6204:, 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°.
5697:
1005:
5686:
5675:
837:
359:
5303:
2774:
56:
5670:{\displaystyle {\begin{aligned}{\frac {\operatorname {versin} (\theta )+\operatorname {coversin} (\theta )}{\operatorname {versin} (\theta )-\operatorname {coversin} (\theta )}}-{\frac {\operatorname {exsec} (\theta )+\operatorname {excsc} (\theta )}{\operatorname {exsec} (\theta )-\operatorname {excsc} (\theta )}}&={\frac {2\operatorname {versin} (\theta )\operatorname {coversin} (\theta )}{\operatorname {versin} (\theta )-\operatorname {coversin} (\theta )}}\\\,&=\sin(\theta )\cos(\theta )\end{aligned}}}
2322:
2769:{\displaystyle {\begin{aligned}\mathrm {versin} (\theta )&=\mathrm {coversin} \left(\theta +{\frac {\pi }{2}}\right)=\mathrm {vercosin} \left(\theta +\pi \right)=\mathrm {covercosin} \left(\theta +{\frac {3\pi }{2}}\right)\\\mathrm {haversin} (\theta )&=\mathrm {hacoversin} \left(\theta +{\frac {\pi }{2}}\right)=\mathrm {havercosin} \left(\theta +\pi \right)=\mathrm {hacovercosin} \left(\theta +{\frac {3\pi }{2}}\right)\end{aligned}}}
6180:
the use of all classes of practical mechanics. Together with the
Engineer's Field Book: Containing formulæ for the various of running and changing lines, locating side tracks and switches, &c., &c. Tables of radii and their logarithms, natural and logarithmic versed sines and external secants, natural sines and tangents to every degree and minute of the quadrant, and logarithms from the natural numbers from 1 to 10,000
2303:
2198:
2061:
1819:
1956:
1714:
1611:
1506:
5229:
782:
2192:
1950:
6179:
The
Mechanic's, Machinist's, Engineer's Practical Book of Reference: Containing tables and formulæ for use in superficial and solid mensuration; strength and weight of materials; mechanics; machinery; hydraulics, hydrodynamics; marine engines, chemistry; and miscellaneous recipes. Adapted to and for
4994:
6282:. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 78.
538:
1813:
2297:
1605:
2055:
1708:
4739:
1500:
4832:
4233:
4018:
3806:
3600:
2070:
1828:
5295:
4345:
4124:
3912:
3406:
3221:
3039:
3700:
2863:
5224:{\displaystyle {\begin{aligned}\operatorname {versin} (z)&=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}z^{2k}}{(2k)!}}\\\operatorname {haversin} (z)&=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}z^{2k}}{2(2k)!}}\end{aligned}}}
4966:
4899:
441:
4645:
4581:
3503:
4517:
4453:
5904:
3312:
3130:
2948:
5917:
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
8593:
5308:
1723:
2207:
4999:
2327:
1515:
1965:
5826:
5770:
1620:
4653:
1412:
4747:
8586:
200:
7913:
Stark Tables for
Clearing the Lunar Distance and Finding Universal Time by Sextant Observation Including a Convenient Way to Sharpen Celestial Navigation Skills While On Land
4132:
3920:
3708:
7717:
3511:
2187:{\displaystyle {\textrm {havercosin}}(\theta ):={\frac {{\textrm {vercosin}}(\theta )}{2}}=\cos ^{2}\!\left({\frac {\theta }{2}}\right)={\frac {1+\cos(\theta )}{2}}\,}
8579:
8182:
1945:{\displaystyle {\textrm {haversin}}(\theta ):={\frac {{\textrm {versin}}(\theta )}{2}}=\sin ^{2}\!\left({\frac {\theta }{2}}\right)={\frac {1-\cos(\theta )}{2}}\,}
841:
7883:
7849:
7686:
Memoria sobre algunos métodos nuevos de calcular la longitud por las distancias lunares: y aplicación de su teórica á la solucion de otros problemas de navegacion
5237:
4239:
5842:
196:
4024:
3812:
3320:
3138:
2956:
3606:
206:
5696:
2789:
1004:
6468:
Hall, Arthur Graham; Frink, Fred
Goodrich (January 1909). "Review Exercises Secondary Trigonometric Functions". Written at Ann Arbor, Michigan, USA.
8249:
4907:
4840:
4589:
4525:
8400:
3412:
298:
8662:
4461:
4397:
1257:
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
8384:
6878:
6862:
6835:
6726:
6614:
6427:
6382:
6362:
6287:
5774:
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
3227:
3045:
7969:
5926:
measurement. For a straight line, the versine of any chord is zero, so this measurement characterizes the straightness of the curve. In the
6411:
2869:
5685:
8128:
7318:
8376:
6668:
6510:
8558:
7921:
7631:
7375:
6353:
Oldham, Keith B.; Myland, Jan C.; Spanier, Jerome (2009) . "32.13. The Cosine cos(x) and Sine sin(x) functions - Cognate functions".
8519:
8418:
7998:
6939:
6518:
533:{\displaystyle \operatorname {versin} \theta =1-\cos \theta =2\sin ^{2}{\frac {\theta }{2}}=\sin \theta \,\tan {\frac {\theta }{2}}}
8212:
8087:
7971:
5785:
5727:
1081:
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°).
327:
8688:
8161:
8083:
6009:
120:
8320:
8626:
8465:
8439:
1808:{\displaystyle {\textrm {covercosin}}(\theta ):={\textrm {vercosin}}\!\left({\frac {\pi }{2}}-\theta \right)=1+\sin(\theta )\,}
98:
6332:
2292:{\displaystyle {\textrm {hacovercosin}}(\theta ):={\frac {{\textrm {covercosin}}(\theta )}{2}}={\frac {1+\sin(\theta )}{2}}\,}
7544:
1372:
6565:
358:
7452:
7230:
7070:
1600:{\displaystyle {\textrm {coversin}}(\theta ):={\textrm {versin}}\!\left({\frac {\pi }{2}}-\theta \right)=1-\sin(\theta )\,}
8279:
8241:
8208:
6718:
6647:
6201:
2050:{\displaystyle {\textrm {hacoversin}}(\theta ):={\frac {{\textrm {coversin}}(\theta )}{2}}={\frac {1-\sin(\theta )}{2}}\,}
7277:
7027:
6982:
8636:
7957:
7503:
1074:
866:). The meaning of these terms is apparent if one looks at the functions in the original context for their definition, a
8042:
7680:
1196:
6164:
6140:
sometimes used for the haversine function in signal processing and filtering is also sometimes used for the unrelated
1258:
6561:
5689:
Comparison of the versine function with three approximations to the versine functions, for angles ranging from 0 to 2
1254:
Other high-regarded tables of haversines were those of
Richard Farley in 1856 and John Caulfield Hannyngton in 1876.
391:
already appearing in some of the earliest trigonometric tables. It is symbolized in formulas using the abbreviations
6593:(2000) . "Appendix B: B9. Plane and Spherical Trigonometry: Formulas Expressed in Terms of the Haversine Function".
5700:
Comparison of the versine function with three approximations to the versine functions, for angles ranging from 0 to
8410:
6931:
6827:
6784:
1349:
291:
242:
191:
125:
8120:
7771:
7577:
6921:
6741:
6597:
Mathematical handbook for scientists and engineers: Definitions, theorems, and formulars for reference and review
1305:
1036:
626:
In full analogy to the above-mentioned four functions another set of four "half-value" functions exists as well:
6602:
6034:
104:
7659:
1703:{\displaystyle {\textrm {vercosin}}(\theta ):=2\cos ^{2}\!\left({\frac {\theta }{2}}\right)=1+\cos(\theta )\,}
1317:
836:
7954:"Bruce Stark - Tables for Clearing the Lunar Distance and Finding G.M.T. by Sextant Observation (1995, 1997)"
4734:{\displaystyle \operatorname {archaversin} (y)=2\arcsin \left({\sqrt {y}}\right)=\arccos \left(1-2y\right)\,}
8614:
8499:
8303:
8245:
8025:
7776:
7705:
7582:
7527:
7435:
7346:
7260:
7213:
7053:
7010:
6965:
6478:/ Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. pp. 125–127
6141:
6014:
1495:{\displaystyle {\textrm {versin}}(\theta ):=2\sin ^{2}\!\left({\frac {\theta }{2}}\right)=1-\cos(\theta )\,}
435:
94:
82:
70:
65:
8237:
4827:{\displaystyle \operatorname {archavercos} (y)=2\arccos \left({\sqrt {y}}\right)=\arccos \left(2y-1\right)}
8453:
8317:
8039:
7982:
7541:
7449:
7274:
7227:
7067:
7024:
6979:
6818:
6475:
6469:
1391:
1240:
970:
In further keeping with the interpretation of the sine as "vertical" and the versed sine as "horizontal",
388:
367:
320:
55:
7814:
7792:
7154:
6858:
6808:
6750:
6277:
1376:
982:
284:
6674:
Vorlesungen über
Geschichte der Trigonometrie - Von der Erfindung der Logarithmen bis auf die Gegenwart
6408:
The
Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library
8631:
7623:
7098:
6780:
4228:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {hacovercosin} (x)={\frac {\cos {x}}{2}}}
1345:
257:
211:
130:
75:
4013:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {hacoversin} (x)={\frac {-\cos {x}}{2}}}
3801:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {havercosin} (x)={\frac {-\sin {x}}{2}}}
1192:
directly, but having a table of the haversine removed the need to compute squares and square roots.
7609:
7400:
6606:
6496:
6366:
6029:
5927:
1325:
1203:
1032:
333:
173:
8344:
6183:. New York, USA: James G. Gregory, successor of W. A. Townsend & Co. (Stringer & Townsend)
8176:
7936:
7877:
7843:
7133:
7102:
6500:
6441:
3595:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {haversin} (x)={\frac {\sin {x}}{2}}}
1353:
1296:) of a versine or, more commonly, a haversine (or havercosine) waveform is also commonly used in
1262:
1085:
7190:
6812:
1020:
Historically, the versed sine was considered one of the most important trigonometric functions.
7870:
Haversines, Natural and
Logarithmic, used in Computing Lunar Distances for the Nautical Almanac
7748:
Astronomical and
Nautical Tables with Precepts for finding the Latitude and Longitude of Places
8525:
8515:
8477:
8469:
8414:
8380:
8370:
7994:
7939:
7917:
7627:
7408:
7314:
7158:
6935:
6884:
6874:
6831:
6722:
6610:
6524:
6514:
6433:
6423:
6388:
6378:
6309:
6293:
6283:
6269:
5832:, and a more accurate formula also involving the sagitta was developed two centuries later by
1297:
1239:
to simplify the calculation of distances between two points on the surface of the Earth using
1220:
1149:
1063:
675:
352:
262:
166:
8404:
7590:
6056:
Some
English sources confuse the versed cosine with the coversed sine. Historically (f.e. in
1035:
for the cosine alone would need a very high accuracy to obtain the versine in order to avoid
8554:
8406:
Science and Civilisation in China: Mathematics and the Sciences of the Heavens and the Earth
7726:
7298:
Fulst, Otto (1972). "17, 18". In Lütjen, Johannes; Stein, Walter; Zwiebler, Gerhard (eds.).
7186:
7129:
6925:
6866:
6594:
6503:(The Origins of Classical Mechanics from Aristotle to Newton)". In Clagett, Marshall (ed.).
6415:
6370:
6354:
5980:
4985:
1157:
1070:
874:
822:
38:
7367:
6823:
6305:
8619:
6590:
6328:
6301:
6193:
6192:
Still there would be much labor of computation which may be saved by the use of tables of
6023:
1309:
1266:
1078:
1040:
855:
161:
90:
86:
8571:
7092:
6595:
6355:
5290:{\displaystyle \lim _{\theta \to 0}{\frac {\operatorname {versin} (\theta )}{\theta }}=0}
8200:
8075:
7953:
7123:
6817:(in French). Vol. 1. L'Imprimerie Royale, Debure frères, Libraires du Roi et de la
6679:
Lectures on history of trigonometry - from the invention of logarithms up to the present
6177:
4340:{\displaystyle \int \mathrm {hacovercosin} (x)\,\mathrm {d} x={\frac {x-\cos {x}}{2}}+C}
1206:
was published by James Andrew in 1805, under the name "Squares of Natural Semi-Chords".
1039:, making separate tables for the latter convenient. Even with a calculator or computer,
7987:
7897:
7893:
7180:
6708:
5972:
5968:
5960:
1387:
1383:
1301:
272:
267:
156:
8153:
4119:{\displaystyle \int \mathrm {hacoversin} (x)\,\mathrm {d} x={\frac {x+\cos {x}}{2}}+C}
3907:{\displaystyle \int \mathrm {havercosin} (x)\,\mathrm {d} x={\frac {x+\sin {x}}{2}}+C}
3401:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {covercosin} (x)=\cos {x}}
1050:
Another historical advantage of the versine is that it is always non-negative, so its
8682:
8511:
8307:
7617:
7148:
6682:
5993:
4979:
3216:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {coversin} (x)=-\cos {x}}
3034:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {vercosin} (x)=-\sin {x}}
1341:
252:
8201:"identify.py: An asteroid_server client which identifies measurements in MPC format"
6765:(NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, USA, 2013.)
6445:
3695:{\displaystyle \int \mathrm {haversin} (x)\,\mathrm {d} x={\frac {x-\sin {x}}{2}}+C}
8550:
7613:
7571:
6851:
Bradley, Robert E.; Sandifer, Charles Edward (2010-01-14) . Buchwald, J. Z. (ed.).
6777:"德博士的 Notes About Circles, ज्य, & कोज्य: What in the world is a hacovercosine?"
