Small-Angle Approximation Explorer
See exactly how accurate (or inaccurate) the paraxial approximation is at any angle. Compares first-order and third-order corrections to exact trigonometric values.
Input
Exact Values
θ
0.174533radθ
10.0000°sin θ
0.173648cos θ
0.984808tan θ
0.176327First-Order (Paraxial) Approximation
sin θ ≈ θ
0.174533radError: 0.5095%
cos θ ≈ 1
1Error: 1.5427%
tan θ ≈ θ
0.174533radError: 1.0175%
⚠ Paraxial approximation marginal at 10.000° — consider exact trig or higher-order correction
Third-Order Correction
sin θ ≈ θ − θ³/6
0.173647Error: 0.0008%
cos θ ≈ 1 − θ²/2
0.984769Error: 0.0039%
tan θ ≈ θ + θ³/3
0.176305Error: 0.0124%
Abridged Optics — Small-Angle Approximation Explorer v1.0The paraxial (first-order) approximation is conventionally considered valid when sin θ error is below 0.5%, typically at angles ≲ 10°. Third-order corrections extend accuracy to ~30° for most applications.