2586:
2178:
55:
2581:{\displaystyle {\begin{aligned}{\frac {\cos {\tfrac {1}{2}}(\alpha -\beta )}{\cos {\tfrac {1}{2}}(\alpha +\beta )}}&={\frac {\cot {\tfrac {1}{2}}\alpha \,\cot {\tfrac {1}{2}}\beta +1}{\cot {\tfrac {1}{2}}\alpha \,\cot {\tfrac {1}{2}}\beta -1}}\\&={\frac {\cot {\tfrac {1}{2}}\alpha +\cot {\tfrac {1}{2}}\beta +2\cot {\tfrac {1}{2}}\gamma }{\cot {\tfrac {1}{2}}\alpha +\cot {\tfrac {1}{2}}\beta }}\\&={\frac {4s-a-b-2c}{2s-a-b}}.\end{aligned}}}
36:
20:
2065:
1430:
1843:
1205:
562:
1217:
1753:
1078:
962:
1085:
2695:
2163:
420:
2060:{\displaystyle {\frac {\sin {\tfrac {1}{2}}(\alpha -\beta )}{\sin {\frac {1}{2}}(\alpha +\beta )}}={\frac {\cot {\frac {1}{2}}\beta -\cot {\tfrac {1}{2}}\alpha }{\cot {\frac {1}{2}}\beta +\cot {\tfrac {1}{2}}\alpha }}={\frac {a-b}{2s-a-b}}.}
3014:
656:
In the upper figure, the points of tangency of the incircle with the sides of the triangle break the perimeter into 6 segments, in 3 pairs. In each pair the segments are of equal length. For example, the 2 segments adjacent to vertex
1556:
2183:
1222:
780:
1425:{\displaystyle {\begin{aligned}{\frac {(s-a)}{r}}{\frac {(s-b)}{r}}{\frac {(s-c)}{r}}&={\frac {s-a}{r}}+{\frac {s-b}{r}}+{\frac {s-c}{r}}\\&={\frac {3s-2s}{r}}\\&={\frac {s}{r}}\end{aligned}}}
2799:
768:
647:
969:
2598:
1561:
2070:
2899:
3064:
The
Universal Encyclopaedia of Mathematics, Pan Reference Books, 1976, page 530. English version George Allen and Unwin, 1964. Translated from the German version Meyers Rechenduden, 1960.
1828:
199:
1200:{\displaystyle \cot {\frac {\alpha }{2}}\cot {\frac {\beta }{2}}\cot {\frac {\gamma }{2}}=\cot {\frac {\alpha }{2}}+\cot {\frac {\beta }{2}}+\cot {\frac {\gamma }{2}}.}
2933:
2925:
2822:
1481:
is also divided into 6 smaller triangles, also in 3 pairs, with the triangles in each pair having the same area. For example, the two triangles near vertex
195:
557:{\displaystyle {\frac {\cot {\frac {1}{2}}\alpha }{s-a}}={\frac {\cot {\frac {1}{2}}\beta }{s-b}}={\frac {\cot {\frac {1}{2}}\gamma }{s-c}}={\frac {1}{r}},}
205:
718:
569:
297:
3120:
28:
1748:{\displaystyle {\begin{aligned}S&=r(s-a)+r(s-b)+r(s-c)\\&=r{\bigl (}3s-(a+b+c){\bigr )}\\&=r(3s-2s)\\&=rs\end{aligned}}}
337:
of the triangle (or to its reciprocal, depending on how the law is expressed), so also the law of cotangents relates the radius of the
3140:
1208:
3199:
3174:
3084:
2743:
1758:
774:
119:
1073:{\displaystyle \cot \left({\tfrac {1}{2}}\alpha +{\tfrac {1}{2}}\beta +{\tfrac {1}{2}}\gamma \right)=\cot {\tfrac {\pi }{2}}=0,}
97:
957:{\displaystyle \cot(u+v+w)={\frac {\cot u+\cot v+\cot w-\cot u\cot v\cot w}{1-\cot u\cot v-\cot v\cot w-\cot w\cot u}}.}
2830:
2804:
The law of cosines can be expressed in terms of the cotangent instead of the cosine, which brings the triangle's area
668:. An example of this is the segments shown in color in the figure. The two segments making up the red line add up to
290:
241:
190:
124:
103:
2690:{\displaystyle {\frac {b+a}{c}}={\dfrac {\cos {\tfrac {1}{2}}(\alpha -\beta )}{\sin {\tfrac {1}{2}}\gamma }}}
93:
81:
69:
64:
2158:{\displaystyle {\frac {a-b}{c}}={\dfrac {\sin {\frac {1}{2}}(\alpha -\beta )}{\cos {\frac {1}{2}}\gamma }}}
3040:
2169:
1834:
414:
54:
2590:
Here, an extra step is required to transform a product into a sum, according to the sum/product formula.
