323:
346:
1382:
2241:
394:
20:
1610:
3020:
162:
1392:
2231:
1133:
707:
2852:
1712:
2071:
51:
1371:
998:
595:
1716:
It is readily seen that the second most significant (third-order) term falls off as the cube of the first term; thus, even for a not-so-small argument such as 0.01, the value of the second most significant term is on the order of
322:
1605:{\displaystyle {\begin{aligned}\sin \theta &=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}\theta ^{2n+1}\\&=\theta -{\frac {\theta ^{3}}{3!}}+{\frac {\theta ^{5}}{5!}}-{\frac {\theta ^{7}}{7!}}+\cdots \end{aligned}}}
2667:
944:
833:
1619:
315:
The accuracy of the approximations can be seen below in Figure 1 and Figure 2. As the measure of the angle approaches zero, the difference between the approximation and the original function also approaches 0.
1824:
756:
496:
345:
2062:
1191:
2545:
2857:
1397:
56:
2797:
2710:
of a simple pendulum, the small-angle approximation for sine is used to allow the resulting differential equation to be solved easily by comparison with the differential equation describing
982:
871:
2018:
1775:
1255:
3015:{\displaystyle {\begin{aligned}\sin(0.755)&=\sin(0.75+0.005)\\&\approx \sin(0.75)+(0.005)\cos(0.75)\\&\approx (0.6816)+(0.005)(0.7317)\\&\approx 0.6853.\end{aligned}}}
300:
240:
1898:
1977:
1951:
1864:
157:{\displaystyle {\begin{aligned}\sin \theta &\approx \theta \\\cos \theta &\approx 1-{\frac {\theta ^{2}}{2}}\approx 1\\\tan \theta &\approx \theta \end{aligned}}}
1918:
1284:
2806:
The small-angle approximation also appears in structural mechanics, especially in stability and bifurcation analyses (mainly of axially-loaded columns ready to undergo
2226:{\displaystyle \sin ^{2}(\theta \varepsilon )+\cos ^{2}(\theta \varepsilon )=(\theta \varepsilon )^{2}+1^{2}=\theta ^{2}\varepsilon ^{2}+1=\theta ^{2}\cdot 0+1=1}
1289:
260:
883:
772:
1782:
714:
447:
1128:{\displaystyle \lim _{\theta \to 0}{\frac {\cos(\theta )-1}{\theta ^{2}}}=\lim _{\theta \to 0}{\frac {-\sin(\theta )}{2\theta }}=-{\frac {1}{2}},}
702:{\displaystyle \sin \theta ={\frac {O}{H}}\approx {\frac {O}{A}}=\tan \theta ={\frac {O}{A}}\approx {\frac {s}{A}}={\frac {A\theta }{A}}=\theta .}
3176:
2593:
1198:
2255:
Figure 3 shows the relative errors of the small angle approximations. The angles at which the relative error exceeds 1% are as follows:
3022:
where the values for sin(0.75) and cos(0.75) are obtained from trigonometric table. The result is accurate to the four digits given.
2761:
206:
There are a number of ways to demonstrate the validity of the small-angle approximations. The most direct method is to truncate the
2023:
2329:
1748:
1138:
211:
1779:
By extension, since the cosine of a small angle is very nearly 1, and the tangent is given by the sine divided by the cosine,
3185:
1707:{\displaystyle \sin \theta =\theta -{\frac {\theta ^{3}}{6}}+{\frac {\theta ^{5}}{120}}-{\frac {\theta ^{7}}{5040}}+\cdots }
3104:
3077:
3036:
2499:
949:
838:
3253:
1982:
1203:
3212:
3155:
2810:). This leads to significant simplifications, though at a cost in accuracy and insight into the true behavior.
217:
265:
2826:
has its basis in the small-angle approximation, plus the fact that one radian is approximately 60 degrees.
1869:
1956:
1923:
1840:
992:
2481:(denoted by the symbol ″), so it is well suited to the small angle approximation. The linear size (
1903:
1389:
The
Maclaurin expansion (the Taylor expansion about 0) of the relevant trigonometric function is
3046:
38:
1260:
3228:
2735:
2723:
2711:
1381:
3248:
3202:
3145:
3094:
3067:
3041:
200:
1366:{\textstyle \cos \theta =1-2\sin ^{2}{\frac {\theta }{2}}\approx 1-{\frac {\theta ^{2}}{2}}}
2700:
2065:
8:
2843:
2739:
245:
3208:
3181:
3151:
3100:
3073:
3120:
2692:
207:
196:
180:
3031:
766:
2823:
2248:
2691:
The second-order cosine approximation is especially useful in calculating the
3242:
2839:
382:. It is seen that as the angle approaches 0 the approximation becomes better.
339:. It is seen that as the angle approaches 0 the approximations become better.
