44:
240:
1459:
173:
His Caught A Herring" (tangent, sine, cosine) or "The Old Army
Colonel And His Son Often Hiccup" (tangent, cosine, sine) or "Come And Have Some Oranges Help To Overcome Amnesia" (cosine, sine, tangent). Communities in Chinese circles may choose to remember it as TOA-CAH-SOH, which also means 'big-footed woman' (
172:
Another method is to expand the letters into a sentence, such as "Some Old Horses Chew Apples
Happily Throughout Old Age", "Some Old Hippy Caught Another Hippy Tripping On Acid", or "Studying Our Homework Can Always Help To Obtain Achievement". The order may be switched, as in "Tommy On A Ship Of
941:
1066:
1015:
840:
1232:
1292:
1123:
1649:
1403:
675:
1359:
719:
1182:
876:
626:
766:
1787:
1445:
1563:
3164:
3117:
2960:
2913:
2756:
2709:
2552:
2505:
2348:
2301:
2144:
2097:
1911:
1849:
1701:
484:
3070:
2866:
2662:
2458:
2254:
2050:
3206:
3002:
2798:
2594:
2390:
2186:
571:
544:
517:
3031:
2827:
2623:
2419:
2215:
2011:
882:
1021:
970:
3544:
795:
1188:
1238:
435:
laws. These have the disadvantages of not going sequentially from quadrants 1 to 4 and not reinforcing the numbering convention of the quadrants.
3459:
1072:
3575:
1570:
3431:
3370:
3313:
1365:
637:
270:
of the plane. The letters ASTC signify which of the trigonometric functions are positive, starting in the top right 1st quadrant and moving
1466:
Another mnemonic permits all of the basic identities to be read off quickly. The hexagonal chart can be constructed with a little thought:
1321:
681:
3505:
3391:
3360:
1152:
846:
1567:
Going either clockwise or counter-clockwise, the starting vertex equals the next vertex divided by the vertex after that. For example,
1706:
3250:
148:
577:
3219:
3537:
231:) for O/H, A/H, O/A. Longer mnemonics for these letters include "Oscar Has A Hold On Angie" and "Oscar Had A Heap of Apples."
63:
ratios in a right triangle can be remembered by representing them as strings of letters, for instance SOH-CAH-TOA in
English:
3421:
1705:
The sum of the squares of the two items at the top of a triangle equals the square of the item at the bottom. These are the
3700:
3638:
725:
267:
1718:
3726:
3530:
3562:
1409:
1521:
198:
An alternate way to remember the letters for Sin, Cos, and Tan is to memorize the syllables Oh, Ah, Oh-Ah (i.e.
3123:
3076:
2919:
2872:
2715:
2668:
2511:
2464:
2307:
2260:
2103:
2056:
1855:
1793:
1656:
43:
28:
460:
263:
32:
3037:
2833:
2629:
2425:
2221:
2017:
239:
3170:
2966:
2762:
2558:
2354:
2150:
3721:
3602:
3451:
3664:
3628:
1458:
550:
523:
182:
3592:
1918:
Aside from the last bullet, the specific values for each identity are summarized in this table:
1422:
1256:
1090:
901:
738:
604:
3633:
3481:
3427:
3366:
3330:
3309:
3278:
3256:
3246:
3497:
3399:
502:
442:
still goes counterclockwise but starts in quadrant 4 going through quadrants 4, 1, 2, then 3.
3674:
3010:
2806:
2602:
2398:
2194:
1990:
271:
201:
174:
113:
3580:
1471:
936:{\displaystyle \;\,{\frac {1}{2}}\;{\Big /}{\frac {\sqrt {3}}{2}}={\frac {1}{\sqrt {3}}}}
1061:{\displaystyle {\frac {\sqrt {\mathbf {\color {green}{2}} }}{2}}={\frac {1}{\sqrt {2}}}}
1010:{\displaystyle {\frac {\sqrt {\mathbf {\color {green}{2}} }}{2}}={\frac {1}{\sqrt {2}}}}
1483:
Write the functions without "co" on the three left outer vertices (from top to bottom:
3303:
448:
still starts in quadrant 1 but goes clockwise going through quadrants 1, 4, 3, then 2.
3715:
3570:
3365:. Spectrum series. Washington, DC: Mathematical Association of America. p. 119.
835:{\displaystyle {\frac {\sqrt {\mathbf {\color {teal}{1}} }}{2}}=\;\,{\frac {1}{2}}}
20:
3362:
Twenty years before the blackboard: the lessons and humor of a mathematics teacher
3333:
1653:
The starting corner equals the product of its two nearest neighbors. For example,
3437:
1470:
Draw three triangles pointing down, touching at a single point. This resembles a
1227:{\displaystyle {\frac {\sqrt {\mathbf {\color {teal}{1}} }}{2}}=\;{\frac {1}{2}}}
3669:
3654:
457:
Sines and cosines of common angles 0°, 30°, 45°, 60° and 90° follow the pattern
3281:
1287:{\displaystyle {\frac {\sqrt {3}}{2}}{\Big /}\;{\frac {1}{2}}\;\,={\sqrt {3}}}
3338:
3286:
3260:
1118:{\displaystyle {\frac {1}{\sqrt {2}}}{\Big /}{\frac {1}{\sqrt {2}}}=\;\;1}
3554:
3240:
259:
161:
24:
1644:{\displaystyle \sin A={\frac {\cos A}{\cot A}}={\frac {\tan A}{\sec A}}}
1498:
Write the co-functions on the corresponding three right outer vertices (
110:
One way to remember the letters is to sound them out phonetically (i.e.
