[{"data":1,"prerenderedAt":5583},["ShallowReactive",2],{"subject-categories-math":3,"formula-de\u002Fformulas\u002Fmath\u002Fcalculus\u002Fgrundidentitaet-hyperbelfunktionen":546,"formula-related-de\u002Fformulas\u002Fmath\u002Fcalculus\u002Fgrundidentitaet-hyperbelfunktionen":5573},[4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,173,174,175,176,177,178,179,180,181,182,183,184,185,186,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545],{"category":5},"algebra",{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":65},"arithmetic",{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":101},"calculus",{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":172},"differential-equations",{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":187},"discrete-math",{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":208},"functions",{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":247},"geometry",{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":312},"linear-algebra",{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":368},"multivariable-calculus",{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":391},"numerics",{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":412},"sequences-series",{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":441},"stochastics",{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":518},"trigonometry",{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"id":547,"title":548,"aliases":549,"body":555,"category":101,"checks":5449,"conditions":5474,"conditionsHtml":5474,"deprecated":5474,"description":5442,"extension":5475,"figure":5476,"figureCaption":5477,"figureHtml":5539,"formulaId":5540,"latex":5541,"latexHtml":5542,"level":5543,"meta":5544,"navigation":5545,"path":5546,"redirectFrom":5547,"related":5548,"seo":5552,"stem":5553,"subject":577,"summary":5554,"tags":5555,"variables":5560,"__hash__":5572},"formulas_de\u002Fde\u002Fformulas\u002Fmath\u002Fcalculus\u002Fgrundidentitaet-hyperbelfunktionen.md","Hyperbolischer Pythagoras",[550,551,552,553,554],"cosh^2 - sinh^2 = 1","cosh² − sinh² = 1","Zusammenhang cosh sinh","Grundidentität Hyperbelfunktionen","hyperbolischer Satz des Pythagoras",{"type":556,"value":557,"toc":5441},"minimark",[558,563,1032,1035,1957,2326,2330,2333,2997,3001,3139,3804,4525,4529],[559,560,562],"h2",{"id":561},"erklärung","Erklärung",[564,565,566,567,774,775,779,780,862,863,892,893,1031],"p",{},"Die Identität ist das Gegenstück zum trigonometrischen Pythagoras ",[568,569,572,633],"span",{"className":570},[571],"katex",[568,573,576],{"className":574},[575],"katex-mathml",[577,578,580],"math",{"xmlns":579},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[581,582,583,628],"semantics",{},[584,585,586,603,606,609,620,622,625],"mrow",{},[587,588,589,599],"msup",{},[584,590,591,595],{},[592,593,594],"mi",{},"sin",[596,597,598],"mo",{},"⁡",[600,601,602],"mn",{},"2",[592,604,605],{},"x",[596,607,608],{},"+",[587,610,611,618],{},[584,612,613,616],{},[592,614,615],{},"cos",[596,617,598],{},[600,619,602],{},[592,621,605],{},[596,623,624],{},"=",[600,626,627],{},"1",[629,630,632],"annotation",{"encoding":631},"application\u002Fx-tex","\\sin^2 x + \\cos^2 x = 1",[568,634,638,710,764],{"className":635,"ariaHidden":637},[636],"katex-html","true",[568,639,642,647,690,695,699,703,707],{"className":640},[641],"base",[568,643],{"className":644,"style":646},[645],"strut","height:0.9552em;vertical-align:-0.0833em;",[568,648,651,654],{"className":649},[650],"mop",[568,652,594],{"className":653},[650],[568,655,658],{"className":656},[657],"msupsub",[568,659,662],{"className":660},[661],"vlist-t",[568,663,666],{"className":664},[665],"vlist-r",[568,667,671],{"className":668,"style":670},[669],"vlist","height:0.8719em;",[568,672,674,679],{"style":673},"top:-3.1208em;margin-