2826:
2260:
2821:{\displaystyle {\begin{aligned}\tan {\tfrac {1}{2}}c\,\cos {\tfrac {1}{2}}(\alpha -\beta )&=\tan {\tfrac {1}{2}}(a+\,b)\cos {\tfrac {1}{2}}(\alpha +\beta )\\\tan {\tfrac {1}{2}}c\,\sin {\tfrac {1}{2}}(\alpha -\beta )&=\tan {\tfrac {1}{2}}(a\ \!-\,b)\sin {\tfrac {1}{2}}(\alpha +\beta )\\\cot {\tfrac {1}{2}}\gamma \ \!\cos {\tfrac {1}{2}}(a\ \!-\,b)&=\tan {\tfrac {1}{2}}(\alpha +\beta )\cos {\tfrac {1}{2}}(a+b)\\\cot {\tfrac {1}{2}}\gamma \,\sin {\tfrac {1}{2}}(a\ \!-\,b)&=\tan {\tfrac {1}{2}}(\alpha -\beta )\sin {\tfrac {1}{2}}(a+b).\end{aligned}}}
82:
3947:
6629:
5828:
6178:
3563:
5041:
2136:
3551:
152:
2182:
3137:
5575:
4769:
1898:
3273:
3942:{\displaystyle {\begin{aligned}c&=\arctan {\frac {\sqrt {(\sin a\cos b-\cos a\sin b\cos \gamma )^{2}+(\sin b\sin \gamma )^{2}}}{\cos a\cos b+\sin a\sin b\cos \gamma }},\\\alpha &=\arctan {\frac {\sin a\sin \gamma }{\sin b\cos a-\cos b\sin a\cos \gamma }},\\\beta &=\arctan {\frac {\sin b\sin \gamma }{\sin a\cos b-\cos a\sin b\cos \gamma }},\end{aligned}}}
4762:
6797:
2866:
5550:
5046:
4454:
3142:
3953:
5823:{\displaystyle {\begin{aligned}a&=\arccos {\frac {\cos \alpha +\cos \beta \cos \gamma }{\sin \beta \sin \gamma }},\\b&=\arccos {\frac {\cos \beta +\cos \gamma \cos \alpha }{\sin \gamma \sin \alpha }},\\c&=\arccos {\frac {\cos \gamma +\cos \alpha \cos \beta }{\sin \alpha \sin \beta }}.\end{aligned}}}
4588:
2831:
379:
1861:
3559:
of finding the great circle between two points on the earth specified by their latitude and longitude; in this application, it is important to use formulas which are not susceptible to round-off errors. For this purpose, the following formulas (which may be derived using vector algebra) can be used:
684:
can equal either 30° or 150°. Using the law of cosines avoids this problem: within the interval from 0° to 180° the cosine value unambiguously determines its angle. On the other hand, if the angle is small (or close to 180°), then it is more robust numerically to determine it from its sine than its
5036:{\displaystyle {\begin{aligned}a&=\arctan {\frac {2\sin \alpha }{\cot {\frac {1}{2}}c\,\sin(\beta +\alpha )+\tan {\frac {1}{2}}c\,\sin(\beta -\alpha )}},\\b&=\arctan {\frac {2\sin \beta }{\cot {\frac {1}{2}}c\,\sin(\alpha +\beta )+\tan {\frac {1}{2}}c\,\sin(\alpha -\beta )}}.\end{aligned}}}
888:
1178:
1377:
6781:
6344:
2131:{\displaystyle {\begin{aligned}a&=c\ {\frac {\sin \alpha }{\sin \gamma }}=c\ {\frac {\sin \alpha }{\sin \alpha \cos \beta +\sin \beta \cos \alpha }}\\b&=c\ {\frac {\sin \beta }{\sin \gamma }}=c\ {\frac {\sin \beta }{\sin \alpha \cos \beta +\sin \beta \cos \alpha }}\end{aligned}}}
6491:
699:
3546:{\displaystyle {\begin{aligned}\alpha &=\arctan \ {\frac {2\sin a}{\tan {\frac {1}{2}}\gamma \,\sin(b+a)+\cot {\frac {1}{2}}\gamma \,\sin(b-a)}},\\\beta &=\arctan \ {\frac {2\sin b}{\tan {\frac {1}{2}}\gamma \,\sin(a+b)+\cot {\frac {1}{2}}\gamma \,\sin(a-b)}}.\end{aligned}}}
6988:
638:
5545:
4449:
3132:{\displaystyle {\begin{aligned}\alpha &=\arccos {\frac {\cos a-\cos b\ \cos c}{\sin b\ \sin c}},\\\beta &=\arccos {\frac {\cos b-\cos c\ \cos a}{\sin c\ \sin a}},\\\gamma &=\arccos {\frac {\cos c-\cos a\ \cos b}{\sin a\ \sin b}}.\end{aligned}}}
4575:
168:
723:
6604:
462:
6655:
6218:
6359:
6852:
4757:{\displaystyle {\begin{aligned}a&=\arccos {\frac {\cos \alpha +\cos \beta \cos \gamma }{\sin \beta \sin \gamma }},\\b&=\arccos {\frac {\cos \beta +\cos \alpha \cos \gamma }{\sin \alpha \sin \gamma }},\end{aligned}}}
3260:
688:
We assume that the relative position of specified characteristics is known. If not, the mirror reflection of the triangle will also be a solution. For example, three side lengths uniquely define either a triangle or its
5234:
5142:
4112:
1851:
1168:
4036:
99:). The classical plane trigonometry problem is to specify three of the six characteristics and determine the other three. A triangle can be uniquely determined in this sense when given any of the following:
5580:
3568:
2871:
1352:
522:
5246:
4774:
4593:
4146:
3278:
1903:
728:
2238:
cannot be unequal, so the problem of constructing a triangle with specified three angles has a unique solution. The basic relations used to solve a problem are similar to those of the planar case: see
5241:
4141:
1768:
1267:
1461:
7153:
6857:
4487:
2265:
2145:
The procedure for solving an AAS triangle is same as that for an ASA triangle: First, find the third angle by using the angle sum property of a triangle, then find the other two sides using the
513:
173:
1004:
1396:
This case is not solvable in all cases; a solution is guaranteed to be unique only if the side length adjacent to the angle is shorter than the other side length. Assume that two sides
6162:
6110:
6059:
6003:
5952:
5896:
3967:
This problem is not solvable in all cases; a solution is guaranteed to be unique only if the side length adjacent to the angle is shorter than the other side length. Known: the sides
6508:
1051:
390:
3175:
2207:) and length around the sphere are numerically the same. On other spheres, the angle (in radians) is equal to the length around the sphere divided by the radius.)
1387:
374:{\displaystyle {\begin{aligned}a^{2}&=b^{2}+c^{2}-2bc\cos \alpha \\b^{2}&=a^{2}+c^{2}-2ac\cos \beta \\c^{2}&=a^{2}+b^{2}-2ab\cos \gamma \end{aligned}}}
5165:
139:
For all cases in the plane, at least one of the side lengths must be specified. If only the angles are given, the side lengths cannot be determined, because any
5079:
4049:
1775:
3978:
883:{\displaystyle {\begin{aligned}\alpha &=\arccos {\frac {b^{2}+c^{2}-a^{2}}{2bc}}\\\beta &=\arccos {\frac {a^{2}+c^{2}-b^{2}}{2ac}}.\end{aligned}}}
1272:
6776:{\displaystyle h={\frac {\sin \alpha \,\sin \beta }{\sin(\beta -\alpha )}}\ell ={\frac {\tan \alpha \,\tan \beta }{\tan \beta -\tan \alpha }}\ell .}
6339:{\displaystyle d={\frac {\sin \alpha \,\sin \beta }{\sin(\alpha +\beta )}}\ell ={\frac {\tan \alpha \,\tan \beta }{\tan \alpha +\tan \beta }}\ell .}
7234:
1718:
1198:
6486:{\displaystyle \tan b={\frac {2\sin \beta }{\cot {\frac {1}{2}}\ell \,\sin(\alpha +\beta )+\tan {\frac {1}{2}}\ell \,\sin(\alpha -\beta )}},}
2170:
1415:
1058:
6997:
6983:{\displaystyle {\begin{aligned}a&=90^{\circ }-\lambda _{B},\\b&=90^{\circ }-\lambda _{A},\\\gamma &=L_{A}-L_{B}.\end{aligned}}}
473:
7463:– Free software to solve the spherical triangles, configurable to different practical applications and configured for gnomonic.
7394:
3949:
where the signs of the numerators and denominators in these expressions should be used to determine the quadrant of the arctangent.
7467:
5841:) is the right angle. Such a spherical triangle is fully defined by its two elements, and the other three can be calculated using
7495:
7192:
7410:
7455:– Triangle solver. Solve any plane triangle problem with the minimum of input data. Drawing of the solved triangle.
2157:
In many cases, triangles can be solved given three pieces of information some of which are the lengths of the triangle's
5549:
633:{\displaystyle {\frac {a-b}{a+b}}={\frac {\tan {\frac {1}{2}}(\alpha -\beta )}{\tan {\tfrac {1}{2}}(\alpha +\beta )}}.}
2218:
466:
2169:. Posamentier and Lehmann list the results for the question of solvability using no higher than square roots (i.e.,
7500:
7423:
by Alfred Monroe Kenyon and Louis Ingold, The
Macmillan Company, 1914. In images, full text presented. Google book.
5045:
4453:
1386:
934:
5540:{\displaystyle {\begin{aligned}c&=2\arctan \left,\\\gamma &=2\operatorname {arccot} \left.\end{aligned}}}
4444:{\displaystyle {\begin{aligned}a&=2\arctan \left,\\\alpha &=2\operatorname {arccot} \left.\end{aligned}}}
7349:
6117:
6065:
6014:
6006:
5958:
5907:
5851:
5569:
4481:
3169:
2860:
2239:
3141:
4570:{\displaystyle \gamma =\arccos \!{\bigl (}\sin \alpha \sin \beta \cos c-\cos \alpha \cos \beta {\bigr )}.\,}
3952:
3556:
5899:
5842:
5073:
4043:
2243:
1009:
648:
7420:
2195:
is fully determined by three of its six characteristics (3 sides and 3 angles). The lengths of the sides
7505:
7172:
6791:
2830:
2235:
140:
2254:
1860:
81:
6200:
from shore to a remote ship via triangulation, one marks on the shore two points with known distance
7326:
7304:
7282:
7260:
7238:
6628:
5837:
The above algorithms become much simpler if one of the angles of a triangle (for example, the angle
7378:
7177:
6212:
2162:
1177:
917:
but (as Note 1 above states) there is a risk of confusing an acute angle value with an obtuse one.
7187:
1376:
6211:
From the formulae above (ASA case, assuming planar geometry) one can compute the distance as the
2214:
2210:
2192:
7212:
4579:
We can find the two unknown sides from the spherical law of cosines (using the calculated angle
928:
7434:
1583:(the larger side corresponds to a larger angle). Since no triangle can have two obtuse angles,
91:
A general form triangle has six main characteristics (see picture): three linear (side lengths
7480:
7390:
2250:
2158:
921:
698:
644:
49:(angles and lengths of sides), when some of these are known. The triangle can be located on a
6599:{\displaystyle \sin d=\sin b\sin \alpha ={\frac {\tan b}{\sqrt {1+\tan ^{2}b}}}\sin \alpha .}
457:{\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}}
50:
6177:
7427:
517:
7451:
1195:
between these sides are known. The third side can be determined from the law of cosines:
7444:
7440:
7417:, Princeton University Press, 1998. Ebook version, in PDF format, full text presented.