2734:
The sine and tangent small-angle approximations are used in relation to the
2240:
3171:
2819:
2474:
1834:
332:
188:
172:
19:
2746:
is the distance of a fringe from the center of maximum light intensity,
2477:
or angle subtended by the image of a distant object is often only a few
3072:(2nd ed.), Springer Science & Business Media, pp. 30–32,
939:{\displaystyle \lim _{\theta \to 0}{\frac {\tan(\theta )}{\theta }}=1,}
828:{\displaystyle \lim _{\theta \to 0}{\frac {\sin(\theta )}{\theta }}=1,}
393:
2662:{\displaystyle D=d\tan \left(X{\frac {2\pi }{1\,296\,000{''}}}\right)}
1979:. By using the MacLaurin series of cosine and sine, one can show that
2707:
2478:
2470:
192:
176:
2807:
2696:
23:
Approximately equal behavior of some (trigonometric) functions for
41:, provided that the angle in question is small and is measured in
168:
2722:
In optics, the small-angle approximations form the basis of the
1819:{\displaystyle \tan \theta \approx \sin \theta \approx \theta ,}
751:{\displaystyle \sin \theta \approx \tan \theta \approx \theta .}
491:{\displaystyle \cos {\theta }\approx 1-{\frac {\theta ^{2}}{2}}}
3121:"Small-Angle Approximation | Brilliant Math & Science Wiki"
2566:
184:
42:
880:
A more careful application of the squeeze theorem proves that
167:
These approximations have a wide range of uses in branches of
3147:
Calculus of a Single
Variable: Early Transcendental Functions
2754:
is the distance between the slits and projection screen, and
2057:{\displaystyle \sin(\theta \varepsilon )=\theta \varepsilon }
1186:{\textstyle \cos(\theta )\approx 1-{\frac {\theta ^{2}}{2}}}
401:, is the difference between the lengths of the hypotenuse,
3099:(2nd ed.), McGraw-Hill Higher Education, p. 12,
2742:
to develop simplified equations like the following, where
210:
for each of the trigonometric functions. Depending on the
2332:
reduce to the following when one of the angles is small (
203:
that do not need to be answered with absolute precision.
2565:
is approximately equal to the number of arcseconds in a
504:, is approximately equal to the length of the blue arc,
199:. One reason for this is that they can greatly simplify
3065:
3143:
1292:
1141:
268:
221:
2855:
2764:
2596:
2502:
2074:
2026:
1985:
1959:
1926:
1906:
1872:
1843:
1785:
1751:
1622:
1395:
1263:
1206:
1001:
952:
886:
841:
775:
717:
598:
450:
248:
220:
54:
1385:
The small-angle approximation for the sine function.
835:which is a formal restatement of the approximation
16:
Simplification of the basic trigonometric functions
3092:
3014:
2791:
2703:to find the indirect (energy) equation of motion.
2661:
2539:
2225:
2056:
2012:
1971:
1945:
1912:
1892:
1858:
1818:
1769:
1745:the first term. One can thus safely approximate:
1706:
1604:
1365:
1278:
1249:
1185:
1127:
976:
938:
865:
827:
750:
701:
490:
294:
254:
234:
156:
37:can be used to approximate the values of the main
3240:
2836:addition and subtraction involving a small angle
2792:{\displaystyle y\approx {\frac {m\lambda D}{d}}}
2064:. Furthermore, it is not hard to prove that the
1059:
1003:
888:
777:
3150:(4th ed.), Cengage Learning, p. 85,
2235:
3177:Mathematical Methods in the Physical Sciences
2835:
2584:, or, the number of arcseconds in 1 radian.
2540:{\displaystyle D=X{\frac {d}{206\,265{''}}}}
1616:is the angle in radians. In clearer terms,
977:{\displaystyle \tan(\theta )\approx \theta }
866:{\displaystyle \sin(\theta )\approx \theta }
3096:Engineering Mechanics: Statics and Dynamics
2323:
2013:{\displaystyle \cos(\theta \varepsilon )=1}
1770:{\displaystyle \sin \theta \approx \theta }
1250:{\displaystyle \cos 2A\equiv 1-2\sin ^{2}A}
3207:, Cambridge University Press, p. 19,
3059:
3137:
3086:
3066:Holbrow, Charles H.; et al. (2010),
2672:and the above approximation follows when
2638:
2634:
2521:
1886:
3194:
2239:
1380:
18:
2801:
2686:
2330:angle addition and subtraction theorems
295:{\textstyle 1-{\frac {\theta ^{2}}{2}}}
235:{\displaystyle \textstyle \cos \theta }
3241:
2489:) and the distance from the observer (
3200:
3093:Plesha, Michael; et al. (2012),
3170:
2729:
417:are almost the same length, meaning
2758:is the distance between the slits:
2699:, which can then be applied with a
2251:for the small angle approximations.