3194:
2730:, so the same name is sometimes applied to other triangle identities involving cotangents. For example:
283:
3122:
Cosine and
Cotangent Theorems for a Quadrilateral, two new Formulas for its Area and Their Applications
23:
A triangle, showing the "incircle" and the partitioning of the sides. The angle bisectors meet at the
334:
256:
210:
129:
74:
2734:
172:
3009:{\displaystyle \cot \alpha \,\cot \beta +\cot \alpha \,\cot \gamma +\cot \beta \,\cot \gamma =1.}
3045:
1472:
3170:
3080:
261:
3102:
3101:
338:
2907:
3035:
2727:
2702:
160:
89:
85:
2733:
The sum of the cotangents of two angles equals the ratio of the side between them to the
3030:
2807:
2723:
271:
266:
155:
3188:
715:
By inspection of the figure, using the definition of the cotangent function, we have
662:
374:
251:
357:
Using the usual notations for a triangle (see the figure at the upper right), where
3025:
2719:
330:
311:
150:
46:
35:
19:
236:
134:
246:
226:
323:
346:
342:
319:
181:
24:
661:
are equal. If we pick one segment from each pair, their sum will be the
231:
1468:
A number of other results can be derived from the law of cotangents.
770:
and similarly for the other two angles, proving the first assertion.
2718:
The law of cotangents is not as common or well established as the
34:
18:
16:
Trigonometric identity relating the sides and angles of a triangle
2927:
the sum of the pairwise products of their cotangents is one:
2172:. From the addition formula and the law of cotangents we have
1837:. From the addition formula and the law of cotangents we have
2794:{\displaystyle \cot \alpha +\cot \beta ={\frac {c}{h_{c}}}.}
1214:
Substituting the values obtained in the first part, we get:
763:{\displaystyle \cot {\frac {\alpha }{2}}={\frac {s-a}{r}}\,}
682:. Obviously, the other five segments must also have lengths
329:
Just as three quantities whose equality is expressed by the
773:
For the second one—the inradius formula—we start from the
642:{\displaystyle r={\sqrt {\frac {(s-a)(s-b)(s-c)}{s}}}\,.}
365:
are the vertices opposite those three respective sides,
2669:
2634:
2491:
2467:
2444:
2417:
2393:
2350:
2328:
2299:
2277:
2231:
2196:
1999:
1954:
1857:
1050:
1021:
1003:
985:
318:
is a relationship among the lengths of the sides of a
3141:"Diophantine Laws for Nets of the Highest Symmetries"
3100:
Gilli, Angelo C. (1959). "F-10c. The
Cotangent Law".
2936:
2910:
2833:
2810:
2746:
2624:
2601:
2181:
2096:
2073:
1846:
1761:
1559:
1220:
1088:
972:
783:
721:
572:
423:
1531:. So those two triangles together have an area of
39:
By the above reasoning, all six parts are as shown.
3008:
2919:
2893:
2816:
2793:
2689:
2580:
2157:
2059:
1822:
1747:
1424:
1199:
1072:
956:
762:
641:
556:
413:is the radius of the inscribed circle, the law of
2894:{\displaystyle c^{2}=a^{2}+b^{2}-4S\cot \gamma .}
3073:It is called the 'theorem of the cotangents' in
369:are the corresponding angles at those vertices,
2714:Other identities called the "law of cotangents"
2904:Because the three angles of a triangle sum to
566:and furthermore that the inradius is given by
1686:
1649:
291:
8:
1823:{\displaystyle S={\sqrt {s(s-a)(s-b)(s-c)}}}
3077:Illustrated glossary for school mathematics
298:
284:
42:
3119:Nenkov, V.; St Stefanov, H.; Velchev, A.
2990:
2968:
2946:
2935:
2909:
2864:
2851:
2838:
2832:
2809:
2780:
2771:
2745:
2706:
2668:
2633:
2623:
2602:
2600:
2518:
2490:
2466:
2443:
2416:
2392:
2383:
2349:
2342:
2327:
2298:
2291:
2276:
2267:
2230:
2195:
2186:
2182:
2180:
2138:
2105:
2095:
2074:
2072:
2019:
1998:
1976:
1953:
1931:
1922:
1891:
1856:
1847:
1845:
1768:
1760:
1685:
1684:
1648:
1647:
1560:
1558:
1408:
1374:
1346:
1325:
1304:
1273:
1249:
1225:
1221:
1219:
1184:
1165:
1146:
1127:
1111:
1095:
1087:
1049:
1020:
1002:
984:
971:
814:
782:
759:
741:
728:
720:
635:
579:
571:
541:
511:
502:
472:
463:
433:
424:
422:
3169:. Oxford University Press. p. 313.
672:, so the blue segment must be of length
3057:
1464:Some proofs using the law of cotangents
218:
180:
142:
111:
45:
7:
361:are the lengths of the three sides,
326:of the halves of the three angles.