6273:
5833:
2302:
1380:
1321:
956:
818:
797:. This figure also illustrates the reason why the versine was sometimes called the
151:
47:
7731:
7709:
7531:
2858:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {versin} (x)=\sin {x}}
840:
Graphs of historical trigonometric functions compared with sin and cos – in
8504:
7911:
7684:
7340:
6852:
6717:. Vol. 2 (2 (3rd corrected printing of 1929 issue) ed.). Chicago, USA:
6712:
6672:
6550:
6504:
6279:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
6243:
6228:
6207:
935:
This figure also illustrates the reason why the versine was sometimes called the
7836:
Natural Versed Sines from 0 to 125°, and Logarithmic Versed Sines from 0 to 135°
7439:
7217:
7057:
6745:
6265:
6224:
5964:
2197:
2060:
1818:
1313:
1232:
1009:
867:
790:
363:
237:
135:
8271:
6200:, which have been employed with great success recently by the Engineers on the
8079:
7264:
7014:
6969:
6870:
6419:
6406:
Beebe, Nelson H. F. (2017-08-22). "Chapter 11.1. Sine and cosine properties".
6374:
1955:
1713:
1610:
1357:
1337:
1333:
1282:
1145:
679:
247:
227:
8481:
8473:
7412:
4961:{\displaystyle \operatorname {archacovercos} (y)=\arcsin \left(2y-1\right)\,}
4894:{\displaystyle \operatorname {archacoversin} (y)=\arcsin \left(1-2y\right)\,}
8312:
8034:
8029:
7889:
7855:
7753:
7536:
7444:
7269:
7222:
7062:
7019:
6974:
6643:
5956:
1211:
1161:
1152:, which is used to reasonably accurately compute distances on an astronomic
1051:
900:
from the center of the chord to the center of the arc. Thus, the sum of cos(
6857:. Sources and Studies in the History of Mathematics and Physical Sciences.
6266:"Chapter 4.3.147: Elementary Transcendental Functions - Circular functions"
1505:
1008:
The trigonometric functions can be constructed geometrically in terms of a
6744:(Madrid, 1801, also 1805, 1809), and later in a treatise on navigation of
4640:{\displaystyle \operatorname {arccovercos} (y)=\arcsin \left(y-1\right)\,}
4576:{\displaystyle \operatorname {arccoversin} (y)=\arcsin \left(1-y\right)\,}
1277:
Whilst the usage of the versine, coversine and haversine as well as their
8657:
7495:
6019:
5829:
3498:{\displaystyle \int \mathrm {covercosin} (x)\,\mathrm {d} x=x-\cos {x}+C}
1165:
1153:
975:
324:
182:
17:
6776:
5922:
from the chord to the curve (usually at the chord midpoint) is called a
4512:{\displaystyle \operatorname {arcvercos} (y)=\arccos \left(y-1\right)\,}
4448:{\displaystyle \operatorname {arcversin} (y)=\arccos \left(1-y\right)\,}
1031:) is the difference between two nearly equal quantities, so a user of a
5899:{\displaystyle v\approx {\frac {s^{\frac {3}{2}}L^{\frac {1}{2}}}{8r}}}
3307:{\displaystyle \int \mathrm {coversin} (x)\,\mathrm {d} x=x+\cos {x}+C}
3125:{\displaystyle \int \mathrm {vercosin} (x)\,\mathrm {d} x=x+\sin {x}+C}
1133:
232:
31:
8565:
7407:(227). Ix, Hanno. Portland, ME, USA: Navigator Publishing LLC: 42–43.
1084:
The versine appears as an intermediate step in the application of the
6740:
The haversine first appears in the tables of logarithmic versines of
5997:
5989:
2943:{\displaystyle \int \mathrm {versin} (x)\,\mathrm {d} x=x-\sin {x}+C}
1340:(for haversine) and back to zero. In these applications, it is named
928:, literally "turned against, out-of-place"); both are distances from
337:
8506:
On the Shoulders of Giants: The Great Works of Physics and Astronomy
6313:
7655:
781:
8667:
8529:
6888:
6437:
6392:
6357:
An Atlas of Functions: with Equator, the Atlas Function Calculator
5695:
5684:
1329:
1003:
940:
802:
780:
542:
There are several related functions corresponding to the versine:
412:
6927:
Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry
351:. The latter, half a versine, is of particular importance in the
7302:(in German) (24 ed.). Bremen, Germany: Arthur Geist Verlag.
6297:
6230:
Navigation and Nautical Astronomy: For the Use of British Seamen
6092:). However, in their 2009 English translation of Cauchy's work,
1237:
Navigation and Nautical Astronomy: For the Use of British Seamen
1199:
of what later would be called haversines is documented in 1801.
850:
8575:
7989:
Divine Proportions: Rational Trigonometry to Universal Geometry
6528:
4377:(archaversin, archav, haversin, invhav, ahav, ahvs, ahv, hav),
6176:
Haslett, Charles (September 1855). Hackley, Charles W. (ed.).
8456:(1972). "Track measurement systems—concepts and techniques".
7824:, this journal has a discussion on the origin of haversines.)
7802:, this journal has a discussion on the origin of haversines.)
7656:"Earliest Known Uses of Some of the Words of Mathematics (V)"
5963:, where it describes measurements of the straightness of the
920:(with length 1). Illustrated this way, the sine is vertical (
6233:(3 ed.). London, UK: W. Woodward, C. & J. Rivington
8238:"astrotrig.py: Astronomical trigonometry related functions"
1281:
can be traced back centuries, the names for the other five
892:(half of the chord). On the other hand, the versed sine of
7315:"Semiversus-Verfahren: Logarithmische Berechnung der Höhe"
7094:
Elements of Trigonometry with Logarithmic and Other Tables
6052:
6050:
1243:
for applications in navigation. Inman also used the terms
924:, literally "straight") while the versine is horizontal (
6558:
Precalculus: A Study of Functions and Their Applications
1043:
make it advisable to use the sin formula for small
862:("straight sine"), to contrast it with the versed sine (
7522:
7520:
7048:
7046:
7044:
6549:
Swanson, Todd; Andersen, Janet; Keeley, Robert (1999).
6474:. Vol. Part I: Plane Trigonometry. New York, USA:
7710:"Recent Mathematical Tables : 197[C, D].—
6104:(what is now also known as vercosine) rather than the
4389:(archacovercosin, archacovercos, ahcc) exist as well:
343:
There are several related functions, most notably the
8372:
Geometry - Plane, Solid & Analytic Problem Solver
7764:
7762:
7430:
7428:
7208:
7206:
5845:
5839:
A more accurate approximation used in engineering is
5788:
5730:
5306:
5240:
4997:
4910:
4843:
4750:
4656:
4592:
4528:
4464:
4400:
4242:
4135:
4027:
3923:
3815:
3711:
3609:
3514:
3415:
3323:
3230:
3141:
3048:
2959:
2872:
2792:
2325:
2210:
2073:
1968:
1831:
1726:
1623:
1518:
1415:
444:
37:"versin" redirects here. For the Polish village, see
8436:
Table For Use in Computing Arcs, Chords and Versines
7317:(in German). Hotheim am Taunus, Germany: Astrosail.
5828:
This formula was known to the Chinese mathematician
5716:, it may be approximated from the half-chord length
2316:
The functions are circular rotations of each other.
1069:
In fact, the earliest surviving table of sine (half-
1066:
for multiplications in formulas involving versines.
8645:
8607:
8349:
Wolfram Language & System: Documentation Center
8244:source code). Ljubljana, Slovenia: Telescope Vega,
7372:
Wolfram Language & System: Documentation Center
7005:
7003:
7001:
6999:
6960:
6958:
1054:
is defined everywhere except for the single angle (
438:sine, cosine, and tangent, the versine is equal to
8503:
7986:
7570:
6854:Cauchy's Cours d'analyse: An Annotated Translation
6681:] (in German). Vol. 2. Leipzig, Germany:
5898:
5820:
5764:
5669:
5289:
5223:
4960:
4893:
4826:
4733:
4639:
4575:
4511:
4447:
4339:
4227:
4118:
4012:
3906:
3800:
3694:
3594:
3497:
3400:
3306:
3215:
3124:
3033:
2942:
2857:
2768:
2291:
2186:
2049:
1944:
1807:
1702:
1599:
1494:
532:
8464:(3). International Railway Congress Association,
8298:
8296:
8020:
8018:
8016:
7790:White, J. D. (February 1926). "(unknown title)".
7718:Mathematical Tables and Other Aids to Computation
7394:
7392:
6916:
6914:
6912:
6910:
6259:
6257:
6255:
6253:
6251:
2131:
1889:
1753:
1656:
1545:
1448:
7489:
7255:
7253:
7251:
7249:
7247:
5242:
1062:, …) where it is zero—thus, one could use
8339:
8337:
8069:
8067:
8065:
8063:
8061:
8059:
7487:
7485:
7483:
7481:
7479:
7477:
7475:
7473:
7471:
7469:
7179:Anderegg, Frederick; Roe, Edward Drake (1896).
6846:
6844:
6814:Cours d'Analyse de l'Ecole royale polytechnique
6637:
6635:
6633:
6631:
6076:(what is now also known as coversine) as cosiv(
4373:(arccovercosin, arccovercos, acovercos, acvc),
1144:The haversine, in particular, was important in
8231:
8229:
8194:
8192:
7565:
7563:
7561:
7091:Ludlow, Henry Hunt; Bass, Edgar Wales (1891).
6803:
6801:
6703:
6701:
6699:
6509:(3 ed.). Madison, Milwaukee, and London:
1077:and other Greek authors), calculated from the
8587:
7812:White, J. D. (July 1926). "(unknown title)".
7750:. Vol. T. XIII. London. pp. 29–148.
7604:
7602:
7600:
7147:Kenyon, Alfred Monroe; Ingold, Louis (1913).
5821:{\displaystyle s\approx L+{\frac {v^{2}}{r}}}
5765:{\displaystyle v\approx {\frac {L^{2}}{2r}}.}
844:, hover over or click a graph to highlight it
292:
8:
8114:
8112:
8110:
8108:
8106:
8104:
6219:
6217:
6215:
8181:: CS1 maint: numeric names: authors list (
8147:
8145:
7993:(1 ed.). Australia: Wild Egg Pty Ltd.
7756:table from 0° to 120° in intervals of 10".)
7700:
7698:
7696:
7689:(in Spanish). Madrid, Spain: Imprenta Real.
6506:Critical Problems in the History of Science
6463:
6461:
6459:
6457:
6455:
1265:since 1995 or in a more compact method for
8594:
8580:
8572:
7882:: CS1 maint: location missing publisher (
7848:: CS1 maint: location missing publisher (
1136:, that was used to construct such tables.
884:(representing half of the subtended angle
880:of the unit circle, the sine of the angle
299:
285:
43:
7730:
7339:Rider, Paul Reece; Davis, Alfred (1923).
5874:
5859:
5852:
5844:
5807:
5801:
5787:
5743:
5737:
5729:
5586:
5469:
5388:
5311:
5307:
5305:
5257:
5245:
5239:
5185:
5169:
5153:
5147:
5136:
5077:
5061:
5045:
5039:
5028:
4998:
4996:
4978:These functions can be extended into the
4957:
4909:
4890:
4842:
4782:
4749:
4730:
4688:
4655:
4636:
4591:
4572:
4527:
4508:
4463:
4444:
4399:
4369:(arccoversin, arccovers, acovers, acvs),
4365:(arcvercosin, arcvercos, avercos, avcs),
4320:
4305:
4294:
4293:
4246:
4241:
4214:
4205:
4155:
4144:
4138:
4136:
4134:
4099:
4084:
4073:
4072:
4031:
4026:
3999:
3987:
3943:
3932:
3926:
3924:
3922:
3887:
3872:
3861:
3860:
3819:
3814:
3787:
3775:
3731:
3720:
3714:
3712:
3710:
3675:
3660:
3649:
3648:
3613:
3608:
3581:
3572:
3534:
3523:
3517:
3515:
3513:
3484:
3461:
3460:
3419:
3414:
3393:
3343:
3332:
3326:
3324:
3322:
3293:
3270:
3269:
3234:
3229:
3208:
3161:
3150:
3144:
3142:
3140:
3111:
3088:
3087:
3052:
3047:
3026:
2979:
2968:
2962:
2960:
2958:
2929:
2906:
2905:
2876:
2871:
2850:
2812:
2801:
2795:
2793:
2791:
2742:
2693:
2639:
2621:
2578:
2536:
2512:
2469:
2421:
2403:
2366:
2330:
2326:
2324:
2288:
2258:
2234:
2233:
2230:
2212:
2211:
2209:
2183:
2153:
2136:
2125:
2097:
2096:
2093:
2075:
2074:
2072:
2046:
2016:
1992:
1991:
1988:
1970:
1969:
1967:
1941:
1911:
1894:
1883:
1855:
1854:
1851:
1833:
1832:
1830:
1804:
1759:
1747:
1746:
1728:
1727:
1725:
1699:
1661:
1650:
1625:
1624:
1622:
1596:
1551:
1539:
1538:
1520:
1519:
1517:
1491:
1453:
1442:
1417:
1416:
1414:
520:
513:
491:
482:
443:
336:. The versine of an angle is 1 minus its
8082:source code). Greenbelt, Maryland, USA:
6499:(1969) . "5: Commentary on the Paper of
4391:
2783:
1406:
1202:The first known English equivalent to a
858:) was sometimes historically called the
835:
357:
8401:Needham, Noel Joseph Terence Montgomery
6410:(1 ed.). Salt Lake City, UT, USA:
6157:
6046:
1348:. Likewise, the havercosine is used in
219:
181:
143:
112:
46:
8602:Trigonometric and hyperbolic functions
8174:
7875:
7841:
7182:Trigonometry: For Schools and Colleges
6601:(3 ed.). Mineola, New York, USA:
6329:"Background Notes on Measures: Angles"
6132:
6130:
6108:. Similarly, in their 1968/2000 work,
1278:
7916:(2 ed.). Starpath Publications.
