Versine

The versine or versed sine, versin(θ), is a trigonometric function equal to 1 − cos(θ), or 2sin²(½θ). The function appeared in some of the earliest trigonometric tables. There are several related functions, most notably the haversine, half the versine, known in the haversine formula of navigation.

It is also written as vers(θ) or ver(θ). In Latin, it is known as the sinus versus (flipped sine), versinus, versus or the sagitta (arrow).

Related functions

There are several other related functions:

The versed cosine, or vercosine, written $\operatorname{vercosin}(\theta)$
The coversed sine, coversine, cosinus versus, or coversinus, written $\operatorname{coversin}(\theta)$ and sometimes abbreviated to $\operatorname{cvs}(\theta)$
The coversed cosine, or covercosine, written $\operatorname{covercosin}(\theta)$
The haversed sine, haversine, or semiversus, written $\operatorname{haversin}(\theta)$ or $\operatorname{sem}(\theta)$ , most famous from the haversine formula used historically in navigation
The haversed cosine, or havercosine, written $\operatorname{havercosin}(\theta)$
The hacoversed sine, also called hacoversine or cohaversine and written $\operatorname{hacoversin}(\theta)$
The hacoversed cosine, also called hacovercosine or cohavercosine and written $\operatorname{hacovercosin}(\theta)$
The exsecant, written $\operatorname{exsec}(\theta)$
The excosecant, written $\operatorname{excosec}(\theta)$

The trigonometric functions can be constructed geometrically in terms of a unit circle centered at O. This figure also illustrates the reason why the versine was sometimes called the sagitta, Latin for arrow.^[1] If the arc ADB is viewed as a "bow" and the chord AB as its "string", then the versine CD is clearly the "arrow shaft".

Definitions

$\textrm{versin} (\theta) := 2\sin^2\!\left(\frac{\theta}{2}\right) = 1 - \cos (\theta) \,$
$\textrm{vercosin} (\theta) := 2\cos^2\!\left(\frac{\theta}{2}\right) = 1 + \cos (\theta) \,$
$\textrm{coversin}(\theta) := \textrm{versin}\!\left(\frac{\pi}{2} - \theta\right) = 1 - \sin(\theta) \,$
$\textrm{covercosin}(\theta) := \textrm{vercosin}\!\left(\frac{\pi}{2} - \theta\right) = 1 + \sin(\theta) \,$
$\textrm{haversin}(\theta) := \frac {\textrm{versin}(\theta)} {2} = \frac{1 - \cos (\theta)}{2} \,$
$\textrm{havercosin}(\theta) := \frac {\textrm{vercosin}(\theta)} {2} = \frac{1 + \cos (\theta)}{2} \,$
$\textrm{hacoversin}(\theta) := \frac {\textrm{coversin}(\theta)} {2} = \frac{1 - \sin (\theta)}{2} \,$
$\textrm{hacovercosin}(\theta) := \frac {\textrm{covercosin}(\theta)} {2} = \frac{1 + \sin (\theta)}{2} \,$

Derivatives and Integrals

$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{versin}(x) = \sin{x}$	$\int\mathrm{versin}(x) \,\mathrm{d}x = x - \sin{x} + C$
$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{vercosin}(x) = -\sin{x}$	$\int\mathrm{vercosin}(x) \,\mathrm{d}x = x + \sin{x} + C$
$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{coversin}(x) = -\cos{x}$	$\int\mathrm{coversin}(x) \,\mathrm{d}x = x + \cos{x} + C$
$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{covercosin}(x) = \cos{x}$	$\int\mathrm{covercosin}(x) \,\mathrm{d}x = x - \cos{x} + C$
$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{haversin}(x) = \frac{\sin{x}}{2}$	$\int\mathrm{haversin}(x) \,\mathrm{d}x = \frac{x - \sin{x}}{2} + C$
$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{havercosin}(x) = \frac{-\sin{x}}{2}$	$\int\mathrm{havercosin}(x) \,\mathrm{d}x = \frac{x + \sin{x}}{2} + C$
$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{hacoversin}(x) = \frac{-\cos{x}}{2}$	$\int\mathrm{hacoversin}(x) \,\mathrm{d}x = \frac{x + \cos{x}}{2} + C$
$\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{hacovercosin}(x) = \frac{\cos{x}}{2}$	$\int\mathrm{hacovercosin}(x) \,\mathrm{d}x = \frac{x - \cos{x}}{2} + C$

History and applications

Historically, the versed sine was considered one of the most important trigonometric functions.^[2]^[3]^[4] As θ goes to zero, versin(θ) is the difference between two nearly equal quantities, so a user of a trigonometric table for the cosine alone would need a very high accuracy to obtain the versine in order to avoid catastrophic cancellation, making separate tables for the latter convenient.^[4] Even with a calculator or computer, round-off errors make it advisable to use the sin² formula for small θ. Another historical advantage of the versine is that it is always non-negative, so its logarithm is defined everywhere except for the single angle (θ = 0, 2π,...) where it is zero—thus, one could use logarithmic tables for multiplications in formulas involving versines.