6493:
and insert this into the AAS formula for the right subtriangle that contains the angle
2217:, so the solution of spherical triangles is built on different rules. For example, the
2203:, measured in angular units rather than linear units. (On a unit sphere, the angle (in
2166:
1547:
717:
661:
163:
151:
34:
2181:
672:
for the angle of the triangle does not uniquely determine this angle. For example, if
7489:
7383:
7362:
7182:
6191:
6183:
2200:
6348:
For the spherical case, one can first compute the length of side from the point at
2146:
1409:
914:
665:
385:
66:
42:
121:), if the side length adjacent to the angle is shorter than the other side length.
6652:
be the distance between these points. From the same ASA case formulas we obtain:
6796:
685:
cosine because the arc-cosine function has a divergent derivative at 1 (or −1).
17:
70:
3255:{\displaystyle c=\arccos \left(\cos a\cos b+\sin a\sin b\cos \gamma \right).}
159:
The standard method of solving the problem is to use fundamental relations.
62:
7459:
5229:{\displaystyle b=\pi -\arcsin {\frac {\sin a\,\sin \beta }{\sin \alpha }}.}
6606:(The planar formula is actually the first term of the Taylor expansion of
920:
Another method of calculating the angles from known sides is to apply the
7414:
6618:
46:
643:
There are other (sometimes practically useful) universal relations: the
155:
Overview of particular steps and tools used when solving plane triangles
5056:
A side, one adjacent angle and the opposite angle given (spherical AAS)
2204:
58:
5137:{\displaystyle b=\arcsin {\frac {\sin a\,\sin \beta }{\sin \alpha }}.}
4107:{\displaystyle \gamma =\arcsin {\frac {\sin c\,\sin \beta }{\sin b}}.}
3975:
not between them. A solution exists if the following condition holds:
1846:{\displaystyle a=c\cos \beta \pm {\sqrt {b^{2}-c^{2}\sin ^{2}\beta }}}
1163:{\displaystyle A={\frac {\sqrt {(a+b+c)(b+c-a)(a^{2}-(b-c)^{2})}}{4}}}
7374:
4031:{\displaystyle b>\arcsin \!{\bigl (}\sin c\,\sin \beta {\bigr )}.}
54:
2173:) for each of the 95 distinct cases; 63 of these are constructible.
7443:
by I. Todhunter, M.A., F.R.S. Historical Math
Monograph posted by
7441:
Spherical
Trigonometry — for the use of colleges and schools
7160:
6627:
6208:
be the angles between the baseline and the direction to the ship.
6176:
5548:
5044:
4452:
3951:
3140:
2829:
2180:
1859:
1375:
1176:
697:
150:
80:
6625:
are defined by observation of familiar landmarks from the ship.
2859:(in angular units). The triangle's angles are computed using the
1347:{\displaystyle \alpha =\arccos {\frac {b^{2}+c^{2}-a^{2}}{2bc}}.}
669:
5238:
We can find other characteristics by using Napier's analogies:
4138:
We can find other characteristics by using Napier's analogies:
131:
A side, the angle opposite to it and an angle adjacent to it (
2141:
A side, one adjacent angle and the opposite angle given (AAS)
1512:. For the same reason a solution does not exist if the angle
6795:
3268:
can be calculated as above, or by using Napier's analogies:
1493:(the equation's right side). There are four possible cases:
7437:
Includes discussion of The Napier circle and Napier's rules
6991:
6803:
To calculate the distance between two points on the globe,
1895:
Two unknown sides can be calculated from the law of sines:
27:
Problem of finding unknown lengths and angles of a triangle
7475:
5050:
One side, one adjacent angle and the opposite angle given
1763:{\displaystyle a=b\ {\frac {\sin \alpha }{\sin \beta }}}
1715:
The third side can then be found from the law of sines:
1262:{\displaystyle c={\sqrt {a^{2}+b^{2}-2ab\cos \gamma }}.}
6638:
As another example, if one wants to measure the height
1456:{\displaystyle \sin \gamma ={\frac {c}{b}}\sin \beta .}
7389:. Translated with introduction and commentary. Dover.
5463:
5424:
5319:
5280:
4400:
4365:
4326:
4254:
4219:
4180:
3963:
Two sides and non-included angle given (spherical SSA)
3152:
Two sides and the included angle given (spherical SAS)
2785:
2752:
2707:
2685:
2648:
2615:
2570:
2545:
2508:
2470:
2430:
2408:
2371:
2337:
2297:
2275:
598:
7148:{\displaystyle {\overline {AB}}=R\arccos \!{\Bigr }.}
7000:
6855:
6658:
6646:
from two ground points to the top are specified. Let
6511:
6362:
6221:
6120:
6068:
6017:
5961:
5910:
5854:
5578:
5244:
5168:
5082:
4772:
4591:
4490:
4144:
4052:
3981:
3566:
3276:
3178:
2869:
2263:
2249:
Among other relationships that may be useful are the
1901:
1778:
1721:
1418:
1275:
1201:
1061:
1012:
937:
726:
525:
476:
393:
171:
4464:
A side and two adjacent angles given (spherical ASA)
1269:
Now we use law of cosines to find the second angle:
57:. Applications requiring triangle solutions include
508:{\displaystyle \alpha +\beta +\gamma =180^{\circ }}
7382:
7147:
6982:
6775:
6598:
6485:
6338:
6156:
6104:
6053:
5997:
5946:
5890:
5822:
5539:
5228:
5136:
5035:
4756:
4569:
4443:
4106:
4030:
3941:
3545:
3254:
3131:
2820:
2234:depends on the size of the triangle. In addition,
2130:
1845:
1762:
1455:
1346:
1261:
1162:
1045:
998:
882:
632:
507:
456:
373:
117:Two sides and an angle not included between them (
7137:
7028:
7025:
4500:
3991:
2727:
2590:
2562:
2490:
6849:is the North Pole. Some characteristics are:
6642:of a mountain or a high building, the angles
4558:
4503:
4020:
3994:
8:
7385:The Thirteen Books of the Elements. Volume I
6786:The distance between two points on the globe
1372:Two sides and non-included angle given (SSA)
1173:Two sides and the included angle given (SAS)
999:{\displaystyle A={\sqrt {s(s-a)(s-b)(s-c)}}}
45:problem of finding the characteristics of a
1504:, no such triangle exists because the side
7347:Alfred S. Posamentier and Ingmar Lehmann,
1856:A side and two adjacent angles given (ASA)
124:A side and the two angles adjacent to it (
7136:
7135:
7126:
7113:
7094:
7078:
7059:
7043:
7027:
7026:
7001:
6999:
6967:
6954:
6927:
6914:
6887:
6874:
6856:
6854:
6731:
6719:
6677:
6665:
6657:
6569:
6545:
6510:
6455:
6442:
6411:
6398:
6375:
6361:
6294:
6282:
6240:
6228:
6220:
6157:{\displaystyle \cos c=\cot A\cdot \cot B}
6119:
6105:{\displaystyle \cos A=\cos a\cdot \sin B}
6067:
6054:{\displaystyle \tan b=\tan c\cdot \cos A}
6016:
5998:{\displaystyle \cos c=\cos a\cdot \cos b}
5960:
5947:{\displaystyle \tan a=\sin b\cdot \tan A}
5909:
5891:{\displaystyle \sin a=\sin c\cdot \sin A}
5853:
5757:
5678:
5599:
5579:
5577:
5497:
5462:
5453:
5423:
5353:
5318:
5309:
5279:
5245:
5243:
5199:
5187:
5167:
5107:
5095:
5081:
5001:
4988:
4957:
4944:
4921:
4873:
4860:
4829:
4816:
4793:
4773:
4771:
4691:
4612:
4592:
4590:
4566:
4557:
4556:
4502:
4501:
4489:
4399:
4364:
4355:
4325:
4253:
4218:
4209:
4179:
4145:
4143:
4077:
4065:
4051:
4019:
4018:
4008:
3993:
3992:
3980:
3858:
3761:
3679:
3645:
3587:
3567:
3565:
3511:
3498:
3467:
3454:
3431:
3380:
3367:
3336:
3323:
3300:
3277:
3275:
3177:
3060:
2975:
2890:
2870:
2868:
2784:
2751:
2731:
2706:
2699:
2684:
2647:
2614:
2594:
2569:
2544:
2507:
2494:
2469:
2429:
2422:
2407:
2370:
2357:
2336:
2296:
2289:
2274:
2264:
2262:
2068:
2033:
1957:
1922:
1902:
1900:
1829:
1819:
1806:
1800:
1777:
1734:
1720:
1431:
1417:
1321:
1308:
1295:
1288:
1274:
1227:
1214:
1208:
1200:
1145:
1120:
1068:
1060:
1019:
1011:
944:
936:
853:
840:
827:
820:
780:
767:
754:
747:
727:
725:
597:
564:
555:
526:
524:
499:
475:
436:
415:
394:
392:
340:
327:
310:
275:
262:
245:
210:
197:
180:
172:
170:
6112:(also from the spherical law of cosines)
5833:Solving right-angled spherical triangles
3957:Two sides and a non-included angle given
1870:The known characteristics are the side
1385:
1381:Two sides and a non-included angle given
7204:
6610:of the spherical solution in powers of
6352:to the ship (i.e. the side opposite to
7353:, Prometheus Books, 2012: pp. 201–203.
6992:two sides and the included angle given
4458:One side and two adjacent angles given
3146:Two sides and the included angle given
1865:One side and two adjacent angles given
1641:. The figure on right shows the point
1404:are known. The equation for the angle
1182:Two sides and the included angle given
6196:If one wants to measure the distance
1659:as the first solution, and the point
909:Some sources recommend to find angle
7:
7237:. web.horacemann.org. Archived from
1046:{\displaystyle s={\frac {a+b+c}{2}}}
7471:– Solves spherical triangles.
6841:we consider the spherical triangle
1589:is an acute angle and the solution
6633:How to measure a mountain's height
5560:Three angles given (spherical AAA)
2199:of a spherical triangle are their
1055:Área using Jesús Sánchez formula:
668:. The reason is that the value of
110:Two sides and the included angle (
25:
6204:between them (the baseline). Let
2841:Three sides given (spherical SSS)
712:be specified. To find the angles
4766:or by using Napier's analogies:
86:Standard notation for a triangle
4476:. First we determine the angle
1560:two alternatives are possible.
7132:
7106:
6994:, we obtain from the formulas
6707:
6695:
6474:
6462:
6430:
6418:
6270:
6258:
5519:
5507:
5486:
5474:
5447:
5435:
5375:
5363:
5342:
5330:
5303:
5291:
5020:
5008:
4976:
4964:
4892:
4880:
4848:
4836:
4423:
4411:
4388:
4376:
4349:
4337:
4277:
4265:
4242:
4230:
4203:
4191:
4124:then there are two solutions:
3676:
3654:
3642:
3590:
3530:
3518:
3486:
3474:
3399:
3387:
3355:
3343:
2808:
2796:
2775:
2763:
2735:
2718:
2671:
2659:
2638:
2626:
2598:
2581:
2531:
2519:
2498:
2481:
2453:
2441:
2394:
2382:
2361:
2348:
2320:
2308:
1391:Two solutions for the triangle
1151:
1142:
1129:
1113:
1110:
1092:
1089:
1071:
991:
979:
976:
964:
961:
949:
660:To find an unknown angle, the
621:
609:
586:
574:
1:
7468:Spherical Triangle Calculator
1698:is obtained, the third angle
7479:– Triangles solver by
7011:
5845:or the following relations.