3108:. Prentice-Hall. pp. 266–267.
1550:of the whole triangle is therefore
1475:. Note that the area of triangle
14:
1485:, being right triangles of width
333:are equal to the diameter of the
1460:, proving the second assertion.
712:, as shown in the lower figure.
53:
2705:can also be derived from this (
2657:
2645:
2254:
2242:
2219:
2207:
2127:
2115:
1913:
1901:
1880:
1868:
1815:
1803:
1800:
1788:
1785:
1773:
1722:
1704:
1681:
1663:
1631:
1619:
1610:
1598:
1589:
1577:
1288:
1276:
1264:
1252:
1240:
1228:
808:
790:
625:
613:
610:
598:
595:
583:
1:
27:, which is the center of the
3167:Geometry: Ancient and Modern
3165:Silvester, John R. (2001).
3139:Sheremet'ev, I. A. (2001).
349:) to its sides and angles.
3216:
2737:through the third vertex:
2170:Mollweide's second formula
191:Trigonometric substitution
3075:Apolinar, Efraín (2023).
1835:Mollweide's first formula
1209:triple cotangent identity
3200:Theorems about triangles
775:general addition formula
104:Generalized trigonometry
3148:Crystallography Reports
2593:This gives the result
2067:This gives the result
1755:This gives the result
1501:, each have an area of
1432:Multiplying through by
3010:
2921:
2895:
2818:
2795:
2691:
2582:
2159:
2061:
1824:
1749:
1426:
1201:
1074:
958:
764:
643:
558:
40:
32:
3011:
2922:
2920:{\displaystyle \pi ,}
2896:
2819:
2796:
2692:
2583:
2160:
2062:
1825:
1750:
1427:
1202:
1075:
959:
765:
644:
559:
38:
22:
3079:. pp. 260–261.
2934:
2908:
2831:
2808:
2744:
2599:
2179:
2071:
1844:
1759:
1557:
1218:
1086:
970:
781:
719:
570:
421:
335:circumscribed circle
211:Trigonometric series
3041:Mollweide's formula
2824:into the identity:
1454:gives the value of
173:Pythagorean theorem
3006:
2917:
2891:
2814:
2791:
2687:
2685:
2678:
2643:
2578:
2576:
2500:
2476:
2453:
2426:
2402:
2359:
2337:
2308:
2286:
2240:
2205:
2155:
2153:
2057:
2008:
1963:
1866:
1820:
1745:
1743:
1422:
1420:
1207:(This is also the
1197:
1070:
1059:
1030:
1012:
994:
954:
760:
639:
554:
41:
33:
2817:{\displaystyle S}
2786:
2684:
2677:
2642:
2618:
2569:
2506:
2499:
2475:
2452:
2425:
2401:
2371:
2358:
2336:
2307:
2285:
2258:
2239:
2204:
2152:
2146:
2113:
2090:
2052:
2014:
2007:
1984:
1962:
1939:
1917:
1899:
1865:
1818:
1416:
1396:
1362:
1341:
1320:
1295:
1271:
1247:
1192:
1173:
1154:
1135:
1119:
1103:
1058:
1029:
1011:
993:
949:
757:
736:
633:
632:
549:
536:
519:
497:
480:
458:
441:
316:law of cotangents
308:
307:
200:inverse functions
143:Laws and theorems
3207:
3180:
3156:
3155:
3145:
3136:
3130:
3129:
3127:
3116:
3110:
3109:
3107:
3097:
3091:
3090:
3071:
3065:
3062:
3015:
3013:
3012:
3007:
2926:
2924:
2923:
2918:
2900:
2898:
2897:
2892:
2869:
2868:
2856:
2855:
2843:
2842:
2823:
2821:
2820:
2815:
2800:
2798:
2797:
2792:
2787:
2785:
2784:
2772:
2696:
2694:
2693:
2688:
2686:
2683:
2679:
2670:
2660:
2644:
2635:
2625:
2619:
2614:
2603:
2587:
2585:
2584:
2579:
2577:
2570:
2568:
2548:
2519:
2511:
2507:
2505:
2501:
2492:
2477:
2468:
2458:
2454:
2445:
2427:
2418:
2403:
2394:
2384:
2376:
2372:
2370:
2360:
2351:
2338:
2329:
2319:
2309:
2300:
2287:
