The external secant function (abbreviated exsecant , symbolized exsec ) is a trigonometric function defined in terms of the secant function:
46-512: Sec-1 , SEC-1 , sec-1 , or sec may refer to: sec x −1 = sec( x )−1 = exsec( x ) or exsecant of x , an old trigonometric function sec y = sec( y ) , sometimes interpreted as arcsec( y ) or arcsecant of y , the compositional inverse of the trigonometric function secant (see below for ambiguity) sec x = sec( x ) , sometimes interpreted as (sec( x )) = 1 / sec( x ) = cos( x ) or cosine of x ,
92-411: 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
138-407: A circle with one endpoint on the circumference a secans exterior . The trigonometric secant , named by Thomas Fincke (1583), is more specifically based on a line segment with one endpoint at the center of a circle and the other endpoint outside the circle; the circle divides this segment into a radius and an external secant. The external secant segment was used by Galileo Galilei (1632) under
184-456: A letter–number combination. If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Sec-1&oldid=1172984829 " Category : Letter–number combination disambiguation pages Hidden categories: All articles with unsourced statements Articles with unsourced statements from November 2017 Short description
230-447: A more compact method for sight reduction since 2014. Whilst the usage of the versine, coversine and haversine as well as their inverse functions can be traced back centuries, the names for the other five cofunctions appear to be of much younger origin. One period (0 < θ < 2 π ) of a versine or, more commonly, a haversine (or havercosine) waveform is also commonly used in signal processing and control theory as
276-571: A sufficiently small angle, a circular arc is approximately shaped like a parabola , and the versine and exsecant are approximately equal to each-other and both proportional to the square of the arclength. The inverse of the exsecant function, which might be symbolized arcexsec , is well defined if its argument y ≥ 0 {\displaystyle y\geq 0} or y ≤ − 2 {\displaystyle y\leq -2} and can be expressed in terms of other inverse trigonometric functions (using radians for
322-482: A table of the haversine removed the need to compute squares and square roots. An early utilization by José de Mendoza y Ríos of what later would be called haversines is documented in 1801. The first known English equivalent to a table of haversines was published by James Andrew in 1805, under the name "Squares of Natural Semi-Chords". In 1835, the term haversine (notated naturally as hav. or base-10 logarithmically as log. haversine or log. havers. )
368-443: A wire. In recent years, the availability of calculators and computers has removed the need for trigonometric tables of specialized functions such as this one. Exsecant is generally not directly built into calculators or computing environments (though it has sometimes been included in software libraries ), and calculations in general are much cheaper than in the past, no longer requiring tedious manual labor. Naïvely evaluating
414-399: Is v ≈ s 3 2 L 1 2 8 r {\displaystyle v\approx {\frac {s^{\frac {3}{2}}L^{\frac {1}{2}}}{8r}}} 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,
460-519: Is log exsec 1° ≈ −3.817 220 , all of whose digits are meaningful. If the logarithm of exsecant is calculated by looking up the secant in a six-place trigonometric table and then subtracting 1 , the difference sec 1° − 1 ≈ 0.000 152 has only 3 significant digits , and after computing the logarithm only three digits are correct, log(sec 1° − 1) ≈ −3.81 8 156 . For even smaller angles loss of precision
506-867: Is a trigonometric function already appearing in some of the earliest trigonometric tables. It is symbolized in formulas using the abbreviations versin , sinver , vers , ver or siv . In Latin , it is known as the sinus versus (flipped sine), versinus , versus , or sagitta (arrow). Expressed in terms of common trigonometric functions sine, cosine, and tangent, the versine is equal to versin θ = 1 − cos θ = 2 sin 2 θ 2 = sin θ tan θ 2 {\displaystyle \operatorname {versin} \theta =1-\cos \theta =2\sin ^{2}{\frac {\theta }{2}}=\sin \theta \,\tan {\frac {\theta }{2}}} There are several related functions corresponding to
