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.
The paraxial approximation — sin θ ≈ tan θ ≈ θ (radians), cos θ ≈ 1 — underlies most of geometric optics, including the thin lens equation, Gaussian beam propagation, and the Abbe sine condition. Its accuracy degrades rapidly with angle: sin θ error exceeds 0.5% above roughly 10° and reaches 5% near 17°. This tool computes exact values of sin θ, cos θ, and tan θ alongside the first-order paraxial approximations and the third-order corrections (sin θ ≈ θ − θ³/6, cos θ ≈ 1 − θ²/2, tan θ ≈ θ + θ³/3), reporting the percentage error of each approximation at the entered angle. Input accepts degrees, radians, or milliradians. A geometric diagram illustrates how sin, tan, and arc length diverge as angle increases, making the approximation's breakdown visually apparent.