6193:, 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°.
994:
5686:
5675:
5664:
826:
348:
5292:
2763:
45:
5659:{\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}}}
2311:
2292:
2187:
2050:
1808:
2758:{\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}}}
1945:
1703:
1600:
6169:
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
1495:
771:
5218:
2181:
1939:
6168:
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
4983:
6271:. 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.
527:
1802:
2286:
1594:
2044:
1697:
4728:
1489:
4821:
4222:
4007:
3795:
3589:
2059:
1817:
5284:
4334:
4113:
3901:
3395:
3210:
3028:
3689:
2852:
5213:{\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}}}
4955:
4888:
430:
4634:
4570:
3492:
4506:
4442:
5893:
3301:
3119:
2937:
5906:
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
5297:
1712:
2196:
4988:
2316:
1504:
1954:
5815:
5759:
1609:
8582:
4642:
1401:
4736:
7902:
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
4121:
3909:
3697:
3500:
2176:{\displaystyle {\textrm {havercosin}}(\theta ):={\frac {{\textrm {vercosin}}(\theta )}{2}}=\cos ^{2}\!\left({\frac {\theta }{2}}\right)={\frac {1+\cos(\theta )}{2}}\,}
1934:{\displaystyle {\textrm {haversin}}(\theta ):={\frac {{\textrm {versin}}(\theta )}{2}}=\sin ^{2}\!\left({\frac {\theta }{2}}\right)={\frac {1-\cos(\theta )}{2}}\,}
830:
7675:
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
8442:
5226:
4228:
5831:
4013:
3801:
3309:
8575:
3127:
2945:
189:
3595:
2778:
6457:
Hall, Arthur Graham; Frink, Fred
Goodrich (January 1909). "Review Exercises Secondary Trigonometric Functions". Written at Ann Arbor, Michigan, USA.
8238:
7706:
4896:
4829:
8568:
8171:
4578:
4514:
7872:
7838:
3401:
4450:
4386:
1246:
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
5763:
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
3216:
3034:
7958:
5915:
measurement. For a straight line, the versine of any chord is zero, so this measurement characterizes the straightness of the curve. In the
2858:
185:
8117:
7307:
5998:
195:
7364:
6342:
Oldham, Keith B.; Myland, Jan C.; Spanier, Jerome (2009) . "32.13. The Cosine cos(x) and Sine sin(x) functions - Cognate functions".
522:{\displaystyle \operatorname {versin} \theta =1-\cos \theta =2\sin ^{2}{\frac {\theta }{2}}=\sin \theta \,\tan {\frac {\theta }{2}}}
8201:
8076:
7960:
5774:
5716:
1070:
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°).
8150:
8309:
8389:
1797:{\displaystyle {\textrm {covercosin}}(\theta ):={\textrm {vercosin}}\!\left({\frac {\pi }{2}}-\theta \right)=1+\sin(\theta )\,}
287:
6321:
2281:{\displaystyle {\textrm {hacovercosin}}(\theta ):={\frac {{\textrm {covercosin}}(\theta )}{2}}={\frac {1+\sin(\theta )}{2}}\,}
8373:
7533:
6867:
6851:
6824:
6715:
6603:
6416:
6371:
6351:
6276:
8651:
6554:
7441:
7219:
7059:
6400:
1589:{\displaystyle {\textrm {coversin}}(\theta ):={\textrm {versin}}\!\left({\frac {\pi }{2}}-\theta \right)=1-\sin(\theta )\,}
8268:
6636:
5685:
2039:{\displaystyle {\textrm {hacoversin}}(\theta ):={\frac {{\textrm {coversin}}(\theta )}{2}}={\frac {1-\sin(\theta )}{2}}\,}
993:
8365:
7266:
7016:
6971:
6499:
7946:
7492:
6657:
855:). The meaning of these terms is apparent if one looks at the functions in the original context for their definition, a
8547:
8031:
7910:
7620:
6153:
6129:
sometimes used for the haversine function in signal processing and filtering is also sometimes used for the unrelated
8508:
8407:
7987:
6928:
6507:
5678:
Comparison of the versine function with three approximations to the versine functions, for angles ranging from 0 to 2
1243:
Other high-regarded tables of haversines were those of
Richard Farley in 1856 and John Caulfield Hannyngton in 1876.
380:
already appearing in some of the earliest trigonometric tables. It is symbolized in formulas using the abbreviations
6582:(2000) . "Appendix B: B9. Plane and Spherical Trigonometry: Formulas Expressed in Terms of the Haversine Function".
5689:
Comparison of the versine function with three approximations to the versine functions, for angles ranging from 0 to
8677:
8072:
6773:
109:
316:
8615:
8454:
8428:
8109:
6586:
Mathematical handbook for scientists and engineers: Definitions, theorems, and formulars for reference and review
615:
In full analogy to the above-mentioned four functions another set of four "half-value" functions exists as well:
87:
5674:
1361:
7648:
1692:{\displaystyle {\textrm {vercosin}}(\theta ):=2\cos ^{2}\!\left({\frac {\theta }{2}}\right)=1+\cos(\theta )\,}
825:
8230:
8197:
7943:"Bruce Stark - Tables for Clearing the Lunar Distance and Finding G.M.T. by Sextant Observation (1995, 1997)"
6707:
6190:
4723:{\displaystyle \operatorname {archaversin} (y)=2\arcsin \left({\sqrt {y}}\right)=\arccos \left(1-2y\right)\,}
6467:/ Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. pp. 125–127
1484:{\displaystyle {\textrm {versin}}(\theta ):=2\sin ^{2}\!\left({\frac {\theta }{2}}\right)=1-\cos(\theta )\,}
8625:
8226:
4816:{\displaystyle \operatorname {archavercos} (y)=2\arccos \left({\sqrt {y}}\right)=\arccos \left(2y-1\right)}
1063:
6458:
959:
In further keeping with the interpretation of the sine as "vertical" and the versed sine as "horizontal",
7669:
6550:
1247:
1185:
83:
8399:
6920:
6816:
6663:
Vorlesungen über
Geschichte der Trigonometrie - Von der Erfindung der Logarithmen bis auf die Gegenwart
6397:
The
Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library
1338:
280:
231:
180:
114:
7760:
7566:
6910:
4217:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {hacovercosin} (x)={\frac {\cos {x}}{2}}}
1294:
1025:
347:
6730:
4002:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {hacoversin} (x)={\frac {-\cos {x}}{2}}}
3790:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {havercosin} (x)={\frac {-\sin {x}}{2}}}
1181:
directly, but having a table of the haversine removed the need to compute squares and square roots.
7389:
6595:
6591:
6355:
6023:
93:
8333:
6172:. New York, USA: James G. Gregory, successor of W. A. Townsend & Co. (Stringer & Townsend)
8603:
8488:
8292:
8234:
8014:
7765:
7694:
7571:
7516:
7424:
7335:
7249:
7202:
7122:
7091:
7042:
6999:
6954:
6130:
6003:
3584:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {haversin} (x)={\frac {\sin {x}}{2}}}
1306:
1285:) of a versine or, more commonly, a haversine (or havercosine) waveform is also commonly used in
424:
71:
59:
54:
7179:
1009:
Historically, the versed sine was considered one of the most important trigonometric functions.
8306:
8028:
7971:
7859:
Haversines, Natural and
Logarithmic, used in Computing Lunar Distances for the Nautical Almanac
7737:
Astronomical and
Nautical Tables with Precepts for finding the Latitude and Longitude of Places
7530:
7438:
7263:
7216:
7056:
7013:
6968:
6464:
1380:
1229:
377:
356:
309:
8359:
7303:
7147:
6807:
5821:, and a more accurate formula also involving the sagitta was developed two centuries later by
1228:
to simplify the calculation of distances between two points on the surface of the Earth using
8393:
7803:
7781:
7579:
7143:
6847:
6797:
6739:
6266:
6045:
Some
English sources confuse the versed cosine with the coversed sine. Historically (f.e. in
1365:
1024:
for the cosine alone would need a very high accuracy to obtain the versine in order to avoid
971:
273:
8543:
8395:
Science and Civilisation in China: Mathematics and the Sciences of the Heavens and the Earth
7287:
Fulst, Otto (1972). "17, 18". In Lütjen, Johannes; Stein, Walter; Zwiebler, Gerhard (eds.).
6914:
6583:
6492:(The Origins of Classical Mechanics from Aristotle to Newton)". In Clagett, Marshall (ed.).
6343:
8620:
7612:
7356:
7087:
6812:
6769:
6294:
1334:
844:
200:
119:
64:
6317:
6181:
Still there would be much labor of computation which may be saved by the use of tables of
8:
8560:
7598:
7081:
6584:
6485:
6344:
6018:
5916:
5279:{\displaystyle \lim _{\theta \to 0}{\frac {\operatorname {versin} (\theta )}{\theta }}=0}
1314:
1192:
1021:
322:
246:
162:
8189:
8064:
7942:
7112:
6806:(in French). Vol. 1. L'Imprimerie Royale, Debure frères, Libraires du Roi et de la
6668:
Lectures on history of trigonometry - from the invention of logarithms up to the present
6166:
4329:{\displaystyle \int \mathrm {hacovercosin} (x)\,\mathrm {d} x={\frac {x-\cos {x}}{2}}+C}
1195:
was published by James Andrew in 1805, under the name "Squares of Natural Semi-Chords".
1028:, making separate tables for the latter convenient. Even with a calculator or computer,
8165:
7976:
7925:
7866:
7832:
7169:
6801:
6489:
6430:
1342:
1251:
1074:
44:
8142:
4108:{\displaystyle \int \mathrm {hacoversin} (x)\,\mathrm {d} x={\frac {x+\cos {x}}{2}}+C}
3896:{\displaystyle \int \mathrm {havercosin} (x)\,\mathrm {d} x={\frac {x+\sin {x}}{2}}+C}
3390:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {covercosin} (x)=\cos {x}}
1039:
Another historical advantage of the versine is that it is always non-negative, so its
8514:
8504:
8466:
8458:
8403:
8369:
8296:
7983:
7928:
7906:
7616:
7606:
7397:
7137:
6924:
6873:
6863:
6820:
6711:
6599:
6513:
6503:
6422:
6412:
6377:
6367:
6298:
6282:
6272:
6258:
3205:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {coversin} (x)=-\cos {x}}
3023:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {vercosin} (x)=-\sin {x}}
1286:
1209:
1138:
1052:
664:
341:
251:
155:
8190:"identify.py: An asteroid_server client which identifies measurements in MPC format"
6754:(NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, USA, 2013.)
6434:
3684:{\displaystyle \int \mathrm {haversin} (x)\,\mathrm {d} x={\frac {x-\sin {x}}{2}}+C}
7715:
7560:
7175:
7118:
6855:
6840:
Bradley, Robert E.; Sandifer, Charles Edward (2010-01-14) . Buchwald, J. Z. (ed.).
6766:"德博士的 Notes About Circles, ज्य, & कोज्य: What in the world is a hacovercosine?"
6404:
6359:
5969:
4974:
1146:
1059:
863:
811:
786:. This figure also illustrates the reason why the versine was sometimes called the
27:
7720:
7698:
7520:
2847:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {versin} (x)=\sin {x}}
829:
Graphs of historical trigonometric functions compared with sin and cos – in
8608:
8493:
7900:
7673:
7329:
6841:
6706:. Vol. 2 (2 (3rd corrected printing of 1929 issue) ed.). Chicago, USA:
6701:
6661:
6579:
6539:
6493:
6290:
6268:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
6232:
6217:
6196:
6182:
6012:
1298:
1255:
1067:
1029:
924:
This figure also illustrates the reason why the versine was sometimes called the
150:
79:
75:
7825:
Natural Versed Sines from 0 to 125°, and Logarithmic Versed Sines from 0 to 135°
7428:
7206:
7046:
6254:
8260:
7886:
7882:
6697:
6189:, which have been employed with great success recently by the Engineers on the
5961:
5957:
5949:
1376:
1372:
1290:
261:
256:
145:
7253:
7003:
6958:
6859:
6408:
6395:
Beebe, Nelson H. F. (2017-08-22). "Chapter 11.1. Sine and cosine properties".
6363:
8671:
8500:
8470:
8462:
7401:
6671:
5982:
4968:
4950:{\displaystyle \operatorname {archacovercos} (y)=\arcsin \left(2y-1\right)\,}
4883:{\displaystyle \operatorname {archacoversin} (y)=\arcsin \left(1-2y\right)\,}
1330:
241:
8018:
6632:
1141:, which is used to reasonably accurately compute distances on an astronomic
889:
from the center of the chord to the center of the arc. Thus, the sum of cos(
8539:
7602:
6846:. Sources and Studies in the History of Mathematics and Physical Sciences.