1772:or from the law of cosines:
1539:, a unique solution exists:
5162:, another solution exists:
3555:This problem arises in the
2177:Solving spherical triangles
7522:
7445:Cornell University Library
6789:
6189:
5146:If the angle for the side
4114:As for the plane case, if
1187:Here the lengths of sides
7193:Snellius–Pothenot problem
1408:can be implied from the
7435:Intro to Spherical Trig.
7350:The Secrets of Triangles
6182:Distance measurement by
6007:spherical law of cosines
5570:spherical law of cosines
4482:spherical law of cosines
3170:spherical law of cosines
2861:spherical law of cosines
2240:Spherical law of cosines
1546:, i.e., the triangle is
143:triangle is a solution.
7327:"Solving ASA Triangles"
7305:"Solving SSA Triangles"
7283:"Solving SAS Triangles"
7261:"Solving SSS Triangles"
6617:This method is used in
3164:between them. The side
2219:sum of the three angles
1686:as the second solution.
708:Let three side lengths
694:Three sides given (SSS)
77:Solving plane triangles
7496:Spherical trigonometry
7428:Spherical trigonometry
7411:Trigonometric Delights
7149:
6984:
6800:
6777:
6635:
6600:
6487:
6356:) via the ASA formula
6340:
6187:
6158:
6106:
6055:
5999:
5948:
5900:spherical law of sines
5892:
5824:
5556:
5541:
5230:
5138:
5074:spherical law of sines
5072:can be found from the
5052:
5037:
4758:
4571:
4460:
4445:
4108:
4044:spherical law of sines
4042:can be found from the
4032:
3959:
3943:
3547:
3256:
3168:can be found from the
3148:
3133:
2837:
2822:
2244:Spherical law of sines
2188:
2132:
1867:
1847:
1764:
1457:
1393:
1383:
1348:
1263:
1184:
1164:
1047:
1000:
884:
705:
634:
509:
458:
375:
156:
88:
38:
7150:
6985:
6799:
6792:Great-circle distance
6778:
6631:
6601:
6488:
6341:
6180:
6159:
6107:
6056:
6000:
5949:
5893:
5825:
5552:
5542:
5231:
5139:
5048:
5038:
4759:
4572:
4456:
4446:
4109:
4033:
3955:
3944:
3548:
3257:
3144:
3134:
2833:
2823:
2184:
2133:
1863:
1848:
1765:
1458:
1389:
1379:
1349:
1264:
1180:
1165:
1048:
1001:
885:
701:
635:
510:
459:
376:
154:
95:) and three angular (
84:
31:Solution of triangles
6998:
6853:
6656:
6509:
6360:
6219:
6118:
6066:
6015:
5959:
5908:
5852:
5576:
5242:
5166:
5080:
4770:
4589:
4488:
4142:
4050:
3979:
3564:
3274:
3176:
2867:
2261:
2213:differs from planar
1899:
1776:
1719:
1508:does not reach line
1416:
1273:
1199:
1059:
1010:
935:
724:
523:
474:
391:
169:
147:Trigonomic relations
39:solutio triangulorum
7235:"Solving Triangles"
7213:"Solving Triangles"
2153:Other given lengths
649:Mollweide's formula
7363:Napier's Analogies
7145:
6980:
6978:
6824:Point B: latitude
6807:Point A: latitude
6801:
6773:
6636:
6596:
6483:
6336:
6188:
6154:
6102:
6051:
5995:
5944:
5888:
5820:
5818:
5564:Known: the angles
5557:
5554:Three angles given
5537:
5535:
5472:
5433:
5328:
5289:
5226:
5134:
5053:
5033:
5031:
4754:
4752:
4567:
4461:
4441:
4439:
4409:
4374:
4335:
4263:
4228:
4189:
4104:
4028:
3960:
3939:
3937:
3557:navigation problem
3543:
3541:
3252:
3149:
3129:
3127:
2838:
2818:
2816:
2794:
2761:
2716:
2694:
2657:
2624:
2579:
2554:
2517:
2479:
2439:
2417:
2380:
2346:
2306:
2284:
2255:Napier's analogies
2215:Euclidean geometry
2211:Spherical geometry
2193:spherical triangle
2189:
2186:Spherical triangle
2128:
2126:
1878:. The third angle
1868:
1843:
1760:
1463:We denote further
1453:
1394:
1384:
1344:
1259:
1185:
1160:
1043:
996:
880:
878:
706:
664:is safer than the
630:
607:
505:
454:
371:
369:
157:
114:, side-angle-side)
89:
7501:Triangle problems
7241:on 7 January 2014
7014:
6765:
6711:
6582:
6581:
6478:
6450:
6406:
6328:
6274:
6168:Some applications
5843:Napier's Pentagon
5811:
5732:
5653:
5523:
5505:
5471:
5452:
5432:
5379:
5361:
5327:
5308:
5288:
5221:
5129:
5024:
4996:
4952:
4896:
4868:
4824:
4745:
4666:
4427:
4408:
4373:
4354:
4334:
4281:
4262:
4227:
4208:
4188:
4099:
3930:
3833:
3736:
3685:
3534:
3506:
3462:
3430:
3403:
3375:
3331:
3299:
3156:Known: the sides
3120:
3109:
3086:
3035:
3024:
3001:
2950:
2939:
2916:
2845:Known: the sides
2835:Three sides given
2793:
2760:
2726:
2715:
2693:
2656:
2623:
2589:
2578:
2561:
2553:
2516:
2489:
2478:
2438:
2416:
2379:
2345:
2305:
2283:
2251:half-side formula
2236:similar triangles
2122:
2067:
2057:
2032:
2011:
1956:
1946:
1921:
1841:
1758:
1733:
1439:
1339:
1254:
1158:
1154:
1041:
994:
922:law of cotangents
871:
798:
703:Three sides given
645:law of cotangents
625:
606:
572:
550:
452:
431:
410:
16:(Redirected from
7513:
7400:
7388:
7379:Sir Thomas Heath
7366:
7360:
7354:
7345:
7339:
7338:
7336:
7334:
7323:
7317:
7316:
7314:
7312:
7301:
7295:
7294:
7292:
7290:
7279:
7273:
7272:
7270:
7268:
7257:
7251:
7250:
7248:
7246:
7231:
7225:
7224:
7222:
7220:
7209:
7178:Hansen's problem
7158:
7154:
7152:
7151:
7146:
7141:
7140:
7131:
7130:
7118:
7117:
7099:
7098:
7083:
7082:
7064:
7063:
7048:
7047:
7032:
7031:
7015:
7010:
7002:
6989:
6987:
6986:
6981:
6979:
6972:
6971:
6959:
6958:
6932:
6931:
6919:
6918:
6892:
6891:
6879:
6878:
6848:
6844:
6837:
6830:
6820:
6813:
6782:
6780:
6779:
6774:
6766:
6764:
6741:
6720:
6712:
6710:
6687:
6666:
6651:
6645:
6641:
6624:
6613:
6609:
6605:
6603:
6602:
6597:
6583:
6574:
6573:
6558:
6557:
6546:
6504:
6500:
6496:
6492:
6490:
6489:
6484:
6479:
6477:
6451:
6443:
6407:
6399:
6390:
6376:
6355:
6351:
6345:
6343:
6342:
6337:
6329:
6327:
6304:
6283:
6275:
6273:
6250:
6229:
6207:
6203:
6199:
6163:
6161:
6160:
6155:
6111:
6109:
6108:
6103:
6060:
6058:
6057:
6052:
6004:
6002:
6001:
5996:
5953:
5951:
5950:
5945:
5897:
5895:
5894:
5889:
5840:
5829:
5827:
5826:
5821:
5819:
5812:
5810:
5790:
5758:
5733:
5731:
5711:
5679:
5654:
5652:
5632:
5600:
5567:
5546:
5544:
5543:
5538:
5536:
5529:
5525:
5524:
5522:
5506:
5498:
5489:
5473:
5464:
5454:
5450:
5434:
5425:
5385:
5381:
5380:
5378:
5362:
5354:
5345:
5329:
5320:
5310:
5306:
5290:
5281:
5235:
5233:
5232:
5227:
5222:
5220:
5209:
5188:
5161:
5151:
5143:
5141:
5140:
5135:
5130:
5128:
5117:
5096:
5071:
5067:
5063:
5060:Known: the side
5042:
5040:
5039:
5034:
5032:
5025:
5023:
4997:
4989:
4953:
4945:
4936:
4922:
4897:
4895:
4869:
4861:
4825:
4817:
4808:
4794:
4763:
4761:
4760:
4755:
4753:
4746:
4744:
4724:
4692:
4667:
4665:
4645:
4613:
4582:
4576:
4574:
4573:
4568:
4562:
4561:
4507:
4506:
4479:
4475:
4471:
4468:Known: the side
4450:
4448:
4447:
4442:
4440:
4433:
4429:
4428:
4426:
4410:
4401:
4391:
4375:
4366:
4356:
4352:
4336:
4327:
4287:
4283:
4282:
4280:
4264:
4255:
4245:
4229:
4220:
4210:
4206:
4190:
4181:
4134:
4127:
4123:
4113:
4111:
4110:
4105:
4100:
4098:
4087:
4066:
4041:
4037:
4035:
4034:
4029:
4024:
4023:
3998:
3997:
3974:
3970:
3948:
3946:
3945:
3940:
3938:
3931:
3929:
3879:
3859:
3834:
3832:
3782:
3762:
3737:
3735:
3684:
3683:
3650:
3649:
3589:
3588:
3552:
3550:
3549:
3544:
3542:
3535:
3533:
3507:
3499:
3463:
3455:
3446:
3432:
3428:
3404:
3402:
3376:
3368:
3332:
3324:
3315:
3301:
3297:
3267:
3261:
3259:
3258:
3253:
3248:
3244:
3167:
3163:
3159:
3138:
3136:
3135:
3130:
3128:
3121:
3119:
3107:
3096:
3084:
3061:
3036:
3034:
3022:
3011:
2999:
2976:
2951:
2949:
2937:
2926:
2914:
2891:
2858:
2827:
2825:
2824:
2819:
2817:
2795:
2786:
2762:
2753:
2724:
2717:
2708:
2695:
2686:
2658:
2649:
2625:
2616:
2587:
2580:
2571:
2559:
2555:
2546:
2518:
2509:
2487:
2480:
2471:
2440:
2431:
2418:
2409:
2381:
2372:
2347:
2338:
2307:
2298:
2285:
2276:
2233:
2198:
2171:constructibility
2137:
2135:
2134:
2129:
2127:
2123:
2121:
2080:
2069:
2065:
2058:
2056:
2045:
2034:
2030:
2012:
2010:
1969:
1958:
1954:
1947:
1945:
1934:
1923:
1919:
1891:
1877:
1873:
1852:
1850:
1849:
1844:
1842:
1834:
1833:
1824:
1823:
1811:
1810:
1801:
1769:
1767:
1766:
1761:
1759:
1757:
1746:
1735:
1731:
1711:
1697:
1685:
1683:
1676:
1674:
1667:
1665:
1658:
1652:
1646:
1640:
1634:
1627:
1617:
1611:
1598:
1588:
1582:
1572:
1559:
1545:
1538:
1528:
1518:
1511:
1507:
1503:
1492:
1487:
1485:
1484:
1479:
1476:
1462:
1460:
1459:
1454:
1440:
1432:
1407:
1403:
1399:
1367:
1353:
1351:
1350:
1345:
1340:
1338:
1327:
1326:
1325:
1313:
1312:
1300:
1299:
1289:
1268:
1266:
1265:
1260:
1255:
1232:
1231:
1219:
1218:
1209:
1194:
1190:
1169:
1167:
1166:
1161:
1159:
1150:
1149:
1125:
1124:
1070:
1069:
1052:
1050:
1049:
1044:
1042:
1037:
1020:
1005:
1003:
1002:
997:
995:
945:
912:
905:
889:
887:
886:
881:
879:
872:
870:
859:
858:
857:
845:
844:
832:
831:
821:
799:
797:
786:
785:
784:
772:
771:
759:
758:
748:
715:
711:
683:
679:
639:
637:
636:
631:
626:
624:
608:
599:
589:
573:
565:
556:
551:
549:
538:
527:
514:
512:
511:
506:
504:
503:
463:
461:
460:
455:
453:
451:
437:
432:
430:
416:
411:
409:
395:
380:
378:
377:
372:
370:
345:
344:
332:
331:
315:
314:
280:
279:
267:
266:
250:
249:
215:
214:
202:
201:
185:
184:
98:
94:
21:
7521:
7520:
7516:
7515:
7514:
7512:
7511:
7510:
7486:
7485:
7407:
7397:
7373:
7370:
7369:
7361:
7357:
7346:
7342:
7332:
7330:
7325:
7324:
7320:
7310:
7308:
7303:
7302:
7298:
7288:
7286:
7281:
7280:
7276:
7266:
7264:
7259:
7258:
7254:
7244:
7242:
7233:
7232:
7228:
7218:
7216:
7211:
7210:
7206:
7201:
7169:
7156:
7122:
7109:
7090:
7074:
7055:
7039:
7003:
6996:
6995:
6977:
6976:
6963:
6950:
6943:
6937:
6936:
6923:
6910:
6903:
6897:
6896:
6883:
6870:
6863:
6851:
6850:
6846:
6842:
6836:
6832:
6829:
6825:
6819:
6815:
6812:
6808:
6794:
6788:
6742:
6721:
6688:
6667:
6654:
6653:
6647:
6643:
6639:
6634:
6622:
6611:
6607:
6565:
6547:
6507:
6506:
6502:
6498:
6494:
6391:
6377:
6358:
6357:
6353:
6349:
6305:
6284:
6251:
6230:
6217:
6216:
6213:triangle height
6205:
6201:
6197:
6194:
6186:
6175:
6170:
6116:
6115:
6064:
6063:
6013:
6012:
5957:
5956:
5906:
5905:
5850:
5849:
5838:
5835:
5817:
5816:
5791:
5759:
5744:
5738:
5737:
5712:
5680:
5665:
5659:
5658:
5633:
5601:
5586:
5574:
5573:
5565:
5562:
5555:
5534:
5533:
5490:
5455:
5416:
5412:
5396:
5390:
5389:
5346:
5311:
5272:
5268:
5252:
5240:
5239:
5210:
5189:
5164:
5163:
5153:
5147:
5118:
5097:
5078:
5077:
5069:
5065:
5064:and the angles
5061:
5058:
5051:
5030:
5029:
4937:
4923:
4908:
4902:
4901:
4809:
4795:
4780:
4768:
4767:
4751:
4750:
4725:
4693:
4678:
4672:
4671:
4646:
4614:
4599:
4587:
4586:
4580:
4486:
4485:
4477:
4473:
4472:and the angles
4469:
4466:
4459:
4438:
4437:
4392:
4357:
4318:
4314:
4298:
4292:
4291:
4246:
4211:
4172:
4168:
4152:
4140:
4139:
4129:
4125:
4115:
4088:
4067:
4048:
4047:
4039:
3977:
3976:
3972:
3968:
3965:
3958:
3936:
3935:
3880:
3860:
3845:
3839:
3838:
3783:
3763:
3748:
3742:
3741:
3686:
3675:
3641:
3574:
3562:
3561:
3540:
3539:
3447:
3433:
3415:
3409:
3408:
3316:
3302:
3284:
3272:
3271:
3265:
3195:
3191:
3174:
3173:
3165:
3161:
3157:
3154:
3147:
3126:
3125:
3097:
3062:
3047:
3041:
3040:
3012:
2977:
2962:
2956:
2955:
2927:
2892:
2877:
2865:
2864:
2846:
2843:
2836:
2815:
2814:
2738:
2675:
2674:
2601:
2535:
2534:
2456:
2398:
2397:
2323:
2259:
2258:
2221:
2196:
2187:
2179:
2167:angle bisectors
2155:
2143:
2125:
2124:
2081:
2070:
2046:
2035:
2020:
2014:
2013:
1970:
1959:
1935:
1924:
1909:
1897:
1896:
1879:
1875:
1874:and the angles
1871:
1866:
1858:
1825:
1815:
1802:
1774:
1773:
1747:
1736:
1717:
1716:
1699:
1693:
1681:
1678:
1672:
1669:
1663:
1660:
1654:
1648:
1642:
1632:
1629:
1619:
1613:
1603:
1590:
1584:
1574:
1564:
1554:
1540:
1533:
1520:
1513:
1509:
1505:
1498:
1480:
1477:
1472:
1471:
1469:
1464:
1414:
1413:
1405:
1401:
1397:
1392:
1382:
1374:
1355:
1328:
1317:
1304:
1291:
1290:
1271:
1270:
1223:
1210:
1197:
1196:
1192:
1188:
1183:
1175:
1141:
1116:
1057:
1056:
1021:
1008:
1007:
933:
932:
929:Heron's formula
910:
893:
877:
876:
860:
849:
836:
823:
822:
807:
801:
800:
787:
776:
763:
750:
749:
734:
722:
721:
713:
709:
704:
696:
681:
673:
657:
590:
557:
539:
528:
521:
520:
518:Law of tangents
495:
472:
471:
441:
420:
399:
389:
388:
368:
367:
336:
323:
316:
306:
303:
302:
271:
258:
251:
241:
238:
237:
206:
193:
186:
176:
167:
166:
149:
96:
92:
87:
79:
28:
23:
22:
18:Side-angle-side
15:
12:
11:
5:
7519:
7517:
7509:
7508:
7503:
7498:
7488:
7487:
7484:
7483:
7472:
7464:
7456:
7448:
7438:
7432:
7431:on Math World.
7424:
7418:
7406:
7405:External links
7403:
7402:
7401:
7395:
7368:
7367:
7355:
7340:
7329:. Maths is Fun
7318:
7307:. Maths is Fun
7296:
7285:. Maths is Fun
7274:
7263:. Maths is Fun
7252:
7226:
7215:. Maths is Fun
7203:
7202:
7200:
7197:
7196:
7195:
7190:
7185:
7180:
7175:
7168:
7165:
7161:Earth's radius
7144:
7139:
7134:
7129:
7125:
7121:
7116:
7112:
7108:
7105:
7102:
7097:
7093:
7089:
7086:
7081:
7077:
7073:
7070:
7067:
7062:
7058:
7054:
7051:
7046:
7042:
7038:
7035:
7030:
7024:
7021:
7018:
7013:
7009:
7006:
6975:
6970:
6966:
6962:
6957:
6953:
6949:
6946:
6944:
6942:
6939:
6938:
6935:
6930:
6926:
6922:
6917:
6913:
6909:
6906:
6904:
6902:
6899:
6898:
6895:
6890:
6886:
6882:
6877:
6873:
6869:
6866:
6864:
6862:
6859:
6858:
6839:
6838:
6834:
6827:
6822:
6817:
6810:
6790:Main article:
6787:
6784:
6772:
6769:
6763:
6760:
6757:
6754:
6751:
6748:
6745:
6740:
6737:
6734:
6730:
6727:
6724:
6718:
6715:
6709:
6706:
6703:
6700:
6697:
6694:
6691:
6686:
6683:
6680:
6676:
6673:
6670:
6664:
6661:
6632:
6595:
6592:
6589:
6586:
6580:
6577:
6572:
6568:
6564:
6561:
6556:
6553:
6550:
6544:
6541:
6538:
6535:
6532:
6529:
6526:
6523:
6520:
6517:
6514:
6497:and the sides
6482:
6476:
6473:
6470:
6467:
6464:
6461:
6458:
6454:
6449:
6446:
6441:
6438:
6435:
6432:
6429:
6426:
6423:
6420:
6417:
6414:
6410:
6405:
6402:
6397:
6394:
6389:
6386:
6383:
6380:
6374:
6371:
6368:
6365:
6335:
6332:
6326:
6323:
6320:
6317:
6314:
6311:
6308:
6303:
6300:
6297:
6293:
6290:
6287:
6281:
6278:
6272:
6269:
6266:
6263:
6260:
6257:
6254:
6249:
6246:
6243:
6239:
6236:
6233:
6227:
6224:
6190:Main article:
6181:
6174:
6171:
6169:
6166:
6165:
6164:
6153:
6150:
6147:
6144:
6141:
6138:
6135:
6132:
6129:
6126:
6123:
6113:
6101:
6098:
6095:
6092:
6089:
6086:
6083:
6080:
6077:
6074:
6071:
6061:
6050:
6047:
6044:
6041:
6038:
6035:
6032:
6029:
6026:
6023:
6020:
6010:
5994:
5991:
5988:
5985:
5982:
5979:
5976:
5973:
5970:
5967:
5964:
5954:
5943:
5940:
5937:
5934:
5931:
5928:
5925:
5922:
5919:
5916:
5913:
5903:
5887:
5884:
5881:
5878:
5875:
5872:
5869:
5866:
5863:
5860:
5857:
5834:
5831:
5815:
5809:
5806:
5803:
5800:
5797:
5794:
5789:
5786:
5783:
5780:
5777:
5774:
5771:
5768:
5765:
5762:
5756:
5753:
5750:
5747:
5745:
5743:
5740:
5739:
5736:
5730:
5727:
5724:
5721:
5718:
5715:
5710:
5707:
5704:
5701:
5698:
5695:
5692:
5689:
5686:
5683:
5677:
5674:
5671:
5668:
5666:
5664:
5661:
5660:
5657:
5651:
5648:
5645:
5642:
5639:
5636:
5631:
5628:
5625:
5622:
5619:
5616:
5613:
5610:
5607:
5604:
5598:
5595:
5592:
5589:
5587:
5585:
5582:
5581:
5561:
5558:
5553:
5532:
5528:
5521:
5518:
5515:
5512:
5509:
5504:
5501:
5496:
5493:
5488:
5485:
5482:
5479:
5476:
5470:
5467:
5461:
5458:
5449:
5446:
5443:
5440:
5437:
5431:
5428:
5422:
5419:
5415:
5411:
5408:
5405:
5402:
5399:
5397:
5395:
5392:
5391:
5388:
5384:
5377:
5374:
5371:
5368:
5365:
5360:
5357:
5352:
5349:
5344:
5341:
5338:
5335:
5332:
5326:
5323:
5317:
5314:
5305:
5302:
5299:
5296:
5293:
5287:
5284:
5278:
5275:
5271:
5267:
5264:
5261:
5258:
5255:
5253:
5251:
5248:
5247:
5225:
5219:
5216:
5213:
5208:
5205:
5202:
5198:
5195:
5192:
5186:
5183:
5180:
5177:
5174:
5171:
5133:
5127:
5124:
5121:
5116:
5113:
5110:
5106:
5103:
5100:
5094:
5091:
5088:
5085:
5057:
5054:
5049:
5028:
5022:
5019:
5016:
5013:
5010:
5007:
5004:
5000:
4995:
4992:
4987:
4984:
4981:
4978:
4975:
4972:
4969:
4966:
4963:
4960:
4956:
4951:
4948:
4943:
4940:
4935:
4932:
4929:
4926:
4920:
4917:
4914:
4911:
4909:
4907:
4904:
4903:
4900:
4894:
4891:
4888:
4885:
4882:
4879:
4876:
4872:
4867:
4864:
4859:
4856:
4853:
4850:
4847:
4844:
4841:
4838:
4835:
4832:
4828:
4823:
4820:
4815:
4812:
4807:
4804:
4801:
4798:
4792:
4789:
4786:
4783:
4781:
4779:
4776:
4775:
4749:
4743:
4740:
4737:
4734:
4731:
4728:
4723:
4720:
4717:
4714:
4711:
4708:
4705:
4702:
4699:
4696:
4690:
4687:
4684:
4681:
4679:
4677:
4674:
4673:
4670:
4664:
4661:
4658:
4655:
4652:
4649:
4644:
4641:
4638:
4635:
4632:
4629:
4626:
4623:
4620:
4617:
4611:
4608:
4605:
4602:
4600:
4598:
4595:
4594:
4565:
4560:
4555:
4552:
4549:
4546:
4543:
4540:
4537:
4534:
4531:
4528:
4525:
4522:
4519:
4516:
4513:
4510:
4505:
4499:
4496:
4493:
4465:
4462:
4457:
4436:
4432:
4425:
4422:
4419:
4416:
4413:
4407:
4404:
4398:
4395:
4390:
4387:
4384:
4381:
4378:
4372:
4369:
4363:
4360:
4351:
4348:
4345:
4342:
4339:
4333:
4330:
4324:
4321:
4317:
4313:
4310:
4307:
4304:
4301:
4299:
4297:
4294:
4293:
4290:
4286:
4279:
4276:
4273:
4270:
4267:
4261:
4258:
4252:
4249:
4244:
4241:
4238:
4235:
4232:
4226:
4223:
4217:
4214:
4205:
4202:
4199:
4196:
4193:
4187:
4184:
4178:
4175:
4171:
4167:
4164:
4161:
4158:
4155:
4153:
4151:
4148:
4147:
4103:
4097:
4094:
4091:
4086:
4083:
4080:
4076:
4073:
4070:
4064:
4061:
4058:
4055:
4027:
4022:
4017:
4014:
4011:
4007:
4004:
4001:
3996:
3990:
3987:
3984:
3971:and the angle
3964:
3961:
3956:
3934:
3928:
3925:
3922:
3919:
3916:
3913:
3910:
3907:
3904:
3901:
3898:
3895:
3892:
3889:
3886:
3883:
3878:
3875:
3872:
3869:
3866:
3863:
3857:
3854:
3851:
3848:
3846:
3844:
3841:
3840:
3837:
3831:
3828:
3825:
3822:
3819:
3816:
3813:
3810:
3807:
3804:
3801:
3798:
3795:
3792:
3789:
3786:
3781:
3778:
3775:
3772:
3769:
3766:
3760:
3757:
3754:
3751:
3749:
3747:
3744:
3743:
3740:
3734:
3731:
3728:
3725:
3722:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3682:
3678:
3674:
3671:
3668:
3665:
3662:
3659:
3656:
3653:
3648:
3644:
3640:
3637:
3634:
3631:
3628:
3625:
3622:
3619:
3616:
3613:
3610:
3607:
3604:
3601:
3598:
3595:
3592:
3586:
3583:
3580:
3577:
3575:
3573:
3570:
3569:
3538:
3532:
3529:
3526:
3523:
3520:
3517:
3514:
3510:
3505:
3502:
3497:
3494:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3466:
3461:
3458:
3453:
3450:
3445:
3442:
3439:
3436:
3427:
3424:
3421:
3418:
3416:
3414:
3411:
3410:
3407:
3401:
3398:
3395:
3392:
3389:
3386:
3383:
3379:
3374:
3371:
3366:
3363:
3360:
3357:
3354:
3351:
3348:
3345:
3342:
3339:
3335:
3330:
3327:
3322:
3319:
3314:
3311:
3308:
3305:
3296:
3293:
3290:
3287:
3285:
3283:
3280:
3279:
3251:
3247:
3243:
3240:
3237:
3234:
3231:
3228:
3225:
3222:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3194:
3190:
3187:
3184:
3181:
3160:and the angle
3153:
3150:
3145:
3124:
3118:
3115:
3112:
3106:
3103:
3100:
3095:
3092:
3089:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3059:
3056:
3053:
3050:
3048:
3046:
3043:
3042:
3039:
3033:
3030:
3027:
3021:
3018:
3015:
3010:
3007:
3004:
2998:
2995:
2992:
2989:
2986:
2983:
2980:
2974:
2971:
2968:
2965:
2963:
2961:
2958:
2957:
2954:
2948:
2945:
2942:
2936:
2933:
2930:
2925:
2922:
2919:
2913:
2910:
2907:
2904:
2901:
2898:
2895:
2889:
2886:
2883:
2880:
2878:
2876:
2873:
2872:
2842:
2839:
2834:
2813:
2810:
2807:
2804:
2801:
2798:
2792:
2789:
2783:
2780:
2777:
2774:
2771:
2768:
2765:
2759:
2756:
2750:
2747:
2744:
2741:
2739:
2737:
2734:
2730:
2723:
2720:
2714:
2711:
2705:
2702:
2698:
2692:
2689:
2683:
2680:
2677:
2676:
2673:
2670:
2667:
2664:
2661:
2655:
2652:
2646:
2643:
2640:
2637:
2634:
2631:
2628:
2622:
2619:
2613:
2610:
2607:
2604:
2602:
2600:
2597:
2593:
2586:
2583:
2577:
2574:
2568:
2565:
2558:
2552:
2549:
2543:
2540:
2537:
2536:
2533:
2530:
2527:
2524:
2521:
2515:
2512:
2506:
2503:
2500:
2497:
2493:
2486:
2483:
2477:
2474:
2468:
2465:
2462:
2459:
2457:
2455:
2452:
2449:
2446:
2443:
2437:
2434:
2428:
2425:
2421:
2415:
2412:
2406:
2403:
2400:
2399:
2396:
2393:
2390:
2387:
2384:
2378:
2375:
2369:
2366:
2363:
2360:
2356:
2353:
2350:
2344:
2341:
2335:
2332:
2329:
2326:
2324:
2322:
2319:
2316:
2313:
2310:
2304:
2301:
2295:
2292:
2288:
2282:
2279:
2273:
2270:
2267:
2266:
2201:central angles
2185:
2178:
2175:
2154:
2151:
2142:
2139:
2120:
2117:
2114:
2111:
2108:
2105:
2102:
2099:
2096:
2093:
2090:
2087:
2084:
2079:
2076:
2073:
2064:
2061:
2055:
2052:
2049:
2044:
2041:
2038:
2029:
2026:
2023:
2021:
2019:
2016:
2015:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1968:
1965:
1962:
1953:
1950:
1944:
1941:
1938:
1933:
1930:
1927:
1918:
1915:
1912:
1910:
1908:
1905:
1904:
1864:
1857:
1854:
1840:
1837:
1832:
1828:
1822:
1818:
1814:
1809:
1805:
1799:
1796:
1793:
1790:
1787:
1784:
1781:
1756:
1753:
1750:
1745:
1742:
1739:
1730:
1727:
1724:
1690:
1689:
1688:
1687:
1677:and the angle
1653:and the angle
1618:may be acute:
1600:
1551:
1530:
1452:
1449:
1446:
1443:
1438:
1435:
1430:
1427:
1424:
1421:
1400:and the angle
1390:
1380:
1373:
1370:
1343:
1337:
1334:
1331:
1324:
1320:
1316:
1311:
1307:
1303:
1298:
1294:
1287:
1284:
1281:
1278:
1258:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1230:
1226:
1222:
1217:
1213:
1207:
1204:
1191:and the angle
1181:
1174:
1171:
1157:
1153:
1148:
1144:
1140:
1137:
1134:
1131:
1128:
1123:
1119:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1067:
1064:
1040:
1036:
1033:
1030:
1027:
1024:
1018:
1015:
993:
990:
987:
984:
981:
978:
975:
972:
969:
966:
963:
960:
957:
954:
951:
948:
943:
940:
875:
869:
866:
863:
856:
852:
848:
843:
839:
835:
830:
826:
819:
816:
813:
810:
808:
806:
803:
802:
796:
793:
790:
783:
779:
775:
770:
766:
762:
757:
753:
746:
743:
740:
737:
735:
733:
730:
729:
718:law of cosines
702:
695:
692:
691:
690:
686:
662:law of cosines
656:
653:
641:
640:
629:
623:
620:
617:
614:
611:
605:
602:
596:
593:
588:
585:
582:
579:
576:
571:
568:
563:
560:
554:
548:
545:
542:
537:
534:
531:
515:
502:
498:
494:
491:
488:
485:
482:
479:
469:
464:
450:
447:
444:
440:
435:
429:
426:
423:
419:
414:
408:
405:
402:
398:
382:
381:
366:
363:
360:
357:
354:
351:
348:
343:
339:
335:
330:
326:
322:
319:
317:
313:
309:
305:
304:
301:
298:
295:
292:
289:
286:
283:
278:
274:
270:
265:
261:
257:
254:
252:
248:
244:
240:
239:
236:
233:
230:
227:
224:
221:
218:
213:
209:
205:
200:
196:
192:
189:
187:
183:
179:
175:
174:
164:Law of cosines
148:
145:
137:
136:
129:
122:
115:
108:
85:
78:
75:
41:) is the main
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7518:
7507:
7504:
7502:
7499:
7497:
7494:
7493:
7491:
7482:
7478:
7477:
7473:
7470:
7469:
7465:
7462:
7461:
7457:
7454:
7453:
7449:
7446:
7442:
7439:
7436:
7433:
7430:
7429:
7425:
7422:
7419:
7416:
7412:
7409:
7408:
7404:
7398:
7396:0-486-60088-2
7392:
7387:
7386:
7380:
7376:
7372:
7371:
7364:
7359:
7356:
7352:
7351:
7344:
7341:
7328:
7322:
7319:
7306:
7300:
7297:
7284:
7278:
7275:
7262:
7256:
7253:
7240:
7236:
7230:
7227:
7214:
7208:
7205:
7198:
7194:
7191:
7189:
7188:Lénárt sphere
7186:
7184:
7183:Hinge theorem
7181:
7179:
7176:
7174:
7171:
7170:
7166:
7164:
7162:
7142:
7127:
7123:
7119:
7114:
7110:
7103:
7100:
7095:
7091:
7087:
7084:
7079:
7075:
7071:
7068:
7065:
7060:
7056:
7052:
7049:
7044:
7040:
7036:
7033:
7022:
7019:
7016:
7007:
7004:
6993:
6973:
6968:
6964:
6960:
6955:
6951:
6947:
6945:
6940:
6933:
6928:
6924:
6920:
6915:
6911:
6907:
6905:
6900:
6893:
6888:
6884:
6880:
6875:
6871:
6867:
6865:
6860:
6823:
6806:
6805:
6804:
6798:
6793:
6785:
6783:
6770:
6767:
6761:
6758:
6755:
6752:
6749:
6746:
6743:
6738:
6735:
6732:
6728:
6725:
6722:
6716:
6713:
6704:
6701:
6698:
6692:
6689:
6684:
6681:
6678:
6674:
6671:
6668:
6662:
6659:
6650:
6630:
6626:
6621:. The angles
6620:
6615:
6593:
6590:
6587:
6584:
6578:
6575:
6570:
6566:
6562:
6559:
6554:
6551:
6548:
6542:
6539:
6536:
6533:
6530:
6527:
6524:
6521:
6518:
6515:
6512:
6480:
6471:
6468:
6465:
6459:
6456:
6452:
6447:
6444:
6439:
6436:
6433:
6427:
6424:
6421:
6415:
6412:
6408:
6403:
6400:
6395:
6392:
6387:
6384:
6381:
6378:
6372:
6369:
6366:
6363:
6346:
6333:
6330:
6324:
6321:
6318:
6315:
6312:
6309:
6306:
6301:
6298:
6295:
6291:
6288:
6285:
6279:
6276:
6267:
6264:
6261:
6255:
6252:
6247:
6244:
6241:
6237:
6234:
6231:
6225:
6222:
6214:
6209:
6193:
6192:Triangulation
6185:
6184:triangulation
6179:
6173:Triangulation
6172:
6167:
6151:
6148:
6145:
6142:
6139:
6136:
6133:
6130:
6127:
6124:
6121:
6114:
6099:
6096:
6093:
6090:
6087:
6084:
6081:
6078:
6075:
6072:
6069:
6062:
6048:
6045:
6042:
6039:
6036:
6033:
6030:
6027:
6024:
6021:
6018:
6011:
6008:
5992:
5989:
5986:
5983:
5980:
5977:
5974:
5971:
5968:
5965:
5962:
5955:
5941:
5938:
5935:
5932:
5929:
5926:
5923:
5920:
5917:
5914:
5911:
5904:
5901:
5885:
5882:
5879:
5876:
5873:
5870:
5867:
5864:
5861:
5858:
5855:
5848:
5847:
5846:
5844:
5832:
5830:
5813:
5807:
5804:
5801:
5798:
5795:
5792:
5787:
5784:
5781:
5778:
5775:
5772:
5769:
5766:
5763:
5760:
5754:
5751:
5748:
5746:
5741:
5734:
5728:
5725:
5722:
5719:
5716:
5713:
5708:
5705:
5702:
5699:
5696:
5693:
5690:
5687:
5684:
5681:
5675:
5672:
5669:
5667:
5662:
5655:
5649:
5646:
5643:
5640:
5637:
5634:
5629:
5626:
5623:
5620:
5617:
5614:
5611:
5608:
5605:
5602:
5596:
5593:
5590:
5588:
5583:
5571:
5559:
5551:
5547:
5530:
5526:
5516:
5513:
5510:
5502:
5499:
5494:
5491:
5483:
5480:
5477:
5468:
5465:
5459:
5456:
5444:
5441:
5438:
5429:
5426:
5420:
5417:
5413:
5409:
5406:
5403:
5400:
5398:
5393:
5386:
5382:
5372:
5369:
5366:
5358:
5355:
5350:
5347:
5339:
5336:
5333:
5324:
5321:
5315:
5312:
5300:
5297:
5294:
5285:
5282:
5276:
5273:
5269:
5265:
5262:
5259:
5256:
5254:
5249:
5236:
5223:
5217:
5214:
5211:
5206:
5203:
5200:
5196:
5193:
5190:
5184:
5181:
5178:
5175:
5172:
5169:
5160:
5156:
5152:is acute and
5150:
5144:
5131:
5125:
5122:
5119:
5114:
5111:
5108:
5104:
5101:
5098:
5092:
5089:
5086:
5083:
5075:
5055:
5047:
5043:
5026:
5017:
5014:
5011:
5005:
5002:
4998:
4993:
4990:
4985:
4982:
4979:
4973:
4970:
4967:
4961:
4958:
4954:
4949:
4946:
4941:
4938:
4933:
4930:
4927:
4924:
4918:
4915:
4912:
4910:
4905:
4898:
4889:
4886:
4883:
4877:
4874:
4870:
4865:
4862:
4857:
4854:
4851:
4845:
4842:
4839:
4833:
4830:
4826:
4821:
4818:
4813:
4810:
4805:
4802:
4799:
4796:
4790:
4787:
4784:
4782:
4777:
4764:
4747:
4741:
4738:
4735:
4732:
4729:
4726:
4721:
4718:
4715:
4712:
4709:
4706:
4703:
4700:
4697:
4694:
4688:
4685:
4682:
4680:
4675:
4668:
4662:
4659:
4656:
4653:
4650:
4647:
4642:
4639:
4636:
4633:
4630:
4627:
4624:
4621:
4618:
4615:
4609:
4606:
4603:
4601:
4596:
4584:
4577:
4563:
4553:
4550:
4547:
4544:
4541:
4538:
4535:
4532:
4529:
4526:
4523:
4520:
4517:
4514:
4511:
4508:
4497:
4494:
4491:
4483:
4463:
4455:
4451:
4434:
4430:
4420:
4417:
4414:
4405:
4402:
4396:
4393:
4385:
4382:
4379:
4370:
4367:
4361:
4358:
4346:
4343:
4340:
4331:
4328:
4322:
4319:
4315:
4311:
4308:
4305:
4302:
4300:
4295:
4288:
4284:
4274:
4271:
4268:
4259:
4256:
4250:
4247:
4239:
4236:
4233:
4224:
4221:
4215:
4212:
4200:
4197:
4194:
4185:
4182:
4176:
4173:
4169:
4165:
4162:
4159:
4156:
4154:
4149:
4136:
4133:
4122:
4118:
4101:
4095:
4092:
4089:
4084:
4081:
4078:
4074:
4071:
4068:
4062:
4059:
4056:
4053:
4045:
4025:
4015:
4012:
4009:
4005:
4002:
3999:
3988:
3985:
3982:
3962:
3954:
3950:
3932:
3926:
3923:
3920:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3876:
3873:
3870:
3867:
3864:
3861:
3855:
3852:
3849:
3847:
3842:
3835:
3829:
3826:
3823:
3820:
3817:
3814:
3811:
3808:
3805:
3802:
3799:
3796:
3793:
3790:
3787:
3784:
3779:
3776:
3773:
3770:
3767:
3764:
3758:
3755:
3752:
3750:
3745:
3738:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3702:
3699:
3696:
3693:
3690:
3687:
3680:
3672:
3669:
3666:
3663:
3660:
3657:
3651:
3646:
3638:
3635:
3632:
3629:
3626:
3623:
3620:
3617:
3614:
3611:
3608:
3605:
3602:
3599:
3596:
3593:
3584:
3581:
3578:
3576:
3571:
3558:
3553:
3536:
3527:
3524:
3521:
3515:
3512:
3508:
3503:
3500:
3495:
3492:
3489:
3483:
3480:
3477:
3471:
3468:
3464:
3459:
3456:
3451:
3448:
3443:
3440:
3437:
3434:
3425:
3422:
3419:
3417:
3412:
3405:
3396:
3393:
3390:
3384:
3381:
3377:
3372:
3369:
3364:
3361:
3358:
3352:
3349:
3346:
3340:
3337:
3333:
3328:
3325:
3320:
3317:
3312:
3309:
3306:
3303:
3294:
3291:
3288:
3286:
3281:
3269:
3262:
3249:
3245:
3241:
3238:
3235:
3232:
3229:
3226:
3223:
3220:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3192:
3188:
3185:
3182:
3179:
3171:
3151:
3143:
3139:
3122:
3116:
3113:
3110:
3104:
3101:
3098:
3093:
3090:
3087:
3081:
3078:
3075:
3072:
3069:
3066:
3063:
3057:
3054:
3051:
3049:
3044:
3037:
3031:
3028:
3025:
3019:
3016:
3013:
3008:
3005:
3002:
2996:
2993:
2990:
2987:
2984:
2981:
2978:
2972:
2969:
2966:
2964:
2959:
2952:
2946:
2943:
2940:
2934:
2931:
2928:
2923:
2920:
2917:
2911:
2908:
2905:
2902:
2899:
2896:
2893:
2887:
2884:
2881:
2879:
2874:
2862:
2857:
2853:
2849:
2840:
2832:
2828:
2811:
2805:
2802:
2799:
2790:
2787:
2781:
2778:
2772:
2769:
2766:
2757:
2754:
2748:
2745:
2742:
2740:
2732:
2728:
2721:
2712:
2709:
2703:
2700:
2696:
2690:
2687:
2681:
2678:
2668:
2665:
2662:
2653:
2650:
2644:
2641:
2635:
2632:
2629:
2620:
2617:
2611:
2608:
2605:
2603:
2595:
2591:
2584:
2575:
2572:
2566:
2563:
2556:
2550:
2547:
2541:
2538:
2528:
2525:
2522:
2513:
2510:
2504:
2501:
2495:
2491:
2484:
2475:
2472:
2466:
2463:
2460:
2458:
2450:
2447:
2444:
2435:
2432:
2426:
2423:
2419:
2413:
2410:
2404:
2401:
2391:
2388:
2385:
2376:
2373:
2367:
2364:
2358:
2354:
2351:
2342:
2339:
2333:
2330:
2327:
2325:
2317:
2314:
2311:
2302:
2299:
2293:
2290:
2286:
2280:
2277:
2271:
2268:
2256:
2252:
2247:
2245:
2241:
2237:
2232:
2228:
2224:
2220:
2216:
2212:
2208:
2206:
2202:
2194:
2183:
2176:
2174:
2172:
2168:
2164:
2160:
2152:
2150:
2148:
2140:
2138:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2097:
2094:
2091:
2088:
2085:
2082:
2077:
2074:
2071:
2062:
2059:
2053:
2050:
2047:
2042:
2039:
2036:
2027:
2024:
2022:
2017:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1966:
1963:
1960:
1951:
1948:
1942:
1939:
1936:
1931:
1928:
1925:
1916:
1913:
1911:
1906:
1893:
1890:
1886:
1882:
1862:
1855:
1853:
1838:
1835:
1830:
1826:
1820:
1816:
1812:
1807:
1803:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1770:
1754:
1751:
1748:
1743:
1740:
1737:
1728:
1725:
1722:
1713:
1710:
1706:
1702:
1696:
1684:
1675:
1666:
1657:
1651:
1645:
1639:
1635:
1626:
1622:
1616:
1610:
1606:
1601:
1597:
1593:
1587:
1581:
1577:
1571:
1567:
1562:
1561:
1557:
1552:
1549:
1543:
1536:
1531:
1527:
1523:
1516:
1501:
1496:
1495:
1494:
1491:
1483:
1475:
1467:
1450:
1447:
1444:
1441:
1436:
1433:
1428:
1425:
1422:
1419:
1411:
1388:
1378:
1371:
1369:
1366:
1362:
1358:
1341:
1335:
1332:
1329:
1322:
1318:
1314:
1309:
1305:
1301:
1296:
1292:
1285:
1282:
1279:
1276:
1256:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1228:
1224:
1220:
1215:
1211:
1205:
1202:
1179:
1172:
1170:
1155:
1146:
1138:
1135:
1132:
1126:
1121:
1117:
1107:
1104:
1101:
1098:
1095:
1086:
1083:
1080:
1077:
1074:
1065:
1062:
1053:
1038:
1034:
1031:
1028:
1025:
1022:
1016:
1013:
988:
985:
982:
973:
970:
967:
958:
955:
952:
946:
941:
938:
930:
925:
923:
918:
916:
907:
904:
900:
896:
890:
873:
867:
864:
861:
854:
850:
846:
841:
837:
833:
828:
824:
817:
814:
811:
809:
804:
794:
791:
788:
781:
777:
773:
768:
764:
760:
755:
751:
744:
741:
738:
736:
731:
720:can be used:
719:
700:
693:
687:
677:
671:
667:
663:
659:
658:
654:
652:
650:
646:
627:
618:
615:
612:
603:
600:
594:
591:
583:
580:
577:
569:
566:
561:
558:
552:
546:
543:
540:
535:
532:
529:
519:
516:
500:
496:
492:
489:
486:
483:
480:
477:
470:
468:
467:Sum of angles
465:
448:
445:
442:
438:
433:
427:
424:
421:
417:
412:
406:
403:
400:
396:
387:
384:
383:
364:
361:
358:
355:
352:
349:
346:
341:
337:
333:
328:
324:
320:
318:
311:
307:
299:
296:
293:
290:
287:
284:
281:
276:
272:
268:
263:
259:
255:
253:
246:
242:
234:
231:
228:
225:
222:
219:
216:
211:
207:
203:
198:
194:
190:
188:
181:
177:
165:
162:
161:
160:
153:
146:
144:
142:
134:
130:
127:
123:
120:
116:
113:
109:
106:
103:Three sides (
102:
101:
100:
83:
76:
74:
72:
68:
64:
60:
56:
52:
48:
44:
43:trigonometric
40:
36:
32:
19:
7506:Trigonometry
7474:
7466:
7458:
7452:Triangulator
7450:
7426:
7421:Trigonometry
7384:
7365:at MathWorld
7358:
7348:
7343:
7331:. Retrieved
7321:
7309:. Retrieved
7299:
7287:. Retrieved
7277:
7265:. Retrieved
7255:
7243:. Retrieved
7239:the original
7229:
7217:. Retrieved
7207:
6840:
6831:, longitude
6814:, longitude
6802:
6648:
6637:
6616:
6347:
6210:
6195:
5836:
5563:
5237:
5158:
5154:
5148:
5145:
5059:
4765:
4585:
4578:
4467:
4137:
4131:
4120:
4116:
3966:
3554:
3270:
3263:
3155:
2855:
2851:
2847:
2844:
2248:
2230:
2226:
2222:
2209:
2191:The general
2190:
2156:
2147:law of sines
2144:
1894:
1888:
1884:
1880:
1869:
1771:
1714:
1708:
1704:
1700:
1694:
1691:
1679:
1670:
1661:
1655:
1649:
1643:
1637:
1630:
1624:
1620:
1614:
1612:, the angle
1608:
1604:
1595:
1591:
1585:
1579:
1575:
1569:
1565:
1555:
1548:right-angled
1541:
1534:
1525:
1521:
1514:
1499:
1489:
1481:
1473:
1465:
1410:law of sines
1395:
1364:
1360:
1356:
1186:
1054:
926:
919:
915:law of sines
908:
902:
898:
894:
891:
707:
680:, the angle
675:
666:law of sines
642:
386:Law of sines
158:
138:
132:
125:
118:
111:
104:
90:
67:construction
30:
29:
5568:. From the
5068:. The side
3264:The angles
1647:, the side
1628:or obtuse:
927:Area using
892:Then angle
689:reflection.