right:0.05em;",[568,675],{"className":676,"style":678},[677],"pstrut","height:2.7em;",[568,680,686],{"className":681},[682,683,684,685],"sizing","reset-size6","size3","mtight",[568,687,602],{"className":688},[689,685],"mord",[568,691],{"className":692,"style":694},[693],"mspace","margin-right:0.1667em;",[568,696,605],{"className":697},[689,698],"mathnormal",[568,700],{"className":701,"style":702},[693],"margin-right:0.2222em;",[568,704,608],{"className":705},[706],"mbin",[568,708],{"className":709,"style":702},[693],[568,711,713,717,747,750,753,757,761],{"className":712},[641],[568,714],{"className":715,"style":716},[645],"height:0.8141em;",[568,718,720,723],{"className":719},[650],[568,721,615],{"className":722},[650],[568,724,726],{"className":725},[657],[568,727,729],{"className":728},[661],[568,730,732],{"className":731},[665],[568,733,735],{"className":734,"style":716},[669],[568,736,738,741],{"style":737},"top:-3.063em;margin-right:0.05em;",[568,739],{"className":740,"style":678},[677],[568,742,744],{"className":743},[682,683,684,685],[568,745,602],{"className":746},[689,685],[568,748],{"className":749,"style":694},[693],[568,751,605],{"className":752},[689,698],[568,754],{"className":755,"style":756},[693],"margin-right:0.2778em;",[568,758,624],{"className":759},[760],"mrel",[568,762],{"className":763,"style":756},[693],[568,765,767,771],{"className":766},[641],[568,768],{"className":769,"style":770},[645],"height:0.6444em;",[568,772,627],{"className":773},[689],", aber mit ",[776,777,778],"strong",{},"Minuszeichen",". Sie besagt, dass der Punkt ",[568,781,783,819],{"className":782},[571],[568,784,786],{"className":785},[575],[577,787,788],{"xmlns":579},[581,789,790,816],{},[584,791,792,796,799,801,803,806,809,811,813],{},[596,793,795],{"stretchy":794},"false","(",[592,797,798],{},"cosh",[596,800,598],{},[592,802,605],{},[596,804,805],{"separator":637},",",[592,807,808],{},"sinh",[596,810,598],{},[592,812,605],{},[596,814,815],{"stretchy":794},")",[629,817,818],{"encoding":631},"(\\cosh x, \\sinh x)",[568,820,822],{"className":821,"ariaHidden":637},[636],[568,823,825,829,833,836,839,842,846,849,852,855,858],{"className":824},[641],[568,826],{"className":827,"style":828},[645],"height:1em;vertical-align:-0.25em;",[568,830,795],{"className":831},[832],"mopen",[568,834,798],{"className":835},[650],[568,837],{"className":838,"style":694},[693],[568,840,605],{"className":841},[689,698],[568,843,805],{"className":844},[845],"mpunct",[568,847],{"className":848,"style":694},[693],[568,850,808],{"className":851},[650],[568,853],{"className":854,"style":694},[693],[568,856,605],{"className":857},[689,698],[568,859,815],{"className":860},[861],"mclose"," für jedes ",[568,864,866,879],{"className":865},[571],[568,867,869],{"className":868},[575],[577,870,871],{"xmlns":579},[581,872,873,877],{},[584,874,875],{},[592,876,605],{},[629,878,605],{"encoding":631},[568,880,882],{"className":881,"ariaHidden":637},[636],[568,883,885,889],{"className":884},[641],[568,886],{"className":887,"style":888},[645],"height:0.4306em;",[568,890,605],{"className":891},[689,698]," auf der Einheitshyperbel ",[568,894,896,929],{"className":895},[571],[568,897,899],{"className":898},[575],[577,900,901],{"xmlns":579},[581,902,903,926],{},[584,904,905,912,915,922,924],{},[587,906,907,910],{},[592,908,909],{},"X",[600,911,602],{},[596,913,914],{},"−",[587,916,917,920],{},[592,918,919],{},"Y",[600,921,602],{},[596,923,624],{},[600,925,627],{},[629,927,928],{"encoding":631},"X^2 - Y^2 = 