6262:
6255:"Chapter 4.3.147: Elementary Transcendental Functions - Circular functions"
5822:
1369:
1310:
997:
The trigonometric functions can be constructed geometrically in terms of a
945:
807:
140:
36:
6733:(Madrid, 1801, also 1805, 1809), and later in a treatise on navigation of
4629:{\displaystyle \operatorname {arccovercos} (y)=\arcsin \left(y-1\right)\,}
4565:{\displaystyle \operatorname {arccoversin} (y)=\arcsin \left(1-y\right)\,}
1266:
Whilst the usage of the versine, coversine and haversine as well as their
7484:
6734:
6213:
5953:
3487:{\displaystyle \int \mathrm {covercosin} (x)\,\mathrm {d} x=x-\cos {x}+C}
1302:
1221:
998:
856:
779:
352:
226:
124:
6765:
5911:
from the chord to the curve (usually at the chord midpoint) is called a
4501:{\displaystyle \operatorname {arcvercos} (y)=\arccos \left(y-1\right)\,}
4437:{\displaystyle \operatorname {arcversin} (y)=\arccos \left(1-y\right)\,}
1020:) is the difference between two nearly equal quantities, so a user of a
8068:
5888:{\displaystyle v\approx {\frac {s^{\frac {3}{2}}L^{\frac {1}{2}}}{8r}}}
3296:{\displaystyle \int \mathrm {coversin} (x)\,\mathrm {d} x=x+\cos {x}+C}
3114:{\displaystyle \int \mathrm {vercosin} (x)\,\mathrm {d} x=x+\sin {x}+C}
1346:
1326:
1322:
1271:
1134:
668:
236:
216:
20:
8554:
7396:(227). Ix, Hanno. Portland, ME, USA: Navigator Publishing LLC: 42–43.
2291:
1073:
The versine appears as an intermediate step in the application of the
8301:
8023:
7878:
7844:
7742:
7525:
7433:
7258:
7211:
7051:
7008:
6963:
6729:
The haversine first appears in the tables of logarithmic versines of
5945:
2932:{\displaystyle \int \mathrm {versin} (x)\,\mathrm {d} x=x-\sin {x}+C}
1329:(for haversine) and back to zero. In these applications, it is named
1200:
1150:
1040:
917:, literally "turned against, out-of-place"); both are distances from
8495:
On the Shoulders of Giants: The Great Works of Physics and Astronomy
6302:
2186:
2049:
1807:
8646:
7644:
6008:
5818:
1154:
1142:
964:
313:
171:
8518:
6877:
6426:
6381:
6346:
An Atlas of Functions: with Equator, the Atlas Function Calculator
1944:
1702:
1599:
531:
There are several related functions corresponding to the versine:
6916:
Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry
1122:
340:. The latter, half a versine, is of particular importance in the
221:
7291:(in German) (24 ed.). Bremen, Germany: Arthur Geist Verlag.
6286:
6219:
Navigation and Nautical Astronomy: For the Use of British Seamen
6081:). However, in their 2009 English translation of Cauchy's work,
1494:
1226:
Navigation and Nautical Astronomy: For the Use of British Seamen
1188:
of what later would be called haversines is documented in 1801.
7978:
Divine Proportions: Rational Trigonometry to Universal Geometry
6517:
5986:
5978:
4366:(archaversin, archav, haversin, invhav, ahav, ahvs, ahv, hav),
326:
6165:
Haslett, Charles (September 1855). Hackley, Charles W. (ed.).
8656:
8445:(1972). "Track measurement systems—concepts and techniques".
7813:, this journal has a discussion on the origin of haversines.)
7791:, this journal has a discussion on the origin of haversines.)
7645:"Earliest Known Uses of Some of the Words of Mathematics (V)"
5952:, where it describes measurements of the straightness of the
1318:
929:
909:(with length 1). Illustrated this way, the sine is vertical (
791:
401:
6222:(3 ed.). London, UK: W. Woodward, C. & J. Rivington
8227:"astrotrig.py: Astronomical trigonometry related functions"
1270:
can be traced back centuries, the names for the other five
881:(half of the chord). On the other hand, the versed sine of
839:
8590:
7304:"Semiversus-Verfahren: Logarithmische Berechnung der Höhe"
7083:
Elements of Trigonometry with Logarithmic and Other Tables
6041:
6039:
1232:
for applications in navigation. Inman also used the terms
770:
913:, literally "straight") while the versine is horizontal (
6547:
Precalculus: A Study of Functions and Their Applications
1032:
make it advisable to use the sin formula for small
851:("straight sine"), to contrast it with the versed sine (
7511:
7509:
7037:
7035:
7033:
6538:
Swanson, Todd; Andersen, Janet; Keeley, Robert (1999).
6463:. Vol. Part I: Plane Trigonometry. New York, USA:
6036:
7699:"Recent Mathematical Tables : 197[C, D].—
6093:(what is now also known as vercosine) rather than the
4378:(archacovercosin, archacovercos, ahcc) exist as well:
332:
There are several related functions, most notably the
8361:
Geometry - Plane, Solid & Analytic Problem Solver
7753:
7751:
7419:
7417:
7197:
7195:
6537:
5834:
5828:
A more accurate approximation used in engineering is
5777:
5719:
5295:
5229:
4986:
4899:
4832:
4739:
4645:
4581:
4517:
4453:
4389:
4231:
4124:
4016:
3912:
3804:
3700:
3598:
3503:
3404:
3312:
3219:
3130:
3037:
2948:
2861:
2781:
2314:
2199:
2062:
1957:
1820:
1715:
1612:
1507:
1404:
433:
26:"versin" redirects here. For the Polish village, see
8425:
Table For Use in Computing Arcs, Chords and Versines
7757:
7506:
7306:(in German). Hotheim am Taunus, Germany: Astrosail.
7030:
5817:
This formula was known to the Chinese mathematician
5705:, it may be approximated from the half-chord length
2305:
The functions are circular rotations of each other.
1058:
In fact, the earliest surviving table of sine (half-
1055:
for multiplications in formulas involving versines.
8338:
Wolfram Language & System: Documentation Center
8233:source code). Ljubljana, Slovenia: Telescope Vega,
7361:
Wolfram Language & System: Documentation Center
6994:
6992:
6990:
6988:
6949:
6947:
6656:
6341:
1043:
is defined everywhere except for the single angle (
427:sine, cosine, and tangent, the versine is equal to
8492:
7975:
7748:
7559:
7414:
7192:
6843:Cauchy's Cours d'analyse: An Annotated Translation
6670:] (in German). Vol. 2. Leipzig, Germany:
5887:
5809:
5753:
5658:
5278:
5212:
4949:
4882:
4815:
4722:
4628:
4564:
4500:
4436:
4328:
4216:
4107:
4001:
3895:
3789:
3683:
3583:
3486:
3389:
3295:
3204:
3113:
3022:
2931:
2846:
2757:
2280:
2175:
2038:
1933:
1796:
1691:
1588:
1483:
521:
8453:(3). International Railway Congress Association,
8287:
8285:
8009:
8007:
8005:
7779:White, J. D. (February 1926). "(unknown title)".
7707:Mathematical Tables and Other Aids to Computation
7668:
7558:
7383:
7381:
6905:
6903:
6901:
6899:
6248:
6246:
6244:
6242:
6240:
2120:
1878:
1742:
1645:
1534:
1437:
8669:
7478:
7244:
7242:
7240:
7238:
7236:
6985:
6944:
5231:
1051:, …) where it is zero—thus, one could use
8328:
8326:
8058:
8056:
8054:
8052:
8050:
8048:
7964:
7850:
7476:
7474:
7472:
7470:
7468:
7466:
7464:
7462:
7460:
7458:
7168:Anderegg, Frederick; Roe, Edward Drake (1896).
6839:
6835:
6833:
6803:Cours d'Analyse de l'Ecole royale polytechnique
6626:
6624:
6622:
6620:
6065:(what is now also known as coversine) as cosiv(
4362:(arccovercosin, arccovercos, acovercos, acvc),
1133:The haversine, in particular, was important in
8282:
8220:
8218:
8183:
8181:
8002:
7554:
7552:
7550:
7378:
7104:
7080:Ludlow, Henry Hunt; Bass, Edgar Wales (1891).
6896:
6792:
6790:
6692:
6690:
6688:
6650:
6498:(3 ed.). Madison, Milwaukee, and London:
6237:
5897:
1066:and other Greek authors), calculated from the
8576:
8416:
8351:
7934:
7801:White, J. D. (July 1926). "(unknown title)".
7794:
7772:
7739:. Vol. T. XIII. London. pp. 29–148.
7597:
7593:
7591:
7589:
7233:
7161:
7136:Kenyon, Alfred Monroe; Ingold, Louis (1913).
6909:
5810:{\displaystyle s\approx L+{\frac {v^{2}}{r}}}
5754:{\displaystyle v\approx {\frac {L^{2}}{2r}}.}
833:, hover over or click a graph to highlight it
281:
8382:
8323:
8107:
8103:
8101:
8099:
8097:
8095:
8093:
8045:
7455:
7135:
6830:
6617:
6531:
6208:
6206:
6204:
2768:
8252:
8215:
8178:
8170:: CS1 maint: numeric names: authors list (
8136:
8134:
7982:(1 ed.). Australia: Wild Egg Pty Ltd.
7816:
7745:table from 0° to 120° in intervals of 10".)
7728:
7689:
7687:
7685:
7678:(in Spanish). Madrid, Spain: Imprenta Real.
7636:
7547:
7482:
7349:
7167:
7129:
7073:
6787:
6685:
6495:Critical Problems in the History of Science
6478:
6452:
6450:
6448:
6446:
6444:
6158:
1254:since 1995 or in a more compact method for
760:
8583:
8569:
7970:
7892:
7871:: CS1 maint: location missing publisher (
7856:
7837:: CS1 maint: location missing publisher (
7586:
7327:
7321:
7295:
7280:
7079:
6757:
6309:
1386:
1125:, that was used to construct such tables.
873:(representing half of the subtended angle
869:of the unit circle, the sine of the angle
288:
274:
8435:
8090:
7719:
7693:
7662:
7387:
7328:Rider, Paul Reece; Davis, Alfred (1923).
7110:
6571:
6456:
6388:
6335:
6201:
5575:
4967:These functions can be extended into the
4946:
4879:
4719:
4625:
4561:
4497:
4433:
4358:(arccoversin, arccovers, acovers, acvs),
4354:(arcvercosin, arcvercos, avercos, avcs),
4282:
4061:
3849:
3637:
3449:
3258:
3076:
2894:
2277:
2172:
2035:
1930:
1793:
1688:
1585:
1480:
502:
325:. The versine of an angle is 1 minus its
8422:
8357:
8131:
8071:source code). Greenbelt, Maryland, USA:
7940:
7682:
6577:
6488:(1969) . "5: Commentary on the Paper of
6441:
5684:
5673:
1191:The first known English equivalent to a
992:
847:) was sometimes historically called the
824:
769:
765:
346:
8487:
8390:Needham, Noel Joseph Terence Montgomery
8388:
8062:
6630:
6399:(1 ed.). Salt Lake City, UT, USA:
6164:
1337:. Likewise, the havercosine is used in
8670:
8591:Trigonometric and hyperbolic functions
8258:
8224:
8187:
7822:
7734:
7642:
7171:Trigonometry: For Schools and Colleges
6796:
6763:
6696:
6590:(3 ed.). Mineola, New York, USA:
6318:"Background Notes on Measures: Angles"
6315:
6252:
6121:
6119:
6097:. Similarly, in their 1968/2000 work,
1267:
8564:
8538:
8291:
8140:
8013:
7905:(2 ed.). Starpath Publications.
7898:
7800:
7778:
7515:
7423:
7301:
7286:
7248:
7201:
7041:
6998:
6953:
6484:
6394:
6212:
5948:. This usage is especially common in
5701:is small in comparison to the radius
2300:
1274:appear to be of much younger origin.
8441:
8366:Research & Education Association
7580:participating institution membership
6852:Springer Science+Business Media, LLC
6401:Springer International Publishing AG
6352:Springer Science+Business Media, LLC
5771:in the figure above) by the formula
4370:(archavercosin, archavercos, ahvc),
4341:
1356:) the haversine of the double-angle
963:is also an obsolete synonym for the
8364:. Problem Solvers Solution Guides.
7857:Hannyngton, John Caulfield (1876).
6890:
6703:A History of Mathematical Notations
6500:University of Wisconsin Press, Ltd.
6116:
4962:
4350:(arcversin, arcvers, avers, aver),
774:Sine, cosine, and versine of angle
13:
8548:The Wolfram Demonstrations Project
8481:
8358:Woodward, Ernest (December 1978).
8114:JASS - Java Audio Synthesis System
7945:(Review). Prague, Czech Republic.
7701:Natural and Logarithmic Haversines
7489:Eenheden, constanten en conversies
7483:van Vlijmen, Oscar (2005-12-28) .
7111:Wentworth, George Albert (1903) .