5959:. This usage is especially common in
5712:is small in comparison to the radius
1285:appear to be of much younger origin.
7:
8377:Research & Education Association
7591:participating institution membership
6863:Springer Science+Business Media, LLC
6412:Springer International Publishing AG
6363:Springer Science+Business Media, LLC
5782:in the figure above) by the formula
4381:(archavercosin, archavercos, ahvc),
1367:) the haversine of the double-angle
974:is also an obsolete synonym for the
8375:. Problem Solvers Solution Guides.
7868:Hannyngton, John Caulfield (1876).
6901:
6714:A History of Mathematical Notations
6511:University of Wisconsin Press, Ltd.
4361:(arcversin, arcvers, avers, aver),
785:Sine, cosine, and versine of angle
8559:The Wolfram Demonstrations Project
8369:Woodward, Ernest (December 1978).
8125:JASS - Java Audio Synthesis System
7956:(Review). Prague, Czech Republic.
7712:Natural and Logarithmic Haversines
7500:Eenheden, constanten en conversies
7494:van Vlijmen, Oscar (2005-12-28) .
7122:Wentworth, George Albert (1903) .
6759:
6755:
5148:
5040:
4295:
4280:
4277:
4274:
4271:
4268:
4265:
4262:
4259:
4256:
4253:
4250:
4247:
4189:
4186:
4183:
4180:
4177:
4174:
4171:
4168:
4165:
4162:
4159:
4156:
4145:
4139:
4074:
4059:
4056:
4053:
4050:
4047:
4044:
4041:
4038:
4035:
4032:
3971:
3968:
3965:
3962:
3959:
3956:
3953:
3950:
3947:
3944:
3933:
3927:
3862:
3847:
3844:
3841:
3838:
3835:
3832:
3829:
3826:
3823:
3820:
3759:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3735:
3732:
3721:
3715:
3650:
3635:
3632:
3629:
3626:
3623:
3620:
3617:
3614:
3556:
3553:
3550:
3547:
3544:
3541:
3538:
3535:
3524:
3518:
3462:
3447:
3444:
3441:
3438:
3435:
3432:
3429:
3426:
3423:
3420:
3371:
3368:
3365:
3362:
3359:
3356:
3353:
3350:
3347:
3344:
3333:
3327:
3271:
3256:
3253:
3250:
3247:
3244:
3241:
3238:
3235:
3183:
3180:
3177:
3174:
3171:
3168:
3165:
3162:
3151:
3145:
3089:
3074:
3071:
3068:
3065:
3062:
3059:
3056:
3053:
3001:
2998:
2995:
2992:
2989:
2986:
2983:
2980:
2969:
2963:
2907:
2892:
2889:
2886:
2883:
2880:
2877:
2828:
2825:
2822:
2819:
2816:
2813:
2802:
2796:
2727:
2724:
2721:
2718:
2715:
2712:
2709:
2706:
2703:
2700:
2697:
2694:
2667:
2664:
2661:
2658:
2655:
2652:
2649:
2646:
2643:
2640:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2585:
2582:
2579:
2558:
2555:
2552:
2549:
2546:
2543:
2540:
2537:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2422:
2388:
2385:
2382:
2379:
2376:
2373:
2370:
2367:
2346:
2343:
2340:
2337:
2334:
2331:
978:(the horizontal axis of a graph).
963:as its "string", then the versine
828:as its "string", then the versine
30:For the archaic unit of mass, see
25:
8119:van den Doel, Kees (2010-01-25).
7658:. New Port Richey, Florida, USA.
6642:Calvert, James B. (2007-09-14) .
6093:
5992:, for describing the surfaces of
1235:in the third edition of his work
1160:) given angular positions (e.g.,
8154:"Lost but lovely: The haversine"
8084:NASA Goddard Space Flight Center
8074:Simpson, David G. (2001-11-08).
7821:
7799:
6622:
6057:
2301:
2196:
2059:
1954:
1817:
1712:
1609:
1504:
54:
8466:International Union of Railways
8440:Chicago Bridge and Iron Company
8323:from the original on 2008-06-08
8282:from the original on 2007-02-08
8252:from the original on 2015-11-28
8215:from the original on 2008-11-20
8164:from the original on 2014-07-18
8131:from the original on 2007-09-02
8090:from the original on 2008-06-16
8045:from the original on 2010-03-31
7960:from the original on 2004-01-12
7662:from the original on 2015-09-05
7547:from the original on 2014-03-29
7506:from the original on 2009-10-28
7455:from the original on 2014-03-29
7401:"Ultra compact sight reduction"
7378:from the original on 2014-09-01
7321:from the original on 2013-09-17
7280:from the original on 2005-03-10
7233:from the original on 2014-03-28
7073:from the original on 2014-03-29
7030:from the original on 2005-11-27
6985:from the original on 2014-03-24
6891:. 1441905499, 978-1-4419-0549-9
6787:from the original on 2015-09-19
6650:from the original on 2007-10-02
6571:from the original on 2003-06-17
6335:from the original on 2007-02-09
1371:describes the relation between
912:) (equal to the length of line
904:) (equal to the length of line
323:found in some of the earliest (
6811:(1821). "Analyse Algébrique".
6109:
5660:
5654:
5645:
5639:
5623:
5620:
5614:
5602:
5596:
5587:
5583:
5580:
5574:
5562:
5556:
5547:
5537:
5531:
5519:
5513:
5502:
5496:
5487:
5481:
5456:
5450:
5438:
5432:
5421:
5415:
5403:
5397:
5379:
5373:
5361:
5355:
5344:
5338:
5326:
5320:
5272:
5266:
5249:
5208:
5199:
5166:
5156:
5122:
5116:
5097:
5088:
5058:
5048:
5014:
5008:
4923:
4917:
4856:
4850:
4763:
4757:
4669:
4663:
4605:
4599:
4541:
4535:
4477:
4471:
4413:
4407:
4290:
4284:
4199:
4193:
4069:
4063:
3981:
3975:
3857:
3851:
3769:
3763:
3645:
3639:
3566:
3560:
3457:
3451:
3381:
3375:
3266:
3260:
3193:
3187:
3084:
3078:
3011:
3005:
2902:
2896:
2838:
2832:
2568:
2562:
2356:
2350:
2279:
2273:
2246:
2240:
2224:
2218:
2174:
2168:
2109:
2103:
2087:
2081:
2037:
2031:
2004:
1998:
1982:
1976:
1932:
1926:
1867:
1861:
1845:
1839:
1801:
1795:
1740:
1734:
1696:
1690:
1637:
1631:
1593:
1587:
1532:
1526:
1488:
1482:
1429:
1423:
1379:, a proposed reformulation of
967:is clearly the "arrow shaft".
27:1 minus the cosine of an angle
1:
8663:Jyā, koti-jyā and utkrama-jyā
8555:"Sagitta, Apothem, and Chord"
7732:10.1090/S0025-5718-45-99080-6
7622:(2 ed.). New York, USA:
7399:Rudzinski, Greg (July 2015).
6719:Open court publishing company
6551:"5 (Trigonometric Functions)"
6202:Ohio and Mississippi Railroad
6064:(versine) was defined as siv(
832:is clearly the "arrow shaft".
434:Expressed in terms of common
7952:Kalivoda, Jan (2003-07-30).
7858:table from 0° to 125°/135°.)
6824:access-date=2015-11-07-->
6562:Harcourt Brace & Company
6165:The Āryabhaṭīya by Āryabhaṭa
5724:shown above) by the formula
1073:) values (as opposed to the
8270:Ballew, Pat (2007-02-08) .
8236:Skvarc, Jure (2014-10-27).
8199:Skvarc, Jure (1999-03-01).
7654:Miller, Jeff (2007-09-10).
7128:(2 ed.). Boston, USA:
7097:(3 ed.). Boston, USA:
6748:(1821). See J. D. White in
6084:), and the vercosine as vcs
5967:and it is the basis of the
5909:Arbitrary curves and chords
1350:raised-cosine distributions
1075:chords tabulated by Ptolemy
793:with radius 1, centered at
8705:
8411:Cambridge University Press
6932:Princeton University Press
6922:van Brummelen, Glen Robert
6828:Cambridge University Press
6669:Edler von Braunmühl, Anton
5955:goes to the instantaneous
1148:because it appears in the
192:Trigonometric substitution
36:
29:
7935:(NB. Contains a table of
7910:Stark, Bruce D. (1997) .
7772:Oxford English Dictionary
7681:de Mendoza y Ríos, Joseph
7578:Oxford English Dictionary
6871:10.1007/978-1-4419-0549-9
6420:10.1007/978-3-319-64110-2
6375:10.1007/978-0-387-48807-3
4385:(archacoversin, ahcv) or
2780:Derivatives and integrals
1168:). One could also use sin
1158:Earth's radius vs. sphere
1037:catastrophic cancellation
8500:Hawking, Stephen William
8434:Boardman, Harry (1930).
8304:Weisstein, Eric Wolfgang
8121:"jass.utils Class Fmath"
8026:Weisstein, Eric Wolfgang
7834:Farley, Richard (1856).
7706:Archibald, Raymond Clare
7619:A History of Mathematics
7528:Weisstein, Eric Wolfgang
7436:Weisstein, Eric Wolfgang
7261:Weisstein, Eric Wolfgang
7214:Weisstein, Eric Wolfgang
7054:Weisstein, Eric Wolfgang
7011:Weisstein, Eric Wolfgang
6966:Weisstein, Eric Wolfgang
6603:Dover Publications, Inc.
6197:
6035:Natural logarithm plus 1
6010:Trigonometric identities
5934:goes to zero, the ratio
1324:), because it smoothly (
1195:An early utilization by
772:History and applications
105:Generalized trigonometry
8689:Trigonometric functions
8566:Trigonometric Functions
8246:University of Ljubljana
7983:Wildberger, Norman John
7892:table from 0° to 180°,
7777:Oxford University Press
7583:Oxford University Press
7347:D. Van Nostrand Company
6589:Korn, Grandino Arthur;
6142:Heaviside step function
6015:Exsecant and excosecant
5988:) is used similarly in
4357:Inverse functions like
1398:Mathematical identities
1221:base-10 logarithmically
674:, most famous from the
436:trigonometric functions
368:trigonometric functions
8318:Wolfram Research, Inc.
8040:Wolfram Research, Inc.
7746:Andrew, James (1805).
7542:Wolfram Research, Inc.
7450:Wolfram Research, Inc.
7313:Sauer, Frank (2015) .
7275:Wolfram Research, Inc.
7228:Wolfram Research, Inc.
7068:Wolfram Research, Inc.
7025:Wolfram Research, Inc.
6980:Wolfram Research, Inc.
6859:Cauchy, Augustin-Louis
6809:Cauchy, Augustin-Louis
6742:José de Mendoza y Rios
6476:Henry Holt and Company
6327:Tapson, Frank (2004).
6264:Zucker, Ruth (1983) .
5900:
5822:
5766:
5705:
5693:
5671:
5291:
5225:
5152:
5044:
4962:
4895:
4828:
4735:
4641:
4577:
4513:
4449:
4341:
4229:
4120:
4014:
3908:
3802:
3696:
3596:
3499:
3402:
3308:
3217:
3126:
3035:
2944:
2859:
2770:
2293:
2188:
2051:
1946:
1809:
1704:
1601:
1496:
1392:Norman John Wildberger
1241:spherical trigonometry
1215:(notated naturally as
1197:José de Mendoza y Ríos
1017:
993:) for the versine and
845:
833:
534:
389:trigonometric function
371:
321:trigonometric function
8627:Inverse trigonometric
8510:. Philadelphia, USA:
8454:Nair, P. N. Bhaskaran
7900:at intervals of 10".)
7896:at intervals of 15",
7624:John Wiley & Sons
7155:The Macmillan Company
7099:John Wiley & Sons
6331:. 1.4. Cleave Books.
6112:associate the covers(
6100:(and cosiv) with the
5901:
5823:
5767:
5699:
5688:
5672:
5292:
5226:
5132:
5024:
4963:
4896:
4829:
4736:
4642:
4578:
4514:
4450:
4342:
4230:
4121:
4015:
3909:
3803:
3697:
3597:
3500:
3403:
3309:
3218:
3127:
3036:
2945:
2860:
2771:
2294:
2189:
2052:
1947:
1810:
1705:
1602:
1497:
1377:rational trigonometry
1156:(see issues with the
1027:goes to zero, versin(
1007:
1001:) for the coversine.
856:see note on etymology
839:
784:
777:Versine and coversine
678:used historically in
535:
415:, it is known as the
361:
8379:(REA). p. 359.
8152:mf344 (2014-07-04).
7610:Boyer, Carl Benjamin
6865:. pp. 10, 285.
6781:University of Hawaii
6497:Boyer, Carl Benjamin
6116:) function with the
6094:Bradley and Sandifer
6080:) = 1−sin(
6068:) = 1−cos(
5930:as the chord length
5843:
5786:
5728:
5304:
5238:
4995:
4908:
4841:
4748:
4654:
4590:
4526:
4462:
4398:
4240:
4133:
4025:
3921:
3813:
3709:
3607:
3512:
3413:
3321:
3228:
3139:
3046:
2957:
2870:
2790:
2323:
2208:
2071:
1966:
1829:
1724:
1621:
1516:
1413:
1346:raised-cosine filter
947:of the double-angle
809:of the double-angle
442:
334:trigonometric tables
212:Trigonometric series
8308:"Inverse Haversine"
7937:Gaussian logarithms
7581:(Online ed.).
6819:Bibliothèque du Roi
6775:Shaneyfelt, Ted V.