2278:
2268:
2259:
2257:
2241:
2232:
2222:
2206:
2197:
2187:
2164:
2162:
2161:
2156:
2154:
2151:
2147:
2139:
2130:
2114:
2106:
2097:
2091:
2086:
2075:
2066:
2064:
2063:
2058:
2053:
2051:
2031:
2020:
2015:
2013:
2009:
2000:
1985:
1977:
1968:
1964:
1955:
1940:
1932:
1923:
1918:
1916:
1900:
1892:
1883:
1867:
1858:
1848:
1829:
1827:
1826:
1821:
1819:
1769:
1754:
1752:
1751:
1746:
1744:
1728:
1694:
1690:
1689:
1653:
1652:
1637:
1549:
1545:
1530:
1517:
1515:
1514:
1511:
1508:
1500:
1494:
1484:
1480:
1459:
1453:
1452:
1450:
1449:
1444:
1441:
1431:
1429:
1428:
1423:
1421:
1417:
1409:
1401:
1397:
1392:
1375:
1367:
1363:
1358:
1347:
1342:
1337:
1326:
1321:
1316:
1305:
1296:
1291:
1274:
1272:
1267:
1250:
1248:
1243:
1226:
1206:
1204:
1203:
1198:
1193:
1185:
1174:
1166:
1155:
1147:
1136:
1128:
1120:
1112:
1104:
1096:
1079:
1077:
1076:
1071:
1060:
1051:
1039:
1035:
1031:
1022:
1013:
1004:
995:
986:
963:
961:
960:
955:
950:
948:
880:
815:
769:
767:
766:
761:
758:
753:
742:
737:
729:
711:
701:
691:
681:
671:
667:
660:
648:
646:
645:
640:
634:
628:
581:
580:
563:
561:
560:
555:
550:
542:
537:
535:
524:
520:
512:
503:
498:
496:
485:
481:
473:
464:
459:
457:
446:
442:
434:
425:
412:
408:
407:
405:
404:
401:
398:
372:
368:
364:
360:
339:inscribed circle
300:
293:
286:
57:
43:
3215:
3214:
3210:
3209:
3208:
3206:
3205:
3204:
3185:
3184:
3183:
3177:
3164:
3160:
3159:
3143:
3138:
3137:
3133:
3125:
3118:
3117:
3113:
3099:
3098:
3094:
3087:
3074:
3072:
3068:
3063:
3059:
3054:
3046:Heron's formula
3036:Law of tangents
3022:
2932:
2931:
2906:
2905:
2860:
2847:
2834:
2829:
2828:
2806:
2805:
2776:
2742:
2741:
2716:
2703:law of tangents
2661:
2626:
2604:
2597:
2596:
2575:
2574:
2549:
2520:
2509:
2508:
2459:
2385:
2374:
2373:
2320:
2269:
2260:
2223:
2188:
2177:
2176:
2131:
2098:
2076:
2069:
2068:
2032:
2021:
1969:
1924:
1884:
1849:
1842:
1841:
1757:
1756:
1742:
1741:
1726:
1725:
1692:
1691:
1635:
1634:
1567:
1555:
1554:
1547:
1546:, and the area
1532:
1512:
1509:
1506:
1505:
1503:
1502:
1496:
1486:
1482:
1476:
1473:Heron's formula
1466:
1455:
1445:
1442:
1437:
1436:
1434:
1433:
1419:
1418:
1399:
1398:
1376:
1365:
1364:
1348:
1327:
1306:
1297:
1275:
1251:
1227:
1216:
1215:
1084:
1083:
983:
979:
968:
967:
881:
816:
779:
778:
743:
717:
716:
703:
693:
683:
673:
669:
665:
658:
654:
582:
568:
567:
525:
504:
486:
465:
447:
426:
419:
418:
410:
402:
399:
386:
385:
383:
378:
370:
366:
362:
358:
355:
304:
125:Exact constants
17:
12:
11:
5:
3213:
3211:
3203:
3202:
3197:
3187:
3186:
3182:
3181:
3175:
3161:
3158:
3157:
3131:
3111:
3092:
3085:
3066:
3056:
3055:
3053:
3050:
3049:
3048:
3043:
3038:
3033:
3031:Law of cosines
3028:
3021:
3018:
3017:
3016:
3005:
3002:
2999:
2996:
2993:
2989:
2986:
2983:
2980:
2977:
2974:
2971:
2967:
2964:
2961:
2958:
2955:
2952:
2949:
2945:
2942:
2939:
2916:
2913:
2902:
2901:
2890:
2887:
2884:
2881:
2878:
2875:
2872:
2867:
2863:
2859:
2854:
2850:
2846:
2841:
2837:
2813:
2802:
2801:
2790:
2783:
2779:
2775:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2715:
2712:
2711:
2710:
2709:, p. 99).
