Versine
The versed sine, also called the versine and, in
Latin, the
sinus versus or the
sagitta , is a
trigonometric function versin defined by the equation:
There are also three corresponding functions:
* the coversed sine :
* the haversed sine or haversine :
* the hacoversed sine :
Another similar function is the
exsecant .
Encyclopedia
The
versed sine, also called the
versine and, in
Latin, the
sinus versus or the
sagitta , is a
trigonometric function versin defined by the equation:
There are also three corresponding functions:
- the haversed sine or haversine :
Another similar function is the
exsecant .
History and applications
Historically, the versed sine was considered one of the most important trigonometric functions, but it has fallen from popularity in modern times due to the availability of
computers and scientific
calculators. As θ goes to zero, versin 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, making separate tables for the latter convenient. Another historical advantage of the versine is that it is always non-negative, so its
logarithm is defined everywhere except for the single angle 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 accurately compute distances on a sphere given angular positions . The term
haversine was, apparently, coined in a navigation text for just such an application .
In fact, the earliest surviving trigonometric table, from the
4th–
5th century Siddhantas from
India, was a table of values for the sine and versed sine only . This is, perhaps, even less surprising considering that the versine appears as an intermediate step in the application of the half-angle formula sin
2 = versin/2, derived by
Ptolemy, that was used to construct such tables.
As for sine, the etymology derives from a
12th century mistranslation of the
Sanskrit jiva via
Arabic. To contrast it with the versed sine , the ordinary sine function was sometimes historically called the
sinus rectus . 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 is the distance
AC . 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. Illustrated this way, the sine is vertical while the versine is flipped on its side ; 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, from the Arabic usage
sahem of the same meaning. 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 .
One period 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 "turns on" from zero to one and back to zero.
"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 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 8
v/
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 .
References
- Carl B. Boyer, A History of Mathematics, 2nd ed. .
- "sagitta", Oxford English Dictionary.
- J. Miller, .
- James B. Calvert, .
- "haversine", Oxford English Dictionary. Cites coinage by Prof. Jas. Inman, D. D., in his Navigation and Nautical Astronomy, 3rd ed. .
- Bhaskaran Nair, "Track measurement systems—concepts and techniques," Rail International 3 , 159-166 .