6088: = 1+cos(
6030:Exponential minus 1
5984:(often abbreviated
1363:In the form of sin(
1288:One period (0 <
1263:Gaussian logarithms
1204:table of haversines
1033:trigonometric table
174:Pythagorean theorem
8637:Inverse hyperbolic
8458:Rail International
8345:"InverseHaversine"
8276:Math Words, page 4
7820:(NB. According to
7798:(NB. According to
7342:Plane Trigonometry
7125:Plane Trigonometry
6513:pp. 185–190.
6501:E. J. Dijksterhuis
6270:Abramowitz, Milton
5896:
5818:
5762:
5706:
5694:
5667:
5665:
5287:
5256:
5221:
5219:
4958:
4891:
4824:
4731:
4637:
4573:
4509:
4445:
4337:
4225:
4116:
4010:
3904:
3798:
3692:
3592:
3495:
3398:
3304:
3213:
3122:
3031:
2940:
2855:
2766:
2764:
2312:Circular rotations
2289:
2184:
2047:
1942:
1805:
1700:
1597:
1492:
1354:probability theory
1332:) "turns on" from
1304:as the shape of a
1209:In 1835, the term
1086:half-angle formula
1064:logarithmic tables
1018:
888:) is the distance
846:
834:
530:
372:
8676:
8675:
8386:978-0-87891-510-1
7815:Nautical Magazine
7793:Nautical Magazine
7589:(Subscription or
7345:. New York, USA:
7153:. New York, USA:
6880:978-1-4419-0548-2
6836:978-1-108-00208-0
6751:Nautical Magazine
6728:978-1-60206-714-1
6616:978-0-486-41147-7
6429:978-3-319-64109-6
6384:978-0-387-48806-6
6289:978-0-486-61272-0
6274:Stegun, Irene Ann
6242:(Fourth edition:
6136:The abbreviation
5894:
5882:
5867:
5816:
5757:
5708:When the versine
5541:
5460:
5383:
5279:
5241:
5215:
5104:
4971:
4970:
4787:
4693:
4353:Inverse functions
4350:
4349:
4329:
4223:
4153:
4108:
4008:
3941:
3896:
3796:
3729:
3684:
3590:
3532:
3341:
3159:
2977:
2810:
2755:
2629:
2525:
2411:
2309:
2308:
2286:
2253:
2237:
2215:
2181:
2144:
2116:
2100:
2078:
2044:
2011:
1995:
1973:
1939:
1902:
1874:
1858:
1836:
1767:
1750:
1731:
1669:
1628:
1559:
1542:
1523:
1461:
1420:
1298:signal processing
1279:inverse functions
1150:haversine formula
746:hacoversed cosine
676:haversine formula
528:
499:
353:haversine formula
309:
308:
201:inverse functions
144:Laws and theorems
16:(Redirected from
8696:
8596:
8589:
8582:
8573:
8562:
8539:
8537:
8536:
8509:
8486:
8485:
8450:
8444:
8443:
8431:
8425:
8424:
8397:
8391:
8390:
8366:
8360:
8359:
8357:
8356:
8341:
8332:
8331:
8329:
8328:
8300:
8291:
8290:
8288:
8287:
8267:
8261:
8260:
8258:
8257:
8233:
8224:
8223:
8221:
8220:
8196:
8187:
8186:
8180:
8172:
8170:
8169:
8149:
8140:
8139:
8137:
8136:
8116:
8099:
8098:
8096:
8095:
8071:
8054:
8053:
8051:
8050:
8022:
8011:
8010:
8008:
8007:
7992:
7979:
7973:
7968:
7966:
7965:
7949:
7943:
7934:
7932:
7931:
7907:
7901:
7887:
7881:
7873:
7865:
7859:
7853:
7847:
7839:
7831:
7825:
7819:
7809:
7803:
7797:
7787:
7781:
7780:
7775:(2nd ed.).
7766:
7757:
7751:
7743:
7737:
7736:
7734:
7702:
7691:
7690:
7677:
7671:
7670:
7668:
7667:
7651:
7645:
7644:
7642:
7641:
7614:Merzbach, Uta C.
7606:
7595:
7594:
7586:
7574:
7567:
7556:
7555:
7553:
7552:
7524:
7515:
7514:
7512:
7511:
7491:
7464:
7463:
7461:
7460:
7432:
7423:
7422:
7420:
7419:
7396:
7387:
7386:
7384:
7383:
7364:
7358:
7357:
7355:
7354:
7336:
7330:
7329:
7327:
7326:
7310:
7304:
7303:
7300:Nautische Tafeln
7295:
7289:
7288:
7286:
7285:
7257:
7242:
7241:
7239:
7238:
7210:
7201:
7200:
7198:
7197:
7187:Ginn and Company
7176:
7170:
7169:
7167:
7166:
7144:
7138:
7137:
7130:Ginn and Company
7119:
7113:
7112:
7110:
7109:
7088:
7082:
7081:
7079:
7078:
7050:
7039:
7038:
7036:
7035:
7007:
6994:
6993:
6991:
6990:
6962:
6953:
6952:
6950:
6949:
6918:
6905:
6899:
6897:
6896:
6848:
6839:
6822:
6805:
6796:
6795:
6793:
6792:
6779:. Hilo, Hawaii:
6772:
6766:
6764:
6737:
6736:
6705:
6694:
6693:
6691:
6690:
6665:
6659:
6658:
6656:
6655:
6639:
6626:
6620:
6600:
6591:Korn, Theresa M.
6586:
6580:
6579:
6577:
6576:
6570:
6555:
6546:
6540:
6539:
6537:
6536:
6493:
6487:
6486:
6484:
6483:
6465:
6450:
6449:
6403:
6397:
6396:
6360:
6350:
6344:
6343:
6341:
6340:
6324:
6318:
6317:
6261:
6246:
6241:
6239:
6238:
6221:
6210:
6206:
6194:external secants
6189:
6188:
6173:
6167:
6162:
6145:
6134:
6125:
6054:
5954:
5952:
5951:
5946:
5943:
5905:
5903:
5902:
5897:
5895:
5893:
5885:
5884:
5883:
5875:
5869:
5868:
5860:
5853:
5827:
5825:
5824:
5819:
5817:
5812:
5811:
5802:
5771:
5769:
5768:
5763:
5758:
5756:
5748:
5747:
5738:
5676:
5674:
5673:
5668:
5666:
5542:
5540:
5505:
5470:
5461:
5459:
5424:
5389:
5384:
5382:
5347:
5312:
5296:
5294:
5293:
5288:
5280:
5275:
5258:
5255:
5230:
5228:
5227:
5222:
5220:
5216:
5214:
5194:
5193:
5192:
5180:
5179:
5154:
5151:
5146:
5105:
5103:
5086:
5085:
5084:
5072:
5071:
5046:
5043:
5038:
4986:Maclaurin series
4974:Other properties
4967:
4965:
4964:
4959:
4956:
4952:
4900:
4898:
4897:
4892:
4889:
4885:
4833:
4831:
4830:
4825:
4823:
4819:
4792:
4788:
4783:
4740:
4738:
4737:
4732:
4729:
4725:
4698:
4694:
4689:
4646:
4644:
4643:
4638:
4635:
4631:
4582:
4580:
4579:
4574:
4571:
4567:
4518:
4516:
4515:
4510:
4507:
4503:
4454:
4452:
4451:
4446:
4443:
4439:
4392:
4387:archacovercosine
4346:
4344:
4343:
4338:
4330:
4325:
4324:
4306:
4298:
4283:
4234:
4232:
4231:
4226:
4224:
4219:
4218:
4206:
4192:
4154:
4152:
4148:
4142:
4137:
4125:
4123:
4122:
4117:
4109:
4104:
4103:
4085:
4077:
4062:
4019:
4017:
4016:
4011:
4009:
4004:
4003:
3988:
3974:
3942:
3940:
3936:
3930:
3925:
3913:
3911:
3910:
3905:
3897:
3892:
3891:
3873:
3865:
3850:
3807:
3805:
3804:
3799:
3797:
3792:
3791:
3776:
3762:
3730:
3728:
3724:
3718:
3713:
3701:
3699:
3698:
3693:
3685:
3680:
3679:
3661:
3653:
3638:
3601:
3599:
3598:
3593:
3591:
3586:
3585:
3573:
3559:
3533:
3531:
3527:
3521:
3516:
3504:
3502:
3501:
3496:
3488:
3465:
3450:
3407:
3405:
3404:
3399:
3397:
3374:
3342:
3340:
3336:
3330:
3325:
3313:
3311:
3310:
3305:
3297:
3274:
3259:
3222:
3220:
3219:
3214:
3212:
3186:
3160:
3158:
3154:
3148:
3143:
3131:
3129:
3128:
3123:
3115:
3092:
3077:
3040:
3038:
3037:
3032:
3030:
3004:
2978:
2976:
2972:
2966:
2961:
2949:
2947:
2946:
2941:
2933:
2910:
2895:
2864:
2862:
2861:
2856:
2854:
2831:
2811:
2809:
2805:
2799:
2794:
2784:
2775:
2773:
2772:
2767:
2765:
2761:
2757:
2756:
2751:
2743:
2730:
2689:
2685:
2670:
2635:
2631:
2630:
2622:
2609:
2561:
2531:
2527:
2526:
2521:
2513:
2500:
2465:
2461:
2446:
2417:
2413:
2412:
2404:
2391:
2349:
2305:
2298:
2296:
2295:
2290:
2287:
2282:
2259:
2254:
2249:
2239:
2238:
2235:
2231:
2217:
2216:
2213:
2200:
2193:
2191:
2190:
2185:
2182:
2177:
2154:
2149:
2145:
2137:
2130:
2129:
2117:
2112:
2102:
2101:
2098:
2094:
2080:
2079:
2076:
2063:
2056:
2054:
2053:
2048:
2045:
2040:
2017:
2012:
2007:
1997:
1996:
1993:
1989:
1975:
1974:
1971:
1958:
1951:
1949:
1948:
1943:
1940:
1935:
1912:
1907:
1903:
1895:
1888:
1887:
1875:
1870:
1860:
1859:
1856:
1852:
1838:
1837:
1834:
1821:
1814:
1812:
1811:
1806:
1779:
1775:
1768:
1760:
1752:
1751:
1748:
1733:
1732:
1729:
1716:
1709:
1707:
1706:
1701:
1674:
1670:
1662:
1655:
1654:
1630:
1629:
1626:
1613:
1606:
1604:
1603:
1598:
1571:
1567:
1560:
1552:
1544:
1543:
1540:
1525:
1524:
1521:
1508:
1501:
1499:
1498:
1493:
1466:
1462:
1454:
1447:
1446:
1422:
1421:
1418:
1407:
1388:solid geometries
1231:) was coined by
1191:
1188:
1186:
1185:
1182:
1179:
1171:
1127:
1125:
1124:
1121:
1118:
1111:
1108:
1106:
1105:
1102:
1099:
1091:
1079:Surya Siddhantha
1061:
1041:round-off errors
959:" and the chord
955:is viewed as a "
916:) is the radius
896:is the distance
817:is viewed as a "
767:
763:
759:
741:
737:
733:
729:
725:
707:
703:
699:
695:
673:
669:
665:
661:
657:
653:
649:
645:
622:
618:
614:
600:
596:
592:
588:
565:
561:
557:
539:
537:
536:
531:
529:
521:
500:
492:
487:
486:
419:(flipped sine),
410:
406:
402:
398:
394:
301:
294:
287:
58:
44:
39:Versin (village)
32:Versine in WP:de
21:
8704:
8703:
8699:
8698:
8697:
8695:
8694:
8693:
8679:
8678:
8677:
8672:
8641:
8620:Sine and cosine
8603:
8600:
8568:at GeoGebra.org
8549:
8546:
8534:
8532:
8522:
8498:
8495:
8493:Further reading
8490:
8489:
8452:
8451:
8447:
8433:
8432:
8428:
8421:
8409:. Vol. 3.
8399:
8398:
8394:
8387:
8368:
8367:
8363:
8354:
8352:
8343:
8342:
8335:
8326:
8324:
8302:
8301:
8294:
8285:
8283:
8269:
8268:
8264:
8255:
8253:
8235:
8234:
8227:
8218:
8216:
8198:
8197:
8190:
8173:
8167:
8165:
8151:
8150:
8143:
8134:
8132:
8118:
8117:
8102:
8093:
8091:
8073:
8072:
8057:
8048:
8046:
8024:
8023:
8014:
8005:
8003:
8001:
7981:
7980:
7976:
7963:
7961:
7951:
7950:
7946:
7929:
7927:
7924:
7909:
7908:
7904:
7898:nat. haversines
7894:log. haversines
7874:
7867:
7866:
7862:
7840:
7833:
7832:
7828:
7811:
7810:
7806:
7789:
7788:
7784:
7768:
7767:
7760:
7745:
7744:
7740:
7725:(11): 421–422.
7704:
7703:
7694:
7679:
7678:
7674:
7665:
7663:
7653:
7652:
7648:
7639:
7637:
7634:
7616:(1991-03-06) .
7608:
7607:
7598:
7588:
7569:
7568:
7559:
7550:
7548:
7532:"Hacovercosine"
7526:
7525:
7518:
7509:
7507:
7493:
7492:
7467:
7458:
7456:
7434:
7433:
7426:
7417:
7415:
7405:Ocean Navigator
7398:
7397:
7390:
7381:
7379:
7366:
7365:
7361:
7352:
7350:
7338:
7337:
7333:
7324:
7322:
7312:
7311:
7307:
7297:
7296:
7292:
7283:
7281:
7259:
7258:
7245:
7236:
7234:
7212:
7211:
7204:
7195:
7193:
7185:. Boston, USA:
7178:
7177:
7173:
7164:
7162:
7146:
7145:
7141:
7121:
7120:
7116:
7107:
7105:
7090:
7089:
7085:
7076:
7074:
7052:
7051:
7042:
7033:
7031:
7009:
7008:
6997:
6988:
6986:
6964:
6963:
6956:
6947:
6945:
6942:
6920:
6919:
6908:
6894:
6892:
6881:
6850:
6849:
6842:
6807:
6806:
6799:
6790:
6788:
6774:
6773:
6769:
6734:
6732:
6729:
6721:. p. 172.
