English
Noun
 The versed
sine.
The versed sine, also called the versine and, in
Latin, the
sinus versus ("flipped sine") or the sagitta ("arrow"), is a
trigonometric
function versin(θ) (sometimes further abbreviated
"vers") defined by the equation:
 \textrm (\theta) = 1  \cos (\theta) = 2 \sin^2\left(\frac
\right) \,
There are also three corresponding
functions:
 the coversed sine (the versed sine of the complementary angle
π/2 − θ, or
coversine):

 \textrm(\theta) = \textrm\left(\frac  \theta\right) = 1 
\sin(\theta) \,
 the haversed sine or haversine (half the versed sine):

 \textrm(\theta) = \frac = \sin^2\left(\frac
\right)
 the hacoversed sine (half the coversed sine, also called the
hacoversine, the cohaversine, and the havercosine):

 \textrm(\theta) = \textrm\left(\frac  \theta\right) =
\frac
Another similar function is the
exsecant
(sec θ − 1).
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(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, making separate tables for the latter
convenient. (Even with a computer,
roundoff
errors make it advisable to use the sin2 formula for small
θ.) Another historical advantage of the versine is that
it is always nonnegative, 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 accurately compute distances on a
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.) The term
haversine was, apparently, coined in a navigation text for just
such an application (see references).
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 (in 3.75°
increments from 0 to 90°). This is, perhaps, even less surprising
considering that the versine appears as an intermediate step in the
application of the halfangle formula sin2(θ/2) =
versin(θ)/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 (sinus versus), the ordinary
sine function was sometimes historically called the sinus rectus
("vertical sine"). 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) while the
versine is flipped on its side (versus); 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 (the horizontal axis
of a graph).
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:
raisedcosine
filter.
"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
(Nair, 1972).
See also
References
 A History of Mathematics
 Cites coinage by Prof. Jas. Inman, D. D., in his Navigation and
Nautical Astronomy, 3rd ed. (1835).
External links
versine in Catalan: Versinus
versine in Esperanto: Rivolua sinuso
versine in French: Sinus verse
versine in Italian: Senoverso
versine in Dutch: Sinus versus
versine in Portuguese: Seno verso
versine in Serbian: Синус
версус