1893:{\displaystyle a,b\in \mathbb {R} }
13:
2485:) is related to the angular size (
1972:{\displaystyle \varepsilon \neq 0}
1946:{\displaystyle \varepsilon ^{2}=0}
1432:
14:
3265:
3144:Larson, Ron; et al. (2006),
1837:, defined as numbers in the form
508:. Gathering facts from geometry,
305:
3037:Small oscillations of a pendulum
2829:
2459:
2319:at about 0.6620 radians (37.93°)
2292:at about 0.2441 radians (13.99°)
1197:. Alternatively, we can use the
392:
344:
321:
2279:at about 0.1730 radians (9.91°)
2266:at about 0.1408 radians (8.07°)
1859:{\displaystyle a+b\varepsilon }
1828:
3221:
3164:
3113:
2992:
2986:
2983:
2977:
2971:
2965:
2952:
2946:
2937:
2931:
2925:
2919:
2900:
2888:
2872:
2866:
2141:
2131:
2125:
2116:
2097:
2088:
2042:
2033:
2001:
1992:
1476:
1461:
1450:
1440:
1154:
1148:
1092:
1086:
1066:
1033:
1027:
1010:
965:
959:
918:
912:
895:
854:
848:
807:
801:
784:
397:The red section on the right,
1:
3052:
2750:is the order of the fringe,
2464:
1913:{\displaystyle \varepsilon }
1376:
946:from which we conclude that
387:
7:
3069:Modern Introductory Physics
3025:
2813:
2554:is measured in arcseconds.
2236:Error of the approximations
760:
335:trigonometric functions to
10:
3270:
1279:{\displaystyle \theta =2A}
331:A comparison of the basic
310:
242:is approximated as either
212:order of the approximation
35:small-angle approximations
2717:
2493:) by the simple formula:
1920:satisfying by definition
444:helps trim the red away.
405:, and the adjacent side,
3201:Green, Robin M. (1985),
2324:Angle sum and difference
572:, and from the picture,
3047:Exsecant and excosecant
39:trigonometric functions
3254:Equations of astronomy
3016:
2793:
2736:double-slit experiment
2724:paraxial approximation
2712:simple harmonic motion
2663:
2541:
2252:
2227:
2058:
2014:
1973:
1947:
1914:
1894:
1860:
1820:
1771:
1708:
1606:
1436:
1386:
1367:
1280:
1251:
1187:
1129:
978:
940:
867:
829:
752:
703:
492:
296:
256:
236:
201:differential equations
158:
30:
3180:. Wiley. p. 26.
3042:Versine and haversine
3017:
2794:
2706:When calculating the
2664:
2587:The exact formula is
2542:
2243:
2228:
2059:
2015:
1974:
1948:
1915:
1895:
1861:
1821:
1772:
1709:
1607:
1416:
1384:
1368:
1281:
1252:
1188:
1130:
979:
941:
868:
830:
753:
704:
518:, from trigonometry,
493:
297:
257:
237:
159:
22:
2853:
2849:Example: sin(0.755)
2802:Structural mechanics
2762:
2687:Motion of a pendulum
2594:
2500:
2072:
2066:Pythagorean identity
2024:
1983:
1957:
1924:
1904:
1870:
1841:
1783:
1749:
1620:
1393:
1290:
1261:
1204:
1199:double angle formula
1193:for small values of
1139:
1135:which rearranges to
999:
984:for small values of
950:
884:
873:for small values of
839:
773:
769:, we can prove that
715:
711:Simplifying leaves,
596:
448:
266:
246:
218:
52:
3229:"Slit Interference"
3204:Spherical Astronomy
2844:trigonometric table
2740:diffraction grating
3012:
3010:
2789:
2659:
2537:
2253:
2223:
2054:
2010:
1969:
1943:
1910:
1890:
1856:
1816:
1767:
1704:
1602:
1600:
1387:
1363:
1276:
1247:
1183:
1125:
1073:
1017:
974:
936:
902:
863:
825:
791:
748:
699:
500:The opposite leg,
488:
424:is close to 1 and
292:
252:
232:
231:
154:
152:
31:
3187:978-0-471-19826-0
2834:The formulas for
2787:
2730:Wave Interference
2652:
2535:
2455:
2454:
1833:One may also use
1696:
1676:
1656:
1590:
1565:
1540:
1483:
1361:
1335:
1181:
1120:
1104:
1058:
1053:
1002:
925:
887:
814:
776:
688:
670:
657:
632:
619:
486:
290:
255:{\displaystyle 1}
119:
3261:
3233:
3232:
3225:
3219:
3218:
3198:
3192:
3191:
3168:
3162:
3161:
3141:
3135:
3134:
3132:
3131:
3117:
3111:
3110:
3090:
3084:
3083:
3063:
3021:
3019:
3018:
3013:
3011:
2998:
2958:
2906:
2838:may be used for
2798:
2796:
2795:
2790:
2788:
2783:
2772:
2757:
2753:
2749:
2745:
2693:potential energy