2707:Silvester 2001
2682:
2676:
2673:
2667:
2664:
2659:
2656:
2653:
2650:
2647:
2641:
2638:
2632:
2629:
2622:
2617:
2613:
2610:
2607:
2573:
2567:
2564:
2561:
2558:
2555:
2552:
2547:
2544:
2541:
2538:
2535:
2532:
2529:
2526:
2523:
2517:
2514:
2512:
2510:
2504:
2498:
2495:
2489:
2486:
2483:
2480:
2474:
2471:
2465:
2462:
2457:
2451:
2448:
2442:
2439:
2436:
2433:
2430:
2424:
2421:
2415:
2412:
2409:
2406:
2400:
2397:
2391:
2388:
2382:
2379:
2377:
2375:
2369:
2366:
2363:
2357:
2354:
2348:
2345:
2341:
2335:
2332:
2326:
2323:
2318:
2315:
2312:
2306:
2303:
2297:
2294:
2290:
2284:
2281:
2275:
2272:
2266:
2263:
2261:
2256:
2253:
2250:
2247:
2244:
2238:
2235:
2229:
2226:
2221:
2218:
2215:
2212:
2209:
2203:
2200:
2194:
2191:
2185:
2184:
2174:
2173:
2150:
2145:
2142:
2137:
2134:
2129:
2126:
2123:
2120:
2117:
2112:
2109:
2104:
2101:
2094:
2089:
2085:
2082:
2079:
2056:
2050:
2047:
2044:
2041:
2038:
2035:
2030:
2027:
2024:
2018:
2012:
2006:
2003:
1997:
1994:
1991:
1988:
1983:
1980:
1975:
1972:
1967:
1961:
1958:
1952:
1949:
1946:
1943:
1938:
1935:
1930:
1927:
1921:
1915:
1912:
1909:
1906:
1903:
1898:
1895:
1890:
1887:
1882:
1879:
1876:
1873:
1870:
1864:
1861:
1855:
1852:
1839:
1838:
1817:
1814:
1811:
1808:
1805:
1802:
1799:
1796:
1793:
1790:
1787:
1784:
1781:
1778:
1775:
1772:
1767:
1764:
1740:
1737:
1734:
1731:
1729:
1727:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1697:
1695:
1693:
1688:
1683:
1680:
1677:
1674:
1671:
1668:
1665:
1662:
1659:
1656:
1651:
1646:
1643:
1640:
1638:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1570:
1568:
1566:
1563:
1562:
1552:
1551:
1465:
1462:
1415:
1412:
1407:
1404:
1402:
1400:
1395:
1391:
1388:
1385:
1382:
1379:
1373:
1370:
1368:
1366:
1361:
1357:
1354:
1351:
1345:
1340:
1336:
1333:
1330:
1324:
1319:
1315:
1312:
1309:
1303:
1300:
1298:
1294:
1290:
1287:
1284:
1281:
1278:
1270:
1266:
1263:
1260:
1257:
1254:
1246:
1242:
1239:
1236:
1233:
1230:
1224:
1223:
1196:
1191:
1188:
1183:
1180:
1177:
1172:
1169:
1164:
1161:
1158:
1153:
1150:
1145:
1142:
1139:
1134:
1131:
1126:
1123:
1118:
1115:
1110:
1107:
1102:
1099:
1094:
1091:
1069:
1066:
1063:
1057:
1054:
1048:
1045:
1042:
1038:
1034:
1028:
1025:
1019:
1016:
1010:
1007:
1001:
998:
992:
989:
982:
978:
975:
953:
947:
944:
941:
938:
935:
932:
929:
926:
923:
920:
917:
914:
911:
908:
905:
902:
899:
896:
893:
890:
887:
884:
879:
876:
873:
870:
867:
864:
861:
858:
855:
852:
849:
846:
843:
840:
837:
834:
831:
828:
825:
822:
819:
813:
810:
807:
804:
801:
798:
795:
792:
789:
786:
756:
752:
749:
746:
740:
735:
732:
727:
724:
653:
650:
638:
631:
627:
624:
621:
618:
615:
612:
609:
606:
603:
600:
597:
594:
591:
588:
585:
578:
575:
553:
548:
545:
540:
534:
531:
528:
523:
518:
515:
510:
507:
501:
495:
492:
489:
484:
479:
476:
471:
468:
462:
456:
453:
450:
445:
440:
437:
432:
429:
354:
351:
306:
305:
303:
302:
295:
288:
280:
277:
276:
275:
274:
269:
264:
259:
254:
249:
244:
239:
234:
229:
221:
220:
219:Mathematicians
216:
215:
214:
213:
208:
203:
193:
185:
184:
178:
177:
176:
175:
169:
168:
163:
158:
153:
145:
144:
140:
139:
138:
137:
132:
127:
122:
114:
113:
109:
108:
107:
106:
101:
78:
77:
72:
67:
59:
58:
50:
49:
15:
13:
10:
9:
6:
4:
3:
2:
3212:
3201:
3198:
3196:
3193:
3192:
3190:
3178:
3176:9780198508250
3172:
3168:
3163:
3162:
3154:(2): 161–166.