The haversine, in particular, was important in navigation because it appears in the haversine formula, which is used to reasonably accurately compute distances on a sphere (see issues with the Earth`s radius vs. sphere) given angular positions (e.g., longitude and latitude). One could also use sin²(θ/2) directly, but having a table of the haversine removed the need to compute squares and square roots.^[4] The term haversine was, apparently, coined in a navigation text for just such an application.^[5]

In fact, the earliest surviving table of sine (half-chord) values (as opposed to the chords tabulated by Ptolemy and other Greek authors), calculated from the Surya Siddhantha of India dated back to the 3rd century BC, was a table of values for the sine and versed sine (in 3.75° increments from 0 to 90°).^[2] The versine appears as an intermediate step in the application of the half-angle formula sin²(θ/2) = versin(θ)/2, derived by Ptolemy, that was used to construct such tables.

Sine, cosine, and versine of θ in terms of a unit circle, centered at O

The ordinary sine function (see note on etymology) was sometimes historically called the sinus rectus ("vertical sine"), to contrast it with the versed sine (sinus versus).^[2] The meaning of these terms is apparent if one looks at the functions in the original context for their definition, a unit circle, shown at right. For a vertical chord AB of the unit circle, the sine of the angle θ (half the subtended angle) is the distance AC (half of the chord). On the other hand, the versed sine of θ is the distance CD from the center of the chord to the center of the arc. Thus, the sum of cos(θ) = OC and versin(θ) = CD is the radius OD = 1. Illustrated this way, the sine is vertical (rectus, lit. "straight") while the versine is horizontal (versus, lit. "turned against, out-of-place"); both are distances from C to the circle.

This figure also illustrates the reason why the versine was sometimes called the sagitta, Latin for arrow,^[1] from the Arabic usage sahem^[3] of the same meaning. This itself comes from the Indian word 'sara' (arrow) that was commonly used to refer to "utkrama-jya". If the arc ADB is viewed as a "bow" and the chord AB as its "string", then the versine CD is clearly the "arrow shaft".

In further keeping with the interpretation of the sine as "vertical" and the versed sine as "horizontal", sagitta is also an obsolete synonym for the abscissa (the horizontal axis of a graph).^[1]

One period (0 < θ < π/2) of a versine or, more commonly, a haversine waveform is also commonly used in signal processing and control theory as the shape of a pulse or a window function, because it smoothly (continuous in value and slope) "turns on" from zero to one (for haversine) and back to zero. In these applications, it is given yet another name: raised-cosine filter or Hann function.

Approximations

Comparison of the versine function with three approximations to the versine functions, for angles ranging from 0 to 2 Pi

Comparison of the versine function with three approximations to the versine functions, for angles ranging from 0 to Pi/2

When the versine v is small in comparison to the radius r, it may be approximated from the half-chord length L (the distance AC shown above) by the formula

v \approx \frac{L^2}{2r}

.^[6]

Alternatively, if the versine is small and the versine, radius, and half-chord length are known, they may be used to estimate the arc length s (AD in the figure above) by the formula

s\approx L+\frac{v^2}{r}

This formula was known to the Chinese mathematician Shen Kuo, and a more accurate formula also involving the sagitta was developed two centuries later by Guo Shoujing.^[7]

A more accurate approximation used in engineering^[8] is

v\approx \frac{s^\frac{3}{2} L^\frac{1}{2}}{8r}

"Versines" of arbitrary curves and chords

The term versine is also sometimes used to describe deviations from straightness in an arbitrary planar curve, of which the above circle is a special case. Given a chord between two points in a curve, the perpendicular distance v from the chord to the curve (usually at the chord midpoint) is called a versine measurement. For a straight line, the versine of any chord is zero, so this measurement characterizes the straightness of the curve. In the limit as the chord length L goes to zero, the ratio 8v/L² goes to the instantaneous curvature. This usage is especially common in rail transport, where it describes measurements of the straightness of the rail tracks^[9] and it is the basis of the Hallade method for rail surveying. The term 'sagitta' (often abbreviated sag) is used similarly in optics, for describing the surfaces of lenses and mirrors.

References

↑ 1.0 1.1 1.2 "sagitta". Oxford English Dictionary (3rd ed.). Oxford University Press. September 2005.
↑ 2.0 2.1 2.2 Boyer, Carl B. (1991). A History of Mathematics (2nd ed.). New York: Wiley.
↑ 3.0 3.1 Miller, J. "Earliest known uses of some of the words of mathematics (v)".
↑ 4.0 4.1 4.2 Calvert, James B. "Trigonometry".
↑ "haversine". Oxford English Dictionary (3rd ed.). Oxford University Press. September 2005. Cites coinage by Prof. Jas. Inman, D. D., in his Navigation and Nautical Astronomy, 3rd ed. (1835).
↑ Woodward, Ernest (1978), Geometry - Plane, Solid & Analytic Problem Solver, Research & Education Assoc., p. 359, ISBN 9780878915101 .
↑ Needham, Joseph (1959), Science and Civilisation in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth, Cambridge University Press, p. 39, ISBN 9780521058018 .
↑ Boardman, Harry (1930), Table For Use in Computing Arcs, Chords and Versines, Chicago Bridge and Iron Company, p. 32 .
↑ Nair, Bhaskaran (1972). "Track measurement systems—concepts and techniques". Rail International 3 (3): 159–166. ISSN 0020-8442.

External links

Weisstein, Eric W., "Versine", MathWorld.
Weisstein, Eric W., "Haversine", MathWorld.
Sagitta, Apothem, and Chord by Ed Pegg, Jr., The Wolfram Demonstrations Project.