6748:
6744:
5137:
5029:
4284:
4269:
4266:
4263:
4260:
4257:
4254:
4251:
4248:
4245:
4242:
4239:
4236:
4178:
4175:
4172:
4169:
4166:
4163:
4160:
4157:
4154:
4151:
4148:
4145:
4134:
4128:
4063:
4048:
4045:
4042:
4039:
4036:
4033:
4030:
4027:
4024:
4021:
3960:
3957:
3954:
3951:
3948:
3945:
3942:
3939:
3936:
3933:
3922:
3916:
3851:
3836:
3833:
3830:
3827:
3824:
3821:
3818:
3815:
3812:
3809:
3748:
3745:
3742:
3739:
3736:
3733:
3730:
3727:
3724:
3721:
3710:
3704:
3639:
3624:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3545:
3542:
3539:
3536:
3533:
3530:
3527:
3524:
3513:
3507:
3451:
3436:
3433:
3430:
3427:
3424:
3421:
3418:
3415:
3412:
3409:
3360:
3357:
3354:
3351:
3348:
3345:
3342:
3339:
3336:
3333:
3322:
3316:
3260:
3245:
3242:
3239:
3236:
3233:
3230:
3227:
3224:
3172:
3169:
3166:
3163:
3160:
3157:
3154:
3151:
3140:
3134:
3078:
3063:
3060:
3057:
3054:
3051:
3048:
3045:
3042:
2990:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2958:
2952:
2896:
2881:
2878:
2875:
2872:
2869:
2866:
2817:
2814:
2811:
2808:
2805:
2802:
2791:
2785:
2716:
2713:
2710:
2707:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2547:
2544:
2541:
2538:
2535:
2532:
2529:
2526:
2486:
2483:
2480:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2411:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2335:
2332:
2329:
2326:
2323:
2320:
967:(the horizontal axis of a graph).
952:as its "string", then the versine
817:as its "string", then the versine
19:For the archaic unit of mass, see
14:
8689:
8532:
8108:van den Doel, Kees (2010-01-25).
7647:. New Port Richey, Florida, USA.
6631:Calvert, James B. (2007-09-14) .
6082:
5981:, for describing the surfaces of
5669:
1224:in the third edition of his work
1149:) given angular positions (e.g.,
8143:"Lost but lovely: The haversine"
8073:NASA Goddard Space Flight Center
8063:Simpson, David G. (2001-11-08).
7810:
7788:
6611:
6046:
2290:
2185:
2048:
1943:
1806:
1701:
1598:
1493:
43:
8455:International Union of Railways
8429:Chicago Bridge and Iron Company
8312:from the original on 2008-06-08
8271:from the original on 2007-02-08
8241:from the original on 2015-11-28
8204:from the original on 2008-11-20
8153:from the original on 2014-07-18
8120:from the original on 2007-09-02
8079:from the original on 2008-06-16
8034:from the original on 2010-03-31
7949:from the original on 2004-01-12
7651:from the original on 2015-09-05
7536:from the original on 2014-03-29
7495:from the original on 2009-10-28
7444:from the original on 2014-03-29
7390:"Ultra compact sight reduction"
7367:from the original on 2014-09-01
7310:from the original on 2013-09-17
7269:from the original on 2005-03-10
7222:from the original on 2014-03-28
7062:from the original on 2014-03-29
7019:from the original on 2005-11-27
6974:from the original on 2014-03-24
6880:. 1441905499, 978-1-4419-0549-9
6776:from the original on 2015-09-19
6639:from the original on 2007-10-02
6560:from the original on 2003-06-17
6324:from the original on 2007-02-09
1360:describes the relation between
901:) (equal to the length of line
893:) (equal to the length of line
312:found in some of the earliest (
6800:(1821). "Analyse Algébrique".
6147:
6098:
5649:
5643:
5634:
5628:
5612:
5609:
5603:
5591:
5585:
5576:
5572:
5569:
5563:
5551:
5545:
5536:
5526:
5520:
5508:
5502:
5491:
5485:
5476:
5470:
5445:
5439:
5427:
5421:
5410:
5404:
5392:
5386:
5368:
5362:
5350:
5344:
5333:
5327:
5315:
5309:
5261:
5255:
5238:
5197:
5188:
5155:
5145:
5111:
5105:
5086:
5077:
5047:
5037:
5003:
4997:
4912:
4906:
4845:
4839:
4752:
4746:
4658:
4652:
4594:
4588:
4530:
4524:
4466:
4460:
4402:
4396:
4279:
4273:
4188:
4182:
4058:
4052:
3970:
3964:
3846:
3840:
3758:
3752:
3634:
3628:
3555:
3549:
3446:
3440:
3370:
3364:
3255:
3249:
3182:
3176:
3073:
3067:
3000:
2994:
2891:
2885:
2827:
2821:
2557:
2551:
2345:
2339:
2268:
2262:
2235:
2229:
2213:
2207:
2163:
2157:
2098:
2092:
2076:
2070:
2026:
2020:
1993:
1987:
1971:
1965:
1921:
1915:
1856:
1850:
1834:
1828:
1790:
1784:
1729:
1723:
1685:
1679:
1626:
1620:
1582:
1576:
1521:
1515:
1477:
1471:
1418:
1412:
1391:
1368:, a proposed reformulation of
1261:
956:is clearly the "arrow shaft".
16:1 minus the cosine of an angle
1:
8652:Jyā, koti-jyā and utkrama-jyā
8544:"Sagitta, Apothem, and Chord"
7721:10.1090/S0025-5718-45-99080-6
7611:(2 ed.). New York, USA:
7388:Rudzinski, Greg (July 2015).
6708:Open court publishing company
6540:"5 (Trigonometric Functions)"
6191:Ohio and Mississippi Railroad
6140:
6053:(versine) was defined as siv(
821:is clearly the "arrow shaft".
423:Expressed in terms of common
7941:Kalivoda, Jan (2003-07-30).
7847:table from 0° to 125°/135°.)
6813:access-date=2015-11-07-->
6551:Harcourt Brace & Company
6154:The Āryabhaṭīya by Āryabhaṭa
5713:shown above) by the formula
1128:
1062:) values (as opposed to the
7:
8259:Ballew, Pat (2007-02-08) .
8225:Skvarc, Jure (2014-10-27).
8188:Skvarc, Jure (1999-03-01).
7643:Miller, Jeff (2007-09-10).
7117:(2 ed.). Boston, USA:
7086:(3 ed.). Boston, USA:
6737:(1821). See J. D. White in
6073:), and the vercosine as vcs
5992:
5956:and it is the basis of the
5898:Arbitrary curves and chords
1339:raised-cosine distributions
1064:chords tabulated by Ptolemy
782:with radius 1, centered at
363:
10:
8694:
8400:Cambridge University Press
6921:Princeton University Press
6911:van Brummelen, Glen Robert
6817:Cambridge University Press
6658:Edler von Braunmühl, Anton
5944:goes to the instantaneous
1137:because it appears in the
181:Trigonometric substitution
25:
18:
8634:
8596:
7924:(NB. Contains a table of
7899:Stark, Bruce D. (1997) .
7761:Oxford English Dictionary
7670:de Mendoza y Ríos, Joseph
7567:Oxford English Dictionary
6860:10.1007/978-1-4419-0549-9
6409:10.1007/978-3-319-64110-2
6364:10.1007/978-0-387-48807-3
4374:(archacoversin, ahcv) or
2769:Derivatives and integrals
1157:). One could also use sin
1147:Earth's radius vs. sphere
1026:catastrophic cancellation
8489:Hawking, Stephen William
8423:Boardman, Harry (1930).
8293:Weisstein, Eric Wolfgang
8110:"jass.utils Class Fmath"
8015:Weisstein, Eric Wolfgang
7823:Farley, Richard (1856).
7695:Archibald, Raymond Clare
7608:A History of Mathematics
7517:Weisstein, Eric Wolfgang
7425:Weisstein, Eric Wolfgang
7250:Weisstein, Eric Wolfgang
7203:Weisstein, Eric Wolfgang
7043:Weisstein, Eric Wolfgang
7000:Weisstein, Eric Wolfgang
6955:Weisstein, Eric Wolfgang
6592:Dover Publications, Inc.
6186:
6029:
6024:Natural logarithm plus 1
5999:Trigonometric identities
5923:goes to zero, the ratio
1313:), because it smoothly (
1184:An early utilization by
761:History and applications
94:Generalized trigonometry
8678:Trigonometric functions
8555:Trigonometric Functions
8235:University of Ljubljana
7972:Wildberger, Norman John
7881:table from 0° to 180°,
7766:Oxford University Press
7572:Oxford University Press
7336:D. Van Nostrand Company
6578:Korn, Grandino Arthur;
6131:Heaviside step function
6004:Exsecant and excosecant
5977:) is used similarly in
4346:Inverse functions like
1387:Mathematical identities
1210:base-10 logarithmically
663:, most famous from the
425:trigonometric functions
357:trigonometric functions
8307:Wolfram Research, Inc.
8029:Wolfram Research, Inc.
7735:Andrew, James (1805).
7531:Wolfram Research, Inc.
7439:Wolfram Research, Inc.
7302:Sauer, Frank (2015) .
7264:Wolfram Research, Inc.
7217:Wolfram Research, Inc.
7057:Wolfram Research, Inc.
7014:Wolfram Research, Inc.
6969:Wolfram Research, Inc.
6848:Cauchy, Augustin-Louis
6798:Cauchy, Augustin-Louis
6731:José de Mendoza y Rios
6465:Henry Holt and Company
6316:Tapson, Frank (2004).
6253:Zucker, Ruth (1983) .
5889:
5811:
5755:
5694:
5682:
5660:
5280:
5214:
5141:
5033:
4951:
4884:
4817:
4724:
4630:
4566:
4502:
4438:
4330:
4218:
4109:
4003:
3897:
3791:
3685:
3585:
3488:
3391:
3297:
3206:
3115:
3024:
2933:
2848:
2759:
2282:
2177:
2040:
1935:
1798:
1693:
1590:
1485:
1381:Norman John Wildberger
1230:spherical trigonometry
1204:(notated naturally as
1186:José de Mendoza y Ríos
1006:
982:) for the versine and
834:
822:
523:
378:trigonometric function
360:
310:trigonometric function
8616:Inverse trigonometric
8499:. Philadelphia, USA:
8443:Nair, P. N. Bhaskaran
7889:at intervals of 10".)
7885:at intervals of 15",
7613:John Wiley & Sons
7144:The Macmillan Company
7088:John Wiley & Sons
6320:. 1.4. Cleave Books.
6101:associate the covers(
6089:(and cosiv) with the
5890:
5812:
5756:
5688:
5677:
5661:
5281:
5215:
5121:
5013:
4952:
4885:
4818:
4725:
4631:
4567:
4503:
4439:
4331:
4219:
4110:
4004:
3898:
3792:
3686:
3586:
3489:
3392:
3298:
3207:
3116:
3025:
2934:
2849:
2760:
2283:
2178:
2041:
1936:
1799:
1694:
1591:
1486:
1366:rational trigonometry
1145:(see issues with the
1016:goes to zero, versin(
996:
990:) for the coversine.
845:see note on etymology
828:
773:
766:Versine and coversine
667:used historically in
524:
404:, it is known as the
350:
8368:(REA). p. 359.
8141:mf344 (2014-07-04).
7599:Boyer, Carl Benjamin
6854:. pp. 10, 285.
6770:University of Hawaii
6486:Boyer, Carl Benjamin
6105:) function with the
6083:Bradley and Sandifer
6069:) = 1−sin(
6057:) = 1−cos(
5919:as the chord length
5832:
5775:
5717:
5293:
5227:
4984:
4897:
4830:
4737:
4643:
4579:
4515:
4451:
4387:
4229:
4122:
4014:
3910:
3802:
3698:
3596:
3501:
3402:
3310:
3217:
3128:
3035:
2946:
2859:
2779:
2312:
2197:
2060:
1955:
1818:
1713:
1610:
1505:
1402:
1335:raised-cosine filter
936:of the double-angle
798:of the double-angle
431:
323:trigonometric tables
201:Trigonometric series
8297:"Inverse Haversine"
7926:Gaussian logarithms
7570:(Online ed.).
6808:Bibliothèque du Roi
6764:Shaneyfelt, Ted V.
6077: = 1+cos(
6019:Exponential minus 1
5973:(often abbreviated
1352:In the form of sin(
1277:One period (0 <
1252:Gaussian logarithms
1193:table of haversines
1022:trigonometric table
163:Pythagorean theorem
8626:Inverse hyperbolic
8447:Rail International
8334:"InverseHaversine"
8265:Math Words, page 4
7809:(NB. According to
7787:(NB. According to
7331:Plane Trigonometry
7114:Plane Trigonometry
6502:pp. 185–190.
