Small-angle approximation

The small-angle approximation replaces the tangent of an angle, measured in radians, with the angle itself: tan θ ≈ θ. It makes trigonometric calculations simpler when the angle is close to zero.

Taylor series: A function representation as an infinite sum of terms involving powers of its input and derivatives at a point. Expanding tangent near zero reveals its leading term and the corrections omitted by the approximation.
Paraxial approximation: An optical approximation that treats rays as close to an axis and at small angles to it. It uses small-angle trigonometry to simplify ray paths and imaging equations.
Right triangle: A triangle with one interior angle equal to 90 degrees. The opposite-to-adjacent ratio in the gloss is defined using a right triangle.
Exact trigonometric functions: Trigonometric functions evaluated without replacing them by simplified approximate expressions. Exact tangent values remain necessary when angles are too large for the approximation.
Tangent function: The trigonometric function equal to sine divided by cosine, or opposite side divided by adjacent side. The approximation replaces this function’s value with the angle in radians.
Simple harmonic motion: Periodic motion in which acceleration is proportional to displacement and directed toward equilibrium. Replacing sin θ with θ turns the pendulum’s angular equation into a linear oscillator model.
Opposite side: In a right triangle, the side across from a specified acute angle. Its length forms the numerator of the tangent ratio being approximated.
Degree (angle): An angle unit equal to one three-hundred-sixtieth of a full turn. Using degree values directly makes tan θ ≈ θ invalid without converting units.
Radian: An angle unit defined by the ratio of an arc’s length to a circle’s radius. The approximation works in this unit because the angle then equals the leading term of tangent’s expansion.
Simple pendulum: An idealized point mass suspended from a fixed pivot by a massless, inextensible string. Its small-oscillation period follows from approximating the restoring torque’s sine term.

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