Small oscillations

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:








sin
⁡
θ



≈
θ




cos
⁡
θ



≈
1
−



θ

2


2


≈
1




tan
⁡
θ



≈
θ






{\displaystyle {\begin{aligned}\sin \theta &\approx \theta \\\cos \theta &\approx 1-{\frac {\theta ^{2}}{2}}\approx 1\\\tan \theta &\approx \theta \end{aligned}}}
These approximations have a wide range of uses in branches of physics and engineering, including mechanics, electromagnetism, optics, cartography, astronomy, and computer science. 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,




cos
⁡
θ



{\displaystyle \textstyle \cos \theta }
is approximated as either



1


{\displaystyle 1}
or as



1
−



θ

2


2




{\textstyle 1-{\frac {\theta ^{2}}{2}}}
.

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