2682:
2678:
2668:
2666:
2665:
2660:
2658:
2654:
2653:
2651:
2650:
2649:
2629:
2621:
2583:
2579:
2577:
2574:
2564:
2562:
2553:
2546:
2544:
2543:
2538:
2536:
2534:
2533:
2532:
2513:
2492:
2488:
2484:
2341:
2340:
2318:
2317:
2315:
2314:
2311:
2308:
2291:
2278:
2265:
2232:
2230:
2229:
2224:
2204:
2203:
2185:
2184:
2175:
2174:
2162:
2161:
2149:
2148:
2112:
2111:
2084:
2083:
2063:
2061:
2060:
2055:
2019:
2017:
2016:
2011:
1978:
1976:
1975:
1970:
1952:
1950:
1949:
1944:
1936:
1935:
1919:
1917:
1916:
1911:
1899:
1897:
1896:
1891:
1889:
1865:
1863:
1862:
1857:
1825:
1823:
1822:
1817:
1776:
1774:
1773:
1768:
1744:
1742:
1741:
1740:
1739:
1733:
1730:
1723:
1722:
1713:
1711:
1710:
1705:
1697:
1692:
1691:
1682:
1677:
1672:
1671:
1662:
1657:
1652:
1651:
1642:
1615:
1611:
1609:
1608:
1603:
1601:
1591:
1589:
1581:
1580:
1571:
1566:
1564:
1556:
1555:
1546:
1541:
1539:
1531:
1530:
1521:
1507:
1503:
1502:
1484:
1482:
1459:
1458:
1457:
1438:
1435:
1430:
1372:
1370:
1369:
1364:
1362:
1357:
1356:
1347:
1336:
1328:
1323:
1322:
1285:
1283:
1282:
1277:
1256:
1254:
1253:
1248:
1240:
1239:
1192:
1190:
1189:
1184:
1182:
1177:
1176:
1167:
1134:
1132:
1131:
1126:
1121:
1113:
1105:
1103:
1095:
1075:
1072:
1054:
1052:
1051:
1042:
1019:
1016:
993:L'Hôpital's rule
983:
981:
980:
975:
945:
943:
942:
937:
926:
921:
904:
901:
872:
870:
869:
864:
834:
832:
831:
826:
815:
810:
793:
790:
757:
755:
754:
749:
708:
706:
705:
700:
689:
684:
676:
671:
663:
658:
650:
633:
625:
620:
612:
591:
581:
571:
570:
568:
567:
562:
559:
544:
543:
541:
540:
535:
532:
517:
507:
503:
497:
495:
494:
489:
487:
482:
481:
472:
461:
443:
442:
440:
439:
436:
433:
423:
416:
412:
408:
404:
400:
396:
381:
380:
378:
377:
374:
371:
360:
354:A comparison of
348:
338:
325:
301:
299:
298:
293:
291:
286:
285:
276:
261:
259:
258:
253:
241:
239:
238:
233:
208:Maclaurin series
197:computer science
181:electromagnetism
163:
161:
160:
155:
153:
120:
115:
114:
105:
29:
3269:
3268:
3264:
3263:
3262:
3260:
3259:
3258:
3239:
3238:
3237:
3236:
3227:
3226:
3222:
3215:
3199:
3195:
3188:
3169:
3165:
3158:
3142:
3138:
3129:
3127:
3119:
3118:
3114:
3107:
3091:
3087:
3080:
3064:
3060:
3055:
3032:Skinny triangle
3028:
3009:
3008:
2996:
2995:
2956:
2955:
2904:
2903:
2875:
2856:
2854:
2851:
2850:
2832:
2816:
2804:
2773:
2771:
2763:
2760:
2759:
2755:
2751:
2747:
2743:
2732:
2720:
2689:
2680:
2679:is replaced by
2673:
2643:
2642:
2630:
2622:
2620:
2616:
2612:
2595:
2592:
2591:
2581:
2575:
2572:
2570:
2560:
2558:
2551:
2526:
2525:
2517:
2512:
2501:
2498:
2497:
2490:
2486:
2482:
2467:
2462:
2326:
2312:
2309:
2304:
2303:
2301:
2295:
2282:
2269:
2259:
2249:relative errors
2247:A graph of the
2238:
2199:
2195:
2180:
2176:
2170:
2166:
2157:
2153:
2144:
2140:
2107:
2103:
2079:
2075:
2073:
2070:
2069:
2025:
2022:
2021:
1984:
1981:
1980:
1958:
1955:
1954:
1931:
1927:
1925:
1922:
1921:
1905:
1902:
1901:
1885:
1871:
1868:
1867:
1842:
1839:
1838:
1831:
1784:
1781:
1780:
1750:
1747:
1746:
1737:
1735:
1734:
1731:
1728:
1727:
1725:
1720:
1718:
1687:
1683:
1681:
1667:
1663:
1661:
1647:
1643:
1641:
1621:
1618:
1617:
1613:
1599:
1598:
1582:
1576:
1572:
1570:
1557:
1551:
1547:
1545:
1532:
1526:
1522:
1520:
1505:
1504:
1489:
1485:
1460:
1453:
1449:
1439:
1437:
1431:
1420:
1409:
1396:
1394:
1391:
1390:
1379:
1352:
1348:
1346:
1327:
1318:
1314:
1291:
1288:
1287:
1262:
1259:
1258:
1235:
1231:
1205:
1202:
1201:
1172:
1168:
1166:
1140:
1137:
1136:
1112:
1096:
1076:
1074:
1062:
1047:
1043:
1020:
1018:
1006:
1000:
997:
996:
951:
948:
947:
905:
903:
891:
885:
882:
881:
840:
837:
836:
794:
792:
780:
774:
771:
770:
767:squeeze theorem
763:
716:
713:
712:
677:
675:
662:
649:
624:
611:
597:
594:
593:
583:
573:
563:
560:
555:
554:
552:
546:
536:
533:
528:
527:
525:
519:
509:
505:
501:
477:
473:
471:
457:
449:
446:
445:
437:
434:
429:
428:
426:
425:
418:
414:
410:
409:. As is shown,
406:
402:
398:
390:
383:
375:
372:
367:
366:
364:
362:
355:
349:
340:
336:
326:
313:
308:
281:
277:
275:
267:
264:
263:
247:
244:
243:
219:
216:
215:
151:
150:
140:
128:
127:
110:
106:
104:
91:
79:
78:
68:
55:
53:
50:
49:
24:
17:
12:
11:
5:
3267:
3257:
3256:
3251:
3235:
3234:
3220:
3213:
3193:
3186:
3163:
3156:
3136:
3112:
3106:978-0077570613
3105:
3085:
3079:978-0387790794
3078:
3057:
3056:
3054:
3051:
3050:
3049:
3044:
3039:
3034:
3027:
3024:
3007:
3004:
3001:
2999:
2997:
2994:
2991:
2988:
2985:
2982:
2979:
2976:
2973:
2970:
2967:
2964:
2961:
2959:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2936:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2907:
2905:
2902:
2899:
2896:
2893:
2890:
2887:
2884:
2881:
2878:
2876:
2874:
2871:
2868:
2865:
2862:
2859:
2858:
2831:
2828:
2824:air navigation
2815:
2812:
2803:
2800:
2786:
2782:
2779:
2776:
2770:
2767:
2731:
2728:
2719:
2716:
2688:
2685:
2670:
2669:
2657:
2648:
2645:
2641:
2637:
2633:
2628:
2625:
2619:
2615:
2611:
2608:
2605:
2602:
2599:
2580:), divided by
2548:
2547:
2531:
2528:
2524:
2520:
2516:
2511:
2508:
2505:
2466:
2463:
2461:
2458:
2457:
2456:
2453:
2452:
2437:
2425:
2424:
2409:
2397:
2396:
2381:
2369:
2368:
2353:
2325:
2322:
2321:
2320:
2293:
2280:
2267:
2237:
2234:
2222:
2219:
2216:
2213:
2210:
2207:
2202:
2198:
2194:
2191:
2188:
2183:
2179:
2173:
2169:
2165:
2160:
2156:
2152:
2147:
2143:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2110:
2106:
2102:
2099:
2096:
2093:
2090:
2087:
2082:
2078:
2053:
2050:
2047:
2044:
2041:
2038:
2035:
2032:
2029:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1968:
1965:
1962:
1942:
1939:
1934:
1930:
1909:
1888:
1884:
1881:
1878:
1875:
1855:
1852:
1849:
1846:
1830:
1827:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1766:
1763:
1760:
1757:
1754:
1703:
1700:
1695:
1690:
1686:
1680:
1675:
1670:
1666:
1660:
1655:
1650:
1646:
1640:
1637:
1634:
1631:
1628:
1625:
1597:
1594:
1588:
1585:
1579:
1575:
1569:
1563:
1560:
1554:
1550:
1544:
1538:
1535:
1529:
1525:
1519:
1516:
1513:
1510:
1508:
1506:
1501:
1498:
1495:
1492:
1488:
1481:
1478:
1475:
1472:
1469:
1466:
1463:
1456:
1452:
1448:
1445:
1442:
1434:
1429:
1426:
1423:
1419:
1415:
1412:
1410:
1408:
1405:
1402:
1399:
1398:
1378:
1375:
1360:
1355:
1351:
1345:
1342:
1339:
1334:
1331:
1326:
1321:
1317:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1286:, we get that
1275:
1272:
1269:
1266:
1246:
1243:
1238:
1234:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1180:
1175:
1171:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1124:
1119:
1116:
1111:
1108:
1102:
1099:
1094:
1091:
1088:
1085:
1082:
1079:
1071:
1068:
1065:
1061:
1057:
1050:
1046:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1015:
1012:
1009:
1005:
995:tells us that
973:
970:
967:
964:
961:
958:
955:
935:
932:
929:
924:
920:
917:
914:
911:
908:
900:
897:
894:
890:
862:
859:
856:
853:
850:
847:
844:
824:
821:
818:
813:
809:
806:
803:
800:
797:
789:
786:
783:
779:
762:
759:
747:
744:
741:
738:
735:
732:
729:
726:
723:
720:
698:
695:
692:
687:
683:
680:
674:
669:
666:
661:
656:
653:
648:
645:
642:
639:
636:
631:
628:
623:
618:
615:
610:
607:
604:
601:
485:
480:
476:
470:
467:
464:
460:
456:
453:
389:
386:
385:
384:
350:
343:
341:
327:
320:
312:
309:
307:
306:Justifications
304:
289:
284:
280:
274:
271:
251:
230:
227:
224:
165:
164:
149:
146:
143:
141:
139:
136:
133:
130:
129:
126:
123:
118:
113:
109:
103:
100:
97:
94:
92:
90:
87:
84:
81:
80:
77:
74:
71:
69:
67:
64:
61:
58:
57:
15:
9:
6:
4:
3:
2:
3266:
3255:
3252:
3250:
3247:
3246:
3244:
3230:
3224:
3216:
3210:
3206:
3205:
3197:
3189:
3183:
3179:
3178:
3173:
3172:Boas, Mary L.
