31:
365:
are often developed by starting with one of the above methods and adapting it to a situation other than the real numbers of
Euclidean geometry. Generally, trigonometry can be the study of triples of points in any kind of
326:
751:
Herranz, Francisco J.; Ortega, Ramón; Santander, Mariano (2000), "Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry",
634:
175:
367:
571:
171:
812:
358:
354:
181:
418:
are studied. The spherical triangle identities are written in terms of the ordinary trigonometric functions but differ from the plane
1168:
342:
273:
1180:
645:
346:
1094:
550:
619:
350:
591:
95:
73:
688:
378:
with the smallest number of vertices, so one direction to generalize is to study higher-dimensional analogs of
675:
560:
69:
266:
217:
166:
100:
302:
926:
473:
615:. Trigonometric functions can be defined on an arbitrary time scale (a subset of the real numbers).
843:"A computational trigonometry, and related contributions by Russians Kantorovich, Krein, Kaporin"
334:
57:
45:
40:
525:
1173:-order polynomials and generalizations of trigonometry, oscillator model and Hamilton dynamics"
921:
821:
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411:
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604:
259:
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734:
627:
623:
443:
437:
186:
105:
50:
709:
8:
1213:
1117:
Harkin, Anthony A.; Harkin, Joseph B. (2004), "Geometry of generalized complex numbers",
990:
Masala, G. (1999), "Regular triangles and isoclinic triangles in the
Grassmann manifolds
663:
608:
600:
565:
489:
481:
433:
429:
419:
371:
232:
148:
916:
Aslaksen, Helmer; Huynh, Hsueh-Ling (1997), "Laws of trigonometry in symmetric spaces",
774:
738:
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1134:
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30:
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903:
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131:
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1086:
1273:
668:
542:
227:
1081:, Institute for Nonlinear Science, New York: Springer-Verlag, p. 101,
811:
Liu, Honghai; Coghill, George M. (2005), "Fuzzy
Qualitative Trigonometry",
287:
126:
22:
447:
391:
383:
338:
212:
110:
765:
1138:
1055:
961:
578:
222:
202:
1006:
Rendiconti del
Seminario Matematico Università e Politecnico di Torino
864:
Karpenkov, Oleg (2008), "Elementary notions of lattice trigonometry",
611:
are unified into dynamic equations on time scales which also includes
948:
Leuzinger, Enrico (1992), "On the trigonometry of symmetric spaces",
878:
459:
395:
387:
1047:
1031:
291:
157:
814:
2005 IEEE International
Conference on Systems, Man and Cybernetics
729:
375:
207:
415:
1239:
International
Journal of Mathematics and Mathematical Sciences
379:
568:– a Pythagorean theorem for a tetrahedron with a cube corner
16:
Study of triangles in other spaces than the
Euclidean plane
1077:
West, Bruce J.; Bologna, Mauro; Grigolini, Paolo (2003),
671:– trigonometric identities for multiple distinct angles
750:
330:. There are a number of ways of defining the ordinary
305:
1076:
528:- right simplexes (right triangles generalized to
320:
820:, vol. 2, pp. 1291–1296, archived from
689:The Pythagorean theorem in non-Euclidean geometry
1271:
918:Geometry from the Pacific Rim (Singapore, 1994)
1232:"The combinatorial structure of trigonometry"
915:
622:of sin and cos define these functions on any
267:
1116:
707:
590:Trigonometric functions can be defined for
536:who called the generalized trigonometry of
1029:
810:
584:
363:Generalizations of trigonometric functions
274:
260:
1250:
947:
925:
877:
863:
837:
764:
728:
308:
1166:
1229:
1272:
989:
920:, Berlin: de Gruyter, pp. 23–36,
554:-simplices with an "orthogonal corner"
359:definitions using functional equations
355:definitions via differential equations
1181:Advances in Applied Clifford Algebras
1032:"The Trigonometry of the Tetrahedron"
476:: A form of trigonometry used in the
519:
13:
14:
1291:
950:Commentarii Mathematici Helvetici
710:"Taxicab angles and trigonometry"
592:fractional differential equations
343:right-angled triangle definitions
515:Trigonometry on symmetric spaces
462:is parameterized by (cosh
414:, triangles on the surface of a
321:{\displaystyle \mathbb {R} ^{2}}
29:
1223:
1160:
1110:
1070:
708:Thompson, K.; Dray, T. (2000),
450:is parameterized by (cos
404:
1131:10.1080/0025570X.2004.11953236
1023:
983:
941:
909:
857:
831:
804:
744:
701:
506:Fuzzy qualitative trigonometry
1:
1030:Richardson, G. (1902-03-01).