1",[568,930,932,978,1022],{"className":931,"ariaHidden":637},[636],[568,933,935,939,969,972,975],{"className":934},[641],[568,936],{"className":937,"style":938},[645],"height:0.8974em;vertical-align:-0.0833em;",[568,940,942,946],{"className":941},[689],[568,943,909],{"className":944,"style":945},[689,698],"margin-right:0.0785em;",[568,947,949],{"className":948},[657],[568,950,952],{"className":951},[661],[568,953,955],{"className":954},[665],[568,956,958],{"className":957,"style":716},[669],[568,959,960,963],{"style":737},[568,961],{"className":962,"style":678},[677],[568,964,966],{"className":965},[682,683,684,685],[568,967,602],{"className":968},[689,685],[568,970],{"className":971,"style":702},[693],[568,973,914],{"className":974},[706],[568,976],{"className":977,"style":702},[693],[568,979,981,984,1013,1016,1019],{"className":980},[641],[568,982],{"className":983,"style":716},[645],[568,985,987,990],{"className":986},[689],[568,988,919],{"className":989,"style":702},[689,698],[568,991,993],{"className":992},[657],[568,994,996],{"className":995},[661],[568,997,999],{"className":998},[665],[568,1000,1002],{"className":1001,"style":716},[669],[568,1003,1004,1007],{"style":737},[568,1005],{"className":1006,"style":678},[677],[568,1008,1010],{"className":1009},[682,683,684,685],[568,1011,602],{"className":1012},[689,685],[568,1014],{"className":1015,"style":756},[693],[568,1017,624],{"className":1018},[760],[568,1020],{"className":1021,"style":756},[693],[568,1023,1025,1028],{"className":1024},[641],[568,1026],{"className":1027,"style":770},[645],[568,1029,627],{"className":1030},[689]," liegt.",[564,1033,1034],{},"Daraus folgen weitere Formeln:",[1036,1037,1038,1383,1664],"ul",{},[1039,1040,1041,1042,1115,1116,1382],"li",{},"Teilt man durch ",[568,1043,1045,1069],{"className":1044},[571],[568,1046,1048],{"className":1047},[575],[577,1049,1050],{"xmlns":579},[581,1051,1052,1066],{},[584,1053,1054,1064],{},[587,1055,1056,1062],{},[584,1057,1058,1060],{},[592,1059,798],{},[596,1061,598],{},[600,1063,602],{},[592,1065,605],{},[629,1067,1068],{"encoding":631},"\\cosh^2 x",[568,1070,1072],{"className":1071,"ariaHidden":637},[636],[568,1073,1075,1079,1109,1112],{"className":1074},[641],[568,1076],{"className":1077,"style":1078},[645],"height:0.8984em;",[568,1080,1082,1085],{"className":1081},[650],[568,1083,798],{"className":1084},[650],[568,1086,1088],{"className":1087},[657],[568,1089,1091],{"className":1090},[661],[568,1092,1094],{"className":1093},[665],[568,1095,1097],{"className":1096,"style":1078},[669],[568,1098,1100,1103],{"style":1099},"top:-3.1473em;margin-right:0.05em;",[568,1101],{"className":1102,"style":678},[677],[568,1104,1106],{"className":1105},[682,683,684,685],[568,1107,602],{"className":1108},[689,685],[568,1110],{"className":1111,"style":694},[693],[568,1113,605],{"className":1114},[689,698],": ",[568,1117,1119,1169],{"className":1118},[571],[568,1120,1122],{"className":1121},[575],[577,1123,1124],{"xmlns":579},[581,1125,1126,1166],{},[584,1127,1128,1130,1132,1143,1145,1147],{},[600,1129,627],{},[596,1131,914],{},[587,1133,1134,1141],{},[584,1135,1136,1139],{},[592,1137,1138],{},"tanh",[596,1140,598],{},[600,1142,602],{},[592,1144,605],{},[596,1146,624],{},[1148,1149,1150,1152],"mfrac",{},[600,1151,627],{},[584,1153,1154,1164],{},[587,1155,1156,1162],{},[584,1157,1158,1160],{},[592,1159,798],{},[596,1161,598],{},[600,1163,602],{},[592,1165,605],{},[629,1167,1168],{"encoding":631},"1 - \\tanh^2 x = \\frac{1}{\\cosh^2 x}",[568,1170,1172,1191,1241],{"className":1171,"ariaHidden":637},[636],[568,1173,1175,1179,1182,1185,1188],{"className":1174},[641],[568,1176],{"className":1177,"style":1178},[645],"height:0.7278em;vertical-align:-0.0833em;",[568,1180,627],{"className":1181},[689],[568,1183],{"className":1184,"style":702},[693],[568,1186,914],{"className":1187},[706],[568,1189],{"className":1190,"style":702},[693],[568,1192,1194,1197,1226,1229,1232,1235,1238],{"className":1193},[641],[568,1195],{"className":1196,"style":1078},[645],[568,1198,1200,1203],{"className":1199},[650],[568,1201,1138],{"className":1202},[650],[568,1204,1206],{"className