6490:E. J. Dijksterhuis
6259:Abramowitz, Milton
5885:
5807:
5751:
5695:
5683:
5656:
5654:
5276:
5245:
5210:
5208:
4947:
4880:
4813:
4720:
4626:
4562:
4498:
4434:
4326:
4214:
4105:
3999:
3893:
3787:
3681:
3581:
3484:
3387:
3293:
3202:
3111:
3020:
2929:
2844:
2755:
2753:
2301:Circular rotations
2278:
2173:
2036:
1931:
1794:
1689:
1586:
1481:
1343:probability theory
1321:) "turns on" from
1293:as the shape of a
1198:In 1835, the term
1075:half-angle formula
1053:logarithmic tables
1007:
877:) is the distance
835:
823:
519:
361:
8665:
8664:
8375:978-0-87891-510-1
7804:Nautical Magazine
7782:Nautical Magazine
7578:(Subscription or
7334:. New York, USA:
7142:. New York, USA:
6869:978-1-4419-0548-2
6825:978-1-108-00208-0
6740:Nautical Magazine
6717:978-1-60206-714-1
6605:978-0-486-41147-7
6418:978-3-319-64109-6
6373:978-0-387-48806-6
6278:978-0-486-61272-0
6263:Stegun, Irene Ann
6231:(Fourth edition:
6125:The abbreviation
5883:
5871:
5856:
5805:
5746:
5697:When the versine
5530:
5449:
5372:
5268:
5230:
5204:
5093:
4960:
4959:
4776:
4682:
4342:Inverse functions
4339:
4338:
4318:
4212:
4142:
4097:
3997:
3930:
3885:
3785:
3718:
3673:
3579:
3521:
3330:
3148:
2966:
2799:
2744:
2618:
2514:
2400:
2298:
2297:
2275:
2242:
2226:
2204:
2170:
2133:
2105:
2089:
2067:
2033:
2000:
1984:
1962:
1928:
1891:
1863:
1847:
1825:
1756:
1739:
1720:
1658:
1617:
1548:
1531:
1512:
1450:
1409:
1287:signal processing
1268:inverse functions
1139:haversine formula
735:hacoversed cosine
665:haversine formula
517:
488:
342:haversine formula
298:
297:
190:inverse functions
133:Laws and theorems
8685:
8585:
8578:
8571:
8562:
8561:
8551:
8528:
8526:
8525:
8498:
8475:
8474:
8439:
8433:
8432:
8420:
8414:
8413:
8386:
8380:
8379:
8355:
8349:
8348:
8346:
8345:
8330:
8321:
8320:
8318:
8317:
8289:
8280:
8279:
8277:
8276:
8256:
8250:
8249:
8247:
8246:
8222:
8213:
8212:
8210:
8209:
8185:
8176:
8175:
8169:
8161:
8159:
8158:
8138:
8129:
8128:
8126:
8125:
8105:
8088:
8087:
8085:
8084:
8060:
8043:
8042:
8040:
8039:
8011:
8000:
7999:
7997:
7996:
7981:
7968:
7962:
7957:
7955:
7954:
7938:
7932:
7923:
7921:
7920:
7896:
7890:
7876:
7870:
7862:
7854:
7848:
7842:
7836:
7828:
7820:
7814:
7808:
7798:
7792:
7786:
7776:
7770:
7769:
7764:(2nd ed.).
7755:
7746:
7740:
7732:
7726:
7725:
7723:
7691:
7680:
7679:
7666:
7660:
7659:
7657:
7656:
7640:
7634:
7633:
7631:
7630:
7603:Merzbach, Uta C.
7595:
7584:
7583:
7575:
7563:
7556:
7545:
7544:
7542:
7541:
7513:
7504:
7503:
7501:
7500:
7480:
7453:
7452:
7450:
7449:
7421:
7412:
7411:
7409:
7408:
7385:
7376:
7375:
7373:
7372:
7353:
7347:
7346:
7344:
7343:
7325:
7319:
7318:
7316:
7315:
7299:
7293:
7292:
7289:Nautische Tafeln
7284:
7278:
7277:
7275:
7274:
7246:
7231:
7230:
7228:
7227:
7199:
7190:
7189:
7187:
7186:
7176:Ginn and Company
7165:
7159:
7158:
7156:
7155:
7133:
7127:
7126:
7119:Ginn and Company
7108:
7102:
7101:
7099:
7098:
7077:
7071:
7070:
7068:
7067:
7039:
7028:
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6996:
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6979:
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6894:
6888:
6886:
6885:
6837:
6828:
6811:
6794:
6785:
6784:
6782:
6781:
6768:. Hilo, Hawaii:
6761:
6755:
6753:
6726:
6725:
6694:
6683:
6682:
6680:
6679:
6654:
6648:
6647:
6645:
6644:
6628:
6615:
6609:
6589:
6580:Korn, Theresa M.
6575:
6569:
6568:
6566:
6565:
6559:
6544:
6535:
6529:
6528:
6526:
6525:
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6439:
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6392:
6386:
6385:
6349:
6339:
6333:
6332:
6330:
6329:
6313:
6307:
6306:
6250:
6235:
6230:
6228:
6227:
6210:
6199:
6195:
6183:external secants
6178:
6177:
6162:
6156:
6151:
6134:
6123:
6114:
6043:
5943:
5941:
5940:
5935:
5932:
5894:
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5872:
5864:
5858:
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5800:
5791:
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5758:
5757:
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5665:
5663:
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5657:
5655:
5531:
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5459:
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5413:
5378:
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5336:
5301:
5285:
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5219:
5217:
5216:
5211:
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5205:
5203:
5183:
5182:
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5169:
5168:
5143:
5140:
5135:
5094:
5092:
5075:
5074:
5073:
5061:
5060:
5035:
5032:
5027:
4975:Maclaurin series
4963:Other properties
4956:
4954:
4953:
4948:
4945:
4941:
4889:
4887:
4886:
4881:
4878:
4874:
4822:
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4814:
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4777:
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4721:
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4505:
4504:
4499:
4496:
4492:
4443:
4441:
4440:
4435:
4432:
4428:
4381:
4380:
4376:archacovercosine
4335:
4333:
4332:
4327:
4319:
4314:
4313:
4295:
4287:
4272:
4223:
4221:
4220:
4215:
4213:
4208:
4207:
4195:
4181:
4143:
4141:
4137:
4131:
4126:
4114:
4112:
4111:
4106:
4098:
4093:
4092:
4074:
4066:
4051:
4008:
4006:
4005:
4000:
3998:
3993:
3992:
3977:
3963:
3931:
3929:
3925:
3919:
3914:
3902:
3900:
3899:
3894:
3886:
3881:
3880:
3862:
3854:
3839:
3796:
3794:
3793:
3788:
3786:
3781:
3780:
3765:
3751:
3719:
3717:
3713:
3707:
3702:
3690:
3688:
3687:
3682:
3674:
3669:
3668:
3650:
3642:
3627:
3590:
3588:
3587:
3582:
3580:
3575:
3574:
3562:
3548:
3522:
3520:
3516:
3510:
3505:
3493:
3491:
3490:
3485:
3477:
3454:
3439:
3396:
3394:
3393:
3388:
3386:
3363:
3331:
3329:
3325:
3319:
3314:
3302:
3300:
3299:
3294:
3286:
3263:
3248:
3211:
3209:
3208:
3203:
3201:
3175:
3149:
3147:
3143:
3137:
3132:
3120:
3118:
3117:
3112:
3104:
3081:
3066:
3029:
3027:
3026:
3021:
3019:
2993:
2967:
2965:
2961:
2955:
2950:
2938:
2936:
2935:
2930:
2922:
2899:
2884:
2853:
2851:
2850:
2845:
2843:
2820:
2800:
2798:
2794:
2788:
2783:
2773:
2772:
2764:
2762:
2761:
2756:
2754:
2750:
2746:
2745:
2740:
2732:
2719:
2678:
2674:
2659:
2624:
2620:
2619:
2611:
2598:
2550:
2520:
2516:
2515:
2510:
2502:
2489:
2454:
2450:
2435:
2406:
2402:
2401:
2393:
2380:
2338:
2294:
2287:
2285:
2284:
2279:
2276:
2271:
2248:
2243:
2238:
2228:
2227:
2224:
2220:
2206:
2205:
2202:
2189:
2182:
2180:
2179:
2174:
2171:
2166:
2143:
2138:
2134:
2126:
2119:
2118:
2106:
2101:
2091:
2090:
2087:
2083:
2069:
2068:
2065:
2052:
2045:
2043:
2042:
2037:
2034:
2029:
2006:
2001:
1996:
1986:
1985:
1982:
1978:
1964:
1963:
1960:
1947:
1940:
1938:
1937:
1932:
1929:
1924:
1901:
1896:
1892:
1884:
1877:
1876:
1864:
1859:
1849:
1848:
1845:
1841:
1827:
1826:
1823:
1810:
1803:
1801:
1800:
1795:
1768:
1764:
1757:
1749:
1741:
1740:
1737:
1722:
1721:
1718:
1705:
1698:
1696:
1695:
1690:
1663:
1659:
1651:
1644:
1643:
1619:
1618:
1615:
1602:
1595:
1593:
1592:
1587:
1560:
1556:
1549:
1541:
1533:
1532:
1529:
1514:
1513:
1510:
1497:
1490:
1488:
1487:
1482:
1455:
1451:
1443:
1436:
1435:
1411:
1410:
1407:
1396:
1395:
1377:solid geometries
1220:) was coined by
1180:
1177:
1175:
1174:
1171:
1168:
1160:
1116:
1114:
1113:
1110:
1107:
1100:
1097:
1095:
1094:
1091:
1088:
1080:
1068:Surya Siddhantha
1050:
1030:round-off errors
948:" and the chord
944:is viewed as a "
905:) is the radius
885:is the distance
806:is viewed as a "
756:
752:
748:
730:
726:
722:
718:
714:
696:
692:
688:
684:
662:
658:
654:
650:
646:
642:
638:
634:
611:
607:
603:
589:
585:
581:
577:
554:
550:
546:
528:
526:
525:
520:
518:
510:
489:
481:
476:
475:
408:(flipped sine),
399:
395:
391:
387:
383:
290:
283:
276:
47:
33:
32:
28:Versin (village)
21:Versine in WP:de
8693:
8692:
8688:
8687:
8686:
8684:
8683:
8682:
8668:
8667:
8666:
8661:
8630:
8609:Sine and cosine
8592:
8589:
8557:at GeoGebra.org
8535:
8523:
8521:
8511:
8484:
8482:Further reading
8479:
8478:
8440:
8436:
8421:
8417:
8410:
8398:. Vol. 3.
8387:
8383:
8376:
8356:
8352:
8343:
8341:
8332:
8331:
8324:
8315:
8313:
8290:
8283:
8274:
8272:
8257:
8253:
8244:
8242:
8223:
8216:
8207:
8205:
8186:
8179:
8163:
8162:
8156:
8154:
8139:
8132:
8123:
8121:
8106:
8091:
8082:
8080:
8061:
8046:
8037:
8035:
8012:
8003:
7994:
7992:
7990:
7969:
7965:
7952:
7950:
7939:
7935:
7918:
7916:
7913:
7897:
7893:
7887:nat. haversines
7883:log. haversines
7864:
7863:
7855:
7851:
7830:
7829:
7821:
7817:
7799:
7795:
7777:
7773:
7756:
7749:
7733:
7729:
7714:(11): 421–422.
7692:
7683:
7667:
7663:
7654:
7652:
7641:
7637:
7628:
7626:
7623:
7605:(1991-03-06) .
7596:
7587:
7577:
7557:
7548:
7539:
7537:
7521:"Hacovercosine"
7514:
7507:
7498:
7496:
7481:
7456:
7447:
7445:
7422:
7415:
7406:
7404:
7394:Ocean Navigator
7386:
7379:
7370:
7368:
7355:
7354:
7350:
7341:
7339:
7326:
7322:
7313:
7311:
7300:
7296:
7285:
7281:
7272:
7270:
7247:
7234:
7225:
7223:
7200:
7193:
7184:
7182:
7174:. Boston, USA:
7166:
7162:
7153:
7151:
7134:
7130:
7109:
7105:
7096:
7094:
7078:
7074:
7065:
7063:
7040:
7031:
7022:
7020:
6997:
6986:
6977:
6975:
6952:
6945:
6936:
6934:
6931:
6908:
6897:
6883:
6881:
6870:
6838:
6831:
6795:
6788:
6779:
6777:
6762:
6758:
6723:
6721:
6718:
6710:. p. 172.
6698:Cajori, Florian
6695:
6686:
6677:
6675:
6655:
6651:
6642:
6640:
6629:
6618:
6606:
6576:
6572:
6563:
6561:
6557:
6553:. p. 344.
6542:
6536:
6532:
6523:
6521:
6520:. 9780299018740
6510:
6483:
6479:
6470:
6468:
6455:
6442:
6419:
6403:. p. 301.