694:
676:Lemniscate elliptic functions
662:Polar/Trigonometric forms of
572:A law of sines for tetrahedra
561:Trigonometry of a tetrahedron
1167:Yamaleev, Robert M. (2005),
1079:Physics of fractal operators
7:
783:10.1088/0305-4470/33/24/309
682:
446:in Euclidean geometry: The
10:
1296:
888:10.7146/math.scand.a-15058
458:) whereas the equilateral
167:Trigonometric substitution
1252:10.1155/S0161171203106230
1230:Antippa, Adel F. (2003),
1194:10.1007/s00006-005-0007-y
1087:10.1007/978-0-387-21746-8
847:Вычислительные технологии
549:Pythagorean theorems for
532:dimensions) - studied by
425:Hyperbolic trigonometry:
1036:The Mathematical Gazette
866:Mathematica Scandinavica
656:
503:Spacetime trigonometries
80:Generalized trigonometry
585:Trigonometric functions
484:, with applications to
347:unit circle definitions
335:trigonometric functions
613:q-difference equations
605:differential equations
412:spherical trigonometry
322:
1169:"Complex algebras on
717:Pi Mu Epsilon Journal
540:Euclidean dimensions
526:Schläfli orthoschemes
509:Operator trigonometry
323:
1119:Mathematics Magazine
753:Journal of Physics A
664:hypercomplex numbers
609:difference equations
512:Lattice trigonometry
444:Hyperbolic functions
438:hyperbolic functions
430:hyperbolic triangles
374:. A triangle is the
303:
187:Trigonometric series
775:2000JPhA...33.4525H
739:2011arXiv1101.2917T
678:, sinlem and coslem
601:time scale calculus
490:quantum computation
482:hyperbolic geometry
434:hyperbolic geometry
420:triangle identities
332:Euclidean geometric
149:Pythagorean theorem
962:10.1007/BF02566499
620:series definitions
486:special relativity
466:, sinh
351:series definitions
318:
759:(24): 4525–4551,
520:Higher dimensions
497:Trigonometry for
454:, sin
284:
283:
176:inverse functions
119:Laws and theorems
1287:
1264:
1263:
1254:
1236:
1227:
1221:
1220:
1218:
1212:, archived from
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1172:
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1027:
1021:
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907:
906:
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861:
855:
854:
839:Gustafson, K. E.
835:
829:
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826:
819:
808:
802:
801:
768:
748:
742:
741:
732:
714:
705:
566:De Gua's theorem
499:taxicab geometry
478:gyrovector space
474:Gyrotrigonometry
398:
329:
327:
325:
324:
319:
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33:
19:
18:
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1097:
1075:
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1048:10.2307/3603090
1042:(32): 149–158.