3153:
3149:
3142:
3135:
3132:
3124:
3123:
3115:
3112:
3106:
3105:
3096:
3093:
3088:
3086:9786072941311
3082:
3078:
3070:
3067:
3061:
3058:
3051:
3047:
3044:
3042:
3039:
3037:
3034:
3032:
3029:
3027:
3024:
3023:
3019:
3003:
3000:
2997:
2994:
2991:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2965:
2962:
2959:
2956:
2953:
2950:
2947:
2943:
2940:
2937:
2930:
2929:
2928:
2914:
2911:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2865:
2861:
2857:
2852:
2848:
2844:
2839:
2835:
2827:
2826:
2825:
2811:
2788:
2781:
2777:
2773:
2768:
2765:
2762:
2759:
2756:
2753:
2750:
2747:
2740:
2739:
2738:
2736:
2731:
2729:
2725:
2721:
2720:laws of sines
2713:
2708:
2704:
2700:
2699:
2698:
2697:as required.
2680:
2674:
2671:
2665:
2662:
2654:
2651:
2648:
2639:
2636:
2630:
2627:
2620:
2615:
2611:
2608:
2605:
2594:
2591:
2588:
2571:
2565:
2562:
2559:
2556:
2553:
2550:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2521:
2515:
2513:
2502:
2496:
2493:
2487:
2484:
2481:
2478:
2472:
2469:
2463:
2460:
2455:
2449:
2446:
2440:
2437:
2434:
2431:
2428:
2422:
2419:
2413:
2410:
2407:
2404:
2398:
2395:
2389:
2386:
2380:
2378:
2367:
2364:
2361:
2355:
2352:
2346:
2343:
2339:
2333:
2330:
2324:
2321:
2316:
2313:
2310:
2304:
2301:
2295:
2292:
2288:
2282:
2279:
2273:
2270:
2264:
2262:
2251:
2248:
2245:
2236:
2233:
2227:
2224:
2216:
2213:
2210:
2201:
2198:
2192:
2189:
2171:
2168:
2167:
2166:
2165:as required.
2148:
2143:
2140:
2135:
2132:
2124:
2121:
2118:
2110:
2107:
2102:
2099:
2092:
2087:
2083:
2080:
2077:
2054:
2048:
2045:
2042:
2039:
2036:
2033:
2028:
2025:
2022:
2016:
2010:
2004:
2001:
1995:
1992:
1989:
1986:
1981:
1978:
1973:
1970:
1965:
1959:
1956:
1950:
1947:
1944:
1941:
1936:
1933:
1928:
1925:
1919:
1910:
1907:
1904:
1896:
1893:
1888:
1885:
1877:
1874:
1871:
1862:
1859:
1853:
1850:
1836:
1833:
1832:
1831:
1830:as required.
1812:
1809:
1806:
1797:
1794:
1791:
1782:
1779:
1776:
1770:
1765:
1762:
1738:
1735:
1732:
1730:
1719:
1716:
1713:
1710:
1707:
1701:
1698:
1696:
1678:
1675:
1672:
1669:
1666:
1660:
1657:
1654:
1644:
1641:
1639:
1628:
1625:
1622:
1616:
1613:
1607:
1604:
1601:
1595:
1592:
1586:
1583:
1580:
1574:
1571:
1569:
1564:
1543:
1539:
1535:
1528:
1524:
1520:
1499:
1493:
1489:
1479:
1474:
1471:
1470:
1469:
1463:
1461:
1458:
1448:
1440:
1413:
1410:
1405:
1403:
1393:
1389:
1386:
1383:
1380:
1377:
1371:
1369:
1359:
1355:
1352:
1349:
1343:
1338:
1334:
1331:
1328:
1322:
1317:
1313:
1310:
1307:
1301:
1299:
1292:
1285:
1282:
1279:
1268:
1261:
1258:
1255:
1244:
1237:
1234:
1231:
1212:
1210:
1194:
1189:
1186:
1181:
1178:
1175:
1170:
1167:
1162:
1159:
1156:
1151:
1148:
1143:
1140:
1137:
1132:
1129:
1124:
1121:
1116:
1113:
1108:
1105:
1100:
1097:
1092:
1089:
1081:
1067:
1064:
1061:
1055:
1052:
1046:
1043:
1040:
1036:
1032:
1026:
1023:
1017:
1014:
1008:
1005:
999:
996:
990:
987:
980:
976:
973:
964:
951:
945:
942:
939:
936:
933:
930:
927:
924:
921:
918:
915:
912:
909:
906:
903:
900:
897:
894:
891:
888:
885:
882:
877:
874:
871:
868:
865:
862:
859:
856:
853:
850:
847:
844:
841:
838:
835:
832:
829:
826:
823:
820:
817:
811:
805:
802:
799:
796:
793:
787:
784:
776:
771:
754:
750:
747:
744:
738:
733:
730:
725:
722:
713:
710:
706:
700:
696:
690:
686:
680:
676:
664:
663:semiperimeter
651:
649:
636:
629:
622:
619:
616:
607:
604:
601:
592:
589:
586:
576:
573:
564:
551:
546:
543:
538:
532:
529:
526:
521:
516:
513:
508:
505:
499:
493:
490:
487:
482:
477:
474:
469:
466:
460:
454:
451:
448:
443:
438:
435:
430:
427:
416:
397:
393:
389:
381:
376:
375:semiperimeter
352:
350:
348:
344:
340:
336:
332:
327:
325:
321:
317:
313:
301:
296:
294:
289:
287:
282:
281:
279:
278:
273:
270:
268:
265:
263:
260:
258:
255:
253:
252:Regiomontanus
250:
248:
245:
243:
240:
238:
235:
233:
230:
228:
225:
224:
223:
222:
217:
212:
209:
207:
204:
201:
197:
194:
192:
189:
188:
187:
186:
183:
179:
174:
171:
170:
167:
164:
162:
159:
157:
154:
152:
149:
148:
147:
146:
141:
136:
133:
131:
128:
126:
123:
121:
118:
117:
116:
115:
110:
105:
102:
99:
95:
91:
87:
83:
80:
79:
76:
73:
71:
68:
66:
63:
62:
61:
60:
56:
52:
51:
48:
44:
37:
30:
26:
21:
3195:Trigonometry
3166:
3151:
3147:
3134:
3121:
3114:
3103:
3095:
3076:
3069:
3060:
3026:Law of sines
2903:
2803:
2732:
2717:
2595:
2592:
2589:
2175:
1840:
1553:
1541:
1537:
1533:
1526:
1522:
1518:
1497:
1491:
1487:
1477:
1467:
1456:
1446:
1438:
1213:
1082:
966:Applying to
965:
772:
714:
708:
704:
698:
694:
688:
684:
678:
674:
655:
565:
417:states that
395:
391:
387:
379:
356:
331:law of sines
328:
315:
312:trigonometry
309:
165:
47:Trigonometry
3128:(Preprint).
3104:Transistors
1495:and height
1080:we obtain:
377:, that is,
237:Brahmagupta
206:Derivatives
135:Unit circle
3189:Categories
3052:References
415:cotangents
324:cotangents
247:al-Battani
227:Hipparchus
166:Cotangents
120:Identities
2998:γ
2995:
2988:β
2985:
2976:γ
2973:
2966:α
2963:
2954:β
2951:
2944:α
2941:
2912:π
2886:γ
2883:
2871:−
2766:β
2763:
2754:α
2751:
2681:γ
2666:
2655:β
2652:−
2649:α
2631:
2563:−
2557:−
2540:−
2534:−
2528:−
2503:β
2488:
2479:α
2464:
2456:γ
2441:
2429:β
2414:
2405:α
2390:
2365:−
2362:β
2347:
2340:α
2325:
2311:β
2296:
2289:α
2274:
2252:β
2246:α
2228:
2217:β
2214:−
2211:α
2193:
2149:γ
2136:
2125:β
2122:−
2119:α
2103:
2081:−
2046:−
2040:−
2026:−
2011:α
1996:
1987:β
1974:
1966:α
1951:
1945:−
1942:β
1929:
1911:β
1905:α
1889:
1878:β
1875:−
1872:α
1854:
1810:−
1795:−
1780:−
1714:−
1661:−
1626:−
1605:−
1584:−
1384:−
1353:−
1332:−
1311:−
1283:−
1259:−
1235:−
1187:γ
1182:
1168:β
1163:
1149:α
1144:
1130:γ
1125:
1114:β
1109:
1098:α
1093:
1053:π
1047:
1033:γ
1015:β
997:α
977:
943:
934:
928:−
922:
913:
907:−
901:
892:
886:−
875:
866:
857:
851:−
845:
833:
821:
788:
748:−
731:α
726:
620:−
605:−
590:−
530:−
522:γ
509:
491:−
483:β
470:
452:−
444:α
431:
353:Statement
262:de Moivre
196:Integrals
112:Reference
82:Functions
3020:See also
2735:altitude
2728:tangents
347:inradius
343:triangle
322:and the
320:triangle
242:al-Hasib
182:Calculus
161:Tangents
29:incircle
25:incenter
2724:cosines
1516:
1504:
1451:
1435:
406:
384:
373:is the
367:α, β, γ
363:A, B, C
359:a, b, c
272:Fourier
232:Ptolemy
198: (
156:Cosines
98:inverse
84: (
70:History
65:Outline
3173:
3083:
409:, and
314:, the
130:Tables
3144:(PDF)
3126:(PDF)
2726:, or
702:, or
652:Proof
345:(the
341:of a
267:Euler
257:Viète
151:Sines
75:Usage
3171:ISBN
3081:ISBN
2701:The
2992:cot
2982:cot
2970:cot
2960:cot
2948:cot
2938:cot
2880:cot
2760:cot
2748:cot
2663:sin
2628:cos
2485:cot
2461:cot
2438:cot
2411:cot
2387:cot
2344:cot
2322:cot
2293:cot
2271:cot
2225:cos
2190:cos
2133:cos
2100:sin
1993:cot
1971:cot
1948:cot
1926:cot
1886:sin
1851:sin
1478:ABC
1211:.)
