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:

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

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

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

The following key characteristic can be derived from the above for a right triangle with sides A, B, and H, where H is the hypothenuse and A is the side opposite angle x:

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

${\displaystyle {\frac {A}{H}}\simeq {\frac {A}{B}}}$

${\displaystyle H\simeq B}$

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