6709:Cajori, Florian
6707:
6706:
6697:
6688:
6686:
6667:
6666:
6662:
6653:
6651:
6641:
6640:
6629:
6617:
6588:
6587:
6583:
6574:
6572:
6568:
6564:. p. 344.
6553:
6548:
6547:
6543:
6534:
6532:
6531:. 9780299018740
6521:
6495:
6494:
6490:
6481:
6479:
6467:
6466:
6453:
6430:
6414:. p. 301.
6405:
6404:
6400:
6385:
6352:
6351:
6347:
6338:
6336:
6326:
6325:
6321:
6290:
6263:
6262:
6249:
6236:
6234:
6223:
6222:
6213:
6186:
6184:
6175:
6174:
6170:
6163:
6159:
6154:
6149:
6148:
6135:
6128:
6120:instead of the
6055:
6048:
6043:
6024:Witch of Agnesi
6006:
5947:
5944:
5938:
5937:
5935:
5911:
5886:
5870:
5855:
5854:
5841:
5840:
5803:
5784:
5783:
5749:
5739:
5726:
5725:
5683:
5664:
5663:
5626:
5544:
5543:
5506:
5471:
5462:
5425:
5390:
5348:
5313:
5302:
5301:
5259:
5236:
5235:
5218:
5217:
5195:
5181:
5165:
5155:
5125:
5107:
5106:
5087:
5073:
5057:
5047:
5017:
4993:
4992:
4976:
4939:
4935:
4906:
4905:
4872:
4868:
4839:
4838:
4806:
4802:
4778:
4746:
4745:
4712:
4708:
4684:
4652:
4651:
4621:
4617:
4588:
4587:
4557:
4553:
4524:
4523:
4493:
4489:
4460:
4459:
4429:
4425:
4396:
4395:
4355:
4307:
4238:
4237:
4207:
4143:
4131:
4130:
4086:
4023:
4022:
3989:
3931:
3919:
3918:
3874:
3811:
3810:
3777:
3719:
3707:
3706:
3662:
3605:
3604:
3574:
3522:
3510:
3509:
3411:
3410:
3331:
3319:
3318:
3226:
3225:
3149:
3137:
3136:
3044:
3043:
2967:
2955:
2954:
2868:
2867:
2800:
2788:
2787:
2782:
2763:
2762:
2744:
2735:
2731:
2675:
2671:
2614:
2610:
2571:
2533:
2532:
2514:
2505:
2501:
2451:
2447:
2396:
2392:
2359:
2321:
2320:
2314:
2260:
2232:
2206:
2205:
2155:
2132:
2121:
2095:
2069:
2068:
2018:
1990:
1964:
1963:
1913:
1890:
1879:
1853:
1827:
1826:
1758:
1754:
1722:
1721:
1657:
1646:
1619:
1618:
1550:
1546:
1514:
1513:
1449:
1438:
1411:
1410:
1405:
1400:
1310:window function
1275:
1267:sight reduction
1259:lunar distances
1189:
1183:
1180:
1175:
1174:
1172:
1169:
1142:
1122:
1119:
1116:
1115:
1113:
1109:
1103:
1100:
1095:
1094:
1092:
1089:
1059:
985:used the terms
932:to the circle.
873:For a vertical
779:
774:
765:
761:
757:
739:
735:
731:
727:
723:
712:hacoversed sine
705:
701:
697:
693:
686:haversed cosine
671:
667:
663:
659:
655:
651:
647:
643:
642:), abbreviated
620:
616:
612:
605:coversed cosine
598:
594:
590:
586:
585:), abbreviated
563:
559:
555:
478:
440:
439:
408:
404:
400:
396:
392:
377:
355:of navigation.
305:
126:Exact constants
42:
35:
28:
23:
22:
15:
12:
11:
5:
8702:
8700:
8692:
8691:
8681:
8680:
8674:
8673:
8671:
8670:
8665:
8660:
8655:
8649:
8647:
8643:
8642:
8640:
8639:
8634:
8629:
8624:
8623:
8622:
8611:
8609:
8605:
8604:
8601:
8599:
8598:
8591:
8584:
8576:
8570:
8569:
8563:
8545:
8544:External links
8542:
8541:
8540:
8520:
8502:, ed. (2002).
8494:
8491:
8488:
8487:
8445:
8426:
8419:
8413:. p. 39.
8392:
8385:
8361:
8333:
8292:
8262:
8225:
8211:source code).
8188:
8141:
8100:
8055:
8012:
7999:
7974:
7944:
7923:978-0914025214
7922:
7902:
7860:
7826:
7804:
7782:
7758:
7738:
7692:
7672:
7646:
7633:978-0471543978
7632:
7596:
7557:
7516:
7465:
7424:
7388:
7359:
7331:
7305:
7290:
7243:
7202:
7171:
7139:
7114:
7083:
7040:
6995:
6954:
6940:
6906:
6879:
6840:
6797:
6767:
6727:
6695:
6660:
6644:"Trigonometry"
6627:
6615:
6581:
6541:
6519:
6488:
6451:
6428:
6398:
6383:
6361:(2 ed.).
6345:
6319:
6288:
6247:
6211:
6168:
6156:
6155:
6153:
6150:
6147:
6146:
6126:
6098:cosinus versus
6096:associate the
6074:cosinus versus
6045:
6044:
6042:
6039:
6038:
6037:
6032:
6027:
6017:
6012:
6005:
6002:
5973:rail surveying
5969:Hallade method
5961:rail transport
5910:
5907:
5892:
5889:
5881:
5878:
5873:
5866:
5863:
5858:
5851:
5848:
5815:
5810:
5806:
5800:
5797:
5794:
5791:
5761:
5755:
5752:
5746:
5742:
5736:
5733:
5720:(the distance
5682:
5681:Approximations
5679:
5678:
5677:
5662:
5659:
5656:
5653:
5650:
5647:
5644:
5641:
5638:
5635:
5632:
5629:
5627:
5625:
5622:
5619:
5616:
5613:
5610:
5607:
5604:
5601:
5598:
5595:
5592:
5589:
5585:
5582:
5579:
5576:
5573:
5570:
5567:
5564:
5561:
5558:
5555:
5552:
5549:
5546:
5545:
5539:
5536:
5533:
5530:
5527:
5524:
5521:
5518:
5515:
5512:
5509:
5504:
5501:
5498:
5495:
5492:
5489:
5486:
5483:
5480:
5477:
5474:
5468:
5465:
5463:
5458:
5455:
5452:
5449:
5446:
5443:
5440:
5437:
5434:
5431:
5428:
5423:
5420:
5417:
5414:
5411:
5408:
5405:
5402:
5399:
5396:
5393:
5387:
5381:
5378:
5375:
5372:
5369:
5366:
5363:
5360:
5357:
5354:
5351:
5346:
5343:
5340:
5337:
5334:
5331:
5328:
5325:
5322:
5319:
5316:
5310:
5309:
5298:
5297:
5286:
5283:
5278:
5274:
5271:
5268:
5265:
5262:
5254:
5251:
5248:
5244:
5232:
5231:
5213:
5210:
5207:
5204:
5201:
5198:
5191:
5188:
5184:
5178:
5175:
5172:
5168:
5164:
5161:
5158:
5150:
5145:
5142:
5139:
5135:
5131:
5128:
5126:
5124:
5121:
5118:
5115:
5112:
5109:
5108:
5102:
5099:
5096:
5093:
5090:
5083:
5080:
5076:
5070:
5067:
5064:
5060:
5056:
5053:
5050:
5042:
5037:
5034:
5031:
5027:
5023:
5020:
5018:
5016:
5013:
5010:
5007:
5004:
5001:
5000:
4975:
4972:
4969:
4968:
4955:
4951:
4948:
4945:
4942:
4938:
4934:
4931:
4928:
4925:
4922:
4919:
4916:
4913:
4902:
4901:
4888:
4884:
4881:
4878:
4875:
4871:
4867:
4864:
4861:
4858:
4855:
4852:
4849:
4846:
4835:
4834:
4822:
4818:
4815:
4812:
4809:
4805:
4801:
4798:
4795:
4791:
4786:
4781:
4777:
4774:
4771:
4768:
4765:
4762:
4759:
4756:
4753:
4742:
4741:
4728:
4724:
4721:
4718:
4715:
4711:
4707:
4704:
4701:
4697:
4692:
4687:
4683:
4680:
4677:
4674:
4671:
4668:
4665:
4662:
4659:
4648:
4647:
4634:
4630:
4627:
4624:
4620:
4616:
4613:
4610:
4607:
4604:
4601:
4598:
4595:
4584:
4583:
4570:
4566:
4563:
4560:
4556:
4552:
4549:
4546:
4543:
4540:
4537:
4534:
4531:
4520:
4519:
4506:
4502:
4499:
4496:
4492:
4488:
4485:
4482:
4479:
4476:
4473:
4470:
4467:
4456:
4455:
4442:
4438:
4435:
4432:
4428:
4424:
4421:
4418:
4415:
4412:
4409:
4406:
4403:
4383:archacoversine
4379:archavercosine
4371:arccovercosine
4354:
4351:
4348:
4347:
4336:
4333:
4328:
4323:
4319:
4316:
4313:
4310:
4304:
4301:
4297:
4292:
4289:
4286:
4282:
4279:
4276:
4273:
4270:
4267:
4264:
4261:
4258:
4255:
4252:
4249:
4245:
4235:
4222:
4217:
4213:
4210:
4204:
4201:
4198:
4195:
4191:
4188:
4185:
4182:
4179:
4176:
4173:
4170:
4167:
4164:
4161:
4158:
4151:
4147:
4141:
4127:
4126:
4115:
4112:
4107:
4102:
4098:
4095:
4092:
4089:
4083:
4080:
4076:
4071:
4068:
4065:
4061:
4058:
4055:
4052:
4049:
4046:
4043:
4040:
4037:
4034:
4030:
4020:
4007:
4002:
3998:
3995:
3992:
3986:
3983:
3980:
3977:
3973:
3970:
3967:
3964:
3961:
3958:
3955:
3952:
3949:
3946:
3939:
3935:
3929:
3915:
3914:
3903:
3900:
3895:
3890:
3886:
3883:
3880:
3877:
3871:
3868:
3864:
3859:
3856:
3853:
3849:
3846:
3843:
3840:
3837:
3834:
3831:
3828:
3825:
3822:
3818:
3808:
3795:
3790:
3786:
3783:
3780:
3774:
3771:
3768:
3765:
3761:
3758:
3755:
3752:
3749:
3746:
3743:
3740:
3737:
3734:
3727:
3723:
3717:
3703:
3702:
3691:
3688:
3683:
3678:
3674:
3671:
3668:
3665:
3659:
3656:
3652:
3647:
3644:
3641:
3637:
3634:
3631:
3628:
3625:
3622:
3619:
3616:
3612:
3602:
3589:
3584:
3580:
3577:
3571:
3568:
3565:
3562:
3558:
3555:
3552:
3549:
3546:
3543:
3540:
3537:
3530:
3526:
3520:
3506:
3505:
3494:
3491:
3487:
3483:
3480:
3477:
3474:
3471:
3468:
3464:
3459:
3456:
3453:
3449:
3446:
3443:
3440:
3437:
3434:
3431:
3428:
3425:
3422:
3418:
3408:
3396:
3392:
3389:
3386:
3383:
3380:
3377:
3373:
3370:
3367:
3364:
3361:
3358:
3355:
3352:
3349:
3346:
3339:
3335:
3329:
3315:
3314:
3303:
3300:
3296:
3292:
3289:
3286:
3283:
3280:
3277:
3273:
3268:
3265:
3262:
3258:
3255:
3252:
3249:
3246:
3243:
3240:
3237:
3233:
3223:
3211:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3185:
3182:
3179:
3176:
3173:
3170:
3167:
3164:
3157:
3153:
3147:
3133:
3132:
3121:
3118:
3114:
3110:
3107:
3104:
3101:
3098:
3095:
3091:
3086:
3083:
3080:
3076:
3073:
3070:
3067:
3064:
3061:
3058:
3055:
3051:
3041:
3029:
3025:
3022:
3019:
3016:
3013:
3010:
3007:
3003:
3000:
2997:
2994:
2991:
2988:
2985:
2982:
2975:
2971:
2965:
2951:
2950:
2939:
2936:
2932:
2928:
2925:
2922:
2919:
2916:
2913:
2909:
2904:
2901:
2898:
2894:
2891:
2888:
2885:
2882:
2879:
2875:
2865:
2853:
2849:
2846:
2843:
2840:
2837:
2834:
2830:
2827:
2824:
2821:
2818:
2815:
2808:
2804:
2798:
2781:
2778:
2777:
2776:
2760:
2754:
2750:
2747:
2741:
2738:
2734:
2729:
2726:
2723:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2692:
2688:
2684:
2681:
2678:
2674:
2669:
2666:
2663:
2660:
2657:
2654:
2651:
2648:
2645:
2642:
2638:
2634:
2628:
2625:
2620:
2617:
2613:
2608:
2605:
2602:
2599:
2596:
2593:
2590:
2587:
2584:
2581:
2577:
2574:
2572:
2570:
2567:
2564:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2539:
2535:
2534:
2530:
2524:
2520:
2517:
2511:
2508:
2504:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2472:
2468:
2464:
2460:
2457:
2454:
2450:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2420:
2416:
2410:
2407:
2402:
2399:
2395:
2390:
2387:
2384:
2381:
2378:
2375:
2372:
2369:
2365:
2362:
2360:
2358:
2355:
2352:
2348:
2345:
2342:
2339:
2336:
2333:
2329:
2328:
2313:
2310:
2307:
2306:
2299:
2285:
2281:
2278:
2275:
2272:
2269:
2266:
2263:
2257:
2252:
2248:
2245:
2242:
2229:
2226:
2223:
2220:
2202:
2201:
2194:
2180:
2176:
2173:
2170:
2167:
2164:
2161:
2158:
2152:
2148:
2143:
2140:
2135:
2128:
2124:
2120:
2115:
2111:
2108:
2105:
2092:
2089:
2086:
2083:
2065:
2064:
2057:
2043:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2015:
2010:
2006:
2003:
2000:
1987:
1984:
1981:
1978:
1960:
1959:
1952:
1938:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1910:
1906:
1901:
1898:
1893:
1886:
1882:
1878:
1873:
1869:
1866:
1863:
1850:
1847:
1844:
1841:
1823:
1822:
1815:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1778:
1774:
1771:
1766:
1763:
1757:
1745:
1742:
1739:
1736:
1718:
1717:
1710:
1698:
1695:
1692:
1689:
1686:
1683:
1680:
1677:
1673:
1668:
1665:
1660:
1653:
1649:
1645:
1642:
1639:
1636:
1633:
1615:
1614:
1607:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1570:
1566:
1563:
1558:
1555:
1549:
1537:
1534:
1531:
1528:
1510:
1509:
1502:
1490:
1487:
1484:
1481:
1478:
1475:
1472:
1469:
1465:
1460:
1457:
1452:
1445:
1441:
1437:
1434:
1431:
1428:
1425:
1404:
1401:
1399:
1396:
1375:and angles in
1302:control theory
1274:
1271:
1251:for versines.