3695:
3623:
3587:
1474:
192:
47:
Image mnemonic to help remember the ratios of sides of a right triangle
1518:
The starting vertex equals one over the opposite vertex. For example,
1398:{\displaystyle {\frac {\sqrt {\mathbf {\color {blue}{0}} }}{2}}=\;\,0}
670:{\displaystyle {\frac {\sqrt {\mathbf {\color {blue}{0}} }}{2}}=\;\;0}
3690:
3618:
1354:{\displaystyle {\frac {\sqrt {\mathbf {\color {red}{4}} }}{2}}=\;\,1}
714:{\displaystyle {\frac {\sqrt {\mathbf {\color {red}{4}} }}{2}}=\;\;1}
3522:
3245:. Fiona McGill. Cheltenham: Stanley Thornes Publishers. p. 51.
299:
Quadrant 4 (angles from 270 to 360 degrees, or 3π/2 to 2π radians):
292:
Quadrant 3 (angles from 180 to 270 degrees, or π to 3π/2 radians):
3597:
1457:
1177:{\displaystyle {\frac {\sqrt {\mathbf {\color {orange}{3}} }}{2}}}
871:{\displaystyle {\frac {\sqrt {\mathbf {\color {orange}{3}} }}{2}}}
238:
42:
285:
Quadrant 2 (angles from 90 to 180 degrees, or π/2 to π radians):
3526:
278:
Quadrant 1 (angles from 0 to 90 degrees, or 0 to π/2 radians):
621:{\displaystyle \tan \theta =\sin \theta {\Big /}\cos \theta }
296:
angent and cotangent functions are positive in this quadrant.
3483:
Precalculus with Limits: A Graphing
Approach, Texas Edition
223:
217:
209:
204:
140:
137:
128:
122:
303:
osine and secant functions are positive in this quadrant.
289:
ine and cosecant functions are positive in this quadrant.
1480:
Write a 1 in the middle where the three triangles touch
282:
trigonometric functions are positive in this quadrant.
3173:
3126:
3079:
3040:
3013:
2969:
2922:
2875:
2836:
2809:
2765:
2718:
2671:
2632:
2605:
2561:
2514:
2467:
2428:
2401:
2357:
2310:
2263:
2224:
2197:
2153:
2106:
2059:
2020:
1993:
1858:
1796:
1721:
1659:
1573:
1524:
1412:
1368:
1324:
1241:
1191:
1155:
1075:
1024:
973:
885:
849:
798:
728:
684:
640:
580:
553:
526:
505:
463:
149:
125:
119:
116:
134:
3683:
3647:
3611:
3561:
214:
131:
3200:
3158:
3111:
3064:
3025:
2996:
2954:
2907:
2860:
2821:
2792:
2750:
2703:
2656:
2617:
2588:
2546:
2499:
2452:
2413:
2384:
2342:
2295:
2248:
2209:
2180:
2138:
2091:
2044:
2005:
1905:
1843:
1781:
1695:
1643:
1557:
1439:
1397:
1353:
1286:
1226:
1176:
1117:
1060:
1009:
935:
870:
834:
760:
713:
669:
620:
565:
538:
511:
478:
243:Signs of trigonometric functions in each quadrant.
3420:Heng, H. H.; Cheng, Khoo; Talbert, J. F. (2005).
1514:Starting at any vertex of the resulting hexagon:
1983:... equals the product of two nearest neighbors
761:{\displaystyle \;\;0\;\;{\Big /}\;\;1\;\;=\;\;0}
1782:{\displaystyle \sin ^{2}A+\cos ^{2}A=1^{2}=1\ }
3538:
3426:. Pearson Education South Asia. p. 228.
8:
3392:"Sine, Cosine and Tangent in Four Quadrants"
3452:"Math Mnemonics and Songs for Trigonometry"
3272:
3270:
1440:{\displaystyle \;\;1\;\;{\Big /}\;\;0\;\;=}
220:
3545:
3531:
3523:
3359:Stueben, Michael; Sandford, Diane (1998).
1558:{\displaystyle \sin A={\frac {1}{\csc A}}}
1433:
1432:
1428:
1427:
1419:
1418:
1414:
1413:
1390:
1346:
1272:
1261:
1213:
1111:
1110:
898:
886:
820:
754:
753:
749:
748:
744:
743:
735:
734:
730:
729:
707:
706:
663:
662:
31:and the relationships between the various
3386:
3384:
3382:
3354:
3352:
3350:
3172:
3159:{\displaystyle ={\frac {\csc A}{\cot A}}}
3130:
3125:
3112:{\displaystyle ={\frac {\tan A}{\sin A}}}
3083:
3078:
3044:
3039:
3012:
2968:
2955:{\displaystyle ={\frac {\cot A}{\cos A}}}
2926:
2921:
2908:{\displaystyle ={\frac {\sec A}{\tan A}}}
2879:
2874:
2840:
2835:
2808:
2764:
2751:{\displaystyle ={\frac {\cos A}{\sin A}}}
2722:
2717:
2704:{\displaystyle ={\frac {\csc A}{\sec A}}}
2675:
2670:
2636:
2631:
2604:
2560:
2547:{\displaystyle ={\frac {\sin A}{\tan A}}}
2518:
2513:
2500:{\displaystyle ={\frac {\cot A}{\csc A}}}
2471:
2466:
2432:
2427:
2400:
2356:
2343:{\displaystyle ={\frac {\tan A}{\sec A}}}
2314:
2309:
2296:{\displaystyle ={\frac {\cos A}{\cot A}}}
2267:
2262:
2228:
2223:
2196:
2152:
2139:{\displaystyle ={\frac {\sec A}{\csc A}}}
2110:
2105:
2092:{\displaystyle ={\frac {\sin A}{\cos A}}}
2063:
2058:
2024:
2019:
1992:
1906:{\displaystyle \tan ^{2}A+1=\sec ^{2}A\ }
1888:
1863:
1857:
1844:{\displaystyle 1+\cot ^{2}A=\csc ^{2}A\ }
1826:
1807:
1795:
1764:
1745:
1726:
1720:
1696:{\displaystyle \sin A=\cos A\cdot \tan A}
1658:
1615:
1586:
1572:
1537:
1523:
1421:
1420:
1411:
1391:
1374:
1372:
1369:
1367:
1347:
1330:
1328:
1325:
1323:
1277:
1273:
1262:
1255:
1254:
1242:
1240:
1214:
1197:
1195:
1192:
1190:
1161:
1159:
1156:
1154:
1095:
1089:
1088:
1076:
1074:
1046:
1030:
1028:
1025:
1023:
995:
979:
977:
974:
972:
921:
906:
900:
899:
888:
887:
884:
855:
853:
850:
848:
822:
821:
804:
802:
799:
797:
737:
736:
727:
690:
688:
685:
683:
646:
644:
641:
639:
603:
602:
579:
552:
525:
504:
464:
462:
427:Other easy-to-remember mnemonics are the
1920:
496:
3231:
479:{\displaystyle {\frac {\sqrt {n}}{2}}}
1029:
978:
645:
7:
3065:{\displaystyle ={\frac {1}{\cos A}}}
2861:{\displaystyle ={\frac {1}{\sin A}}}
2657:{\displaystyle ={\frac {1}{\tan A}}}
2453:{\displaystyle ={\frac {1}{\sec A}}}
2249:{\displaystyle ={\frac {1}{\csc A}}}
2045:{\displaystyle ={\frac {1}{\cot A}}}
1707:trigonometric Pythagorean identities
1160:
854:
3498:"Magic Hexagon for Trig Identities"
3201:{\displaystyle =\csc A\cdot \tan A}
2997:{\displaystyle =\cot A\cdot \sec A}
2793:{\displaystyle =\cos A\cdot \csc A}
2589:{\displaystyle =\sin A\cdot \cot A}
2385:{\displaystyle =\cos A\cdot \tan A}
2181:{\displaystyle =\sin A\cdot \sec A}
453:Sines and cosines of special angles
1373:
1196:
803:
14:
3305:Memory: A Very Short Introduction
1462:Trigonometric identities mnemonic
1329:
689:
3220:List of trigonometric identities
1375:
1331:
1198:
1162:
1031:
980:
856:
805:
691:
647:
200:
112:
3508:from the original on 2018-02-05
3486:(6 ed.). Cengage Learning.
3462:from the original on 2019-10-17
1980:counter-clockwise/anticlockwise
187:
3639:List of biochemistry mnemonics
1:
3242:Key Maths : GCSE, Higher
3302:Foster, Jonathan K. (2008).
566:{\displaystyle \cos \theta }
539:{\displaystyle \sin \theta }
3701:List of chemistry mnemonics
3743:
494:for cosine, respectively:
274:through quadrants 2 to 4.
235:All Students Take Calculus
3660:Mnemonics in trigonometry
307:Other mnemonics include:
178:
29:trigonometric identities
16:Mathematical memory aids
3308:. Oxford. p. 128.
512:{\displaystyle \theta }
264:trigonometric functions
33:trigonometric functions
3423:Additional Mathematics
3239:Humble, Chris (2001).
3202:
3160:
3113:
3066:
3027:
3026:{\displaystyle \sec A}
2998:
2956:
2909:
2862:
2823:
2822:{\displaystyle \csc A}
2794:
2752:
2705:
2658:
2619:
2618:{\displaystyle \cot A}
2590:
2548:
2501:
2454:
2415:
2414:{\displaystyle \cos A}
2386:
2344:
2297:
2250:
2211:
2210:{\displaystyle \sin A}
2182:
2140:
2093:
2046:
2007:
2006:{\displaystyle \tan A}
1907:
1845:
1783:
1697:
1645:
1559:
1463:
1441:
1399:
1355:
1288:
1228:
1178:
1119:
1062:
1011:
937:
872:
836:
762:
715:
671:
622:
567:
540:
513:
480:
244:
48:
23:, it is common to use
3203:
3161:
3114:
3067:
3028:
2999:
2957:
2910:
2863:
2824:
2795:
2753:
2706:
2659:
2620:
2591:
2549:
2502:
2455:
2416:
2387:
2345:
2298:
2251:
2212:
2183:
2141:
2094:
2047:
2008:
1908:
1846:
1784:
1698:
1646:
1560:
1461:
1442:
1400:
1356:
1289:
1229:
1179:
1120:
1063:
1012:
938:
873:
837:
763:
716:
672:
623:
568:
541:
514:
481:
262:for the sign of each
242:
46:
3603:Thermodynamic square
3480:Larson, Ron (2014).
3456:Online Math Learning
3171:
3124:
3077:
3038:
3011:
2967:
2920:
2873:
2834:
2807:
2763:
2716:
2669:
2630:
2603:
2559:
2512:
2465:
2426:
2399:
2355:
2308:
2261:
2222:
2195:
2151:
2104:
2057:
2018:
1991:
1856:
1794:
1719:
1657:
1571:
1522:
1410:
1366:
1322:
1239:
1189:
1153:
1073:
1022:
971:
883:
847:
796:
726:
682:
638:
578:
551:
524:
503:
461:
3593:Planetary mnemonic
3331:Weisstein, Eric W.
3279:Weisstein, Eric W.