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#1732880964082552-393: Is a trigonometric function found in some of the earliest ( Sanskrit Aryabhatia , Section I) trigonometric tables . The versine of an angle is 1 minus its cosine . 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
598-402: Is different from Wikidata All article disambiguation pages All disambiguation pages Exsecant exsec θ = sec θ − 1 = 1 cos θ − 1. {\displaystyle \operatorname {exsec} \theta =\sec \theta -1={\frac {1}{\cos \theta }}-1.} It
644-509: Is the natural logarithm . See also Integral of the secant function . The exsecant of twice an angle is: exsec 2 θ = 2 sin 2 θ 1 − 2 sin 2 θ . {\displaystyle \operatorname {exsec} 2\theta ={\frac {2\sin ^{2}\theta }{1-2\sin ^{2}\theta }}.} Van Sickle, Jenna (2011). "The history of one definition: Teaching trigonometry in
690-1369: Is worse. If a table or computer implementation of the exsecant function is not available, the exsecant can be accurately computed as exsec θ = tan θ tan 1 2 θ | , {\textstyle \operatorname {exsec} \theta =\tan \theta \,\tan {\tfrac {1}{2}}\theta {\vphantom {\Big |}},} or using versine, exsec θ = vers θ sec θ , {\textstyle \operatorname {exsec} \theta =\operatorname {vers} \theta \,\sec \theta ,} which can itself be computed as vers θ = 2 ( sin 1 2 θ ) ) 2 | = {\textstyle \operatorname {vers} \theta =2{\bigl (}{\sin {\tfrac {1}{2}}\theta }{\bigr )}{\vphantom {)}}^{2}{\vphantom {\Big |}}={}} sin θ tan 1 2 θ | {\displaystyle \sin \theta \,\tan {\tfrac {1}{2}}\theta \,{\vphantom {\Big |}}} ; Haslett used these identities to compute his 1855 exsecant and versine tables. For
736-472: The complex plane . Maclaurin series : 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 ≈ L 2 2 r . {\displaystyle v\approx {\frac {L^{2}}{2r}}.} Alternatively, if the versine is small and the versine, radius, and half-chord length are known, they may be used to estimate
782-1248: The Extruding Power of Terrestrial Rotation". Synthese . 134 (1–2, Logic and Mathematical Reasoning): 217–244. doi : 10.1023/A:1022143816001 . JSTOR 20117331 . Searles, William Henry; Ives, Howard Chapin (1915) [1880]. Field Engineering: A Handbook of the Theory and Practice of Railway Surveying, Location and Construction (17th ed.). New York: John Wiley & Sons . Meyer, Carl F. (1969) [1949]. Route Surveying and Design (4th ed.). Scranton, PA: International Textbook Co. "MIT/GNU Scheme – Scheme Arithmetic" ( MIT/GNU Scheme source code). v. 12.1. Massachusetts Institute of Technology . 2023-09-01. exsec function, arith.scm lines 61–63 . Retrieved 2024-04-01 . Review: " Field Manual for Railroad Engineers . By J. C. Nagle" . The Engineer (Review). 84 : 540. 1897-12-03. "MIT/GNU Scheme – Scheme Arithmetic" ( MIT/GNU Scheme source code). v. 12.1. Massachusetts Institute of Technology . 2023-09-01. aexsec function, arith.scm lines 65–71 . Retrieved 2024-04-01 . Versine The versine or versed sine
828-1079: The US before 1900" . International Journal for the History of Mathematics Education . 6 (2): 55–70. Review: Poor, Henry Varnum , ed. (1856-03-22). " Practical Book of Reference, and Engineer's Field Book . By Charles Haslett" . American Railroad Journal (Review). Second Quarto Series. XII (12): 184. Whole No. 1040, Vol. XXIX. Zucker, Ruth (1964). "4.3.147: Elementary Transcendental Functions - Circular functions" . In Abramowitz, Milton ; Stegun, Irene A. (eds.). Handbook of Mathematical Functions . Washington, D.C.: National Bureau of Standards. p. 78. LCCN 64-60036 . van Haecht, Joannes (1784). "Articulus III: De secantibus circuli: Corollarium III: [109]". Geometria elementaria et practica: quam in usum auditorum (in Latin). Lovanii, e typographia academica. p. 24, foldout. Finocchiaro, Maurice A. (2003). "Physical-Mathematical Reasoning: Galileo on
874-643: The United States for railroad and road design , and since the early 20th century has sometimes been briefly mentioned in American trigonometry textbooks and general-purpose engineering manuals. For completeness, a few books also defined a coexsecant or excosecant function (symbolized coexsec or excsc ), coexsec θ = {\displaystyle \operatorname {coexsec} \theta ={}} csc θ − 1 , {\displaystyle \csc \theta -1,}
920-1016: The angle): arcexsec y = arcsec ( y + 1 ) = { arctan ( y 2 + 2 y ) if y ≥ 0 , undefined if − 2 < y < 0 , π − arctan ( y 2 + 2 y ) if y ≤ − 2 ; . {\displaystyle \operatorname {arcexsec} y=\operatorname {arcsec}(y+1)={\begin{cases}{\arctan }{\bigl (}\!{\textstyle {\sqrt {y^{2}+2y}}}\,{\bigr )}&{\text{if}}\ \ y\geq 0,\\[6mu]{\text{undefined}}&{\text{if}}\ \ {-2}<y<0,\\[4mu]\pi -{\arctan }{\bigl (}\!{\textstyle {\sqrt {y^{2}+2y}}}\,{\bigr )}&{\text{if}}\ \ y\leq {-2};\\\end{cases}}_{\vphantom {.}}}