6393:
6389:
6374:
6340:
6336:
6327:
6325:
6314:
6310:
6279:
6251:
6238:
6225:
6223:
6211:
6202:
6175:
6173:
6163:
6159:
6152:
6148:
6143:
6138:
6137:
6124:
6117:
6109:instead of the
6044:
6037:
6032:
6013:Witch of Agnesi
5995:
5936:
5933:
5927:
5926:
5924:
5900:
5875:
5863:
5859:
5848:
5844:
5843:
5841:
5833:
5830:
5829:
5796:
5792:
5790:
5776:
5773:
5772:
5738:
5732:
5728:
5726:
5718:
5715:
5714:
5672:
5653:
5652:
5615:
5533:
5532:
5495:
5460:
5458:
5451:
5414:
5379:
5377:
5337:
5302:
5300:
5296:
5294:
5291:
5290:
5248:
5246:
5234:
5228:
5225:
5224:
5207:
5206:
5184:
5174:
5170:
5158:
5154:
5144:
5142:
5136:
5125:
5114:
5096:
5095:
5076:
5066:
5062:
5050:
5046:
5036:
5034:
5028:
5017:
5006:
4987:
4985:
4982:
4981:
4965:
4928:
4924:
4898:
4895:
4894:
4861:
4857:
4831:
4828:
4827:
4795:
4791:
4771:
4767:
4738:
4735:
4734:
4701:
4697:
4677:
4673:
4644:
4641:
4640:
4610:
4606:
4580:
4577:
4576:
4546:
4542:
4516:
4513:
4512:
4482:
4478:
4452:
4449:
4448:
4418:
4414:
4388:
4385:
4384:
4344:
4309:
4296:
4294:
4283:
4235:
4230:
4227:
4226:
4203:
4196:
4194:
4144:
4133:
4132:
4127:
4125:
4123:
4120:
4119:
4088:
4075:
4073:
4062:
4020:
4015:
4012:
4011:
3988:
3978:
3976:
3932:
3921:
3920:
3915:
3913:
3911:
3908:
3907:
3876:
3863:
3861:
3850:
3808:
3803:
3800:
3799:
3776:
3766:
3764:
3720:
3709:
3708:
3703:
3701:
3699:
3696:
3695:
3664:
3651:
3649:
3638:
3602:
3597:
3594:
3593:
3570:
3563:
3561:
3523:
3512:
3511:
3506:
3504:
3502:
3499:
3498:
3473:
3450:
3408:
3403:
3400:
3399:
3382:
3332:
3321:
3320:
3315:
3313:
3311:
3308:
3307:
3282:
3259:
3223:
3218:
3215:
3214:
3197:
3150:
3139:
3138:
3133:
3131:
3129:
3126:
3125:
3100:
3077:
3041:
3036:
3033:
3032:
3015:
2968:
2957:
2956:
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2947:
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2918:
2895:
2865:
2860:
2857:
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2839:
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2790:
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2784:
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2780:
2777:
2776:
2771:
2752:
2751:
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2731:
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2720:
2682:
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2628:
2610:
2603:
2599:
2567:
2560:
2525:
2522:
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2501:
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2490:
2458:
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2410:
2392:
2385:
2381:
2355:
2348:
2319:
2315:
2313:
2310:
2309:
2303:
2249:
2247:
2223:
2222:
2221:
2219:
2201:
2200:
2198:
2195:
2194:
2144:
2142:
2125:
2121:
2114:
2110:
2086:
2085:
2084:
2082:
2064:
2063:
2061:
2058:
2057:
2007:
2005:
1981:
1980:
1979:
1977:
1959:
1958:
1956:
1953:
1952:
1902:
1900:
1883:
1879:
1872:
1868:
1844:
1843:
1842:
1840:
1822:
1821:
1819:
1816:
1815:
1748:
1747:
1743:
1736:
1735:
1717:
1716:
1714:
1711:
1710:
1650:
1646:
1639:
1635:
1614:
1613:
1611:
1608:
1607:
1540:
1539:
1535:
1528:
1527:
1509:
1508:
1506:
1503:
1502:
1442:
1438:
1431:
1427:
1406:
1405:
1403:
1400:
1399:
1394:
1389:
1299:window function
1264:
1256:sight reduction
1248:lunar distances
1178:
1172:
1169:
1164:
1163:
1161:
1158:
1131:
1111:
1108:
1105:
1104:
1102:
1098:
1092:
1089:
1084:
1083:
1081:
1078:
1048:
974:used the terms
921:to the circle.
862:For a vertical
768:
763:
754:
750:
746:
728:
724:
720:
716:
712:
701:hacoversed sine
694:
690:
686:
682:
675:haversed cosine
660:
656:
652:
648:
644:
640:
636:
632:
631:), abbreviated
609:
605:
601:
594:coversed cosine
587:
583:
579:
575:
574:), abbreviated
552:
548:
544:
509:
480:
471:
467:
432:
429:
428:
397:
393:
389:
385:
381:
366:
344:of navigation.
294:
115:Exact constants
31:
24:
17:
12:
11:
5:
8691:
8681:
8680:
8663:
8662:
8660:
8659:
8654:
8649:
8644:
8638:
8636:
8632:
8631:
8629:
8628:
8623:
8618:
8613:
8612:
8611:
8600:
8598:
8594:
8593:
8588:
8587:
8580:
8573:
8565:
8559:
8558:
8552:
8534:
8533:External links
8531:
8530:
8529:
8509:
8491:, ed. (2002).
8483:
8480:
8477:
8476:
8434:
8415:
8408:
8402:. p. 39.
8381:
8374:
8350:
8322:
8281:
8251:
8214:
8200:source code).
8177:
8130:
8089:
8044:
8001:
7988:
7963:
7933:
7912:978-0914025214
7911:
7891:
7849:
7815:
7793:
7771:
7747:
7727:
7681:
7661:
7635:
7622:978-0471543978
7621:
7585:
7546:
7505:
7454:
7413:
7377:
7348:
7320:
7294:
7279:
7232:
7191:
7160:
7128:
7103:
7072:
7029:
6984:
6943:
6929:
6895:
6868:
6829:
6786:
6756:
6716:
6684:
6649:
6633:"Trigonometry"
6616:
6604:
6570:
6530:
6508:
6477:
6440:
6417:
6387:
6372:
6350:(2 ed.).
6334:
6308:
6277:
6236:
6200:
6157:
6145:
6144:
6142:
6139:
6136:
6135:
6115:
6087:cosinus versus
6085:associate the
6063:cosinus versus
6034:
6033:
6031:
6028:
6027:
6026:
6021:
6016:
6006:
6001:
5994:
5991:
5962:rail surveying
5958:Hallade method
5950:rail transport
5899:
5896:
5881:
5878:
5870:
5867:
5862:
5855:
5852:
5847:
5840:
5837:
5804:
5799:
5795:
5789:
5786:
5783:
5780:
5750:
5744:
5741:
5735:
5731:
5725:
5722:
5709:(the distance
5671:
5670:Approximations
5668:
5667:
5666:
5651:
5648:
5645:
5642:
5639:
5636:
5633:
5630:
5627:
5624:
5621:
5618:
5616:
5614:
5611:
5608:
5605:
5602:
5599:
5596:
5593:
5590:
5587:
5584:
5581:
5578:
5574:
5571:
5568:
5565:
5562:
5559:
5556:
5553:
5550:
5547:
5544:
5541:
5538:
5535:
5534:
5528:
5525:
5522:
5519:
5516:
5513:
5510:
5507:
5504:
5501:
5498:
5493:
5490:
5487:
5484:
5481:
5478:
5475:
5472:
5469:
5466:
5463:
5457:
5454:
5452:
5447:
5444:
5441:
5438:
5435:
5432:
5429:
5426:
5423:
5420:
5417:
5412:
5409:
5406:
5403:
5400:
5397:
5394:
5391:
5388:
5385:
5382:
5376:
5370:
5367:
5364:
5361:
5358:
5355:
5352:
5349:
5346:
5343:
5340:
5335:
5332:
5329:
5326:
5323:
5320:
5317:
5314:
5311:
5308:
5305:
5299:
5298:
5287:
5286:
5275:
5272:
5267:
5263:
5260:
5257:
5254:
5251:
5243:
5240:
5237:
5233:
5221:
5220:
5202:
5199:
5196:
5193:
5190:
5187:
5180:
5177:
5173:
5167:
5164:
5161:
5157:
5153:
5150:
5147:
5139:
5134:
5131:
5128:
5124:
5120:
5117:
5115:
5113:
5110:
5107:
5104:
5101:
5098:
5097:
5091:
5088:
5085:
5082:
5079:
5072:
5069:
5065:
5059:
5056:
5053:
5049:
5045:
5042:
5039:
5031:
5026:
5023:
5020:
5016:
5012:
5009:
5007:
5005:
5002:
4999:
4996:
4993:
4990:
4989:
4964:
4961:
4958:
4957:
4944:
4940:
4937:
4934:
4931:
4927:
4923:
4920:
4917:
4914:
4911:
4908:
4905:
4902:
4891:
4890:
4877:
4873:
4870:
4867:
4864:
4860:
4856:
4853:
4850:
4847:
4844:
4841:
4838:
4835:
4824:
4823:
4811:
4807:
4804:
4801:
4798:
4794:
4790:
4787:
4784:
4780:
4775:
4770:
4766:
4763:
4760:
4757:
4754:
4751:
4748:
4745:
4742:
4731:
4730:
4717:
4713:
4710:
4707:
4704:
4700:
4696:
4693:
4690:
4686:
4681:
4676:
4672:
4669:
4666:
4663:
4660:
4657:
4654:
4651:
4648:
4637:
4636:
4623:
4619:
4616:
4613:
4609:
4605:
4602:
4599:
4596:
4593:
4590:
4587:
4584:
4573:
4572:
4559:
4555:
4552:
4549:
4545:
4541:
4538:
4535:
4532:
4529:
4526:
4523:
4520:
4509:
4508:
4495:
4491:
4488:
4485:
4481:
4477:
4474:
4471:
4468:
4465:
4462:
4459:
4456:
4445:
4444:
4431:
4427:
4424:
4421:
4417:
4413:
4410:
4407:
4404:
4401:
4398:
4395:
4392:
4372:archacoversine
4368:archavercosine
4360:arccovercosine
4343:
4340:
4337:
4336:
4325:
4322:
4317:
4312:
4308:
4305:
4302:
4299:
4293:
4290:
4286:
4281:
4278:
4275:
4271:
4268:
4265:
4262:
4259:
4256:
4253:
4250:
4247:
4244:
4241:
4238:
4234:
4224:
4211:
4206:
4202:
4199:
4193:
4190:
4187:
4184:
4180:
4177:
4174:
4171:
4168:
4165:
4162:
4159:
4156:
4153:
4150:
4147:
4140:
4136:
4130:
4116:
4115:
4104:
4101:
4096:
4091:
4087:
4084:
4081:
4078:
4072:
4069:
4065:
4060:
4057:
4054:
4050:
4047:
4044:
4041:
4038:
4035:
4032:
4029:
4026:
4023:
4019:
4009:
3996:
3991:
3987:
3984:
3981:
3975:
3972:
3969:
3966:
3962:
3959:
3956:
3953:
3950:
3947:
3944:
3941:
3938:
3935:
3928:
3924:
3918:
3904:
3903:
3892:
3889:
3884:
3879:
3875:
3872:
3869:
3866:
3860:
3857:
3853:
3848:
3845:
3842:
3838:
3835:
3832:
3829:
3826:
3823:
3820:
3817:
3814:
3811:
3807:
3797:
3784:
3779:
3775:
3772:
3769:
3763:
3760:
3757:
3754:
3750:
3747:
3744:
3741:
3738:
3735:
3732:
3729:
3726:
3723:
3716:
3712:
3706:
3692:
3691:
3680:
3677:
3672:
3667:
3663:
3660:
3657:
3654:
3648:
3645:
3641:
3636:
3633:
3630:
3626:
3623:
3620:
3617:
3614:
3611:
3608:
3605:
3601:
3591:
3578:
3573:
3569:
3566:
3560:
3557:
3554:
3551:
3547:
3544:
3541:
3538:
3535:
3532:
3529:
3526:
3519:
3515:
3509:
3495:
3494:
3483:
3480:
3476:
3472:
3469:
3466:
3463:
3460:
3457:
3453:
3448:
3445:
3442:
3438:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3407:
3397:
3385:
3381:
3378:
3375:
3372:
3369:
3366:
3362:
3359:
3356:
3353:
3350:
3347:
3344:
3341:
3338:
3335:
3328:
3324:
3318:
3304:
3303:
3292:
3289:
3285:
3281:
3278:
3275:
3272:
3269:
3266:
3262:
3257:
3254:
3251:
3247:
3244:
3241:
3238:
3235:
3232:
3229:
3226:
3222:
3212:
3200:
3196:
3193:
3190:
3187:
3184:
3181:
3178:
3174:
3171:
3168:
3165:
3162:
3159:
3156:
3153:
3146:
3142:
3136:
3122:
3121:
3110:
3107:
3103:
3099:
3096:
3093:
3090:
3087:
3084:
3080:
3075:
3072:
3069:
3065:
3062:
3059:
3056:
3053:
3050:
3047:
3044:
3040:
3030:
3018:
3014:
3011:
3008:
3005:
3002:
2999:
2996:
2992:
2989:
2986:
2983:
2980:
2977:
2974:
2971:
2964:
2960:
2954:
2940:
2939:
2928:
2925:
2921:
2917:
2914:
2911:
2908:
2905:
2902:
2898:
2893:
2890:
2887:
2883:
2880:
2877:
2874:
2871:
2868:
2864:
2854:
2842:
2838:
2835:
2832:
2829:
2826:
2823:
2819:
2816:
2813:
2810:
2807:
2804:
2797:
2793:
2787:
2770:
2767:
2766:
2765:
2749:
2743:
2739:
2736:
2730:
2727:
2723:
2718:
2715:
2712:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2688:
2685:
2681:
2677:
2673:
2670:
2667:
2663:
2658:
2655:
2652:
2649:
2646:
2643:
2640:
2637:
2634:
2631:
2627:
2623:
2617:
2614:
2609:
2606:
2602:
2597:
2594:
2591:
2588:
2585:
2582:
2579:
2576:
2573:
2570:
2566:
2563:
2561:
2559:
2556:
2553:
2549:
2546:
2543:
2540:
2537:
2534:
2531:
2528:
2524:
2523:
2519:
2513:
2509:
2506:
2500:
2497:
2493:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2457:
2453:
2449:
2446:
2443:
2439:
2434:
2431:
2428:
2425:
2422:
2419:
2416:
2413:
2409:
2405:
2399:
2396:
2391:
2388:
2384:
2379:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2354:
2351:
2349:
2347:
2344:
2341:
2337:
2334:
2331:
2328:
2325:
2322:
2318:
2317:
2302:
2299:
2296:
2295:
2288:
2274:
2270:
2267:
2264:
2261:
2258:
2255:
2252:
2246:
2241:
2237:
2234:
2231:
2218:
2215:
2212:
2209:
2191:
2190:
2183:
2169:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2141:
2137:
2132:
2129:
2124:
2117:
2113:
2109:
2104:
2100:
2097:
2094:
2081:
2078:
2075:
2072:
2054:
2053:
2046:
2032:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
2004:
1999:
1995:
1992:
1989:
1976:
1973:
1970:
1967:
1949:
1948:
1941:
1927:
1923:
1920:
1917:
1914:
1911:
1908:
1905:
1899:
1895:
1890:
1887:
1882:
1875:
1871:
1867:
1862:
1858:
1855:
1852:
1839:
1836:
1833:
1830:
1812:
1811:
1804:
1792:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1767:
1763:
1760:
1755:
1752:
1746:
1734:
1731:
1728:
1725:
1707:
1706:
1699:
1687:
1684:
1681:
1678:
1675:
1672:
1669:
1666:
1662:
1657:
1654:
1649:
1642:
1638:
1634:
1631:
1628:
1625:
1622:
1604:
1603:
1596:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1559:
1555:
1552:
1547:
1544:
1538:
1526:
1523:
1520:
1517:
1499:
1498:
1491:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1454:
1449:
1446:
1441:
1434:
1430:
1426:
1423:
1420:
1417:
1414:
1393:
1390:
1388:
1385:
1364:and angles in
1291:control theory
1263:
1260:
1240:for versines.