1225:log. haversine
1141:
1138:
1132:), derived by
995:cosinus versus
789:in terms of a
778:
775:
773:
770:
769:
768:
756:, abbreviated
742:
722:, abbreviated
708:
692:, abbreviated
682:
624:
623:
611:, abbreviated
601:
579:cosinus versus
567:
554:, abbreviated
527:
524:
519:
516:
512:
509:
506:
503:
498:
495:
490:
485:
481:
477:
474:
471:
468:
465:
462:
459:
456:
453:
450:
447:
376:
373:
307:
306:
304:
303:
296:
289:
281:
278:
277:
276:
275:
270:
265:
260:
255:
250:
245:
240:
235:
230:
222:
221:
220:Mathematicians
217:
216:
215:
214:
209:
204:
194:
186:
185:
179:
178:
177:
176:
170:
169:
164:
159:
154:
146:
145:
141:
140:
139:
138:
133:
128:
123:
115:
114:
110:
109:
108:
107:
102:
79:
78:
73:
68:
60:
59:
51:
50:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8701:
8690:
8687:
8686:
8684:
8669:
8666:
8664:
8661:
8659:
8656:
8654:
8651:
8650:
8648:
8644:
8638:
8635:
8633:
8630:
8628:
8625:
8621:
8618:
8617:
8616:
8615:Trigonometric
8613:
8612:
8610:
8606:
8597:
8592:
8590:
8585:
8583:
8578:
8577:
8574:
8567:
8564:
8560:
8556:
8552:
8551:Pegg, Jr., Ed
8548:
8547:
8543:
8531:
8527:
8523:
8521:0-7624-1698-X
8517:
8513:
8512:Running Press
8508:
8507:
8501:
8497:
8496:
8492:
8483:
8479:
8475:
8471:
8467:
8463:
8459:
8455:
8449:
8446:
8442:. p. 32.
8441:
8437:
8430:
8427:
8422:
8420:9780521058018
8416:
8412:
8408:
8407:
8402:
8396:
8393:
8388:
8382:
8378:
8374:
8373:
8365:
8362:
8350:
8346:
8340:
8338:
8334:
8322:
8319:
8315:
8314:
8309:
8305:
8299:
8297:
8293:
8281:
8277:
8273:
8266:
8263:
8251:
8247:
8243:
8239:
8232:
8230:
8226:
8214:
8210:
8206:
8202:
8195:
8193:
8189:
8184:
8178:
8163:
8160:. maths.org.
8159:
8158:Plus magazine
8155:
8148:
8146:
8142:
8130:
8126:
8122:
8115:
8113:
8111:
8109:
8107:
8105:
8101:
8089:
8085:
8081:
8077:
8070:
8068:
8066:
8064:
8062:
8060:
8056:
8044:
8041:
8037:
8036:
8031:
8027:
8021:
8019:
8017:
8013:
8002:
8000:0-9757492-0-X
7996:
7991:
7990:
7984:
7978:
7975:
7972:
7970:
7959:
7955:
7948:
7945:
7941:
7938:
7925:
7919:
7915:
7914:
7906:
7903:
7899:
7895:
7891:
7885:
7879:
7871:
7864:
7861:
7857:
7851:
7845:
7837:
7830:
7827:
7823:
7817:
7816:
7808:
7805:
7801:
7795:
7794:
7786:
7783:
7778:
7774:
7773:
7769:"haversine".
7765:
7763:
7759:
7755:
7749:
7742:
7739:
7733:
7728:
7724:
7720:
7719:
7715:
7713:
7707:
7701:
7699:
7697:
7693:
7688:
7687:
7682:
7676:
7673:
7661:
7657:
7650:
7647:
7635:
7629:
7625:
7621:
7620:
7615:
7611:
7605:
7603:
7601:
7597:
7592:
7584:
7580:
7579:
7573:
7566:
7564:
7562:
7558:
7546:
7543:
7539:
7538:
7533:
7529:
7523:
7521:
7517:
7505:
7501:
7497:
7490:
7488:
7486:
7484:
7482:
7480:
7478:
7476:
7474:
7472:
7470:
7466:
7454:
7451:
7447:
7446:
7441:
7440:"Havercosine"
7437:
7431:
7429:
7425:
7414:
7410:
7406:
7402:
7395:
7393:
7389:
7377:
7374:. 7.0. 2008.
7373:
7369:
7363:
7360:
7348:
7344:
7343:
7335:
7332:
7320:
7316:
7309:
7306:
7301:
7294:
7291:
7279:
7276:
7272:
7271:
7266:
7262:
7256:
7254:
7252:
7250:
7248:
7244:
7232:
7229:
7225:
7224:
7219:
7218:"Covercosine"
7215:
7209:
7207:
7203:
7192:
7188:
7184:
7183:
7175:
7172:
7160:
7156:
7152:
7151:
7143:
7140:
7135:
7131:
7127:
7126:
7118:
7115:
7104:
7100:
7096:
7095:
7087:
7084:
7072:
7069:
7065:
7064:
7059:
7058:"Hacoversine"
7055:
7049:
7047:
7045:
7041:
7029:
7026:
7022:
7021:
7016:
7012:
7006:
7004:
7002:
7000:
6996:
6984:
6981:
6977:
6976:
6971:
6967:
6961:
6959:
6955:
6943:
6941:9780691148922
6937:
6933:
6929:
6928:
6923:
6917:
6915:
6913:
6911:
6907:
6903:
6890:
6886:
6882:
6876:
6872:
6868:
6864:
6860:
6856:
6855:
6847:
6845:
6841:
6837:
6833:
6829:
6826:(reissued by
6825:
6820:
6816:
6815:
6810:
6804:
6802:
6798:
6786:
6782:
6778:
6771:
6768:
6763:
6761:
6757:
6753:
6752:
6747:
6743:
6730:
6724:
6720:
6716:
6715:
6710:
6704:
6702:
6700:
6696:
6685:. p. 231
6684:
6683:B. G. Teubner
6680:
6676:
6675:
6670:
6664:
6661:
6649:
6645:
6638:
6636:
6634:
6632:
6628:
6624:
6618:
6612:
6608:
6604:
6599:
6598:
6592:
6585:
6582:
6567:
6563:
6559:
6552:
6545:
6542:
6530:
6526:
6522:
6520:0-299-01874-1
6516:
6512:
6508:
6507:
6502:
6498:
6492:
6489:
6477:
6473:
6472:
6464:
6462:
6460:
6458:
6456:
6452:
6447:
6443:
6439:
6435:
6431:
6425:
6421:
6417:
6413:
6409:
6402:
6399:
6394:
6390:
6386:
6380:
6376:
6372:
6368:
6364:
6359:
6358:
6349:
6346:
6334:
6330:
6323:
6320:
6315:
6311:
6307:
6303:
6299:
6295:
6291:
6285:
6281:
6280:
6275:
6271:
6267:
6260:
6258:
6256:
6254:
6252:
6248:
6244:
6232:
6231:
6226:
6220:
6218:
6216:
6212:
6209:
6205:
6203:
6199:
6195:
6182:
6181:
6172:
6169:
6166:
6161:
6158:
6151:
6143:
6139:
6133:
6131:
6127:
6123:
6122:coversed sine
6119:
6118:versed cosine
6115:
6111:
6110:Korn and Korn
6107:
6106:coversed sine
6103:
6102:versed cosine
6099:
6095:
6091:
6087:
6083:
6079:
6075:
6071:
6067:
6063:
6059:
6053:
6051:
6047:
6040:
6036:
6033:
6031:
6028:
6025:
6021:
6018:
6016:
6013:
6011:
6008:
6007:
6003:
6001:
5999:
5995:
5991:
5987:
5983:
5982:
5976:
5974:
5970:
5966:
5962:
5958:
5950:
5942:
5933:
5929:
5925:
5921:
5916:
5908:
5906:
5890:
5887:
5879:
5876:
5871:
5864:
5861:
5856:
5849:
5846:
5837:
5835:
5831:
5813:
5808:
5804:
5798:
5795:
5792:
5789:
5781:
5777:
5772:
5759:
5753:
5750:
5744:
5740:
5734:
5731:
5723:
5719:
5715:
5711:
5703:
5698:
5692:
5687:
5680:
5657:
5651:
5648:
5642:
5636:
5633:
5630:
5628:
5617:
5611:
5608:
5605:
5599:
5593:
5590:
5577:
5571:
5568:
5565:
5559:
5553:
5550:
5534:
5528:
5525:
5522:
5516:
5510:
5507:
5499:
5493:
5490:
5484:
5478:
5475:
5472:
5466:
5464:
5453:
5447:
5444:
5441:
5435:
5429:
5426:
5418:
5412:
5409:
5406:
5400:
5394:
5391:
5385:
5376:
5370:
5367:
5364:
5358:
5352:
5349:
5341:
5335:
5332:
5329:
5323:
5317:
5314:
5300:
5299:
5284:
5281:
5276:
5269:
5263:
5260:
5252:
5246:
5234:
5233:
5211:
5205:
5202:
5196:
5189:
5186:
5182:
5176:
5173:
5170:
5162:
5159:
5143:
5140:
5137:
5133:
5129:
5127:
5119:
5113:
5110:
5100:
5094:
5091:
5081:
5078:
5074:
5068:
5065:
5062:
5054:
5051:
5035:
5032:
5029:
5025:
5021:
5019:
5011:
5005:
5002:
4991:
4990:
4989:
4987:
4983:
4981:
4980:complex plane
4973:
4953:
4949:
4946:
4943:
4940:
4936:
4932:
4929:
4926:
4920:
4914:
4912:archacovercos
4911:
4904:
4903:
4886:
4882:
4879:
4876:
4873:
4869:
4865:
4862:
4859:
4853:
4847:
4845:archacoversin
4844:
4837:
4836:
4820:
4816:
4813:
4810:
4807:
4803:
4799:
4796:
4793:
4789:
4784:
4779:
4775:
4772:
4769:
4766:
4760:
4754:
4751:
4744:
4743:
4726:
4722:
4719:
4716:
4713:
4709:
4705:
4702:
4699:
4695:
4690:
4685:
4681:
4678:
4675:
4672:
4666:
4660:
4657:
4650:
4649:
4632:
4628:
4625:
4622:
4618:
4614:
4611:
4608:
4602:
4596:
4593:
4586:
4585:
4568:
4564:
4561:
4558:
4554:
4550:
4547:
4544:
4538:
4532:
4529:
4522:
4521:
4504:
4500:
4497:
4494:
4490:
4486:
4483:
4480:
4474:
4468:
4465:
4458:
4457:
4440:
4436:
4433:
4430:
4426:
4422:
4419:
4416:
4410:
4404:
4401:
4394:
4393:
4390:
4388:
4384:
4380:
4376:
4372:
4368:
4364:
4360:
4352:
4334:
4331:
4326:
4321:
4317:
4314:
4311:
4308:
4302:
4299:
4287:
4243:
4236:
4220:
4215:
4211:
4208:
4202:
4196:
4149:
4129:
4128:
4113:
4110:
4105:
4100:
4096:
4093:
4090:
4087:
4081:
4078:
4066:
4028:
4021:
4005:
4000:
3996:
3993:
3990:
3984:
3978:
3937:
3917:
3916:
3901:
3898:
3893:
3888:
3884:
3881:
3878:
3875:
3869:
3866:
3854:
3816:
3809:
3793:
3788:
3784:
3781:
3778:
3772:
3766:
3725:
3705:
3704:
3689:
3686:
3681:
3676:
3672:
3669:
3666:
3663:
3657:
3654:
3642:
3610:
3603:
3587:
3582:
3578:
3575:
3569:
3563:
3528:
3508:
3507:
3492:
3489:
3485:
3481:
3478:
3475:
3472:
3469:
3466:
3454:
3416:
3409:
3394:
3390:
3387:
3384:
3378:
3337:
3317:
3316:
3301:
3298:
3294:
3290:
3287:
3284:
3281:
3278:
3275:
3263:
3231:
3224:
3209:
3205:
3202:
3199:
3196:
3190:
3155:
3135:
3134:
3119:
3116:
3112:
3108:
3105:
3102:
3099:
3096:
3093:
3081:
3049:
3042:
3027:
3023:
3020:
3017:
3014:
3008:
2973:
2953:
2952:
2937:
2934:
2930:
2926:
2923:
2920:
2917:
2914:
2911:
2899:
2873:
2866:
2851:
2847:
2844:
2841:
2835:
2806:
2786:
2785:
2779:
2758:
2752:
2748:
2745:
2739:
2736:
2732:
2690:
2686:
2682:
2679:
2676:
2672:
2636:
2632:
2626:
2623:
2618:
2615:
2611:
2575:
2573:
2565:
2528:
2522:
2518:
2515:
2509:
2506:
2502:
2466:
2462:
2458:
2455:
2452:
2448:
2418:
2414:
2408:
2405:
2400:
2397:
2393:
2363:
2361:
2353:
2319:
2318:
2317:
2311:
2304:
2300:
2283:
2276:
2270:
2267:
2264:
2261:
2255:
2250:
2243:
2227:
2221:
2204:
2203:
2199:
2195:
2178:
2171:
2165:
2162:
2159:
2156:
2150:
2146:
2141:
2138:
2133:
2126:
2122:
2118:
2113:
2106:
2090:
2084:
2067:
2066:
2062:
2058:
2041:
2034:
2028:
2025:
2022:
2019:
2013:
2008:
2001:
1985:
1979:
1962:
1961:
1957:
1953:
1936:
1929:
1923:
1920:
1917:
1914:
1908:
1904:
1899:
1896:
1891:
1884:
1880:
1876:
1871:
1864:
1848:
1842:
1825:
1824:
1820:
1816:
1798:
1792:
1789:
1786:
1783:
1780:
1776:
1772:
1769:
1764:
1761:
1755:
1743:
1737:
1720:
1719:
1715:
1711:
1693:
1687:
1684:
1681:
1678:
1675:
1671:
1666:
1663:
1658:
1651:
1647:
1643:
1640:
1634:
1617:
1616:
1612:
1608:
1590:
1584:
1581:
1578:
1575:
1572:
1568:
1564:
1561:
1556:
1553:
1547:
1535:
1529:
1512:
1511:
1507:
1503:
1485:
1479:
1476:
1473:
1470:
1467:
1463:
1458:
1455:
1450:
1443:
1439:
1435:
1432:
1426:
1409:
1408:
1402:
1397:
1395:
1393:
1389:
1385:
1382:
1378:
1374:
1370:
1366:
1361:
1359:
1355:
1351:
1347:
1343:
1342:Hann function
1339:
1335:
1331:
1328:in value and
1327:
1323:
1322:Tukey windows
1319:
1315:
1311:
1307:
1303:
1299:
1295:
1291:
1286:
1284:
1280:
1272:
1270:
1268:
1264:
1260:
1255:
1252:
1250:
1246:
1242:
1238:
1234:
1230:
1226:
1222:
1218:
1214:
1213:
1207:
1205:
1200:
1198:
1193:
1178:
1167:
1163:
1159:
1155:
1151:
1147:
1139:
1137:
1135:
1131:
1098:
1087:
1082:
1080:
1076:
1072:
1067:
1065:
1057:
1053:
1048:
1046:
1042:
1038:
1034:
1030:
1026:
1021:
1015:
1011:
1006:
1002:
1000:
996:
992:
988:
984:
979:
977:
973:
968:
966:
962:
958:
954:
950:
946:
943:. If the arc
942:
938:
933:
931:
927:
923:
919:
915:
911:
908:) and versin(
907:
903:
899:
895:
891:
887:
883:
879:
876:
871:
869:
865:
861:
857:
853:
852:
848:The ordinary
843:
838:
831:
827:
824:
820:
816:
812:
808:
805:. If the arc
804:
800:
796:
792:
788:
783:
776:
771:
755:
754:cohavercosine
751:
750:hacovercosine
747:
743:
721:
717:
713:
709:
691:
687:
683:
681:
677:
641:
637:
633:
632:haversed sine
629:
628:
627:
610:
606:
602:
584:
580:
576:
572:
571:coversed sine
568:
553:
549:
548:versed cosine
545:
544:
543:
540:
525:
522:
517:
514:
510:
507:
504:
501:
496:
493:
488:
483:
479:
475:
472:
469:
466:
463:
460:
457:
454:
451:
448:
445:
437:
432:
430:
426:
422:
418:
414:
390:
386:
382:
374:
369:
365:
360:
356:
354:
350:
346:
341:
339:
335:
332:, Section I)
331:
330:
326:
322:
318:
314:
302:
297:
295:
290:
288:
283:
282:
280:
279:
274:
271:
269:
266:
264:
261:
259:
256:
254:
253:Regiomontanus
251:
249:
246:
244:
241:
239:
236:
234:
231:
229:
226:
225:
224:
223:
218:
213:
210:
208:
205:
202:
198:
195:
193:
190:
189:
188:
187:
184:
180:
175:
172:
171:
168:
165:
163:
160:
158:
155:
153:
150:
149:
148:
147:
142:
137:
134:
132:
129:
127:
124:
122:
119:
118:
117:
116:
111:
106:
103:
100:
96:
92:
88:
84:
81:
80:
77:
74:
72:
69:
67:
64:
63:
62:
61:
57:
53:
52:
49:
45:
40:
33:
19:
8652:
8533:. Retrieved
8505:
8461:
8457:
8448:
8435:
8429:
8405:
8395:
8371:
8364:
8353:. Retrieved
8348:
8325:. Retrieved
8311:
8284:. Retrieved
8275:
8265:
8254:. Retrieved
8217:. Retrieved
8204:
8166:. Retrieved
8157:
8133:. Retrieved
8124:
8092:. Retrieved
8047:. Retrieved
8033:
8004:. Retrieved
7988:
7977:
7962:. Retrieved
7947:
7928:. Retrieved
7926:. 091402521X
7912:
7905:
7869:
7863:
7835:
7829:
7822:Cajori, 1929
7813:
7807:
7800:Cajori, 1929
7791:
7785:
7770:
7747:
7741:
7722:
7716:
7711:
7685:
7675:
7664:. Retrieved
7649:
7638:. Retrieved
7636:. 0471543977
7618:
7576:
7549:. Retrieved
7535:
7508:. Retrieved
7499:
7457:. Retrieved
7443:
7416:. Retrieved
7404:
7380:. Retrieved
7371:
7362:
7351:. Retrieved
7349:. p. 42
7341:
7334:
7323:. Retrieved
7308:
7299:
7293:
7282:. Retrieved
7268:
7235:. Retrieved
7221:
7194:. Retrieved
7181:
7174:
7163:. Retrieved
7150:Trigonometry
7149:
7142:
7124:
7117:
7106:. Retrieved
7093:
7086:
7075:. Retrieved
7061:
7032:. Retrieved
7018:
6987:. Retrieved
6973:
6946:. Retrieved
6944:. 0691148929
6926:
6893:. Retrieved
6853:
6813:
6789:. Retrieved
6770:
6749:
6739:
6733:. Retrieved
6731:. 1602067147
6713:
6687:. Retrieved
6678:
6673:
6663:
6652:. Retrieved
6596:
6584:
6573:. Retrieved
6557:
6544:
6533:. Retrieved
6505:
6491:
6480:. Retrieved
6471:Trigonometry
6470:
6407:
6401:
6356:
6348:
6337:. Retrieved
6322:
6278:
6235:. Retrieved
6229:
6225:Inman, James
6208:1856 edition
6198:versed sines
6191:
6185:. Retrieved
6178:
6171:
6160:
6137:
6121:
6117:
6113:
6105:
6101:
6097:
6089:
6085:
6081:
6077:
6073:
6069:
6065:
6062:sinus versus
6061:
6058:Cauchy, 1821
5985:
5979:
5977:
5948:
5940:
5931:
5923:
5919:
5914:
5912:
5838:
5834:Guo Shoujing
5779:
5775:
5773:
5721:
5717:
5713:
5709:
5707:
5701:
5690:
4984:
4977:
4386:
4382:
4378:
4375:archaversine
4374:
4370:
4367:arccoversine
4366:
4363:arcvercosine
4362:
4358:
4356:
2315:
2214:hacovercosin
1394:since 2005.
1368:
1364:
1362:
1318:Hann–Poisson
1293:
1289:
1287:
1276:
1269:since 2014.
1256:
1253:
1248:
1245:nat. versine
1244:
1236:
1229:log. havers.
1228:
1224:
1216:
1210:
1208:
1201:
1194:
1176:
1143:
1129:
1096:
1083:
1068:
1055:
1049:
1044:
1028:
1024:
1022:
1019:
1013:
1012:centered at
998:
994:
990:
987:sinus versus
986:
980:
971:
969:
964:
960:
952:
948:
944:
939:, Latin for
936:
934:
929:
925:
921:
917:
913:
909:
905:
901:
897:
893:
889:
885:
881:
877:
872:
864:sinus versus
863:
860:sinus rectus
859:
849:
847:
842:the SVG file
829:
825:
814:
810:
806:
801:, Latin for
798:
794:
786:
758:hacovercosin
753:
749:
745:
728:semicoversin
719:
715:
711:
689:
685:
652:semiversinus
639:
635:
631:
625:
608:
604:
582:
578:
574:
570:
551:
547:
541:
433:
428:
424:
420:
417:sinus versus
416:
384:
380:
378:
348:
344:
342:
328:
316:
312:
310:
48:Trigonometry
8468:: 159–166.
8351:. 7.0. 2008
8278:. Versine.
7888:(A 7-place
7752:(A 7-place
7496:"Goniology"
7368:"Haversine"
7265:"Haversine"
7157:. pp.
7015:"Coversine"
6970:"Vercosine"
6746:James Inman
5965:rail tracks
4752:archavercos
4658:archaversin
4594:arccovercos
4530:arccoversin
1403:Definitions
1312:(including
1283:cofunctions
1273:Modern uses
1233:James Inman
1010:unit circle
868:unit circle
791:unit circle
720:cohaversine
716:hacoversine
690:havercosine
609:covercosine
577:(in Latin,
385:versed sine
364:unit circle
317:versed sine
238:Brahmagupta
207:Derivatives
136:Unit circle
8632:Hyperbolic
8535:2017-07-31
8530:2002100441
8355:2015-11-05
8327:2015-10-05
8286:2015-11-28
8256:2015-11-28
8219:2015-11-28
8168:2015-11-05
8135:2015-10-26
8094:2015-10-26
8080:Fortran 90
8049:2015-11-05
8006:2015-12-01
7964:2015-12-02
7930:2015-12-02
7666:2015-11-10
7640:2019-08-10
7593:required.)
7551:2015-11-06
7510:2015-11-28
7459:2015-11-06
7418:2015-11-07
7382:2015-11-06
7353:2015-12-08
7325:2015-11-12
7284:2015-11-06
7237:2015-11-06
7196:2015-12-08
7189:. p.
7165:2015-12-08
7132:. p.
7108:2015-12-08
7101:. p.
7077:2015-11-06
7034:2015-11-06
6989:2015-11-06
6948:2015-11-10
6895:2015-11-09
6889:2009932254
6791:2015-11-08
6735:2015-11-11
6689:2015-12-09
6654:2015-11-08
6575:2015-11-12
6535:2015-11-16
6482:2017-08-12
6438:2017947446
6393:2008937525
6365:. p.
6339:2015-11-12
6237:2015-11-09
6187:2017-08-13
6152:References
4359:arcversine
2236:covercosin
2077:havercosin
1972:hacoversin
1730:covercosin
1358:statistics
1326:continuous
1261:utilizing
1249:nat. vers.
1146:navigation
854:function (
821:" and the
762:hacovercos
724:hacoversin
694:havercosin
680:navigation
648:semiversin
640:semiversus
613:covercosin
583:coversinus
329:Aryabhatia
248:al-Battani
228:Hipparchus
167:Cotangents
121:Identities
8482:751627806
8474:0020-8442
8313:MathWorld
8272:"Versine"
8205:Fitsblink
8177:cite news
8076:"AUXTRIG"
8035:MathWorld
8030:"Versine"
7890:haversine
7878:cite book
7872:. London.
7856:haversine
7844:cite book
7838:. London.
7754:haversine
7572:"sagitta"
7537:MathWorld
7445:MathWorld
7413:0886-0149
7270:MathWorld
7223:MathWorld
7063:MathWorld
7020:MathWorld
6975:MathWorld
6760:July 1926
6711:(1952) .
6605:pp.
6227:(1835) .
