|Laws and theorems|
There are several related functions, most notably the coversine and haversine. The latter, half a versine, is of particular importance in the haversine formula of navigation.
The versine or versed sine is a trigonometric function already appearing in some of the earliest trigonometric tables. It is written as versin(θ), sinver(θ), vers(θ), ver(θ) or siv(θ). In Latin, it is known as the sinus versus (flipped sine), versinus, versus or the sagitta (arrow).
There are several related functions corresponding to the versine:
- The versed cosine,[nb 1] or vercosine,[nb 1] written vercosin(θ), vercos(θ) or vcs(θ)
- The coversed sine,[nb 1] coversine, cosinus versus[nb 1] or coversinus, written coversin(θ), covers(θ), cosiv(θ)[nb 1] or cvs(θ)
- The coversed cosine or covercosine, written covercosin(θ) or covercos(θ) or cvc(θ)
In full analogy to the above-mentioned four functions another set of four "half-value" functions exists as well:
- The haversed sine, haversine or semiversus, written haversin(θ), semiversin(θ), semiversinus(θ), havers(θ), hav(θ), hvs(θ),[nb 2] sem(θ) or hv(θ), most famous from the haversine formula used historically in navigation
- The haversed cosine or havercosine, written havercosin(θ), havercos(θ), hac(θ) or hvc(θ)
- The hacoversed sine, also called hacoversine or cohaversine and written hacoversin(θ), semicoversin(θ), hacovers(θ), hacov(θ) or hcv(θ)
- The hacoversed cosine, also called hacovercosine or cohavercosine and written hacovercosin(θ), hacovercos(θ) or hcc(θ)
History and applications
Versine and coversine
The ordinary sine function (see note on etymology) was sometimes historically called the sinus rectus ("straight sine"), to contrast it with the versed sine (sinus versus). The meaning of these terms is apparent if one looks at the functions in the original context for their definition, a unit circle:
For a vertical chord AB of the unit circle, the sine of the angle θ (representing half of the subtended angle Δ) 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) (equal to the length of line OC) and versin(θ) (equal to the length of line CD) is the radius OD (with length 1). Illustrated this way, the sine is vertical (rectus, literally "straight") while the versine is horizontal (versus, literally "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, from the Arabic usage sahem 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 of the double-angle Δ = 2θ is viewed as a "bow" and the chord AB as its "string", then the versine CD is clearly the "arrow shaft".
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. 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.
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°).
The haversine, in particular, was important in navigation because it appears in the haversine formula, which is used to reasonably accurately compute distances on an astronomic spheroid (see issues with the earth's radius vs. sphere) given angular positions (e.g., longitude and latitude). One could also use sin2(θ/) directly, but having a table of the haversine removed the need to compute squares and square roots.
In 1835, the term haversine (notated naturally as hav. or base-10 logarithmically as log. haversine or log. havers.) was coined by James Inman in the third edition of his work Navigation and Nautical Astronomy: For the Use of British Seamen to simplify the calculation of distances between two points on the surface of the earth using spherical trigonometry for applications in navigation. Inman also used the terms nat. versine and nat. vers. for versines.
The haversine continues to be used in navigation and has even found new applications in recent decades, as in Bruce D. Stark's method for clearing lunar distances utilizing Gaussian logarithms since 1995 or in a more compact method for sight reduction since 2014.
One period (0 < θ < π/) of a versine or, more commonly, a haversine (or havercosine) waveform is also commonly used in signal processing and control theory as the shape of a pulse or a window function (including Hann, Hann–Poisson and Tukey windows), because it smoothly (continuous in value and slope) "turns on" from zero to one (for haversine) and back to zero.[nb 2] In these applications, it is named Hann function or raised-cosine filter. Likewise, the havercosine is used in raised-cosine distributions in probability theory and statistics.
In the form of sin2(θ) the haversine of the double-angle Δ describes the relation between spreads and angles in rational trigonometry, a proposed reformulation of metrical planar and solid geometries by Norman John Wildberger since 2005.
As sagitta and cosagitta, double-angle Δ variants of the haversine and havercosine have also found new uses in describing the correlation and anti-correlation of correlated photons in quantum mechanics.
The functions are circular rotations of each other.
Derivatives and integrals
Inverse functions like arcversine (arcversin, arcvers, avers, aver), arcvercosine (arcvercosin, arcvercos, avercos, avcs), arccoversine (arccoversin, arccovers, acovers, acvs), arccovercosine (arccovercosin, arccovercos, acovercos, acvc), archaversine (archaversin, archav, haversin−1, invhav, ahav, ahvs, ahv, hav−1), archavercosine (archavercosin, archavercos, ahvc), archacoversine (archacoversin, ahcv) or archacovercosine (archacovercosin, archacovercos, ahcc) exist as well:
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
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
A more accurate approximation used in engineering is
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/ goes to the instantaneous curvature. This usage is especially common in rail transport, where it describes measurements of the straightness of the rail tracks and it is the basis of the Hallade method for rail surveying.
- Trigonometric identities
- Exsecant and excosecant
- Versiera (Witch of Agnesi)
- Exponential minus 1
- Natural logarithm plus 1
- Some English sources confuse the versed cosine with the coversed sine. Historically (f.e. in Cauchy, 1821), the sinus versus (versine) was defined as siv(θ) = 1−cos(θ), the cosinus versus (what is now also known as coversine) as cosiv(θ) = 1−sin(θ), and the vercosine as vcsθ = 1+cos(θ). However, in their 2009 English translation of Cauchy's work, Bradley and Sandifer associate the cosinus versus (and cosiv) with the versed cosine (what is now also known as vercosine) rather than the coversed sine. Similarly, in their 1968/2000 work, Korn and Korn associate the covers(θ) function with the versed cosine instead of the coversed sine.
- The abbreviation hvs sometimes used for the haversine function in signal processing and filtering is also sometimes used for the unrelated Heaviside step function.
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[…] Still there would be much labor of computation which may be saved by the use of tables of external secants and versed sines, which have been employed with great success recently by the Engineers on the Ohio and Mississippi Railroad, and which, with the formulas and rules necessary for their application to the laying down of curves, drawn up by Mr. Haslett, one of the Engineers of that Road, are now for the first time given to the public. […] In presenting this work to the public, the Author claims for it the adaptation of a new principle in trigonometrical analysis of the formulas generally used in field calculations. Experience has shown, that versed sines and external secants as frequently enter into calculations on curves as sines and tangents; and by their use, as illustrated in the examples given in this work, it is believed that many of the rules in general use are much simplified, and many calculations concerning curves and running lines made less intricate, and results obtained with more accuracy and far less trouble, than by any methods laid down in works of this kind. The examples given have all been suggested by actual practice, and will explain themselves. […] As a book for practical use in field work, it is confidently believed that this is more direct in the application of rules and facility of calculation than any work now in use. In addition to the tables generally found in books of this kind, the author has prepared, with great labor, a Table of Natural and Logarithmic Versed Sines and External Secants, calculated to degrees, for every minute; also, a Table of Radii and their Logarithms, from 1° to 60°. […]1856 edition
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