3198:
3156:
3109:
3062:
3023:
2994:
2952:
2905:
2858:
2819:
2790:
2748:
2701:
2654:
2615:
2586:
2544:
2497:
2450:
2411:
2382:
2340:
2293:
2246:
2207:
2178:
2136:
2089:
2042:
2003:
1903:
1841:
1779:
1693:
1641:
1555:
1464:
1437:
1395:
1379:
1351:
1335:
1284:
1224:
1202:
1174:
1166:
1115:
1058:
1035:
1007:
984:
933:
868:
860:
832:
809:
758:
711:
695:
667:
651:
618:
563:
536:
509:
476:
245:
49:
3727:Science mnemonics
3709:
3708:
3634:Taxonomy mnemonic
3433:978-981-235-211-8
3372:978-0-88385-525-6
3315:978-0-19-280675-8
3211:
3210:
3154:
3107:
3060:
2950:
2903:
2856:
2746:
2699:
2652:
2542:
2495:
2448:
2338:
2291:
2244:
2134:
2087:
2040:
1924:Starting function
1902:
1840:
1778:
1639:
1610:
1553:
1451:
1450:
1385:
1381:
1341:
1337:
1282:
1270:
1252:
1248:
1222:
1208:
1204:
1172:
1168:
1105:
1104:
1086:
1085:
1056:
1055:
1041:
1037:
1005:
1004:
990:
986:
931:
930:
916:
912:
896:
866:
862:
830:
815:
811:
701:
697:
657:
653:
474:
470:
27:to help remember
3734:
3547:
3540:
3533:
3524:
3517:
3516:
3514:
3513:
3494:
3488:
3487:
3477:
3471:
3470:
3468:
3467:
3448:
3442:
3441:
3436:. Archived from
3417:
3411:
3410:
3408:
3407:
3398:. Archived from
3388:
3377:
3376:
3356:
3345:
3344:
3343:
3326:
3320:
3319:
3299:
3293:
3292:
3291:
3274:
3265:
3264:
3236:
3207:
3205:
3204:
3199:
3165:
3163:
3162:
3157:
3155:
3153:
3142:
3131:
3118:
3116:
3115:
3110:
3108:
3106:
3095:
3084:
3071:
3069:
3068:
3063:
3061:
3059:
3045:
3032:
3030:
3029:
3024:
3003:
3001:
3000:
2995:
2961:
2959:
2958:
2953:
2951:
2949:
2938:
2927:
2914:
2912:
2911:
2906:
2904:
2902:
2891:
2880:
2867:
2865:
2864:
2859:
2857:
2855:
2841:
2828:
2826:
2825:
2820:
2799:
2797:
2796:
2791:
2757:
2755:
2754:
2749:
2747:
2745:
2734:
2723:
2710:
2708:
2707:
2702:
2700:
2698:
2687:
2676:
2663:
2661:
2660:
2655:
2653:
2651:
2637:
2624:
2622:
2621:
2616:
2595:
2593:
2592:
2587:
2553:
2551:
2550:
2545:
2543:
2541:
2530:
2519:
2506:
2504:
2503:
2498:
2496:
2494:
2483:
2472:
2459:
2457:
2456:
2451:
2449:
2447:
2433:
2420:
2418:
2417:
2412:
2391:
2389:
2388:
2383:
2349:
2347:
2346:
2341:
2339:
2337:
2326:
2315:
2302:
2300:
2299:
2294:
2292:
2290:
2279:
2268:
2255:
2253:
2252:
2247:
2245:
2243:
2229:
2216:
2214:
2213:
2208:
2187:
2185:
2184:
2179:
2145:
2143:
2142:
2137:
2135:
2133:
2122:
2111:
2098:
2096:
2095:
2090:
2088:
2086:
2075:
2064:
2051:
2049:
2048:
2043:
2041:
2039:
2025:
2012:
2010:
2009:
2004:
1979:
1977:
1976:
1973:
1970:
1960:
1958:
1957:
1954:
1951:
1942:
1940:
1939:
1936:
1933:
1921:
1912:
1910:
1909:
1904:
1900:
1893:
1892:
1868:
1867:
1850:
1848:
1847:
1842:
1838:
1831:
1830:
1812:
1811:
1788:
1786:
1785:
1780:
1776:
1769:
1768:
1750:
1749:
1731:
1730:
1702:
1700:
1699:
1694:
1650:
1648:
1647:
1642:
1640:
1638:
1627:
1616:
1611:
1609:
1598:
1587:
1564:
1562:
1561:
1556:
1554:
1552:
1538:
1446:
1444:
1443:
1438:
1426:
1425:
1404:
1402:
1401:
1396:
1386:
1380:
1378:
1371:
1370:
1360:
1358:
1357:
1352:
1342:
1336:
1334:
1327:
1326:
1315:
1313:
1312:
1309:
1306:
1305:
1293:
1291:
1290:
1285:
1283:
1278:
1271:
1263:
1260:
1259:
1253:
1244:
1243:
1233:
1231:
1230:
1225:
1223:
1215:
1209:
1203:
1201:
1194:
1193:
1183:
1181:
1180:
1175:
1173:
1167:
1165:
1158:
1157:
1146:
1144:
1143:
1140:
1137:
1136:
1124:
1122:
1121:
1116:
1106:
1100:
1096:
1094:
1093:
1087:
1081:
1077:
1067:
1065:
1064:
1059:
1057:
1051:
1047:
1042:
1036:
1034:
1027:
1026:
1016:
1014:
1013:
1008:
1006:
1000:
996:
991:
985:
983:
976:
975:
964:
962:
961:
958:
955:
954:
942:
940:
939:
934:
932:
926:
922:
917:
908:
907:
905:
904:
897:
889:
877:
875:
874:
869:
867:
861:
859:
852:
851:
841:
839:
838:
833:
831:
823:
816:
810:
808:
801:
800:
789:
787:
786:
783:
780:
779:
767:
765:
764:
759:
742:
741:
720:
718:
717:
712:
702:
696:
694:
687:
686:
676:
674:
673:
668:
658:
652:
650:
643:
642:
627:
625:
624:
619:
608:
607:
572:
570:
569:
564:
545:
543:
542:
537:
518:
516:
515:
510:
497:
493:
492:n = 4, 3, ..., 0
489:
488:n = 0, 1, ..., 4
485:
483:
482:
477:
475:
466:
465:
272:counterclockwise
230:
229:
226:
225:
222:
219:
216:
212:
211:
207:
206:
189:
180:
156:
152:
147:
146:
143:
142:
139:
136:
133:
130:
127:
124:
121:
118:
3742:
3741:
3737:
3736:
3735:
3733:
3732:
3731:
3712:
3711:
3710:
3705:
3679:
3643:
3607:
3581:Right-hand rule
3557:
3551:
3521:
3520:
3511:
3509:
3496:
3495:
3491:
3479:
3478:
3474:
3465:
3463:
3450:
3449:
3445:
3434:
3419:
3418:
3414:
3405:
3403:
3390:
3389:
3380:
3373:
3358:
3357:
3348:
3329:
3328:
3327:
3323:
3316:
3301:
3300:
3296:
3277:
3276:
3275:
3268:
3253:
3238:
3237:
3233:
3228:
3216:
3169:
3168:
3143:
3132:
3122:
3121:
3096:
3085:
3075:
3074:
3049:
3036:
3035:
3009:
3008:
2965:
2964:
2939:
2928:
2918:
2917:
2892:
2881:
2871:
2870:
2845:
2832:
2831:
2805:
2804:
2761:
2760:
2735:
2724:
2714:
2713:
2688:
2677:
2667:
2666:
2641:
2628:
2627:
2601:
2600:
2557:
2556:
2531:
2520:
2510:
2509:
2484:
2473:
2463:
2462:
2437:
2424:
2423:
2397:
2396:
2353:
2352:
2327:
2316:
2306:
2305:
2280:
2269:
2259:
2258:
2233:
2220:
2219:
2193:
2192:
2149:
2148:
2123:
2112:
2102:
2101:
2076:
2065:
2055:
2054:
2029:
2016:
2015:
1989:
1988:
1974:
1971:
1968:
1967:
1965:
1955:
1952:
1949:
1948:
1946:
1937:
1934:
1931:
1930:
1928:
1884:
1859:
1854:
1853:
1822:
1803:
1792:
1791:
1760:
1741:
1722:
1717:
1716:
1655:
1654:
1628:
1617:
1599:
1588:
1569:
1568:
1542:
1520:
1519:
1472:fallout shelter
1456:
1408:
1407:
1364:
1363:
1320:
1319:
1310:
1307:
1303:
1302:
1301:
1299:
1237:
1236:
1187:
1186:
1151:
1150:
1141:
1138:
1134:
1133:
1132:
1130:
1071:
1070:
1020:
1019:
969:
968:
959:
956:
952:
951:
950:
948:
881:
880:
845:
844:
794:
793:
784:
781:
777:
776:
775:
773:
724:
723:
680:
679:
636:
635:
632:0° = 0 radians
576:
575:
549:
548:
522:
521:
501:
500:
491:
487:
459:
458:
455:
237:
213:
208:
203:
199:
170:
154:
150:
115:
111:
41:
17:
12:
11:
5:
3740:
3738:
3730:
3729:
3724:
3714:
3713:
3707:
3706:
3704:
3703:
3698:
3693:
3687:
3685:
3681:
3680:
3678:
3677:
3672:
3667:
3662:
3657:
3651:
3649:
3645:
3644:
3642:
3641:
3636:
3631:
3626:
3621:
3615:
3613:
3609:
3608:
3606:
3605:
3600:
3595:
3590:
3585:
3584:
3583:
3578:
3576:Left-hand rule
3567:
3565:
3559:
3558:
3552:
3550:
3549:
3542:
3535:
3527:
3519:
3518:
3489:
3472:
3443:
3440:on 2023-06-10.
3432:
3412:
3378:
3371:
3346:
3334:"Trigonometry"
3321:
3314:
3294:
3266:
3251:
3230:
3229:
3227:
3224:
3223:
3222:
3215:
3212:
3209:
3208:
3197:
3194:
3191:
3188:
3185:
3182:
3179:
3176:
3166:
3152:
3149:
3146:
3141:
3138:
3135:
3129:
3119:
3105:
3102:
3099:
3094:
3091:
3088:
3082:
3072:
3058:
3055:
3052:
3048:
3043:
3033:
3022:
3019:
3016:
3005:
3004:
2993:
2990:
2987:
2984:
2981:
2978:
2975:
2972:
2962:
2948:
2945:
2942:
2937:
2934:
2931:
2925:
2915:
2901:
2898:
2895:
2890:
2887:
2884:
2878:
2868:
2854:
2851:
2848:
2844:
2839:
2829:
2818:
2815:
2812:
2801:
2800:
2789:
2786:
2783:
2780:
2777:
2774:
2771:
2768:
2758:
2744:
2741:
2738:
2733:
2730:
2727:
2721:
2711:
2697:
2694:
2691:
2686:
2683:
2680:
2674:
2664:
2650:
2647:
2644:
2640:
2635:
2625:
2614:
2611:
2608:
2597:
2596:
2585:
2582:
2579:
2576:
2573:
2570:
2567:
2564:
2554:
2540:
2537:
2534:
2529:
2526:
2523:
2517:
2507:
2493:
2490:
2487:
2482:
2479:
2476:
2470:
2460:
2446:
2443:
2440:
2436:
2431:
2421:
2410:
2407:
2404:
2393:
2392:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2360:
2350:
2336:
2333:
2330:
2325:
2322:
2319:
2313:
2303:
2289:
2286:
2283:
2278:
2275:
2272:
2266:
2256:
2242:
2239:
2236:
2232:
2227:
2217:
2206:
2203:
2200:
2189:
2188:
2177:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2146:
2132:
2129:
2126:
2121:
2118:
2115:
2109:
2099:
2085:
2082:
2079:
2074:
2071:
2068:
2062:
2052:
2038:
2035:
2032:
2028:
2023:
2013:
2002:
1999:
1996:
1985:
1984:
1981:
1962:
1943:
1925:
1916:
1915:
1914:
1913:
1899:
1896:
1891:
1887:
1883:
1880:
1877:
1874:
1871:
1866:
1862:
1851:
1837:
1834:
1829:
1825:
1821:
1818:
1815:
1810:
1806:
1802:
1799:
1789:
1775:
1772:
1767:
1763:
1759:
1756:
1753:
1748:
1744:
1740:
1737:
1734:
1729:
1725:
1711:
1710:
1703:
1692:
1689:
1686:
1683:
1680:
1677:
1674:
1671:
1668:
1665:
1662:
1651:
1637:
1634:
1631:
1626:
1623:
1620:
1614:
1608:
1605:
1602:
1597:
1594:
1591:
1585:
1582:
1579:
1576:
1565:
1551:
1548:
1545:
1541:
1536:
1533:
1530:
1527:
1512:
1511:
1496:
1481:
1478:
1455:
1452:
1449:
1448:
1436:
1431:
1424:
1417:
1405:
1394:
1389:
1384:
1377:
1361:
1350:
1345:
1340:
1333:
1317:
1295:
1294:
1281:
1276:
1269:
1266:
1258:
1251:
1247:
1234:
1221:
1218:
1212:
1207:
1200:
1184:
1171:
1164:
1148:
1126:
1125:
1114:
1109:
1103:
1099:
1092:
1084:
1080:
1068:
1054:
1050:
1045:
1040:
1033:
1017:
1003:
999:
994:
989:
982:
966:
944:
943:
929:
925:
920:
915:
911:
903:
895:
892:
878:
865:
858:
842:
829:
826:
819:
814:
807:
791:
769:
768:
757:
752:
747:
740:
733:
721:
710:
705:
700:
693:
677:
666:
661:
656:
649:
633:
629:
628:
617:
614:
611:
606:
601:
598:
595:
592:
589:
586:
583:
573:
562:
559:
556:
546:
535:
532:
529:
519:
508:
473:
469:
454:
451:
450:
449:
443:
425:
424:
406:
389:
372:
367:eachers (are)
355:
337:
323:
305:
304:
297:
290:
283:
236:
233:
169:
166:
108:
107:
93:
79:
40:
37:
15:
13:
10:
9:
6:
4:
3:
2:
3739:
3728:
3725:
3723:
3720:
3719:
3717:
3702:
3699:
3697:
3694:
3692:
3689:
3688:
3686:
3682:
3676:
3673:
3671:
3668:
3666:
3663:
3661:
3658:
3656:
3653:
3652:
3650:
3646:
3640:
3637:
3635:
3632:
3630:
3627:
3625:
3622:
3620:
3617:
3616:
3614:
3610:
3604:
3601:
3599:
3596:
3594:
3591:
3589:
3586:
3582:
3579:
3577:
3574:
3573:
3572:
3569:
3568:
3566:
3564:
3560:
3556:
3548:
3543:
3541:
3536:
3534:
3529:
3528:
3525:
3507:
3503:
3499:
3493:
3490:
3485:
3484:
3476:
3473:
3461:
3457:
3453:
3447:
3444:
3439:
3435:
3429:
3425:
3424:
3416:
3413:
3402:on 2015-01-18
3401:
3397:
3393:
3387:
3385:
3383:
3379:
3374:
3368:
3364:
3363:
3355:
3353:
3351:
3347:
3341:
3340:
3335:
3332:
3325:
3322:
3317:
3311:
3307:
3306:
3298:
3295:
3289:
3288:
3283:
3280:
3273:
3271:
3267:
3262:
3258:
3254:
3252:0-7487-3396-5
3248:
3244:
3243:
3235:
3232:
3225:
3221:
3218:
3217:
3213:
3195:
3192:
3189:
3186:
3183:
3180:
3177:
3174:
3167:
3150:
3147:
3144:
3139:
3136:
3133:
3127:
3120:
3103:
3100:
3097:
3092:
3089:
3086:
3080:
3073:
3056:
3053:
3050:
3046:
3041:
3034:
3020:
3017:
3014:
3007:
3006:
2991:
2988:
2985:
2982:
2979:
2976:
2973:
2970:
2963:
2946:
2943:
2940:
2935:
2932:
2929:
2923:
2916:
2899:
2896:
2893:
2888:
2885:
2882:
2876:
2869:
2852:
2849:
2846:
2842:
2837:
2830:
2816:
2813:
2810:
2803:
2802:
2787:
2784:
2781:
2778:
2775:
2772:
2769:
2766:
2759:
2742:
2739:
2736:
2731:
2728:
2725:
2719:
2712:
2695:
2692:
2689:
2684:
2681:
2678:
2672:
2665:
2648:
2645:
2642:
2638:
2633:
2626:
2612:
2609:
2606:
2599:
2598:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2555:
2538:
2535:
2532:
2527:
2524:
2521:
2515:
2508:
2491:
2488:
2485:
2480:
2477:
2474:
2468:
2461:
2444:
2441:
2438:
2434:
2429:
2422:
2408:
2405:
2402:
2395:
2394:
2379:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2351:
2334:
2331:
2328:
2323:
2320:
2317:
2311:
2304:
2287:
2284:
2281:
2276:
2273:
2270:
2264:
2257:
2240:
2237:
2234:
2230:
2225:
2218:
2204:
2201:
2198:
2191:
2190:
2175:
2172:
2169:
2166:
2163:
2160:
2157:
2154:
2147:
2130:
2127:
2124:
2119:
2116:
2113:
2107:
2100:
2083:
2080:
2077:
2072:
2069:
2066:
2060:
2053:
2036:
2033:
2030:
2026:
2021:
2014:
2000:
1997:
1994:
1987:
1986:
1982:
1963:
1944:
1926:
1923:
1922:
1919:
1897:
1894:
1889:
1885:
1881:
1878:
1875:
1872:
1869:
1864:
1860:
1852:
1835:
1832:
1827:
1823:
1819:
1816:
1813:
1808:
1804:
1800:
1797:
1790:
1773:
1770:
1765:
1761:
1757:
1754:
1751:
1746:
1742:
1738:
1735:
1732:
1727:
1723:
1715:
1714:
1713:
1712:
1708:
1704:
1690:
1687:
1684:
1681:
1678:
1675:
1672:
1669:
1666:
1663:
1660:
1652:
1635:
1632:
1629:
1624:
1621:
1618:
1612:
1606:
1603:
1600:
1595:
1592:
1589:
1583:
1580:
1577:
1574:
1566:
1549:
1546:
1543:
1539:
1534:
1531:
1528:
1525:
1517:
1516:
1515:
1509:
1505:
1501:
1497:
1494:
1490:
1486:
1482:
1479:
1476:
1473:
1469:
1468:
1467:
1460:
1454:Hexagon chart
1453:
1434:
1429:
1415:
1406:
1392:
1387:
1382:
1362:
1348:
1343:
1338:
1318:
1297:
1296:
1279:
1274:
1267:
1264:
1249:
1245:
1235:
1219:
1216:
1210:
1205:
1185:
1169:
1149:
1128:
1127:
1112:
1107:
1101:
1097:
1082:
1078:
1069:
1052:
1048:
1043:
1038:
1018:
1001:
997:
992:
987:
967:
946:
945:
927:
923:
918:
913:
909:
893:
890:
879:
863:
843:
827:
824:
817:
812:
792:
771:
770:
755:
750:
745:
731:
722:
708:
703:
698:
678:
664:
659:
654:
634:
631:
630:
615:
612:
609:
599:
596:
593:
590:
587:
584:
581:
574:
560:
557:
554:
547:
533:
530:
527:
520:
506:
499:
498:
495:
490:for sine and
471:
467:
452:
447:
444:
441:
438:
437:
436:
434:
430:
422:
418:
414:
410:
407:
404:
400:
396:
393:
390:
387:
383:
379:
376:
373:
370:
366:
362:
359:
356:
353:
349:
345:
341:
338:
335:
331:
327:
324:
321:
317:
313:
310:
309:
308:
302:
298:
295:
291:
288:
284:
281:
277:
276:
275:
273:
269:
265:
261:
258:alculus is a
257:
253:
249:
241:
234:
232:
228:
196:
194:
190:
184:
176:
167:
165:
163:
160:, similar to
159:
158:
145:
105:
101:
97:
94:
91:
87:
83:
80:
77:
73:
69:
66:
65:
64:
62:
58:
54:
45:
38:
36:
34:
30:
26:
22:
3722:Trigonometry
3659:
3510:. Retrieved
3501:
3492:
3482:
3475:
3464:. Retrieved
3455:
3446:
3438:the original
3422:
3415:
3404:. Retrieved
3400:the original
3395:
3361:
3337:
3324:
3304:
3297:
3285:
3241:
3234:
1917:
1513:
1507:
1503:
1499:
1492:
1488:
1484:
1465:
456:
445:
439:
432:
428:
426:
420:
416:
412:
408:
402:
398:
394:
391:
385:
381:
377:
374:
368:
364:
360:
357:
351:
347:
343:
339:
333:
329:
325:
319:
315:
311:
306:
300:
293:
286:
279:
255:
251:
247:
246:
197:
186:
171:
109:
103:
99:
95:
89:
85:
81:
75:
71:
67:
60:
56:
52:
50:
21:trigonometry
18:
3670:Piphilology
3648:Mathematics
3619:CHON/CHNOPS
3502:Math is Fun
3396:Math Is Fun
3282:"SOHCAHTOA"
1964:... equals
1945:... equals
1927:... equals
39:SOH-CAH-TOA
3716:Categories
3512:2018-02-04
3466:2019-10-17
3406:2015-01-18
3226:References
1447:undefined
188:tōa-kha-só
183:Pe̍h-ōe-jī
102:pposite ÷
88:djacent ÷
74:pposite ÷
3555:mnemonics
3339:MathWorld
3287:MathWorld
3193:
3187:⋅
3181:
3148:
3137:
3101:
3090:
3054:
3018:
2989:
2983:⋅
2977:
2944:
2933:
2897:
2886:
2850:
2814:
2785:
2779:⋅
2773:
2740:
2729:
2693:
2682:
2646:
2610:
2581:
2575:⋅
2569:
2536:
2525:
2489:
2478:
2442:
2406:
2377:
2371:⋅
2365:
2332:
2321:
2285:
2274:
2238:
2202:
2173:
2167:⋅
2161:
2128:
2117:
2081:
2070:
2034:
1998:
1961:clockwise
1895:
1870:
1833:
1814:
1752:
1733:
1688:
1682:⋅
1676:
1664:
1633:
1622:
1604:
1593:
1578:
1547:
1529:
1506:tangent,
616:θ
613:
600:θ
597:
588:θ
585:
561:θ
558:
534:θ
531:
507:θ
98:angent =
92:ypotenuse
78:ypotenuse
25:mnemonics
3553:Science
3506:Archived
3460:Archived
3261:47985033
3214:See also
1938:opposite
1316:radians
1147:radians
965:radians
790:radians
314:tations
268:quadrant
266:in each
260:mnemonic