966-462: The arc length s ( AD in the figure above) by the formula s ≈ L + v 2 r {\displaystyle 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 . A more accurate approximation used in engineering
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#17328809640821012-893: The arctangent expression is well behaved for small angles. While historical uses of the exsecant did not explicitly involve calculus , its derivative and antiderivative (for x in radians) are: d d x exsec x = tan x sec x , ∫ exsec x d x = ln | sec x + tan x | − x + C , ∫ | {\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {exsec} x&=\tan x\,\sec x,\\[10mu]\int \operatorname {exsec} x\,\mathrm {d} x&=\ln {\bigl |}\sec x+\tan x{\bigr |}-x+C,{\vphantom {\int _{|}}}\end{aligned}}} where ln
1058-586: The central angle between the segment's inner endpoint and the point of tangency for a line through the outer endpoint and tangent to the circle. The word secant comes from Latin for "to cut", and a general secant line "cuts" a circle, intersecting it twice; this concept dates to antiquity and can be found in Book 3 of Euclid's Elements , as used e.g. in the intersecting secants theorem . 18th century sources in Latin called any non- tangential line segment external to
1104-407: The difference between two approximately equal quantities results in catastrophic cancellation : because most of the digits of each quantity are the same, they cancel in the subtraction, yielding a lower-precision result. For example, the secant of 1° is sec 1° ≈ 1.000 152 , with the leading several digits wasted on zeros, while the common logarithm of the exsecant of 1°
1150-415: The double-angle Δ = 2 θ 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). In 1821, Cauchy used the terms sinus versus ( siv ) for
1196-570: 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 versine appears as an intermediate step in the application of the half-angle formula sin ( θ / 2 ) = 1 / 2 versin( θ ), derived by Ptolemy , that
1242-429: The ends of the arc, which equals the radius times the trigonometric exsecant of half the central angle subtended by the arc, R exsec 1 2 Δ . {\displaystyle R\operatorname {exsec} {\tfrac {1}{2}}\Delta .} By comparison, the versed sine of a curved track section is the furthest distance from the long chord (the line segment between endpoints) to
1288-478: The expressions 1 − cos θ {\displaystyle 1-\cos \theta } (versine) and sec θ − 1 {\displaystyle \sec \theta -1} (exsecant) is problematic for small angles where sec θ ≈ cos θ ≈ 1. {\displaystyle \sec \theta \approx \cos \theta \approx 1.} Computing
1334-421: The exsecant of the complementary angle , though it was not used in practice. While the exsecant has occasionally found other applications, today it is obscure and mainly of historical interest. As a line segment , an external secant of a circle has one endpoint on the circumference, and then extends radially outward. The length of this segment is the radius of the circle times the trigonometric exsecant of
1380-872: The form of sin ( 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. The functions are circular rotations of each other. 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 , invhav, ahav, ahvs, ahv, hav ), archavercosine (archavercosin, archavercos, ahvc), archacoversine (archacoversin, ahcv) or archacovercosine (archacovercosin, archacovercos, ahcc) exist as well: These functions can be extended into
1426-554: The late-19th and 20th century, railroads began using arcs of an Euler spiral as a track transition curve between straight or circular sections of differing curvature. These spiral curves can be approximately calculated using exsecants and versines. Solving the same types of problems is required when surveying circular sections of canals and roads, and the exsecant was still used in mid-20th century books about road surveying. The exsecant has sometimes been used for other applications, such as beam theory and depth sounding with
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1472-446: The latter convenient. Even with a calculator or computer, round-off errors make it advisable to use the sin 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, 2 π , …) where it is zero—thus, one could use logarithmic tables for multiplications in formulas involving versines. In fact,
