The small-angle formula, also called the small-angle approximation, replaces trigonometric functions by the leading terms of their Taylor series
near zero. For an angle 
measured in radians (Abramowitz
and Stegun 1972),
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
Consequently, when 
,
one commonly uses 
,
,
and either 
or the coarser approximation 
.
The radian requirement is essential. The approximation error is cubic for sine and tangent and quadratic when cosine is replaced by 1.
For a transverse line segment of length 
perpendicular to the line
of sight and with midpoint at distance 
, the angular diameter
is
 |
(4)
|
When 
is small compared with 
, the astronomical small-angle formula is
 |
(5)
|
where 
is in radians.