7490:Categories
7333:13 January
7289:13 January
7267:13 January
7199:References
7173:Congruence
6005:(from the
5898:(from the
5572:we infer:
4480:using the
4038:The angle
1599:is unique.
71:navigation
7377:(1956) .
7120:−
7104:
7092:λ
7088:
7076:λ
7072:
7057:λ
7053:
7041:λ
7037:
7012:¯
6961:−
6941:γ
6925:λ
6921:−
6916:∘
6885:λ
6881:−
6876:∘
6768:ℓ
6762:α
6759:
6753:−
6750:β
6747:
6739:β
6736:
6729:α
6726:
6714:ℓ
6705:α
6702:−
6699:β
6693:
6685:β
6682:
6675:α
6672:
6591:α
6588:
6576:
6552:
6540:α
6537:
6528:
6516:
6472:β
6469:−
6466:α
6460:
6453:ℓ
6440:
6428:β
6422:α
6416:
6409:ℓ
6396:
6388:β
6385:
6367:
6331:ℓ
6325:β
6322:
6313:α
6310:
6302:β
6299:
6292:α
6289:
6277:ℓ
6268:β
6262:α
6256:
6248:β
6245:
6238:α
6235:
6149:
6143:⋅
6137:
6125:
6097:
6091:⋅
6085:
6073:
6046:
6040:⋅
6034:
6022:
5990:
5984:⋅
5978:
5966:
5939:
5933:⋅
5927:
5915:
5883:
5877:⋅
5871:
5859:
5808:β
5805:
5799:α
5796:
5788:β
5785:
5779:α
5776:
5767:γ
5764:
5755:
5729:α
5726:
5720:γ
5717:
5709:α
5706:
5700:γ
5697:
5688:β
5685:
5676:
5650:γ
5647:
5641:β
5638:
5630:γ
5627:
5621:β
5618:
5609:α
5606:
5597:
5514:−
5495:
5460:
5445:β
5442:−
5439:α
5421:
5410:
5394:γ
5373:β
5370:−
5367:α
5351:
5340:β
5334:α
5316:
5298:−
5277:
5266:
5218:α
5215:
5207:β
5204:
5194:
5185:
5179:−
5176:π
5126:α
5123:
5115:β
5112:
5102:
5093:
5018:β
5015:−
5012:α
5006:
4986:
4974:β
4968:α
4962:
4942:
4934:β
4931:
4919:
4890:α
4887:−
4884:β
4878:
4858:
4846:α
4840:β
4834:
4814:
4806:α
4803:
4791:
4742:γ
4739:
4733:α
4730:
4722:γ
4719:
4713:α
4710:
4701:β
4698:
4689:
4663:γ
4660:
4654:β
4651:
4643:γ
4640:
4634:β
4631:
4622:α
4619:
4610:
4554:β
4551:
4545:α
4542:
4536:−
4530:
4524:β
4521:
4515:α
4512:
4492:γ
4418:−
4397:
4362:
4347:γ
4344:−
4341:β
4323:
4312:
4296:α
4275:γ
4272:−
4269:β
4251:
4240:γ
4234:β
4216:
4198:−
4177:
4166:
4093:
4085:β
4082:
4072:
4063:
4054:γ
4016:β
4013:
4003:
3927:γ
3924:
3915:
3906:
3900:−
3894:
3885:
3877:γ
3874:
3865:
3856:
3843:β
3830:γ
3827:
3818:
3809:
3803:−
3797:
3788:
3780:γ
3777:
3768:
3759:
3746:α
3733:γ
3730:
3721:
3712:
3700:
3691:
3673:γ
3670:
3661:
3639:γ
3636:
3627:
3618:
3612:−
3606:
3597:
3585:
3525:−
3516:
3509:γ
3496:
3472:
3465:γ
3452:
3441:
3426:
3413:β
3394:−
3385:
3378:γ
3365:
3341:
3334:γ
3321:
3310:
3295:
3282:α
3242:γ
3239:
3230:
3221:
3209:
3200:
3189:
3114:
3102:
3091:
3079:
3073:−
3067:
3058:
3045:γ
3029:
3017:
3006:
2994:
2988:−
2982:
2973:
2960:β
2944:
2932:
2921:
2909:
2903:−
2897:
2888:
2875:α
2782:
2773:β
2770:−
2767:α
2749:
2729:−
2704:
2697:γ
2682:
2645:
2636:β
2630:α
2612:
2592:−
2567:
2557:γ
2542:
2529:β
2523:α
2505:
2492:−
2467:
2451:β
2448:−
2445:α
2427:
2405:
2392:β
2386:α
2368:
2334:
2318:β
2315:−
2312:α
2294:
2272:
2163:altitudes
2119:α
2116:
2110:β
2107:
2098:β
2095:
2089:α
2086:
2078:β
2075:
2054:γ
2051:
2043:β
2040:
2008:α
2005:
1999:β
1996:
1987:β
1984:
1978:α
1975:
1967:α
1964:
1943:γ
1940:
1932:α
1929:
1883:= 180° −
1839:β
1836:
1813:−
1798:±
1795:β
1792:
1755:β
1752:
1744:α
1741:
1703:= 180° −
1636:= 180° −
1623:= arcsin
1594:= arcsin
1448:β
1445:
1426:γ
1423:
1359:= 180° −
1354:Finally,
1315:−
1286:
1277:α
1252:γ
1249:
1234:−
1136:−
1127:−
1105:−
986:−
971:−
956:−
913:from the
897:= 180° −
847:−
818:
805:β
774:−
745:
732:α
619:β
613:α
595:
584:β
581:−
578:α
562:
533:−
501:∘
490:γ
484:β
478:α
449:γ
446:
428:β
425:
407:α
404:
365:γ
362:
347:−
300:β
297:
282:−
235:α
232:
217:−
63:astronomy
7481:Jesus S.
7476:TrianCal
7415:Eli Maor
7167:See also
6845:, where
6619:cabotage
53:or on a
47:triangle
7381:(ed.).
7311:9 March
7245:4 April
7219:4 April
7159:is the
5566:α, β, γ
4130:180° -
2205:radians
2197:a, b, c
2159:medians
1668:, side
1573:, then
1486:
1470:
710:a, b, c
141:similar
97:α, β, γ
93:a, b, c
59:geodesy
7460:TriSph
7393:
7375:Euclid
7023:arccos
5752:arccos
5673:arccos
5594:arccos
5451:
5407:arccot
5307:
5263:arctan
5182:arcsin
5090:arcsin
4916:arctan
4788:arctan
4686:arccos
4607:arccos
4498:arccos
4353:
4309:arccot
4207:
4163:arctan
4060:arcsin
3989:arcsin
3853:arctan
3756:arctan
3582:arctan
3429:
3423:arctan
3298:
3292:arctan
3186:arccos
3108:
3085:
3055:arccos
3023:
3000:
2970:arccos
2938:
2915:
2885:arccos
2725:
2588:
2560:
2488:
2066:
2031:
1955:
1920:
1732:
1558:< 1
1502:> 1
1283:arccos
1006:where
815:arccos
742:arccos
716:, the
69:, and
55:sphere
7413:, by
7155:Here
6821:, and
5157:>
4119:<
2165:, or
1692:Once
1607:<
1544:= 90°
1517:≥ 90°
678:= 0.5
655:Notes
51:plane
35:Latin
7391:ISBN
7335:2015
7313:2013
7291:2015
7269:2015
7247:2012
7221:2012
6644:α, β
6623:α, β
6501:and
6206:α, β
5066:α, β
4474:α, β
4128:and
3986:>
3969:b, c
3266:α, β
3158:a, b
2253:and
2242:and
1876:α, β
1519:and
1488:sin
1398:b, c
1189:a, b
714:α, β
674:sin
670:sine
647:and
7101:cos
7085:cos
7069:cos
7050:sin
7034:sin
6990:If
6843:ABC
6756:tan
6744:tan
6733:tan
6723:tan
6690:sin
6679:sin
6669:sin
6614:.)
