approximations. The most direct method is to truncate the Maclaurin series for each of the trigonometric functions. Depending on the order of the approximation, ${\displaystyle \textstyle \cos \theta }$ is approximated as either ${\displaystyle 1}$ or as ${\textstyle 1-{\frac {1}{2}}\theta ^{2}}$.^[3]