5978:The term
5957:curvature
5913:The term
5850:≈
5793:≈
5735:≈
5658:θ
5652:
5643:θ
5637:
5618:θ
5612:
5600:θ
5594:
5578:θ
5572:
5560:θ
5554:
5535:θ
5529:
5523:−
5517:θ
5511:
5500:θ
5494:
5485:θ
5479:
5454:θ
5448:
5442:−
5436:θ
5430:
5419:θ
5413:
5401:θ
5395:
5386:−
5377:θ
5371:
5365:−
5359:θ
5353:
5342:θ
5336:
5324:θ
5318:
5277:θ
5270:θ
5264:
5250:→
5247:θ
5174:−
5160:−
5149:∞
5134:∑
5114:
5066:−
5052:−
5041:∞
5026:∑
5006:
4947:−
4933:
4915:
4877:−
4866:
4848:
4814:−
4800:
4776:
4755:
4717:−
4706:
4682:
4661:
4626:−
4615:
4597:
4562:−
4551:
4533:
4498:−
4487:
4469:
4466:arcvercos
4434:−
4423:
4405:
4402:arcversin
4318:
4312:−
4244:∫
4212:
4097:
4029:∫
3997:
3991:−
3885:
3817:∫
3785:
3779:−
3673:
3667:−
3611:∫
3579:
3482:
3476:−
3417:∫
3391:
3291:
3232:∫
3206:
3200:−
3109:
3050:∫
3024:
3018:−
2927:
2921:−
2874:∫
2848:
2749:π
2737:θ
2683:π
2677:θ
2624:π
2616:θ
2566:θ
2519:π
2507:θ
2459:π
2453:θ
2406:π
2398:θ
2354:θ
2277:θ
2271:
2244:θ
2222:θ
2172:θ
2166:
2139:θ
2107:θ
2085:θ
2035:θ
2029:
2023:−
2002:θ
1980:θ
1930:θ
1924:
1918:−
1897:θ
1865:θ
1843:θ
1799:θ
1793:
1773:θ
1770:−
1762:π
1738:θ
1694:θ
1688:
1664:θ
1635:θ
1591:θ
1585:
1579:−
1565:θ
1562:−
1554:π
1530:θ
1486:θ
1480:
1474:−
1456:θ
1427:θ
1212:haversine
1162:longitude
1140:Haversine
1052:logarithm
981:In 1821,
951: = 2
813: = 2
636:haversine
575:coversine
552:vercosine
523:θ
518:
511:θ
508:
494:θ
489:
470:θ
467:
461:−
452:θ
449:
431:(arrow).
349:haversine
345:coversine
263:de Moivre
197:Integrals
113:Reference
83:Functions
18:Haversine
8683:Category
8658:Exsecant
8403:(1959).
8321:Archived
8280:Archived
8250:Archived
8213:Archived
8162:Archived
8129:Archived
8127:. 1.25.
8088:Archived
8043:Archived
7985:(2005).
7958:Archived
7942:(1+10).)
7708:(1945).
7683:(1795).
7660:Archived
7545:Archived
7504:Archived
7453:Archived
7376:Archived
7319:Archived
7278:Archived
7231:Archived
7071:Archived
7028:Archived
6983:Archived
6924:(2013).
6830:, 2009;
6785:Archived
6756:February
6671:(1903).
6648:Archived
6566:Archived
6446:30244721
6333:Archived
6314:65-12253
6298:64-60036
6276:(eds.).
6020:Versiera
6004:See also
5830:Shen Kuo
5591:coversin
5526:coversin
5491:coversin
5368:coversin
5333:coversin
5111:haversin
2099:vercosin
1994:coversin
1835:haversin
1749:vercosin
1627:vercosin
1522:coversin
1381:metrical
1166:latitude
1154:spheroid
976:abscissa
732:hacovers
698:havercos
644:haversin
617:covercos
587:coversin
556:vercosin
421:versinus
375:Overview
325:Sanskrit
243:al-Hasib
183:Calculus
162:Tangents
8653:Versine
7779:. 1989.
6529:59-5304
6306:0167642
6072:), the
6060:), the
5998:mirrors
5981:sagitta
5953:
5936:
5924:versine
5915:versine
1373:spreads
1187:
1173:
1134:Ptolemy
1128:versin(
1126:
1114:
1107:
1093:
972:sagitta
937:sagitta
799:sagitta
638:(Latin
429:sagitta
381:versine
313:versine
273:Fourier
233:Ptolemy
199: (
157:Cosines
99:inverse
85: (
71:History
66:Outline
8608:Groups
8528:
8518:
8480:
8472:
8417:
8383:
8242:Python
8209:Python
7997:
7920:
7630:
7411:
6938:
6902:errata
6887:
6877:
6834:
6725:
6623:errata
6613:
6609:–893.
6527:
6517:
6444:
6436:
6426:
6391:
6381:
6312:
6304:
6296:
6286:
6114:θ
6090:θ
6086:θ
6082:θ
6078:θ
6070:θ
6066:θ
5994:lenses
5990:optics
5551:versin
5508:versin
5476:versin
5350:versin
5315:versin
5261:versin
5003:versin
4930:arcsin
4863:arcsin
4797:arccos
4773:arccos
4703:arccos
4679:arcsin
4612:arcsin
4548:arcsin
4484:arccos
4420:arccos
1857:versin
1541:versin
1419:versin
1384:planar
1369:Δ
1365:θ
1294:π
1292:< 2
1290:θ
1130:θ
1058:= 0, 2
1056:θ
1045:θ
1029:θ
1025:θ
983:Cauchy
953:θ
949:Δ
926:versus
922:rectus
910:θ
902:θ
894:θ
886:Δ
882:θ
815:θ
811:Δ
787:θ
656:havers
591:covers
560:vercos
446:versin
425:versus
397:sinver
393:versin
338:cosine
131:Tables
8668:atan2
8646:Other
7587:
6900:(See
6677:[
6621:(See
6569:(PDF)
6554:(PDF)
6442:S2CID
6268:. In
6041:Notes
5928:limit
5609:excsc
5569:exsec
5445:excsc
5427:exsec
5410:excsc
5392:exsec
1330:slope
1308:or a
1306:pulse
1071:chord
999:cosiv
941:arrow
875:chord
823:chord
803:arrow
752:, or
736:hacov
718:, or
670:, or
619:, or
597:, or
595:cosiv
562:, or
550:, or
427:, or
413:Latin
411:. In
387:is a
366:with
319:is a
268:Euler
258:Viète
152:Sines
76:Usage
8526:LCCN
8516:ISBN
8478:OCLC
8470:ISSN
8415:ISBN
8381:ISBN
8183:link
7995:ISBN
7918:ISBN
7884:link
7850:link
7714:..."
7628:ISBN
7409:ISSN
6936:ISBN
6885:LCCN
6875:ISBN
6832:ISBN
6758:and
6723:ISBN
6611:ISBN
6525:LCCN
6515:ISBN
6434:LCCN
6424:ISBN
6389:LCCN
6379:ISBN
6310:LCCN
6294:LCCN
6284:ISBN
6196:and
5996:and
5971:for
1386:and
1356:and
1334:zero
1320:and
1314:Hann
1300:and
1247:and
1217:hav.
1164:and
851:sine
744:The
710:The
684:The
630:The
603:The
569:The
546:The
401:vers
379:The
347:and
311:The
7854:(A
7727:doi
6867:doi
6607:892
6416:doi
6371:doi
6367:322
6138:hvs
5986:sag
5649:cos
5634:sin
5243:lim
4315:cos
4209:cos
4094:cos
3994:cos
3882:sin
3782:sin
3670:sin
3576:sin
3479:cos
3388:cos
3288:cos
3203:cos
3106:sin
3021:sin
2924:sin
2845:sin
2268:sin
2163:cos
2123:cos
2026:sin
1921:cos
1881:sin
1790:sin
1685:cos
1648:cos
1582:sin
1477:cos
1440:sin
1390:by
1352:in
1344:or
1338:one
1336:to
1227:or
1223:as
1219:or
1088:sin
1023:As
991:siv
957:bow
945:ADB
819:bow
807:ADB
766:hcc
764:or
740:hcv
738:or
706:hvc
704:or
702:hac
688:or
668:sem
664:hvs
660:hav
634:or
621:cvc
607:or
599:cvs
581:or
573:or
564:vcs
515:tan
505:sin
480:sin
464:cos
409:siv
407:or
405:ver
383:or
315:or
95:tan
91:cos
87:sin
8685::
8557:.
8553:.
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4335:C
4332:+
4327:2
4322:x
4309:x
4303:=
4300:x
4296:d
4291:)
4288:x
4285:(
4281:n
4278:i
4275:s
4272:o
4269:c
4266:r
4263:e
4260:v
4257:o
4254:c
4251:a
4248:h
4221:2
4216:x
4203:=
4200:)
4197:x
4194:(
4190:n
4187:i
4184:s
4181:o
4178:c
4175:r
4172:e
4169:v
4166:o
4163:c
4160:a
4157:h
4150:x
4146:d
4140:d
4114:C
4111:+
4106:2
4101:x
4091:+
4088:x
4082:=
4079:x
4075:d
4070:)
4067:x
4064:(
4060:n
4057:i
4054:s
4051:r
4048:e
4045:v
4042:o
4039:c
4036:a
4033:h
4006:2
4001:x
3985:=
3982:)
3979:x
3976:(
3972:n
3969:i
3966:s
3963:r
3960:e
3957:v
3954:o
3951:c
3948:a
3945:h
3938:x
3934:d
3928:d
3902:C
3899:+
3894:2
3889:x
3879:+
3876:x
3870:=
3867:x
3863:d
3858:)
3855:x
3852:(
3848:n
3845:i
3842:s
3839:o
3836:c
3833:r
3830:e
3827:v
3824:a
3821:h
3794:2
3789:x
3773:=
3770:)
3767:x
3764:(
3760:n
3757:i
3754:s
3751:o
3748:c
3745:r
3742:e
3739:v
3736:a
3733:h
3726:x
3722:d
3716:d
3690:C
3687:+
3682:2
3677:x
3664:x
3658:=
3655:x
3651:d
3646:)
3643:x
3640:(
3636:n
3633:i
3630:s
3627:r
3624:e
3621:v
3618:a
3615:h
3588:2
3583:x
3570:=
3567:)
3564:x
3561:(
3557:n
3554:i
3551:s
3548:r
3545:e
3542:v
3539:a
3536:h
3529:x
3525:d
3519:d
3493:C
3490:+
3486:x
3473:x
3470:=
3467:x
3463:d
3458:)
3455:x
3452:(
3448:n
3445:i
3442:s
3439:o
3436:c
3433:r
3430:e
3427:v
3424:o
3421:c
3395:x
3385:=
3382:)
3379:x
3376:(
3372:n
3369:i
3366:s
3363:o
3360:c
3357:r
3354:e
3351:v
3348:o
3345:c
3338:x
3334:d
3328:d
3302:C
3299:+
3295:x
3285:+
3282:x
3279:=
3276:x
3272:d
3267:)
3264:x
3261:(
3257:n
3254:i
3251:s
3248:r
3245:e
3242:v
3239:o
3236:c
3210:x
3197:=
3194:)
3191:x
3188:(
3184:n
3181:i
3178:s
3175:r
3172:e
3169:v
3166:o
3163:c
3156:x
3152:d
3146:d
3120:C
3117:+
3113:x
3103:+
3100:x
3097:=
3094:x
3090:d
3085:)
3082:x
3079:(
3075:n
3072:i
3069:s
3066:o
3063:c
3060:r
3057:e
3054:v
3028:x
3015:=
3012:)
3009:x
3006:(
3002:n
2999:i
2996:s
2993:o
2990:c
2987:r
2984:e
2981:v
2974:x
2970:d
2964:d
2938:C
2935:+
2931:x
2918:x
2915:=
2912:x
2908:d
2903:)
2900:x
2897:(
2893:n
2890:i
2887:s
2884:r
2881:e
2878:v
2852:x
2842:=
2839:)
2836:x
2833:(
2829:n
2826:i
2823:s
2820:r
2817:e
2814:v
2807:x
2803:d
2797:d
2759:)
2753:2
2746:3
2740:+
2733:(
2728:n
2725:i
2722:s
2719:o
2716:c
2713:r
2710:e
2707:v
2704:o
2701:c
2698:a
2695:h
2691:=
2687:)
2680:+
2673:(
2668:n
2665:i
2662:s
2659:o
2656:c
2653:r
2650:e
2647:v
2644:a
2641:h
2637:=
2633:)
2627:2
2619:+
2612:(
2607:n
2604:i
2601:s
2598:r
2595:e
2592:v
2589:o
2586:c
2583:a
2580:h
2576:=
2569:)
2563:(
2559:n
2556:i
2553:s
2550:r
2547:e
2544:v
2541:a
2538:h
2529:)
2523:2
2516:3
2510:+
2503:(
2498:n
2495:i
2492:s
2489:o
2486:c
2483:r
2480:e
2477:v
2474:o
2471:c
2467:=
2463:)
2456:+
2449:(
2444:n
2441:i
2438:s
2435:o
2432:c
2429:r
2426:e
2423:v
2419:=
2415:)
2409:2
2401:+
2394:(
2389:n
2386:i
2383:s
2380:r
2377:e
2374:v
2371:o
2368:c
2364:=
2357:)
2351:(
2347:n
2344:i
2341:s
2338:r
2335:e
2332:v
2284:2
2280:)
2274:(
2265:+
2262:1
2256:=
2251:2
2247:)
2241:(
2225:)
2219:(
2179:2
2175:)
2169:(
2160:+
2157:1
2151:=
2147:)
2142:2
2134:(
2127:2
2119:=
2114:2
2110:)
2104:(
2088:)
2082:(
2042:2
2038:)
2032:(
2020:1
2014:=
2009:2
2005:)
1999:(
1983:)
1977:(
1937:2
1933:)
1927:(
1915:1
1909:=
1905:)
1900:2
1892:(
1885:2
1877:=
1872:2
1868:)
1862:(
1846:)
1840:(
1802:)
1796:(
1787:+
1784:1
1781:=
1777:)
1765:2
1756:(
1741:)
1735:(
1697:)
1691:(
1682:+
1679:1
1676:=
1672:)
1667:2
1659:(
1652:2
1644:2
1638:)
1632:(
1594:)
1588:(
1576:1
1573:=
1569:)
1557:2
1548:(
1533:)
1527:(
1489:)
1483:(
1471:1
1468:=
1464:)
1459:2
1451:(
1444:2
1436:2
1430:)
1424:(
1190:)
1184:2
1181:/
1177:θ
1170:(
1123:2
1120:/
1117:1
1110:)
1104:2
1101:/
1097:θ
1090:(
1060:π
1016:.
1014:O
997:(
989:(
930:C
795:O
566:.
526:2
502:=
497:2
484:2
476:2
473:=
458:1
455:=
370:.
300:e
293:t
286:v
203:)
101:)
41:.
34:.
20:)
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