1518-400: 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 . If the arc ADB of
1564-485: The logarithm of the exsecant and versine saved significant effort and produced more accurate results compared to calculating the same quantity from values found in previously available trigonometric tables. The same idea was adopted by other authors, such as Searles (1880). By 1913 Haslett's approach was so widely adopted in the American railroad industry that, in that context, "tables of external secants and versed sines [were] more common than [were] tables of secants". In
1610-420: The multiplicative inverse (or reciprocal) of the trigonometric function secant (see above for ambiguity) See also [ edit ] Second (time) Seconds of arc , a unit for angle measurement Inverse function cos (disambiguation) csc (disambiguation) [REDACTED] Topics referred to by the same term This disambiguation page lists articles associated with the same title formed as
1656-410: The multiplicative inverse (or reciprocal) of the trigonometric function secant (see above for ambiguity) sec x , sometimes interpreted as sec( x ) = sec( 1 / x ) , the secant of the multiplicative inverse (or reciprocal) of x (see below for ambiguity) sec x , sometimes interpreted as (sec( x )) = 1 / sec( x ) = cos( x ) or cosine of x ,
1702-459: The name secant . In the 19th century, most railroad tracks were constructed out of arcs of circles , called simple curves . Surveyors and civil engineers working for the railroad needed to make many repetitive trigonometrical calculations to measure and plan circular sections of track. In surveying, and more generally in practical geometry, tables of both "natural" trigonometric functions and their common logarithms were used, depending on
1748-470: 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 8 v / L goes to the instantaneous curvature . This usage is especially common in rail transport , where it describes measurements of
1794-416: 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. 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
1840-404: The specific calculation. Using logarithms converts expensive multiplication of multi-digit numbers to cheaper addition, and logarithmic versions of trigonometric tables further saved labor by reducing the number of necessary table lookups. The external secant or external distance of a curved track section is the shortest distance between the track and the intersection of the tangent lines from
1886-474: The track – cf. Sagitta – which equals the radius times the trigonometric versine of half the central angle, R vers 1 2 Δ . {\displaystyle R\operatorname {vers} {\tfrac {1}{2}}\Delta .} These are both natural quantities to measure or calculate when surveying circular arcs, which must subsequently be multiplied or divided by other quantities. Charles Haslett (1855) found that directly looking up
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1932-451: The versine and cosinus versus ( cosiv ) for the coversine. Historically, the versed sine was considered one of the most important trigonometric functions. 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
1978-428: The versine: In full analogy to the above-mentioned four functions another set of four "half-value" functions exists as well: 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,
2024-835: 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. Other high-regarded tables of haversines were those of Richard Farley in 1856 and John Caulfield Hannyngton in 1876. The haversine continues to be used in navigation and has found new applications in recent decades, as in Bruce D. Stark's method for clearing lunar distances utilizing Gaussian logarithms since 1995 or in
2070-437: Was introduced in 1855 by American civil engineer Charles Haslett , who used it in conjunction with the existing versine function, vers θ = 1 − cos θ , {\displaystyle \operatorname {vers} \theta =1-\cos \theta ,} for designing and measuring circular sections of railroad track. It was adopted by surveyors and civil engineers in
2116-466: Was used to construct such tables. 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 sin ( θ / 2 ) directly, but having
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