Small-angle approximation

Small angle approximation is a way of estimating angles in radians, where the angle is small (less than 3 or 4 degrees).

When x is small, and in radians, the following formulae are true:

${\displaystyle \sin x\simeq x}$

${\displaystyle \cos x\simeq 1-{\frac {x^{2}}{2}}}$

${\displaystyle \tan x\simeq x}$

In other words, utilizing the small-angle approximation assumes that the hypothenuse of a right triangle is approximately equal in length to the side adjacent to the small angle.

These formulas can be derived from the Taylor series expansion of trigonometric functions and chopping off the higher-order terms.

${\displaystyle \sin \left(x\right)=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+...}$

${\displaystyle \cos \left(x\right)=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+...}$