3167:
3159:
3153:
3149:
3148:
3140:
3126:
3125:brilliant.org
3122:
3116:
3108:
3102:
3098:
3097:
3089:
3081:
3075:
3071:
3070:
3062:
3058:
3048:
3045:
3043:
3040:
3038:
3035:
3033:
3030:
3029:
3023:
3005:
3002:
3000:
2989:
2980:
2974:
2968:
2962:
2960:
2949:
2943:
2940:
2934:
2928:
2922:
2916:
2913:
2910:
2908:
2897:
2894:
2891:
2885:
2882:
2879:
2877:
2869:
2863:
2860:
2847:
2845:
2841:
2840:interpolating
2837:
2830:Interpolation
2827:
2825:
2821:
2811:
2809:
2799:
2784:
2780:
2777:
2774:
2768:
2765:
2741:
2737:
2727:
2725:
2715:
2713:
2709:
2704:
2702:
2698:
2694:
2684:
2677:
2655:
2646:
2644:
2639:
2635:
2631:
2626:
2623:
2617:
2613:
2609:
2606:
2603:
2600:
2597:
2590:
2589:
2588:
2585:
2568:
2557:The quantity
2555:
2529:
2527:
2522:
2518:
2514:
2509:
2506:
2503:
2496:
2495:
2494:
2480:
2476:
2472:
2460:Specific uses
2450:
2446:
2442:
2438:
2435:
2431:
2427:
2426:
2422:
2418:
2414:
2410:
2407:
2403:
2399:
2398:
2394:
2390:
2386:
2382:
2379:
2375:
2371:
2370:
2366:
2362:
2358:
2354:
2351:
2347:
2343:
2342:
2339:
2338:
2337:
2335:
2331:
2307:
2299:
2294:
2290:
2286:
2281:
2277:
2273:
2268:
2263:
2258:
2257:
2256:
2250:
2246:
2242:
2233:
2220:
2217:
2214:
2211:
2208:
2205:
2200:
2196:
2192:
2189:
2186:
2181:
2177:
2171:
2167:
2163:
2158:
2154:
2150:
2145:
2137:
2134:
2128:
2122:
2119:
2113:
2108:
2104:
2100:
2094:
2091:
2085:
2080:
2076:
2067:
2051:
2048:
2045:
2039:
2036:
2030:
2027:
2007:
2004:
1998:
1995:
1989:
1986:
1966:
1963:
1960:
1940:
1937:
1932:
1928:
1907:
1882:
1879:
1876:
1873:
1853:
1850:
1847:
1844:
1836:
1826:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1777:
1764:
1761:
1758:
1755:
1752:
1714:
1701:
1698:
1693:
1688:
1684:
1678:
1673:
1668:
1664:
1658:
1653:
1648:
1644:
1638:
1635:
1632:
1629:
1626:
1623:
1595:
1592:
1586:
1583:
1577:
1573:
1567:
1561:
1558:
1552:
1548:
1542:
1536:
1533:
1527:
1523:
1517:
1514:
1511:
1509:
1499:
1496:
1493:
1490:
1486:
1479:
1473:
1470:
1467:
1464:
1454:
1446:
1443:
1427:
1424:
1421:
1417:
1413:
1411:
1406:
1403:
1400:
1383:
1374:
1358:
1353:
1349:
1343:
1340:
1337:
1332:
1329:
1324:
1319:
1315:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1273:
1270:
1267:
1264:
1257:. By letting
1244:
1241:
1236:
1232:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1200:
1196:
1178:
1173:
1169:
1163:
1160:
1157:
1151:
1145:
1142:
1122:
1117:
1114:
1109:
1106:
1100:
1097:
1089:
1083:
1080:
1077:
1069:
1063:
1055:
1048:
1044:
1039:
1036:
1030:
1024:
1021:
1013:
1007:
994:
989:
987:
971:
968:
962:
956:
953:
933:
930:
927:
922:
915:
909:
906:
898:
892:
878:
876:
860:
857:
851:
845:
842:
822:
819:
816:
811:
804:
798:
795:
787:
781:
768:
758:
745:
742:
739:
736:
733:
730:
727:
724:
721:
718:
709:
696:
693:
690:
685:
681:
678:
672:
667:
664:
659:
654:
651:
646:
643:
640:
637:
634:
629:
626:
621:
616:
613:
608:
605:
602:
599:
590:
586:
580:
576:
566:
558:
550:
539:
531:
523:
516:
512:
498:
483:
478:
474:
468:
465:
462:
458:
454:
451:
432:
422:
395:
370:
359:
353:
347:
342:
334:
330:
324:
319:
318:
317:
303:
287:
282:
278:
272:
269:
249:
228:
225:
222:
213:
209:
204:
202:
198:
194:
190:
186:
182:
178:
174:
170:
147:
144:
142:
137:
134:
131:
124:
121:
116:
111:
107:
101:
98:
95:
93:
88:
85:
82:
75:
72:
70:
65:
62:
59:
48:
47:
46:
44:
40:
36:
27:
21:
3249:Trigonometry
3223:
3203:
3196:
3175:
3166:
3146:
3139:
3128:. Retrieved
3124:
3115:
3095:
3088:
3068:
3061:
2848:
2833:
2820:1 in 60 rule
2817:
2805:
2733:
2721:
2705:
2690:
2675:
2671:
2586:
2556:
2549:
2475:angular size
2468:
2448:
2444:
2440:
2433:
2429:
2420:
2416:
2412:
2405:
2401:
2392:
2388:
2384:
2377:
2373:
2364:
2360:
2356:
2349:
2345:
2333:
2327:
2305:
2297:
2288:
2284:
2275:
2271:
2261:
2254:
2244:
1835:dual numbers
1832:
1829:Dual numbers
1778:
1715:
1388:
1194:
990:
985:
879:
874:
764:
710:
588:
584:
578:
574:
564:
556:
548:
537:
529:
521:
514:
510:
499:
430:
420:
391:
368:
357:
351:
328:
314:
205:
175:, including
166:
34:
32:
25:
189:cartography
173:engineering
3243:Categories
3214:0521317797
3157:0618606254
3130:2020-07-22
3053:References
2701:Lagrangian
2479:arcseconds
765:Using the
592:leads to:
3003:≈
2963:≈
2944:
2917:
2911:≈
2886:
2864:
2778:λ
2769:≈
2627:π
2610:
2471:astronomy
2465:Astronomy
2245:Figure 3.
2206:⋅
2197:θ
2178:ε
2168:θ
2138:ε
2135:θ
2123:ε
2120:θ
2114:
2095:ε
2092:θ
2086:
2052:ε
2049:θ
2040:ε
2037:θ
2031:
1999:ε
1996:θ
1990:
1964:≠
1961:ε
1929:ε
1908:ε
1883:∈
1854:ε
1811:θ
1808:≈
1805:θ
1802:
1796:≈
1793:θ
1790:
1765:θ
1762:≈
1759:θ
1756:
1702:⋯
1685:θ
1679:−
1665:θ
1645:θ
1639:−
1636:θ
1630:θ
1627:
1596:⋯
1574:θ
1568:−
1549:θ
1524:θ
1518:−
1515:θ
1487:θ
1444:−
1433:∞
1418:∑
1407:θ
1404:
1377:Algebraic
1350:θ
1344:−
1338:≈
1330:θ
1325:
1309:−
1300:θ
1297:
1265:θ
1242:
1226:−
1220:≡
1211:
1170:θ
1164:−
1158:≈
1152:θ
1146:
1110:−
1101:θ
1090:θ
1084:
1078:−
1067:→
1064:θ
1045:θ
1037:−
1031:θ
1025:
1011:→
1008:θ
991:Finally,
972:θ
969:≈
963:θ
957:
923:θ
916:θ
910:
896:→
893:θ
861:θ
858:≈
852:θ
846:
812:θ
805:θ
799:
785:→
782:θ
743:θ
740:≈
737:θ
734:
728:≈
725:θ
722:
694:θ
682:θ
660:≈
644:θ
641:
622:≈
606:θ
603:
475:θ
469:−
463:≈
459:θ
455:
388:Geometric
352:Figure 2.
329:Figure 1.
279:θ
273:−
229:θ
226:
193:astronomy
177:mechanics
148:θ
145:≈
138:θ
135:
122:≈
108:θ
102:−
96:≈
89:θ
86:
76:θ
73:≈
66:θ
63:
3174:(2006).
3026:See also
2846:values:
2842:between
2822:used in
2814:Piloting
2808:buckling
2697:pendulum
2647:″
2530:″
761:Calculus
3006:0.6853.
2316:
2302:
1866:, with
1743:
1726:
569:
553:
542:
526:
441:
427:
379:
365:
311:Graphic
169:physics
43:radians
3211:
3184:
3154:
3103:
3076:
2990:0.7317
2969:0.6816
2718:Optics
2708:period
2567:circle
2550:where
2473:, the
2439:≈ sin(
2411:≈ sin(
2383:≈ cos(
2355:≈ cos(
2336:≈ 0):
2300:≈ 1 −
2068:holds:
1612:where
262:or as
195:, and
185:optics
2981:0.005
2935:0.005
2898:0.005
2870:0.755
2738:or a
2695:of a
1724:, or
1719:0.000
3209:ISBN
3182:ISBN
3152:ISBN
3101:ISBN
3074:ISBN
2950:0.75
2923:0.75
2892:0.75
2818:The
2674:tan
2447:cos(
2443:) −
2428:sin(
2419:cos(
2415:) +
2400:sin(
2391:sin(
2387:) +
2372:cos(
2363:sin(
2359:) −
2344:cos(
2328:The
2296:cos
2283:sin
2270:tan
2260:cos
2020:and
1953:and
1900:and
1694:5040
582:and
547:tan
545:and
520:sin
419:cos
413:and
363:1 −
356:cos
171:and
33:The
2941:cos
2914:sin
2883:sin
2861:sin
2640:000
2636:296
2607:tan
2576:000
2573:296
2561:265
2559:206
2523:265
2519:206
2469:In
2451:).
