The small-angle formula, also known as the paraxial approximation, is a simplification of trigonometric functions that holds true for small angles. It states that for a small angle θ (measured in radians):

* sin θ ≈ θ

* tan θ ≈ θ

* cos θ ≈ 1

Applications:

The small-angle formula is widely used in various fields, including:

* Optics: To approximate the path of light rays through lenses and mirrors.

* Mechanics: To analyze the motion of pendulums and other oscillating systems.

* Astronomy: To calculate distances and sizes of celestial objects.

* Civil Engineering: To design structures that are stable under small angles of deflection.

Derivation:

The small-angle approximation is derived from the Taylor series expansion of the sine, tangent, and cosine functions. For small angles, the higher-order terms in the Taylor series become negligible, leading to the following approximations:

* sin θ = θ - (θ^3/3!) + (θ^5/5!) - ... ≈ θ

* tan θ = θ + (θ^3/3) + (2θ^5/15) + ... ≈ θ

* cos θ = 1 - (θ^2/2!) + (θ^4/4!) - ... ≈ 1

Note:

The small-angle formula is valid only for angles that are small enough, typically less than 10 degrees (or 0.17 radians). As the angle increases, the approximations become less accurate.