":1205},[657],[568,1207,1209],{"className":1208},[661],[568,1210,1212],{"className":1211},[665],[568,1213,1215],{"className":1214,"style":1078},[669],[568,1216,1217,1220],{"style":1099},[568,1218],{"className":1219,"style":678},[677],[568,1221,1223],{"className":1222},[682,683,684,685],[568,1224,602],{"className":1225},[689,685],[568,1227],{"className":1228,"style":694},[693],[568,1230,605],{"className":1231},[689,698],[568,1233],{"className":1234,"style":756},[693],[568,1236,624],{"className":1237},[760],[568,1239],{"className":1240,"style":756},[693],[568,1242,1244,1248],{"className":1243},[641],[568,1245],{"className":1246,"style":1247},[645],"height:1.2164em;vertical-align:-0.3713em;",[568,1249,1251,1255,1379],{"className":1250},[689],[568,1252],{"className":1253},[832,1254],"nulldelimiter",[568,1256,1258],{"className":1257},[1148],[568,1259,1262,1370],{"className":1260},[661,1261],"vlist-t2",[568,1263,1265,1365],{"className":1264},[665],[568,1266,1269,1339,1350],{"className":1267,"style":1268},[669],"height:0.8451em;",[568,1270,1272,1276],{"style":1271},"top:-2.6287em;",[568,1273],{"className":1274,"style":1275},[677],"height:3em;",[568,1277,1279],{"className":1278},[682,683,684,685],[568,1280,1282,1332,1336],{"className":1281},[689,685],[568,1283,1285,1304],{"className":1284},[650,685],[568,1286,1288,1292,1296,1300],{"className":1287},[650,685],[568,1289,1291],{"className":1290},[685],"c",[568,1293,1295],{"className":1294},[685],"o",[568,1297,1299],{"className":1298},[685],"s",[568,1301,1303],{"className":1302},[685],"h",[568,1305,1307],{"className":1306},[657],[568,1308,1310],{"className":1309},[661],[568,1311,1313],{"className":1312},[665],[568,1314,1317],{"className":1315,"style":1316},[669],"height:0.8019em;",[568,1318,1320,1324],{"style":1319},"top:-2.8416em;margin-right:0.0714em;",[568,1321],{"className":1322,"style":1323},[677],"height:2.5em;",[568,1325,1329],{"className":1326},[682,1327,1328,685],"reset-size3","size1",[568,1330,602],{"className":1331},[689,685],[568,1333],{"className":1334,"style":1335},[693,685],"margin-right:0.1952em;",[568,1337,605],{"className":1338},[689,698,685],[568,1340,1342,1345],{"style":1341},"top:-3.23em;",[568,1343],{"className":1344,"style":1275},[677],[568,1346],{"className":1347,"style":1349},[1348],"frac-line","border-bottom-width:0.04em;",[568,1351,1353,1356],{"style":1352},"top:-3.394em;",[568,1354],{"className":1355,"style":1275},[677],[568,1357,1359],{"className":1358},[682,683,684,685],[568,1360,1362],{"className":1361},[689,685],[568,1363,627],{"className":1364},[689,685],[568,1366,1369],{"className":1367},[1368],"vlist-s","​",[568,1371,1373],{"className":1372},[665],[568,1374,1377],{"className":1375,"style":1376},[669],"height:0.3713em;",[568,1378],{},[568,1380],{"className":1381},[861,1254],".",[1039,1384,1385,1386,1474,1475,1663],{},"Weil ",[568,1387,1389,1418],{"className":1388},[571],[568,1390,1392],{"className":1391},[575],[577,1393,1394],{"xmlns":579},[581,1395,1396,1415],{},[584,1397,1398,1400,1402,1404,1407,1409,1412],{},[592,1399,798],{},[596,1401,598],{},[592,1403,605],{},[596,1405,1406],{},"≥",[600,1408,627],{},[596,1410,1411],{},">",[600,1413,1414],{},"0",[629,1416,1417],{"encoding":631},"\\cosh x \\geq 1 > 0",[568,1419,1421,1446,1465],{"className":1420,"ariaHidden":637},[636],[568,1422,1424,1428,1431,1434,1437,1440,1443],{"className":1423},[641],[568,1425],{"className":1426,"style":1427},[645],"height:0.8304em;vertical-align:-0.136em;",[568,1429,798],{"className":1430},[650],[568,1432],{"className":1433,"style":694},[693],[568,1435,605],{"className":1436},[689,698],[568,1438],{"className":1439,"style":756},[693],[568,1441,1406],{"className":1442},[760],[568,1444],{"className":1445,"style":756},[693],[568,1447,1449,1453,1456,1459,1462],{"className":1448},[641],[568,1450],{"className":1451,"style":1452},[645],"height:0.6835em;vertical-align:-0.0391em;",[568,1454,627],{"className":1455},[689],[568,1457],{"className":1458,"style":756},[693],[568,1460,1411],{"className":1461},[760],[568,1463],{"className":1464,"style":756},[693],[568,1466,1468,1471],{"className":1467},[641],[568,1469],{"className":1470,"style":770},[645],[568,1472,1414],{"className":1473},[689]," ist, gilt ",[568,1476,1478,1519],{"className":1477},[571],[568,1479,1481],{"className":1480},[575],[577,1482,1483],{"xmlns":579},[581,1484,1485,1516],{},[584,1486,1487,1489,1491,1493,1495],{},[592,1488,798],{},[596,1490,598],{},[592,1492,605],{},[596,1494,624],{},[1496,1497,1498],"msqrt",{},[584,1499,1500,1502,1504,1514],{},[600,1501,627],{},[596,1503,608],{},[587,1505,1506,1512],{},[584,1507,1508,1510],{},[592,1509,808],{},[596,1511,598],{},[600,1513,602],{},[592,1515,605],{},[629,1517,1518],{"encoding":631},"\\cosh x = \\sqrt{1 + \\sinh^2 x}",[568,1520,1522,1547],{"className":1521,"ariaHidden":637},[636],[568,1523,1525,1529,1532,1535,1538,1541,1544],{"className":1524},[641],[568,1526],{"className":1527,"style":1528},[645],"height:0.6944em;",[568,1530,798],{"className":1531},[650],[568,1533],{"className":1534,"style":694},[693],[568,1536,605],{"className":1537},[689,698],[568,1539],{"className":1540,"style":756},[693],[568,1542,624],{"className":1543},[760],[568,1545],{"className":1546,"style":756},[693],[568,1548,1550,1554],{"className":1549},[641],[568,1551],{"className":1552,"style":1553},[645],"height:1.24em;vertical-align:-0.1474em;",[568,1555,1558],{"className":1556},[689,1557],"sqrt",[568,1559,1561,1654],{"className":1560},[661,1261],[568,1562,1564,1651],{"className":1563},[665],[568,1565,1568,1628],{"className":1566,"style":1567},[669],"height:1.0926em;",[568,1569,1573,1577],{"className":1570,"style":1572},[1571],"svg-align","top:-3.2em;",[568,1574],{"className":1575,"style":1576},[677],"height:3.2em;",[568,1578,1581,1584,1587,1590,1593,1622,1625],{"className":1579,"style":1580},[689],"padding-left:1em;",[568,1582,627],{"className":1583},[689],[568,1585],{"className":1586,"style":702},[693],[568,1588,608],{"className":1589},[706],[568,1591],{"className":1592,"style":702},[693],[568,1594,1596,1599],{"className":1595},[650],[568,1597,808],{"className":1598},[650],[568,1600,1602],{"className":1601},[657],[568,1603,1605],{"className":1604},[661],[568,1606,1608],{"className":1607},[665],[568,1609,1611],{"className":1610,"style":1078},[669],[568,1612,1613,1616],{"style":1099},[568,1614],{"className":1615,"style":678},[677],[568,1617,1619],{"className":1618},[682,683,684,685],[568,1620,602],{"className":1621},[689,685],[568,1623],{"className":1624,"style":694},[693],[568,1626,605],{"className":1627},[689,698],[568,1629,1631,1634],{"style":1630},"top:-3.0526em;",[568,1632],{"className":1633,"style":1576},[677],[568,1635,1639],{"className":1636,"style":1638},[1637],"hide-tail","min-width:1.02em;height:1.28em;",[1640,1641,1647],"svg",{"xmlns":1642,"width":1643,"height":1644,"viewBox":1645,"preserveAspectRatio":1646},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.28em","0 0 400000 1296","xMinYMin slice",[1648,1649],"path",{"d":1650},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z",[568,1652,1369],{"className":1653},[1368],[568,1655,1657],{"className":1656},[665],[568,1658,1661],{"className":1659,"style":1660},[669],"height:0.1474em;",[568,1662],{}," ohne Fallunterscheidung beim Vorzeichen.",[1039,1665,1666,1667,1847,1848,1382],{},"Doppelte Argumente: ",[568,1668,1670,1722],{"className":1669},[571],[568,1671,1673],{"className":1672},[575],[577,1674,1675],{"xmlns":579},[581,1676,1677,1719],{},[584,1678,1679,1681,1683,1685,1687,1689,1691,1693,1703,1705,1707,1717],{},[592,1680,798],{},[596,1682,598],{},[596,1684,795],{"stretchy":794},[600,1686,602],{},[592,1688,605],{},[596,1690,815],{"stretchy":794},[596,1692,624],{},[587,1694,1695,1701],{},[584,1696,1697,1699],{},[592,1698,798],{},[596,1700,598],{},[600,1702,602],{},[592,1704,605],{},[596,1706,608],{},[587,1708,1709,1715],{},[584,1710,1711,1713],{},[592,1712,808],{},[596,1714,598],{},[600,1716,602],{},[592,1718,605],{},[629,1720,1721],{"encoding":631},"\\cosh(2x) = \\cosh^2 x + \\sinh^2 x",[568,1723,1725,1755,1806],{"className":1724,"ariaHidden":637},[636],[568,1726,1728,1731,1734,1737,1740,1743,1746,1749,1752],{"className":1727},[641],[568,1729],{"className":1730,"style":828},[645],[568,1732,798],{"className":1733},[650],[568,1735,795],{"className":1736},[832],[568,1738,602],{"className":1739},[689],[568,1741,605],{"className":1742},[689,698],[568,1744,815],{"className":1745},[861],[568,1747],{"className":1748,"style":756},[693],[568,1750,624],{"className":1751},[760],[568,1753],{"className":1754,"style":756},[693],[568,1756,1758,1762,1791,1794,1797,1800,1803],{"className":1757},[641],[568,1759],{"className":1760,"style":1761},[645],"height:0.9818em;vertical-align:-0.0833em;",[568,1763,1765,1768],{"className":1764},[650],[568,1766,798],{"className":1767},[650],[568,1769,1771],{"className":1770},[657],[568,1772,1774],{"className":1773},[661],[568,1775,1777],{"className":1776},[665],[568,1778,1780],{"className":1779,"style":1078},[669],[568,1781,1782,1785],{"style":1099},[568,1783],{"className":1784,"style":678},[677],[568,1786,1788],{"className":1787},[682,683,684,685],[568,1789,602],{"className":1790},[689,685],[568,1792],{"className":1793,"style":694},[693],[568,1795,605],{"className":1796},[689,698],[568,1798],{"className":1799,"style":702},[693],[568,1801,608],{"className":1802},[706],[568,1804],{"className":1805,"style":702},[693],[568,1807,1809,1812,1841,1844],{"className":1808},[641],[568,1810],{"className":1811,"style":1078},[645],[568,1813,1815,1818],{"className":1814},[650],[568,1816,808],{"className":1817},[650],[568,1819,1821],{"className":1820},[657],[568,1822,1824],{"className":1823},[661],[568,1825,1827],{"className":1826},[665],[568,1828,1830],{"className":1829,"style":1078},[669],[568,1831,1832,1835],{"style":1099},[568,1833],{"className":1834,"style":678},[677],[568,1836,1838],{"className":1837},[682,683,684,685],[568,1839,602],{"className":1840},[689,685],[568,1842],{"className":1843,"style":694},[693],[568,1845,605],{"className":1846},[689,698]," und ",[568,1849,1851,1891],{"className":1850},[571],[568,1852,1854],{"className":1853},[575],[577,1855,1856],{"xmlns":579},[581,1857,1858,1888],{},[584,1859,1860,1862,1864,1866,1868,1870,1872,1874,1876,1878,1880,1882,1884,1886],{},[592,1861,808],{},[596,1863,598],{},[596,1865,795],{"stretchy":794},[600,1867,602],{},[592,1869,605],{},[596,1871,815],{"stretchy":794},[596,1873,624],{},[600,1875,602],{},[592,1877,808],{},[596,1879,598],{},[592,1881,605],{},[592,1883,798],{},[596,1885,598],{},[592,1887,605],{},[629,1889,1890],{"encoding":631},"\\sinh(2x) = 2 \\sinh x \\cosh x",[568,1892,1894,1924],{"className":1893,"ariaHidden":637},[636],[568,1895,1897,1900,1903,1906,1909,1912,1915,1918,1921],{"className":1896},[641],[568,1898],{"className":1899,"style":828},[645],[568,1901,808],{"className":1902},[650],[568,1904,795],{"className":1905},[832],[568,1907,602],{"className":1908},[689],[568,1910,605],{"className":1911},[689,698],[568,1913,815],{"className":1914},[861],[568,1916],{"className":1917,"style":756},[693],[568,1919,624],{"className":1920},[760],[568,1922],{"className":1923,"style":756},[693],[568,1925,1927,1930,1933,1936,1939,1942,1945,1948,1951,1954],{"className":1926},[641],[568,1928],{"className":1929,"style":1528},[645],[568,1931,602],{"className":1932},[689],[568,1934],{"className":1935,"style":694},[693],[568,1937,808],{"className":1938},[650],[568,1940],{"className":1941,"style":694},[693],[568,1943,605],{"className":1944},[689,698],[568,1946],{"className":1947,"style":694},[693],[568,1949,798],{"className":1950},[650],[568,1952],{"className":1953,"style":694},[693],[568,1955,605],{"className":1956},[689,698],[564,1958,1959,1960,2022,2023,2158,2159,1382],{},"In