1214:log. haversine
1130:
1127:
1121:), derived by
984:cosinus versus
778:in terms of a
767:
764:
762:
759:
758:
757:
745:, abbreviated
731:
711:, abbreviated
697:
681:, abbreviated
671:
613:
612:
600:, abbreviated
590:
568:cosinus versus
556:
543:, abbreviated
516:
513:
508:
505:
501:
498:
495:
492:
487:
484:
479:
474:
470:
466:
463:
460:
457:
454:
451:
448:
445:
442:
439:
436:
365:
362:
296:
295:
293:
292:
285:
278:
270:
267:
266:
265:
264:
259:
254:
249:
244:
239:
234:
229:
224:
219:
211:
210:
209:Mathematicians
206:
205:
204:
203:
198:
193:
183:
175:
174:
168:
167:
166:
165:
159:
158:
153:
148:
143:
135:
134:
130:
129:
128:
127:
122:
117:
112:
104:
103:
99:
98:
97:
96:
91:
68:
67:
62:
57:
49:
48:
40:
39:
15:
9:
6:
4:
3:
2:
8690:
8679:
8676:
8675:
8673:
8658:
8655:
8653:
8650:
8648:
8645:
8643:
8640:
8639:
8637:
8633:
8627:
8624:
8622:
8619:
8617:
8614:
8610:
8607:
8606:
8605:
8604:Trigonometric
8602:
8601:
8599:
8595:
8586:
8581:
8579:
8574:
8572:
8567:
8566:
8563:
8556:
8553:
8549:
8545:
8541:
8540:Pegg, Jr., Ed
8537:
8536:
8520:
8516:
8512:
8510:0-7624-1698-X
8506:
8502:
8501:Running Press
8497:
8496:
8490:
8486:
8485:
8472:
8468:
8464:
8460:
8456:
8452:
8448:
8444:
8438:
8431:. p. 32.
8430:
8426:
8419:
8411:
8409:9780521058018
8405:
8401:
8397:
8396:
8391:
8385:
8377:
8371:
8367:
8363:
8362:
8354:
8339:
8335:
8329:
8327:
8311:
8308:
8304:
8303:
8298:
8294:
8288:
8286:
8270:
8266:
8262:
8255:
8240:
8236:
8232:
8228:
8221:
8219:
8203:
8199:
8195:
8191:
8184:
8182:
8173:
8167:
8152:
8149:. maths.org.
8148:
8147:Plus magazine
8144:
8137:
8135:
8119:
8115:
8111:
8104:
8102:
8100:
8098:
8096:
8094:
8078:
8074:
8070:
8066:
8059:
8057:
8055:
8053:
8051:
8049:
8033:
8030:
8026:
8025:
8020:
8016:
8010:
8008:
8006:
7991:
7989:0-9757492-0-X
7985:
7980:
7979:
7973:
7967:
7961:
7959:
7948:
7944:
7937:
7930:
7927:
7914:
7908:
7904:
7903:
7895:
7888:
7884:
7880:
7874:
7868:
7860:
7853:
7846:
7840:
7834:
7826:
7819:
7812:
7806:
7805:
7797:
7790:
7784:
7783:
7775:
7767:
7763:
7762:
7758:"haversine".
7754:
7752:
7744:
7738:
7731:
7722:
7717:
7713:
7709:
7708:
7704:
7702:
7696:
7690:
7688:
7686:
7677:
7676:
7671:
7665:
7650:
7646:
7639:
7624:
7618:
7614:
7610:
7609:
7604:
7600:
7594:
7592:
7590:
7581:
7573:
7569:
7568:
7562:
7555:
7553:
7551:
7535:
7532:
7528:
7527:
7522:
7518:
7512:
7510:
7494:
7490:
7486:
7479:
7477:
7475:
7473:
7471:
7469:
7467:
7465:
7463:
7461:
7459:
7443:
7440:
7436:
7435:
7430:
7429:"Havercosine"
7426:
7420:
7418:
7403:
7399:
7395:
7391:
7384:
7382:
7366:
7363:. 7.0. 2008.
7362:
7358:
7352:
7337:
7333:
7332:
7324:
7309:
7305:
7298:
7290:
7283:
7268:
7265:
7261:
7260:
7255:
7251:
7245:
7243:
7241:
7239:
7237:
7221:
7218:
7214:
7213:
7208:
7207:"Covercosine"
7204:
7198:
7196:
7181:
7177:
7173:
7172:
7164:
7149:
7145:
7141:
7140:
7132:
7124:
7120:
7116:
7115:
7107:
7093:
7089:
7085:
7084:
7076:
7061:
7058:
7054:
7053:
7048:
7047:"Hacoversine"
7044:
7038:
7036:
7034:
7018:
7015:
7011:
7010:
7005:
7001:
6995:
6993:
6991:
6989:
6973:
6970:
6966:
6965:
6960:
6956:
6950:
6948:
6932:
6930:9780691148922
6926:
6922:
6918:
6917:
6912:
6906:
6904:
6902:
6900:
6892:
6879:
6875:
6871:
6865:
6861:
6857:
6853:
6849:
6845:
6844:
6836:
6834:
6826:
6822:
6818:
6815:(reissued by
6814:
6809:
6805:
6804:
6799:
6793:
6791:
6775:
6771:
6767:
6760:
6752:
6750:
6746:
6742:
6741:
6736:
6732:
6719:
6713:
6709:
6705:
6704:
6699:
6693:
6691:
6689:
6674:. p. 231
6673:
6672:B. G. Teubner
6669:
6665:
6664:
6659:
6653:
6638:
6634:
6627:
6625:
6623:
6621:
6613:
6607:
6601:
6597:
6593:
6588:
6587:
6581:
6574:
6556:
6552:
6548:
6541:
6534:
6519:
6515:
6511:
6509:0-299-01874-1
6505:
6501:
6497:
6496:
6491:
6487:
6481:
6466:
6462:
6461:
6453:
6451:
6449:
6447:
6445:
6436:
6432:
6428:
6424:
6420:
6414:
6410:
6406:
6402:
6398:
6391:
6383:
6379:
6375:
6369:
6365:
6361:
6357:
6353:
6348:
6347:
6338:
6323:
6319:
6312:
6304:
6300:
6296:
6292:
6288:
6284:
6280:
6274:
6270:
6269:
6264:
6260:
6256:
6249:
6247:
6245:
6243:
6241:
6233:
6221:
6220:
6215:
6209:
6207:
6205:
6198:
6194:
6192:
6188:
6184:
6171:
6170:
6161:
6155:
6150:
6146:
6132:
6128:
6122:
6120:
6112:
6111:coversed sine
6108:
6107:versed cosine
6104:
6100:
6099:Korn and Korn
6096:
6095:coversed sine
6092:
6091:versed cosine
6088:
6084:
6080:
6076:
6072:
6068:
6064:
6060:
6056:
6052:
6048:
6042:
6040:
6035:
6025:
6022:
6020:
6017:
6014:
6010:
6007:
6005:
6002:
6000:
5997:
5996:
5990:
5988:
5984:
5980:
5976:
5972:
5971:
5965:
5963:
5959:
5955:
5951:
5947:
5939:
5931:
5922:
5918:
5914:
5910:
5905:
5895:
5879:
5876:
5868:
5865:
5860:
5853:
5850:
5845:
5838:
5835:
5826:
5824:
5820:
5802:
5797:
5793:
5787:
5784:
5781:
5778:
5770:
5766:
5761:
5748:
5742:
5739:
5733:
5729:
5723:
5720:
5712:
5708:
5704:
5700:
5692:
5687:
5681:
5676:
5646:
5640:
5637:
5631:
5625:
5622:
5619:
5617:
5606:
5600:
5597:
5594:
5588:
5582:
5579:
5566:
5560:
5557:
5554:
5548:
5542:
5539:
5523:
5517:
5514:
5511:
5505:
5499:
5496:
5488:
5482:
5479:
5473:
5467:
5464:
5461:
5455:
5453:
5442:
5436:
5433:
5430:
5424:
5418:
5415:
5407:
5401:
5398:
5395:
5389:
5383:
5380:
5374:
5365:
5359:
5356:
5353:
5347:
5341:
5338:
5330:
5324:
5321:
5318:
5312:
5306:
5303:
5289:
5288:
5273:
5270:
5265:
5258:
5252:
5249:
5241:
5235:
5223:
5222:
5200:
5194:
5191:
5185:
5178:
5175:
5171:
5165:
5162:
5159:
5151:
5148:
5132:
5129:
5126:
5122:
5118:
5116:
5108:
5102:
5099:
5089:
5083:
5080:
5070:
5067:
5063:
5057:
5054:
5051:
5043:
5040:
5024:
5021:
5018:
5014:
5010:
5008:
5000:
4994:
4991:
4980:
4979:
4978:
4976:
4972:
4970:
4969:complex plane
4942:
4938:
4935:
4932:
4929:
4925:
4921:
4918:
4915:
4909:
4903:
4901:archacovercos
4900:
4893:
4892:
4875:
4871:
4868:
4865:
4862:
4858:
4854:
4851:
4848:
4842:
4836:
4834:archacoversin
4833:
4826:
4825:
4809:
4805:
4802:
4799:
4796:
4792:
4788:
4785:
4782:
4778:
4773:
4768:
4764:
4761:
4758:
4755:
4749:
4743:
4740:
4733:
4732:
4715:
4711:
4708:
4705:
4702:
4698:
4694:
4691:
4688:
4684:
4679:
4674:
4670:
4667:
4664:
4661:
4655:
4649:
4646:
4639:
4638:
4621:
4617:
4614:
4611:
4607:
4603:
4600:
4597:
4591:
4585:
4582:
4575:
4574:
4557:
4553:
4550:
4547:
4543:
4539:
4536:
4533:
4527:
4521:
4518:
4511:
4510:
4493:
4489:
4486:
4483:
4479:
4475:
4472:
4469:
4463:
4457:
4454:
4447:
4446:
4429:
4425:
4422:
4419:
4415:
4411:
4408:
4405:
4399:
4393:
4390:
4383:
4382:
4379:
4377:
4373:
4369:
4365:
4361:
4357:
4353:
4349:
4323:
4320:
4315:
4310:
4306:
4303:
4300:
4297:
4291:
4288:
4276:
4232:
4225:
4209:
4204:
4200:
4197:
4191:
4185:
4138:
4118:
4117:
4102:
4099:
4094:
4089:
4085:
4082:
4079:
4076:
4070:
4067:
4055:
4017:
4010:
3994:
3989:
3985:
3982:
3979:
3973:
3967:
3926:
3906:
3905:
3890:
3887:
3882:
3877:
3873:
3870:
3867:
3864:
3858:
3855:
3843:
3805:
3798:
3782:
3777:
3773:
3770:
3767:
3761:
3755:
3714:
3694:
3693:
3678:
3675:
3670:
3665:
3661:
3658:
3655:
3652:
3646:
3643:
3631:
3599:
3592:
3576:
3571:
3567:
3564:
3558:
3552:
3517:
3497:
3496:
3481:
3478:
3474:
3470:
3467:
3464:
3461:
3458:
3455:
3443:
3405:
3398:
3383:
3379:
3376:
3373:
3367:
3326:
3306:
3305:
3290:
3287:
3283:
3279:
3276:
3273:
3270:
3267:
3264:
3252:
3220:
3213:
3198:
3194:
3191:
3188:
3185:
3179:
3144:
3124:
3123:
3108:
3105:
3101:
3097:
3094:
3091:
3088:
3085:
3082:
3070:
3038:
3031:
3016:
3012:
3009:
3006:
3003:
2997:
2962:
2942:
2941:
2926:
2923:
2919:
2915:
2912:
2909:
2906:
2903:
2900:
2888:
2862:
2855:
2840:
2836:
2833:
2830:
2824:
2795:
2775:
2774:
2747:
2741:
2737:
2734:
2728:
2725:
2721:
2679:
2675:
2671:
2668:
2665:
2661:
2625:
2621:
2615:
2612:
2607:
2604:
2600:
2564:
2562:
2554:
2517:
2511:
2507:
2504:
2498:
2495:
2491:
2455:
2451:
2447:
2444:
2441:
2437:
2407:
2403:
2397:
2394:
2389:
2386:
2382:
2352:
2350:
2342:
2308:
2307:
2306:
2293:
2289:
2272:
2265:
2259:
2256:
2253:
2250:
2244:
2239:
2232:
2216:
2210:
2193:
2192:
2188:
2184:
2167:
2160:
2154:
2151:
2148:
2145:
2139:
2135:
2130:
2127:
2122:
2115:
2111:
2107:
2102:
2095:
2079:
2073:
2056:
2055:
2051:
2047:
2030:
2023:
2017:
2014:
2011:
2008:
2002:
1997:
1990:
1974:
1968:
1951:
1950:
1946:
1942:
1925:
1918:
1912:
1909:
1906:
1903:
1897:
1893:
1888:
1885:
1880:
1873:
1869:
1865:
1860:
1853:
1837:
1831:
1814:
1813:
1809:
1805:
1787:
1781:
1778:
1775:
1772:
1769:
1765:
1761:
1758:
1753:
1750:
1744:
1732:
1726:
1709:
1708:
1704:
1700:
1682:
1676:
1673:
1670:
1667:
1664:
1660:
1655:
1652:
1647:
1640:
1636:
1632:
1629:
1623:
1606:
1605:
1601:
1597:
1579:
1573:
1570:
1567:
1564:
1561:
1557:
1553:
1550:
1545:
1542:
1536:
1524:
1518:
1501:
1500:
1496:
1492:
1474:
1468:
1465:
1462:
1459:
1456:
1452:
1447:
1444:
1439:
1432:
1428:
1424:
1421:
1415:
1398:
1397:
1384:
1382:
1378:
1374:
1371:
1367:
1363:
1359:
1355:
1350:
1348:
1344:
1340:
1336:
1332:
1331:Hann function
1328:
1324:
1320:
1317:in value and
1316:
1312:
1311:Tukey windows
1308:
1304:
1300:
1296:
1292:
1288:
1284:
1280:
1275:
1273:
1269:
1259:
1257:
1253:
1249:
1244:
1241:
1239:
1235:
1231:
1227:
1223:
1219:
1215:
1211:
1207:
1203:
1202:
1196:
1194:
1189:
1187:
1182:
1167:
1156:
1152:
1148:
1144:
1140:
1136:
1126:
1124:
1120:
1087:
1076:
1071:
1069:
1065:
1061:
1056:
1054:
1046:
1042:
1037:
1035:
1031:
1027:
1023:
1019:
1015:
1010:
1004:
1000:
995:
991:
989:
985:
981:
977:
973:
968:
966:
962:
957:
955:
951:
947:
943:
939:
935:
932:. If the arc
931:
927:
922:
920:
916:
912:
908:
904:
900:
897:) and versin(
896:
892:
888:
884:
880:
876:
872:
868:
865:
860:
858:
854:
850:
846:
842:
841:
837:The ordinary
832:
827:
820:
816:
813:
809:
805:
801:
797:
794:. If the arc
793:
789:
785:
781:
777:
772:
744:
743:cohavercosine
740:
739:hacovercosine
736:
732:
710:
706:
702:
698:
680:
676:
672:
670:
666:
630:
626:
622:
621:haversed sine
618:
617:
616:
599:
595:
591:
573:
569:
565:
561:
560:coversed sine
557:
542:
538:
537:versed cosine
534:
533:
532:
529:
514:
511:
506:
503:
499:
496:
493:
490:
485:
482:
477:
472:
468:
464:
461:
458:
455:
452:
449:
446:
443:
440:
437:
434:
426:
421:
419:
415:
411:
407:
403:
379:
375:
371:
358:
354:
349:
345:
343:
339:
335:
330:
328:
324:
321:, Section I)
320:
319:
315:
311:
307:
303:
291:
286:
284:
279:
277:
272:
271:
269:
268:
263:
260:
258:
255:
253:
250:
248:
245:
243:
242:Regiomontanus
240:
238:
235:
233:
230:
228:
225:
223:
220:
218:
215:
214:
213:
212:
208:
207:
202:
199:
197:
194:
191:
187:
184:
182:
179:
178:
177:
176:
173:
170:
169:
164:
161:
160:
157:
154:
152:
149:
147:
144:
142:
139:
138:
137:
136:
132:
131:
126:
123:
121:
118:
116:
113:
111:
108:
107:
106:
105:
101:
100:
95:
92:
89:
85:
81:
77:
73:
70:
69:
66:
63:
61:
58:
56:
53:
52:
51:
50:
46:
42:
41:
38:
35:
34:
29:
22:
8641:
8522:. Retrieved
8494:
8450:
8446:
8437:
8424:
8418:
8394:
8384:
8360:
8353:
8342:. Retrieved
8337:
8314:. Retrieved
8300:
8273:. Retrieved
8264:
8254:
8243:. Retrieved
8206:. Retrieved
8193:
8155:. Retrieved
8146:
8122:. Retrieved
8113:
8081:. Retrieved
8036:. Retrieved
8022:
7993:. Retrieved
7977:
7966:
7951:. Retrieved
7936:
7917:. Retrieved
7915:. 091402521X
7901:
7894:
7858:
7852:
7824:
7818:
7811:Cajori, 1929
7802:
7796:
7789:Cajori, 1929
7780:
7774:
7759:
7736:
7730:
7711:
7705:
7700:
7674:
7664:
7653:. Retrieved
7638:
7627:. Retrieved
7625:. 0471543977
7607:
7565:
7538:. Retrieved
7524:
7497:. Retrieved
7488:
7446:. Retrieved
7432:
7405:. Retrieved
7393:
7369:. Retrieved
7360:
7351:
7340:. Retrieved
7338:. p. 42
7330:
7323:
7312:. Retrieved
7297:
7288:
7282:
7271:. Retrieved
7257:
7224:. Retrieved
7210:
7183:. Retrieved
7170:
7163:
7152:. Retrieved
7139:Trigonometry
7138:
7131:
7113:
7106:
7095:. Retrieved
7082:
7075:
7064:. Retrieved
7050:
7021:. Retrieved
7007:
6976:. Retrieved
6962:
6935:. Retrieved
6933:. 0691148929
6915:
6882:. Retrieved
6842:
6802:
6778:. Retrieved
6759:
6738:
6728:
6722:. Retrieved
6720:. 1602067147
6702:
6676:. Retrieved
6667:
6662:
6652:
6641:. Retrieved
6585:
6573:
6562:. Retrieved
6546:
6533:
6522:. Retrieved
6494:
6480:
6469:. Retrieved
6460:Trigonometry
6459:
6396:
6390:
6345:
6337:
6326:. Retrieved
6311:
6267:
6224:. Retrieved
6218:
6214:Inman, James
6197:1856 edition
6187:versed sines
6180:
6174:. Retrieved
6167:
6160:
6149:
6126:
6110:
6106:
6102:
6094:
6090:
6086:
6078:
6074:
6070:
6066:
6062:
6058:
6054:
6051:sinus versus
6050:
6047:Cauchy, 1821
5974:
5968:
5966:
5937:
5929:
5920:
5912:
5908:
5903:
5901:
5827:
5823:Guo Shoujing
5768:
5764:
5762:
5710:
5706:
5702:
5698:
5696:
5690:
5679:
4973:
4966:
4375:
4371:
4367:
4364:archaversine
4363:
4359:
4356:arccoversine
4355:
4352:arcvercosine
4351:
4347:
4345:
2304:
2203:hacovercosin
1383:since 2005.
1357:
1353:
1351:
1307:Hann–Poisson
1282:
1278:
1276:
1265:
1258:since 2014.
1245:
1242:
1237:
1234:nat. versine
1233:
1225:
1218:log. havers.
1217:
1213:
1205:
1199:
1197:
1190:
1183:
1165:
1132:
1118:
1085:
1072:
1057:
1044:
1038:
1033:
1017:
1013:
1011:
1008:
1002:
1001:centered at
987:
983:
979:
976:sinus versus
975:
969:
960:
958:
953:
949:
941:
937:
933:
928:, Latin for
925:
923:
918:
914:
910:
906:
902:
898:
894:
890:
886:
882:
878:
874:
870:
866:
861:
853:sinus versus
852:
849:sinus rectus
848:
838:
836:
831:the SVG file
818:
814:
803:
799:
795:
790:, Latin for
787:
783:
775:
747:hacovercosin
742:
738:
734:
717:semicoversin
708:
704:
700:
678:
674:
641:semiversinus
628:
624:
620:
614:
597:
593:
571:
567:
563:
559:
540:
536:
530:
422:
417:
413:
409:
406:sinus versus
405:
373:
369:
367:
337:
333:
331:
317:
305:
301:
299:
37:Trigonometry
8457:: 159–166.
8340:. 7.0. 2008
8267:. Versine.
7877:(A 7-place
7741:(A 7-place
7485:"Goniology"
7357:"Haversine"
7254:"Haversine"
7146:. pp.
7004:"Coversine"
6959:"Vercosine"
6735:James Inman
5954:rail tracks
4741:archavercos
4647:archaversin
4583:arccovercos
4519:arccoversin
1392:Definitions
1301:(including
1272:cofunctions
1262:Modern uses
1222:James Inman
999:unit circle
857:unit circle
780:unit circle
709:cohaversine
705:hacoversine
679:havercosine
598:covercosine
566:(in Latin,
374:versed sine
353:unit circle
306:versed sine
227:Brahmagupta
196:Derivatives
125:Unit circle
8621:Hyperbolic
8524:2017-07-31
8519:2002100441
8344:2015-11-05
8316:2015-10-05
8275:2015-11-28
8245:2015-11-28
8208:2015-11-28
8157:2015-11-05
8124:2015-10-26
8083:2015-10-26
8069:Fortran 90
8038:2015-11-05
7995:2015-12-01
7953:2015-12-02
7919:2015-12-02
7655:2015-11-10
7629:2019-08-10
7582:required.)
7540:2015-11-06
7499:2015-11-28
7448:2015-11-06
7407:2015-11-07
7371:2015-11-06
7342:2015-12-08
7314:2015-11-12
7273:2015-11-06
7226:2015-11-06
7185:2015-12-08
7178:. p.
7154:2015-12-08
7121:. p.
7097:2015-12-08
7090:. p.
7066:2015-11-06
7023:2015-11-06
6978:2015-11-06
6937:2015-11-10
6884:2015-11-09
6878:2009932254
6780:2015-11-08
6724:2015-11-11
6678:2015-12-09
6643:2015-11-08
6564:2015-11-12
6524:2015-11-16
6471:2017-08-12
6427:2017947446
6382:2008937525
6354:. p.
6328:2015-11-12
6226:2015-11-09
6176:2017-08-13
6141:References
4348:arcversine
2225:covercosin
2066:havercosin
1961:hacoversin
1719:covercosin
1347:statistics
1315:continuous
1250:utilizing
1238:nat. vers.
1135:navigation
843:function (
810:" and the
751:hacovercos
713:hacoversin
683:havercosin
669:navigation
637:semiversin
629:semiversus
602:covercosin
572:coversinus
318:Aryabhatia
237:al-Battani
217:Hipparchus
156:Cotangents
110:Identities
8471:751627806
8463:0020-8442
8302:MathWorld
8261:"Versine"
8194:Fitsblink
8166:cite news
8065:"AUXTRIG"
8024:MathWorld
8019:"Versine"
7879:haversine
7867:cite book
7861:. London.
7845:haversine
7833:cite book
7827:. London.
7743:haversine
7561:"sagitta"
7526:MathWorld
7434:MathWorld
7402:0886-0149
7259:MathWorld
7212:MathWorld
7052:MathWorld
7009:MathWorld
6964:MathWorld
6749:July 1926
6700:(1952) .
6594:pp.
6216:(1835) .