250:tudents
162:Krakatoa
84:osine =
3696:SCORPAN
3629:Four Fs
3624:ESCAPPM
3612:Biology
3588:OBAFGKM
3563:Physics
1978:
1966:
1959:
1947:
1941:
1929:
1510:secant)
1489:tangent
1475:trefoil
1314:
1300:
1145:
1131:
963:
949:
788:
774:
405:hildren
401:orture
397:chools
363:cience
193:Hokkien
175:Chinese
168:Phrases
106:djacent
61:tangent
3691:Clorpt
3665:PEMDAS
3430:
3369:
3312:
3259:
3249:
1975:second
1956:second
1901:
1839:
1777:
1502:sine,
1493:secant
1298:90° =
1129:60° =
947:45° =
772:30° =
415:tinky
322:entral
185::
177::
70:ine =
59:, and
57:cosine
3684:Other
3675:Xyzzy
3598:SUVAT
1969:first
1950:first
486:with
423:ourse
411:wful
380:mart
354:offee
346:ugar
328:illy
326:All S
312:All S
248:All S
191:) in
3655:FOIL
3428:ISBN
3367:ISBN
3310:ISBN
3257:OCLC
3247:ISBN
1485:sine
446:ACTS
440:CAST
433:CAST
431:and
429:ACTS
419:rig
388:lass
384:rig
371:razy
254:ake
153:-kə-
53:sine
51:The
3571:FBI
3190:tan
3178:csc
3145:cot
3134:csc
3098:sin
3087:tan
3051:cos
3015:sec
2986:sec
2974:cot
2941:cos
2930:cot
2894:tan
2883:sec
2847:sin
2811:csc
2782:csc
2770:cos
2737:sin
2726:cos
2690:sec
2679:csc
2643:tan
2607:cot
2578:cot
2566:sin
2533:tan
2522:sin
2486:csc
2475:cot
2439:sec
2403:cos
2374:tan
2362:cos
2329:sec
2318:tan
2282:cot
2271:cos
2235:csc
2199:sin
2170:sec
2158:sin
2125:csc
2114:sec
2078:cos
2067:sin
2031:cot
1995:tan
1886:sec
1861:tan
1824:csc
1805:cot
1743:cos
1724:sin
1685:tan
1673:cos
1661:sin
1630:sec
1619:tan
1601:cot
1590:cos
1575:sin
1544:csc
1526:sin
610:cos
594:sin
582:tan
555:cos
528:sin
392:All
358:All
342:dd
336:ats
332:om
280:All
179:大腳嫂
164:).
155:TOH
151:SOH
19:In
3718::
3504:.
3500:.
3458:.
3454:.
3394:.
3381:^
3349:^
3336:.
3284:.
3269:^
3255:.
1508:co
1504:co
1500:co
1491:,
1487:,
350:o
318:o
218:oʊ
205:oʊ
195:.
181:;
157:-ə
138:oʊ
123:oʊ
55:,
35:.
3546:e
3539:t
3532:v
3515:.
3469:.
3409:.
3375:.
3342:.
3318:.
3290:.
3263:.
3196:A
3184:A
3175:=
3151:A
3140:A
3128:=
3104:A
3093:A
3081:=
3057:A
3047:1
3042:=
3021:A
2992:A
2980:A
2971:=
2947:A
2936:A
2924:=
2900:A
2889:A
2877:=
2853:A
2843:1
2838:=
2817:A
2788:A
2776:A
2767:=
2743:A
2732:A
2720:=
2696:A
2685:A
2673:=
2649:A
2639:1
2634:=
2613:A
2584:A
2572:A
2563:=
2539:A
2528:A
2516:=
2492:A
2481:A
2469:=
2445:A
2435:1
2430:=
2409:A
2380:A
2368:A
2359:=
2335:A
2324:A
2312:=
2288:A
2277:A
2265:=
2241:A
2231:1
2226:=
2205:A
2176:A
2164:A
2155:=
2131:A
2120:A
2108:=
2084:A
2073:A
2061:=
2037:A
2027:1
2022:=
2001:A
1972:/
1953:/
1935:/
1932:1
1898:A
1890:2
1882:=
1879:1
1876:+
1873:A
1865:2
1836:A
1828:2
1820:=
1817:A
1809:2
1801:+
1798:1
1774:1
1771:=
1766:2
1762:1
1758:=
1755:A
1747:2
1739:+
1736:A
1728:2
1709::
1691:A
1679:A
1670:=
1667:A
1636:A
1625:A
1613:=
1607:A
1596:A
1584:=
1581:A
1550:A
1540:1
1535:=
1532:A
1495:)
1477:.
1435:=
1430:0
1423:/
1416:1
1393:0
1388:=
1383:2
1376:0
1349:1
1344:=
1339:2
1332:4
1311:2
1308:/
1304:π
1280:3
1275:=
1268:2
1265:1
1257:/
1250:2
1246:3
1220:2
1217:1
1211:=
1206:2
1199:1
1170:2
1163:3
1142:3
1139:/
1135:π
1113:1
1108:=
1102:2
1098:1
1091:/
1083:2
1079:1
1053:2
1049:1
1044:=
1039:2
1032:2
1002:2
998:1
993:=
988:2
981:2
960:4
957:/
953:π
928:3
924:1
919:=
914:2
910:3
902:/
894:2
891:1
864:2
857:3
828:2
825:1
818:=
813:2
806:1
785:6
782:/
778:π
756:0
751:=
746:1
739:/
732:0
709:1
704:=
699:2
692:4
665:0
660:=
655:2
648:0
605:/
591:=
472:2
468:n
421:C
417:T
413:S
409:A
403:C
399:T
395:S
386:C
382:T
378:S
375:A
369:C
365:T
361:S
352:C
348:T
344:S
340:A
334:C
330:T
320:C
316:T
301:C
294:T
287:S
256:C
252:T
227:/
224:ə
221:.
215:ˈ
210:ə
202:/
144:/
141:ə
135:t
132:ˈ
129:ə
126:k
120:s
117:ˌ
114:/
104:A
100:O
96:T
90:H
86:A
82:C
76:H
72:O
68:S
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