1028:
1024:
997:
991:
988:
984:
946:
942:
927:10.1.1.160.1580
914:
910:
862:
858:
836:
832:
824:
817:
809:
805:
766:math-ph/9910041
749:
745:
712:
706:
702:
697:
685:
659:
650:Banach algebras
635:complex numbers
587:
522:
407:
396:
312:
307:
306:
304:
301:
300:
298:
296:Euclidean plane
280:
101:Exact constants
17:
12:
11:
5:
1293:
1283:
1282:
1266:
1265:
1245:(8): 475–500,
1222:
1188:(1): 123–150,
1159:
1125:(2): 118–129,
1109:
1095:
1069:
1022:
995:
982:
956:(2): 252–286,
940:
908:
872:(2): 161–205,
856:
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743:
699:
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696:
693:
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684:
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680:
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672:
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658:
655:
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648:, and various
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406:
403:
382:and polygons:
341:, for example
315:
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197:
196:
195:Mathematicians
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54:
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43:
35:
34:
26:
25:
15:
9:
6:
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3:
2:
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1278:
1277:
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1253:
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1244:
1240:
1233:
1226:
1219:on 2011-07-22
1215:
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1096:0-387-95554-2
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1045:
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1012:(2): 91–104,
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1007:
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844:
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834:
827:on 2011-07-25
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736:
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687:
686:
677:
673:
670:
669:Polygonometry
667:
665:
661:
660:
651:
647:
643:
642:-adic numbers
641:
636:
632:
629:
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621:
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543:polygonometry
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258:
257:
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228:Regiomontanus
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49:
47:
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42:
39:
38:
37:
36:
32:
28:
27:
24:
21:
20:
1280:Trigonometry
1242:
1238:
1225:
1214:the original
1185:
1179:
1162:
1122:
1118:
1112:
1078:
1072:
1039:
1035:
1025:
1009:
1005:
999:
992:
985:
953:
949:
943:
917:
911:
879:math/0604129
869:
865:
859:
850:
846:
833:
822:the original
813:
806:
756:
752:
746:
723:(2): 87–96,
720:
716:
703:
639:
551:
541:
537:
529:
480:approach to
467:
463:
455:
451:
405:Trigonometry
392:tetrahedrons
384:solid angles
362:
339:real numbers
288:trigonometry
285:
79:
23:Trigonometry
448:unit circle
213:Brahmagupta
182:Derivatives
111:Unit circle
853:(3): 73–83
695:References
626:where the
579:Polar sine
399:-simplices
223:al-Battani
203:Hipparchus
142:Cotangents
96:Identities
1210:121144869
1064:125115660
978:123684622
922:CiteSeerX
730:1101.2917
460:hyperbola
428:Study of
388:polytopes
292:triangles
286:Ordinary
238:de Moivre
172:Integrals
88:Reference
58:Functions
1274:Category
904:49911437
841:(1999),
799:15313035
683:See also
646:matrices
633:such as
631:converge
390:such as
368:geometry
290:studies
218:al-Hasib
158:Calculus
137:Tangents
1261:1967890
1202:2236628
1155:7837108
1147:1573734
1139:3219099
1105:1988873
1056:3603090
1018:1974445
970:1161284
936:1468236
896:2437186
791:1768742
771:Bibcode
735:Bibcode
624:algebra
534:Schoute
376:polygon
328:
299:
294:in the
248:Fourier
208:Ptolemy
174: (
132:Cosines
74:inverse
60: (
46:History
41:Outline
1259:
1208:
1200:
1153:
1145:
1137:
1103:
1093:
1062:
1054:
1016:
976:
968:
934:
924:
902:
894:
797:
789:
628:series
416:sphere
380:angles
357:, and
106:Tables
1235:(PDF)
1217:(PDF)
1206:S2CID
1176:(PDF)
1151:S2CID
1135:JSTOR
1060:S2CID
1052:JSTOR
974:S2CID
900:S2CID
874:arXiv
825:(PDF)
818:(PDF)
795:S2CID
761:arXiv
725:arXiv
713:(PDF)
657:Other
436:with
372:space
243:Euler
233:Viète
127:Sines
51:Usage
1243:2003
1091:ISBN
674:The
618:The
607:and
488:and
394:and
386:and
1247:doi
1190:doi
1127:doi
1083:doi
1044:doi
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958:doi
884:doi
870:102
779:doi
599:In
432:in
410:In
370:or
337:on
70:tan
66:cos
62:sin
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