Versine Formula
Calculate the versine of an angle, a classic trigonometric function used in navigation, surveying, and historical computation tables.
The Formula
The versine (short for "versed sine") is a trigonometric function that has been used for over a thousand years. It represents the "flipped" or "turned" version of the cosine function, measuring how far the cosine deviates from its maximum value of 1.
While modern mathematics rarely uses the versine directly, it was historically one of the most important trigonometric functions. Before electronic calculators existed, navigators and astronomers relied on printed tables of versines to perform calculations. The versine was preferred because it is always non-negative for real angles (ranging from 0 to 2), which eliminated sign errors that plagued manual computation with sines and cosines.
The closely related haversine (half versine) is defined as hav(θ) = versin(θ)/2 = (1 - cos(θ))/2. The haversine formula for great-circle distances is still widely used in modern navigation software and GPS applications. It calculates the shortest distance between two points on a sphere, which is essential for aviation and maritime routing.
The versine also connects to fundamental physics and engineering. In simple harmonic motion, the displacement of a spring or pendulum from its extreme position follows a versine curve. In structural engineering, the rise of a circular arch can be expressed as a versine of the subtended angle.
There are several related functions in the versine family. The coversine is defined as 1 - sin(θ), and the vercosine is 1 + cos(θ). The exsecant, another historical function, equals sec(θ) - 1 and is related to the versine through the identity exsec(θ) = versin(θ) × sec(θ).
An important alternate form uses the half-angle identity: versin(θ) = 2sin²(θ/2). This version is numerically more stable for small angles and forms the basis of the haversine formula used in navigation.
Variables
| Symbol | Meaning |
|---|---|
| versin(θ) | The versine of angle θ |
| θ | The angle (in radians or degrees) |
| cos(θ) | The cosine of angle θ |
| hav(θ) | The haversine: versin(θ) / 2 |
Example 1
Calculate the versine of 60°.
versin(60°) = 1 - cos(60°)
cos(60°) = 0.5
versin(60°) = 1 - 0.5
versin(60°) = 0.5
Example 2
Find the versine and haversine of 120°.
versin(120°) = 1 - cos(120°)
cos(120°) = -0.5
versin(120°) = 1 - (-0.5) = 1.5
hav(120°) = versin(120°) / 2 = 1.5 / 2
versin(120°) = 1.5 and hav(120°) = 0.75
When to Use It
The versine appears in navigation, physics, and historical mathematics contexts.
- Great-circle distance calculations via the haversine formula
- Surveying and geodesy for measuring arc heights and sag
- Structural engineering for calculating arch rise from chord length
- Railway engineering for computing curve offsets and superelevation
- Numerical analysis where avoiding subtraction of nearly equal numbers improves accuracy
- Studying historical mathematical tables and navigation methods