5967:The term
5946:curvature
5902:The term
5839:≈
5782:≈
5724:≈
5647:θ
5641:
5632:θ
5626:
5607:θ
5601:
5589:θ
5583:
5567:θ
5561:
5549:θ
5543:
5524:θ
5518:
5512:−
5506:θ
5500:
5489:θ
5483:
5474:θ
5468:
5443:θ
5437:
5431:−
5425:θ
5419:
5408:θ
5402:
5390:θ
5384:
5375:−
5366:θ
5360:
5354:−
5348:θ
5342:
5331:θ
5325:
5313:θ
5307:
5266:θ
5259:θ
5253:
5239:→
5236:θ
5163:−
5149:−
5138:∞
5123:∑
5103:
5055:−
5041:−
5030:∞
5015:∑
4995:
4936:−
4922:
4904:
4866:−
4855:
4837:
4803:−
4789:
4765:
4744:
4706:−
4695:
4671:
4650:
4615:−
4604:
4586:
4551:−
4540:
4522:
4487:−
4476:
4458:
4455:arcvercos
4423:−
4412:
4394:
4391:arcversin
4307:
4301:−
4233:∫
4201:
4086:
4018:∫
3986:
3980:−
3874:
3806:∫
3774:
3768:−
3662:
3656:−
3600:∫
3568:
3471:
3465:−
3406:∫
3380:
3280:
3221:∫
3195:
3189:−
3098:
3039:∫
3013:
3007:−
2916:
2910:−
2863:∫
2837:
2738:π
2726:θ
2672:π
2666:θ
2613:π
2605:θ
2555:θ
2508:π
2496:θ
2448:π
2442:θ
2395:π
2387:θ
2343:θ
2266:θ
2260:
2233:θ
2211:θ
2161:θ
2155:
2128:θ
2096:θ
2074:θ
2024:θ
2018:
2012:−
1991:θ
1969:θ
1919:θ
1913:
1907:−
1886:θ
1854:θ
1832:θ
1788:θ
1782:
1762:θ
1759:−
1751:π
1727:θ
1683:θ
1677:
1653:θ
1624:θ
1580:θ
1574:
1568:−
1554:θ
1551:−
1543:π
1519:θ
1475:θ
1469:
1463:−
1445:θ
1416:θ
1201:haversine
1151:longitude
1129:Haversine
1041:logarithm
970:In 1821,
940: = 2
802: = 2
625:haversine
564:coversine
541:vercosine
512:θ
507:
500:θ
497:
483:θ
478:
459:θ
456:
450:−
441:θ
438:
420:(arrow).
338:haversine
334:coversine
252:de Moivre
186:Integrals
102:Reference
72:Functions
8672:Category
8647:Exsecant
8392:(1959).
8310:Archived
8269:Archived
8239:Archived
8202:Archived
8151:Archived
8118:Archived
8116:. 1.25.
8077:Archived
8032:Archived
7974:(2005).
7947:Archived
7931:(1+10).)
7697:(1945).
7672:(1795).
7649:Archived
7534:Archived
7493:Archived
7442:Archived
7365:Archived
7308:Archived
7267:Archived
7220:Archived
7060:Archived
7017:Archived
6972:Archived
6913:(2013).
6819:, 2009;
6774:Archived
6745:February
6660:(1903).
6637:Archived
6555:Archived
6435:30244721
6322:Archived
6303:65-12253
6287:64-60036
6265:(eds.).
6009:Versiera
5993:See also
5819:Shen Kuo
5580:coversin
5515:coversin
5480:coversin
5357:coversin
5322:coversin
5100:haversin
2088:vercosin
1983:coversin
1824:haversin
1738:vercosin
1616:vercosin
1511:coversin
1370:metrical
1155:latitude
1143:spheroid
965:abscissa
721:hacovers
687:havercos
633:haversin
606:covercos
576:coversin
545:vercosin
410:versinus
364:Overview
314:Sanskrit
232:al-Hasib
172:Calculus
151:Tangents
8642:Versine
7768:. 1989.
6518:59-5304
6295:0167642
6061:), the
6049:), the
5987:mirrors
5970:sagitta
5942:
5925:
5913:versine
5904:versine
1362:spreads
1176:
1162:
1123:Ptolemy
1117:versin(
1115:
1103:
1096:
1082:
961:sagitta
926:sagitta
788:sagitta
627:(Latin
418:sagitta
370:versine
302:versine
262:Fourier
222:Ptolemy
188: (
146:Cosines
88:inverse
74: (
60:History
55:Outline
8597:Groups
8517:
8507:
8469:
8461:
8406:
8372:
8231:Python
8198:Python
7986:
7909:
7619:
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6927:
6891:errata
6876:
6866:
6823:
6714:
6612:errata
6602:
6598:–893.
6516:
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6433:
6425:
6415:
6380:
6370:
6301:
6293:
6285:
6275:
6103:θ
6079:θ
6075:θ
6071:θ
6067:θ
6059:θ
6055:θ
5983:lenses
5979:optics
5540:versin
5497:versin
5465:versin
5339:versin
5304:versin
5250:versin
4992:versin
4919:arcsin
4852:arcsin
4786:arccos
4762:arccos
4692:arccos
4668:arcsin
4601:arcsin
4537:arcsin
4473:arccos
4409:arccos
1846:versin
1530:versin
1408:versin
1373:planar
1358:Δ
1354:θ
1283:π
1281:< 2
1279:θ
1119:θ
1047:= 0, 2
1045:θ
1034:θ
1018:θ
1014:θ
972:Cauchy
942:θ
938:Δ
915:versus
911:rectus
899:θ
891:θ
883:θ
875:Δ
871:θ
804:θ
800:Δ
776:θ
645:havers
580:covers
549:vercos
435:versin
414:versus
386:sinver
382:versin
327:cosine
120:Tables
8657:atan2
8635:Other
7576:
6889:(See
6666:[
6610:(See
6558:(PDF)
6543:(PDF)
6431:S2CID
6257:. In
6030:Notes
5917:limit
5598:excsc
5558:exsec
5434:excsc
5416:exsec
5399:excsc
5381:exsec
1319:slope
1297:or a
1295:pulse
1060:chord
988:cosiv
930:arrow
864:chord
812:chord
792:arrow
741:, or
725:hacov
707:, or
659:, or
608:, or
586:, or
584:cosiv
551:, or
539:, or
416:, or
402:Latin
400:. In
376:is a
355:with
308:is a
257:Euler
247:Viète
141:Sines
65:Usage
8515:LCCN
8505:ISBN
8467:OCLC
8459:ISSN
8404:ISBN
8370:ISBN
8172:link
7984:ISBN
7907:ISBN
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7839:link
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7617:ISBN
7398:ISSN
6925:ISBN
6874:LCCN
6864:ISBN
6821:ISBN
6747:and
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6423:LCCN
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6283:LCCN
6273:ISBN
6185:and
5985:and
5960:for
1375:and
1345:and
1323:zero
1309:and
1303:Hann
1289:and
1236:and
1206:hav.
1153:and
840:sine
733:The
699:The
673:The
619:The
592:The
558:The
535:The
390:vers
368:The
336:and
300:The
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7716:doi
6856:doi
6596:892
6405:doi
6360:doi
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6127:hvs
5975:sag
5638:cos
5623:sin
5232:lim
4304:cos
4198:cos
4083:cos
3983:cos
3871:sin
3771:sin
3659:sin
3565:sin
3468:cos
3377:cos
3277:cos
3192:cos
3095:sin
3010:sin
2913:sin
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2257:sin
2152:cos
2112:cos
2015:sin
1910:cos
1870:sin
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1466:cos
1429:sin
1379:by
1341:in
1333:or
1327:one
1325:to
1216:or
1212:as
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1077:sin
1012:As
980:siv
946:bow
934:ADB
808:bow
796:ADB
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753:or
729:hcv
727:or
695:hvc
693:or
691:hac
677:or
657:sem
653:hvs
649:hav
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610:cvc
596:or
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570:or
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553:vcs
504:tan
494:sin
469:sin
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80:cos
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4267:i
4264:s
4261:o
4258:c
4255:r
4252:e
4249:v
4246:o
4243:c
4240:a
4237:h
4210:2
4205:x
4192:=
4189:)
4186:x
4183:(
4179:n
4176:i
4173:s
4170:o
4167:c
4164:r
4161:e
4158:v
4155:o
4152:c
4149:a
4146:h
4139:x
4135:d
4129:d
4103:C
4100:+
4095:2
4090:x
4080:+
4077:x
4071:=
4068:x
4064:d
4059:)
4056:x
4053:(
4049:n
4046:i
4043:s
4040:r
4037:e
4034:v
4031:o
4028:c
4025:a
4022:h
3995:2
3990:x
3974:=
3971:)
3968:x
3965:(
3961:n
3958:i
3955:s
3952:r
3949:e
3946:v
3943:o
3940:c
3937:a
3934:h
3927:x
3923:d
3917:d
3891:C
3888:+
3883:2
3878:x
3868:+
3865:x
3859:=
3856:x
3852:d
3847:)
3844:x
3841:(
3837:n
3834:i
3831:s
3828:o
3825:c
3822:r
3819:e
3816:v
3813:a
3810:h
3783:2
3778:x
3762:=
3759:)
3756:x
3753:(
3749:n
3746:i
3743:s
3740:o
3737:c
3734:r
3731:e
3728:v
3725:a
3722:h
3715:x
3711:d
3705:d
3679:C
3676:+
3671:2
3666:x
3653:x
3647:=
3644:x
3640:d
3635:)
3632:x
3629:(
3625:n
3622:i
3619:s
3616:r
3613:e
3610:v
3607:a
3604:h
3577:2
3572:x
3559:=
3556:)
3553:x
3550:(
3546:n
3543:i
3540:s
3537:r
3534:e
3531:v
3528:a
3525:h
3518:x
3514:d
3508:d
3482:C
3479:+
3475:x
3462:x
3459:=
3456:x
3452:d
3447:)
3444:x
3441:(
3437:n
3434:i
3431:s
3428:o
3425:c
3422:r
3419:e
3416:v
3413:o
3410:c
3384:x
3374:=
3371:)
3368:x
3365:(
3361:n
3358:i
3355:s
3352:o
3349:c
3346:r
3343:e
3340:v
3337:o
3334:c
3327:x
3323:d
3317:d
3291:C
3288:+
3284:x
3274:+
3271:x
3268:=
3265:x
3261:d
3256:)
3253:x
3250:(
3246:n
3243:i
3240:s
3237:r
3234:e
3231:v
3228:o
3225:c
3199:x
3186:=
3183:)
3180:x
3177:(
3173:n
3170:i
3167:s
3164:r
3161:e
3158:v
3155:o
3152:c
3145:x
3141:d
3135:d
3109:C
3106:+
3102:x
3092:+
3089:x
3086:=
3083:x
3079:d
3074:)
3071:x
3068:(
3064:n
3061:i
3058:s
3055:o
3052:c
3049:r
3046:e
3043:v
3017:x
3004:=
3001:)
2998:x
2995:(
2991:n
2988:i
2985:s
2982:o
2979:c
2976:r
2973:e
2970:v
2963:x
2959:d
2953:d
2927:C
2924:+
2920:x
2907:x
2904:=
2901:x
2897:d
2892:)
2889:x
2886:(
2882:n
2879:i
2876:s
2873:r
2870:e
2867:v
2841:x
2831:=
2828:)
2825:x
2822:(
2818:n
2815:i
2812:s
2809:r
2806:e
2803:v
2796:x
2792:d
2786:d
2748:)
2742:2
2735:3
2729:+
2722:(
2717:n
2714:i
2711:s
2708:o
2705:c
2702:r
2699:e
2696:v
2693:o
2690:c
2687:a
2684:h
2680:=
2676:)
2669:+
2662:(
2657:n
2654:i
2651:s
2648:o
2645:c
2642:r
2639:e
2636:v
2633:a
2630:h
2626:=
2622:)
2616:2
2608:+
2601:(
2596:n
2593:i
2590:s
2587:r
2584:e
2581:v
2578:o
2575:c
2572:a
2569:h
2565:=
2558:)
2552:(
2548:n
2545:i
2542:s
2539:r
2536:e
2533:v
2530:a
2527:h
2518:)
2512:2
2505:3
2499:+
2492:(
2487:n
2484:i
2481:s
2478:o
2475:c
2472:r
2469:e
2466:v
2463:o
2460:c
2456:=
2452:)
2445:+
2438:(
2433:n
2430:i
2427:s
2424:o
2421:c
2418:r
2415:e
2412:v
2408:=
2404:)
2398:2
2390:+
2383:(
2378:n
2375:i
2372:s
2369:r
2366:e
2363:v
2360:o
2357:c
2353:=
2346:)
2340:(
2336:n
2333:i
2330:s
2327:r
2324:e
2321:v
2273:2
2269:)
2263:(
2254:+
2251:1
2245:=
2240:2
2236:)
2230:(
2214:)
2208:(
2168:2
2164:)
2158:(
2149:+
2146:1
2140:=
2136:)
2131:2
2123:(
2116:2
2108:=
2103:2
2099:)
2093:(
2077:)
2071:(
2031:2
2027:)
2021:(
2009:1
2003:=
1998:2
1994:)
1988:(
1972:)
1966:(
1926:2
1922:)
1916:(
1904:1
1898:=
1894:)
1889:2
1881:(
1874:2
1866:=
1861:2
1857:)
1851:(
1835:)
1829:(
1791:)
1785:(
1776:+
1773:1
1770:=
1766:)
1754:2
1745:(
1730:)
1724:(
1686:)
1680:(
1671:+
1668:1
1665:=
1661:)
1656:2
1648:(
1641:2
1633:2
1627:)
1621:(
1583:)
1577:(
1565:1
1562:=
1558:)
1546:2
1537:(
1522:)
1516:(
1478:)
1472:(
1460:1
1457:=
1453:)
1448:2
1440:(
1433:2
1425:2
1419:)
1413:(
1179:)
1173:2
1170:/
1166:θ
1159:(
1112:2
1109:/
1106:1
1099:)
1093:2
1090:/
1086:θ
1079:(
1049:π
1005:.
1003:O
986:(
978:(
919:C
784:O
555:.
515:2
491:=
486:2
473:2
465:2
462:=
447:1
444:=
359:.
289:e
282:t
275:v
192:)
90:)
30:.
23:.
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