1179:cot
1160:cot
1141:cot
1122:cot
1106:cot
1090:cot
1044:cot
974:cot
940:cot
931:cot
919:cot
910:cot
898:cot
889:cot
872:cot
863:cot
854:cot
842:cot
830:cot
818:cot
785:cot
723:cot
506:cot
467:cot
428:cot
310:In
94:tan
90:cos
86:sin
3191::
3152:46
3150:.
3146:.
3004:1.
2722:,
1540:−
1525:−
1490:−
777::
707:−
697:−
692:,
687:−
677:−
394:+
390:+
382:=
96:,
92:,
88:,
3179:.
3089:.
3001:=
2979:+
2957:+
2915:,
2889:.
2877:S
2874:4
2866:2
2862:b
2858:+
2853:2
2849:a
2845:=
2840:2
2836:c
2812:S
2789:.
2782:c
2778:h
2774:c
2769:=
2757:+
2675:2
2672:1
2658:)
2646:(
2640:2
2637:1
2621:=
2616:c
2612:a
2609:+
2606:b
2572:.
2566:b
2560:a
2554:s
2551:2
2546:c
2543:2
2537:b
2531:a
2525:s
2522:4
2516:=
2497:2
2494:1
2482:+
2473:2
2470:1
2450:2
2447:1
2435:2
2432:+
2423:2
2420:1
2408:+
2399:2
2396:1
2381:=
2368:1
2356:2
2353:1
2334:2
2331:1
2317:1
2314:+
2305:2
2302:1
2283:2
2280:1
2265:=
2255:)
2249:+
2243:(
2237:2
2234:1
2220:)
2208:(
2202:2
2199:1
2144:2
2141:1
2128:)
2116:(
2111:2
2108:1
2093:=
2088:c
2084:b
2078:a
2055:.
2049:b
2043:a
2037:s
2034:2
2029:b
2023:a
2017:=
2005:2
2002:1
1990:+
1982:2
1979:1
1960:2
1957:1
1937:2
1934:1
1920:=
1914:)
1908:+
1902:(
1897:2
1894:1
1881:)
1869:(
1863:2
1860:1
1816:)
1813:c
1807:s
1804:(
1801:)
1798:b
1792:s
1789:(
1786:)
1783:a
1777:s
1774:(
1771:s
1766:=
1763:S
1739:s
1736:r
1733:=
1723:)
1720:s
1717:2
1711:s
1708:3
1705:(
1702:r
1699:=
1687:)
1682:)
1679:c
1676:+
1673:b
1670:+
1667:a
1664:(
1658:s
1655:3
1650:(
1645:r
1642:=
1632:)
1629:c
1623:s
1620:(
1617:r
1614:+
1611:)
1608:b
1602:s
1599:(
1596:r
1593:+
1590:)
1587:a
1581:s
1578:(
1575:r
1572:=
1565:S
1548:S
1544:)
1542:a
1538:s
1536:(
1534:r
1529:)
1527:a
1523:s
1521:(
1519:r
1513:2
1510:/
1507:1
1498:r
1492:a
1488:s
1483:A
1457:r
1447:s
1443:/
1439:r
1414:r
1411:s
1406:=
1394:r
1390:s
1387:2
1381:s
1378:3
1372:=
1360:r
1356:c
1350:s
1344:+
1339:r
1335:b
1329:s
1323:+
1318:r
1314:a
1308:s
1302:=
1293:r
1289:)
1286:c
1280:s
1277:(
1269:r
1265:)
1262:b
1256:s
1253:(
1245:r
1241:)
1238:a
1232:s
1229:(
1195:.
1190:2
1176:+
1171:2
1157:+
1152:2
1138:=
1133:2
1117:2
1101:2
1068:,
1065:0
1062:=
1056:2
1041:=
1037:)
1027:2
1024:1
1018:+
1009:2
1006:1
1000:+
991:2
988:1
981:(
952:.
946:u
937:w
925:w
916:v
904:v
895:u
883:1
878:w
869:v
860:u
848:w
839:+
836:v
827:+
824:u
812:=
809:)
806:w
803:+
800:v
797:+
794:u
791:(
755:r
751:a
745:s
739:=
734:2
709:c
705:s
699:b
695:s
689:a
685:s
679:a
675:s
670:a
666:s
659:A
637:.
630:s
626:)
623:c
617:s
614:(
611:)
608:b
602:s
599:(
596:)
593:a
587:s
584:(
577:=
574:r
552:,
547:r
544:1
539:=
533:c
527:s
517:2
514:1
500:=
494:b
488:s
478:2
475:1
461:=
455:a
449:s
439:2
436:1
411:r
403:2
400:/
396:c
392:b
388:a
380:s
371:s
299:e
292:t
285:v
202:)
100:)
31:.
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