2423:),
2395:),
2367:),
2264:≈ 1
2105:cos
2077:sin
2028:sin
1987:cos
1799:sin
1787:tan
1753:sin
1738:000
1721:001
1674:120
1624:sin
1401:sin
1316:sin
1294:cos
1233:sin
1208:cos
1143:cos
1081:sin
1060:lim
1022:cos
1004:lim
954:tan
907:tan
889:lim
843:sin
796:sin
778:lim
731:tan
719:sin
638:tan
600:sin
452:cos
361:to
333:odd
223:cos
132:tan
83:cos
60:sin
28:→ 0
3245::
3123:.
2726:.
2714:.
2683:.
2582:2π
2432:−
2404:+
2376:−
2348:+
2287:≈
2274:≈
1736:10
1373:.
988:.
877:.
587:≈
577:≈
551:=
524:=
515:Aθ
513:=
302:.
214:,
191:,
187:,
183:,
179:,
45::
3231:.
3217:.
3190:.
3160:.
3133:.
3109:.
3082:.
2993:)
2987:(
2984:)
2978:(
2975:+
2972:)
2966:(
2953:)
2947:(
2938:)
2932:(
2929:+
2926:)
2920:(
2901:)
2895:+
2889:(
2880:=
2873:)
2867:(
2785:d
2781:D
2775:m
2766:y
2756:d
2752:D
2748:m
2744:y
2681:X
2676:X
2656:)
2632:1
2624:2
2618:X
2614:(
2604:d
2601:=
2598:D
2578:″
2571:1
2569:(
2563:″
2552:X
2515:d
2510:X
2507:=
2504:D
2491:d
2487:X
2483:D
2449:α
2445:β
2441:α
2436:)
2434:β
2430:α
2421:α
2417:β
2413:α
2408:)
2406:β
2402:α
2393:α
2389:β
2385:α
2380:)
2378:β
2374:α
2365:α
2361:β
2357:α
2352:)
2350:β
2346:α
2334:β
2313:2
2310:/
2306:θ
2298:θ
2289:θ
2285:θ
2276:θ
2272:θ
2262:θ
2221:1
2218:=
2215:1
2212:+
2209:0
2201:2
2193:=
2190:1
2187:+
2182:2
2172:2
2164:=
2159:2
2155:1
2151:+
2146:2
2142:)
2132:(
2129:=
2126:)
2117:(
2109:2
2101:+
2098:)
2089:(
2081:2
2046:=
2043:)
2034:(
2008:1
2005:=
2002:)
1993:(
1967:0
1941:0
1938:=
1933:2
1887:R
1880:b
1877:,
1874:a
1851:b
1848:+
1845:a
1814:,
1732:/
1729:1
1699:+
1689:7
1669:5
1659:+
1654:6
1649:3
1633:=
1614:θ
1593:+
1587:!
1584:7
1578:7
1562:!
1559:5
1553:5
1543:+
1537:!
1534:3
1528:3
1512:=
1500:1
1497:+
1494:n
1491:2
1480:!
1477:)
1474:1
1471:+
1468:n
1465:2
1462:(
1455:n
1451:)
1447:1
1441:(
1428:0
1425:=
1422:n
1414:=
1359:2
1354:2
1341:1
1333:2
1320:2
1312:2
1306:1
1303:=
1274:A
1271:2
1268:=
1245:A
1237:2
1229:2
1223:1
1217:A
1214:2
1195:θ
1179:2
1174:2
1161:1
1155:)
1149:(
1123:,
1118:2
1115:1
1107:=
1098:2
1093:)
1087:(
1070:0
1056:=
1049:2
1040:1
1034:)
1028:(
1014:0
986:θ
966:)
960:(
934:,
931:1
928:=
919:)
913:(
899:0
875:θ
855:)
849:(
823:,
820:1
817:=
808:)
802:(
788:0
746:.
697:.
691:=
686:A
679:A
673:=
668:A
665:s
655:A
652:O
647:=
635:=
630:A
627:O
617:H
614:O
609:=
589:A
585:H
579:s
575:O
565:A
561:/
557:O
549:θ
538:H
534:/
530:O
522:θ
511:s
506:s
502:O
484:2
479:2
466:1
438:2
435:/
431:θ
421:θ
415:A
411:H
407:A
403:H
399:d
376:2
373:/
369:θ
358:θ
337:θ
288:2
283:2
270:1
250:1
125:1
117:2
112:2
99:1
26:x
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