6585:sin
6567:tan
6549:tan
6534:sin
6525:sin
6513:sin
6457:sin
6437:tan
6413:sin
6393:cot
6382:sin
6364:tan
6319:tan
6307:tan
6296:tan
6286:tan
6253:sin
6242:sin
6232:sin
6146:cot
6134:cot
6122:cos
6094:sin
6082:cos
6070:cos
6043:cos
6031:tan
6019:tan
5987:cos
5975:cos
5963:cos
5936:tan
5924:sin
5912:tan
5880:sin
5868:sin
5856:sin
5802:sin
5793:sin
5782:cos
5773:cos
5761:cos
5723:sin
5714:sin
5703:cos
5694:cos
5682:cos
5644:sin
5635:sin
5624:cos
5615:cos
5603:cos
5492:sin
5457:sin
5418:tan
5348:sin
5313:sin
5274:tan
5212:sin
5201:sin
5191:sin
5120:sin
5109:sin
5099:sin
5003:sin
4983:tan
4959:sin
4939:cot
4928:sin
4875:sin
4855:tan
4831:sin
4811:cot
4800:sin
4736:sin
4727:sin
4716:cos
4707:cos
4695:cos
4657:sin
4648:sin
4637:cos
4628:cos
4616:cos
4583:):
4548:cos
4539:cos
4527:cos
4518:sin
4509:sin
4394:sin
4359:sin
4320:tan
4248:sin
4213:sin
4174:tan
4090:sin
4079:sin
4069:sin
4010:sin
4000:sin
3921:cos
3912:sin
3903:cos
3891:cos
3882:sin
3871:sin
3862:sin
3824:cos
3815:sin
3806:cos
3794:cos
3785:sin
3774:sin
3765:sin
3727:cos
3718:sin
3709:sin
3697:cos
3688:cos
3667:sin
3658:sin
3633:cos
3624:sin
3615:cos
3603:cos
3594:sin
3513:sin
3493:cot
3469:sin
3449:tan
3438:sin
3382:sin
3362:cot
3338:sin
3318:tan
3307:sin
3236:cos
3227:sin
3218:sin
3206:cos
3197:cos
3111:sin
3099:sin
3088:cos
3076:cos
3064:cos
3026:sin
3014:sin
3003:cos
2991:cos
2979:cos
2941:sin
2929:sin
2918:cos
2906:cos
2894:cos
2779:sin
2746:tan
2701:sin
2679:cot
2642:cos
2609:tan
2564:cos
2539:cot
2502:sin
2464:tan
2424:sin
2402:tan
2365:cos
2331:tan
2291:cos
2269:tan
2113:cos
2104:sin
2092:cos
2083:sin
2072:sin
2048:sin
2037:sin
2002:cos
1993:sin
1981:cos
1972:sin
1961:sin
1937:sin
1926:sin
1827:sin
1789:cos
1749:sin
1738:sin
1602:If
1563:If
1553:If
1537:= 1
1532:If
1497:If
1442:sin
1420:sin
1246:cos
592:tan
559:tan
497:180
443:sin
422:sin
401:sin
359:cos
294:cos
229:cos
133:AAS
126:ASA
119:SSA
112:SAS
105:SSS
7492::
7163:.
6912:90
6872:90
6505::
6215::
5076::
4484::
4135:.
4046::
3172::
2863::
2854:,
2850:,
2257::
2246:.
2229:+
2225:+
2161:,
2149:.
1892:.
1887:−
1712:.
1707:−
1578:≥
1568:≥
1524:≤
1510:BC
1468:=
1412::
1368:.
1363:−
931::
924:.
906:.
901:−
651:.
135:).
73:.
65:,
61:,
37::
7447:.
7399:.
7337:.
7315:.
7293:.
7271:.
7249:.
7223:.
7157:R
7143:.
7138:]
7133:)
7128:B
7124:L
7115:A
7111:L
7107:(
7096:B
7080:A
7066:+
7061:B
7045:A
7029:[
7020:R
7017:=
7008:B
7005:A
6974:.
6969:B
6965:L
6956:A
6952:L
6948:=
6934:,
6929:A
6908:=
6901:b
6894:,
6889:B
6868:=
6861:a
6847:C
6835:B
6833:L
6828:B
6826:λ
6818:A
6816:L
6811:A
6809:λ
6771:.
6717:=
6708:)
6696:(
6663:=
6660:h
6649:ℓ
6640:h
6612:ℓ
6608:d
6594:.
6579:b
6571:2
6563:+
6560:1
6555:b
6543:=
6531:b
6522:=
6519:d
6503:d
6499:b
6495:α
6481:,
6475:)
6463:(
6448:2
6445:1
6434:+
6431:)
6425:+
6419:(
6404:2
6401:1
6379:2
6373:=
6370:b
6354:β
6350:α
6334:.
6316:+
6280:=
6271:)
6265:+
6259:(
6226:=
6223:d
6202:l
6198:d
6152:B
6140:A
6131:=
6128:c
6100:B
6088:a
6079:=
6076:A
6049:A
6037:c
6028:=
6025:b
6009:)
5993:b
5981:a
5972:=
5969:c
5942:A
5930:b
5921:=
5918:a
5902:)
5886:A
5874:c
5865:=
5862:a
5839:C
5814:.
5770:+
5749:=
5742:c
5735:,
5691:+
5670:=
5663:b
5656:,
5612:+
5591:=
5584:a
5531:.
5527:]
5520:)
5517:b
5511:a
5508:(
5503:2
5500:1
5487:)
5484:b
5481:+
5478:a
5475:(
5469:2
5466:1
5448:)
5436:(
5430:2
5427:1
5414:[
5404:2
5401:=
5387:,
5383:]
5376:)
5364:(
5359:2
5356:1
5343:)
5337:+
5331:(
5325:2
5322:1
5304:)
5301:b
5295:a
5292:(
5286:2
5283:1
5270:[
5260:2
5257:=
5250:c
5224:.
5197:a
5173:=
5170:b
5159:β
5155:α
5149:a
5132:.
5105:a
5087:=
5084:b
5070:b
5062:a
5027:.
5021:)
5009:(
4999:c
4994:2
4991:1
4980:+
4977:)
4971:+
4965:(
4955:c
4950:2
4947:1
4925:2
4913:=
4906:b
4899:,
4893:)
4881:(
4871:c
4866:2
4863:1
4852:+
4849:)
4843:+
4837:(
4827:c
4822:2
4819:1
4797:2
4785:=
4778:a
4748:,
4704:+
4683:=
4676:b
4669:,
4625:+
4604:=
4597:a
4581:γ
4564:.
4559:)
4533:c
4504:(
4495:=
4478:γ
4470:c
4435:.
4431:]
4424:)
4421:c
4415:b
4412:(
4406:2
4403:1
4389:)
4386:c
4383:+
4380:b
4377:(
4371:2
4368:1
4350:)
4338:(
4332:2
4329:1
4316:[
4306:2
4303:=
4289:,
4285:]
4278:)
4266:(
4260:2
4257:1
4243:)
4237:+
4231:(
4225:2
4222:1
4204:)
4201:c
4195:b
4192:(
4186:2
4183:1
4170:[
4160:2
4157:=
4150:a
4132:γ
4126:γ
4121:c
4117:b
4102:.
4096:b
4075:c
4057:=
4040:γ
4026:.
4021:)
4006:c
3995:(
3983:b
3973:β
3933:,
3918:b
3909:a
3897:b
3888:a
3868:b
3850:=
3836:,
3821:a
3812:b
3800:a
3791:b
3771:a
3753:=
3739:,
3724:b
3715:a
3706:+
3703:b
3694:a
3681:2
3677:)
3664:b
3655:(
3652:+
3647:2
3643:)
3630:b
3621:a
3609:b
3600:a
3591:(
3579:=
3572:c
3537:.
3531:)
3528:b
3522:a
3519:(
3504:2
3501:1
3490:+
3487:)
3484:b
3481:+
3478:a
3475:(
3460:2
3457:1
3444:b
3435:2
3420:=
3406:,
3400:)
3397:a
3391:b
3388:(
3373:2
3370:1
3359:+
3356:)
3353:a
3350:+
3347:b
3344:(
3329:2
3326:1
3313:a
3304:2
3289:=
3250:.
3246:)
3233:b
3224:a
3215:+
3212:b
3203:a
3193:(
3183:=
3180:c
3166:c
3162:γ
3123:.
3117:b
3105:a
3094:b
3082:a
3070:c
3052:=
3038:,
3032:a
3020:c
3009:a
2997:c
2985:b
2967:=
2953:,
2947:c
2935:b
2924:c
2912:b
2900:a
2882:=
2856:c
2852:b
2848:a
2812:.
2809:)
2806:b
2803:+
2800:a
2797:(
2791:2
2788:1
2776:)
2764:(
2758:2
2755:1
2743:=
2736:)
2733:b
2722:a
2719:(
2713:2
2710:1
2691:2
2688:1
2672:)
2669:b
2666:+
2663:a
2660:(
2654:2
2651:1
2639:)
2633:+
2627:(
2621:2
2618:1
2606:=
2599:)
2596:b
2585:a
2582:(
2576:2
2573:1
2551:2
2548:1
2532:)
2526:+
2520:(
2514:2
2511:1
2499:)
2496:b
2485:a
2482:(
2476:2
2473:1
2461:=
2454:)
2442:(
2436:2
2433:1
2420:c
2414:2
2411:1
2395:)
2389:+
2383:(
2377:2
2374:1
2362:)
2359:b
2355:+
2352:a
2349:(
2343:2
2340:1
2328:=
2321:)
2309:(
2303:2
2300:1
2287:c
2281:2
2278:1
2231:γ
2227:β
2223:α
2101:+
2063:c
2060:=
2028:c
2025:=
2018:b
1990:+
1952:c
1949:=
1917:c
1914:=
1907:a
1889:β
1885:α
1881:γ
1872:c
1831:2
1821:2
1817:c
1808:2
1804:b
1786:c
1783:=
1780:a
1729:b
1726:=
1723:a
1709:γ
1705:β
1701:α
1695:γ
1682:′
1680:γ
1673:′
1671:b
1664:′
1662:C
1656:γ
1650:b
1644:C
1638:γ
1633:′
1631:γ
1625:D
1621:γ
1615:γ
1609:c
1605:b
1596:D
1592:γ
1586:γ
1580:γ
1576:β
1570:c
1566:b
1556:D
1550:.
1542:γ
1535:D
1529:.
1526:c
1522:b
1515:β
1506:b
1500:D
1490:β
1482:b
1478:/
1474:c
1466:D
1451:.
1437:b
1434:c
1429:=
1406:γ
1402:β
1365:γ
1361:α
1357:β
1342:.
1336:c
1333:b
1330:2
1323:2
1319:a
1310:2
1306:c
1302:+
1297:2
1293:b
1280:=
1257:.
1243:b
1240:a
1237:2
1229:2
1225:b
1221:+
1216:2
1212:a
1206:=
1203:c
1193:γ
1156:4
1152:)
1147:2
1143:)
1139:c
1133:b
1130:(
1122:2
1118:a
1114:(
1111:)
1108:a
1102:c
1099:+
1096:b
1093:(
1090:)
1087:c
1084:+
1081:b
1078:+
1075:a
1072:(
1066:=
1063:A
1039:2
1035:c
1032:+
1029:b
1026:+
1023:a
1017:=
1014:s
992:)
989:c
983:s
980:(
977:)
974:b
968:s
965:(
962:)
959:a
953:s
950:(
947:s
942:=
939:A
911:β
903:β
899:α
895:γ
874:.
868:c
865:a
862:2
855:2
851:b
842:2
838:c
834:+
829:2
825:a
812:=
795:c
792:b
789:2
782:2
778:a
769:2
765:c
761:+
756:2
752:b
739:=
682:β
676:β
628:.
622:)
616:+
610:(
604:2
601:1
587:)
575:(
570:2
567:1
553:=
547:b
544:+
541:a
536:b
530:a
493:=
487:+
481:+
439:c
434:=
418:b
413:=
397:a
356:b
353:a
350:2
342:2
338:b
334:+
329:2
325:a
321:=
312:2
308:c
291:c
288:a
285:2
277:2
273:c
269:+
264:2
260:a
256:=
247:2
243:b
226:c
223:b
220:2
212:2
208:c
204:+
199:2
195:b
191:=
182:2
178:a
128:)
107:)
33:(
20:)
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