der Integralrechnung nutzt man die Identität für die Substitution ",[568,1961,1963,1986],{"className":1962},[571],[568,1964,1966],{"className":1965},[575],[577,1967,1968],{"xmlns":579},[581,1969,1970,1983],{},[584,1971,1972,1974,1976,1978,1980],{},[592,1973,605],{},[596,1975,624],{},[592,1977,808],{},[596,1979,598],{},[592,1981,1982],{},"t",[629,1984,1985],{"encoding":631},"x = \\sinh t",[568,1987,1989,2007],{"className":1988,"ariaHidden":637},[636],[568,1990,1992,1995,1998,2001,2004],{"className":1991},[641],[568,1993],{"className":1994,"style":888},[645],[568,1996,605],{"className":1997},[689,698],[568,1999],{"className":2000,"style":756},[693],[568,2002,624],{"className":2003},[760],[568,2005],{"className":2006,"style":756},[693],[568,2008,2010,2013,2016,2019],{"className":2009},[641],[568,2011],{"className":2012,"style":1528},[645],[568,2014,808],{"className":2015},[650],[568,2017],{"className":2018,"style":694},[693],[568,2020,1982],{"className":2021},[689,698]," bei Wurzeln der Form ",[568,2024,2026,2052],{"className":2025},[571],[568,2027,2029],{"className":2028},[575],[577,2030,2031],{"xmlns":579},[581,2032,2033,2049],{},[584,2034,2035],{},[1496,2036,2037],{},[584,2038,2039,2041,2043],{},[600,2040,627],{},[596,2042,608],{},[587,2044,2045,2047],{},[592,2046,605],{},[600,2048,602],{},[629,2050,2051],{"encoding":631},"\\sqrt{1 + x^2}",[568,2053,2055],{"className":2054,"ariaHidden":637},[636],[568,2056,2058,2062],{"className":2057},[641],[568,2059],{"className":2060,"style":2061},[645],"height:1.04em;vertical-align:-0.1266em;",[568,2063,2065],{"className":2064},[689,1557],[568,2066,2068,2149],{"className":2067},[661,1261],[568,2069,2071,2146],{"className":2070},[665],[568,2072,2075,2129],{"className":2073,"style":2074},[669],"height:0.9134em;",[568,2076,2079,2082],{"className":2077,"style":2078},[1571],"top:-3em;",[568,2080],{"className":2081,"style":1275},[677],[568,2083,2086,2089,2092,2095,2098],{"className":2084,"style":2085},[689],"padding-left:0.833em;",[568,2087,627],{"className":2088},[689],[568,2090],{"className":2091,"style":702},[693],[568,2093,608],{"className":2094},[706],[568,2096],{"className":2097,"style":702},[693],[568,2099,2101,2104],{"className":2100},[689],[568,2102,605],{"className":2103},[689,698],[568,2105,2107],{"className":2106},[657],[568,2108,2110],{"className":2109},[661],[568,2111,2113],{"className":2112},[665],[568,2114,2117],{"className":2115,"style":2116},[669],"height:0.7401em;",[568,2118,2120,2123],{"style":2119},"top:-2.989em;margin-right:0.05em;",[568,2121],{"className":2122,"style":678},[677],[568,2124,2126],{"className":2125},[682,683,684,685],[568,2127,602],{"className":2128},[689,685],[568,2130,2132,2135],{"style":2131},"top:-2.8734em;",[568,2133],{"className":2134,"style":1275},[677],[568,2136,2139],{"className":2137,"style":2138},[1637],"min-width:0.853em;height:1.08em;",[1640,2140,2143],{"xmlns":1642,"width":1643,"height":2141,"viewBox":2142,"preserveAspectRatio":1646},"1.08em","0 0 400000 1080",[1648,2144],{"d":2145},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[568,2147,1369],{"className":2148},[1368],[568,2150,2152],{"className":2151},[665],[568,2153,2156],{"className":2154,"style":2155},[669],"height:0.1266em;",[568,2157],{},", denn dann wird ",[568,2160,2162,2202],{"className":2161},[571],[568,2163,2165],{"className":2164},[575],[577,2166,2167],{"xmlns":579},[581,2168,2169,2199],{},[584,2170,2171,2191,2193,2195,2197],{},[1496,2172,2173],{},[584,2174,2175,2177,2179,2189],{},[600,2176,627],{},[596,2178,608],{},[587,2180,2181,2187],{},[584,2182,2183,2185],{},[592,2184,808],{},[596,2186,598],{},[600,2188,602],{},[592,2190,1982],{},[596,2192,624],{},[592,2194,798],{},[596,2196,598],{},[592,2198,1982],{},[629,2200,2201],{"encoding":631},"\\sqrt{1 + \\sinh^2 t} = \\cosh t",[568,2203,2205,2311],{"className":2204,"ariaHidden":637},[636],[568,2206,2208,2211,2302,2305,2308],{"className":2207},[641],[568,2209],{"className":2210,"style":1553},[645],[568,2212,2214],{"className":2213},[689,1557],[568,2215,2217,2294],{"className":2216},[661,1261],[568,2218,2220,2291],{"className":2219},[665],[568,2221,2223,2279],{"className":2222,"style":1567},[669],[568,2224,2226,2229],{"className":2225,"style":1572},[1571],[568,2227],{"className":2228,"style":1576},[677],[568,2230,2232,2235,2238,2241,2244,2273,2276],{"className":2231,"style":1580},[689],[568,2233,627],{"className":2234},[689],[568,2236],{"className":2237,"style":702},[693],[568,2239,608],{"className":2240},[706],[568,2242],{"className":2243,"style":702},[693],[568,2245,2247,2250],{"className":2246},[650],[568,2248,808],{"className":2249},[650],[568,2251,2253],{"className":2252},[657],[568,2254,2256],{"className":2255},[661],[568,2257,2259],{"className":2258},[665],[568,2260,2262],{"className":2261,"style":1078},[669],[568,2263,2264,2267],{"style":1099},[568,2265],{"className":2266,"style":678},[677],[568,2268,2270],{"className":2269},[682,683,684,685],[568,2271,602],{"className":2272},[689,685],[568,2274],{"className":2275,"style":694},[693],[568,2277,1982],{"className":2278},[689,698],[568,2280,2281,2284],{"style":1630},[568,2282],{"className":2283,"style":1576},[677],[568,2285,2287],{"className":2286,"style":1638},[1637],[1640,2288,2289],{"xmlns":1642,"width":1643,"height":1644,"viewBox":1645,"preserveAspectRatio":1646},[1648,2290],{"d":1650},[568,2292,1369],{"className":2293},[1368],[568,2295,2297],{"className":2296},[665],[568,2298,2300],{"className":2299,"style":1660},[669],[568,2301],{},[568,2303],{"className":2304,"style":756},[693],[568,2306,624],{"className":2307},[760],[568,2309],{"className":2310,"style":756},[693],[568,2312,2314,2317,2320,2323],{"className":2313},[641],[568,2315],{"className":2316,"style":1528},[645],[568,2318,798],{"className":2319},[650],[568,2321],{"className":2322,"style":694},[693],[568,2324,1982],{"className":2325},[689,698],[559,2327,2329],{"id":2328},"herleitung","Herleitung",[564,2331,2332],{},"Mit den Definitionen und den binomischen Formeln:",[568,2334,2337],{"className":2335},[2336],"katex-display",[568,2338,2340,2465],{"className":2339},[571],[568,2341,2343],{"className":2342},[575],[577,2344,2346],{"xmlns":579,"display":2345},"block",[581,2347,2348,2462],{},[584,2349,2350,2360,2362,2364,2374,2376,2378,2414,2416,2450,2452,2458,2460],{},[587,2351,2352,2358],{},[584,2353,2354,2356],{},[592,2355,798],{},[596,2357,598],{},[600,2359,602],{},[592,2361,605],{},[596,2363,914],{},[587,2365,2366,2372],{},[584,2367,2368,2370],{},[592,2369,808],{},[596,2371,598],{},[600,2373,602],{},[592,2375,605],{},[596,2377,624],{},[1148,2379,2380,2411],{},[584,2381,2382,2393,2395,2397,2399],{},[587,2383,2384,2387],{},[592,2385,2386],{},"e",[584,2388,2389,2391],{},[600,2390,602],{},[592,2392,605],{},[596,2394,608],{},[600,2396,602],{},[596,2398,608],{},[587,2400,2401,2403],{},[592,2402,2386],{},[584,2404,2405,2407,2409],{},[596,2406,914],{},[600,2408,602],{},[592,2410,605],{},[600,2412,2413],{},"4",[596,2415,914],{},[1148,2417,2418,2448],{},[584,2419,2420,2430,2432,2434,2436],{},[587,2421,2422,2424],{},[592,2423,2386],{},[584,2425,2426,2428],{},[600,2427,602],{},[592,2429,605],{},[596,2431,914],{},[600,2433,602],{},[596,2435,608],{},[587,2437,2438,2440],{},[592,2439,2386],{},[584,2441,2442,2444,2446],{},[596,2443,914],{},[600,2445,602],{},[592,2447,605],{},[600,2449,2413],{},[596,2451,624],{},[1148,2453,2454,2456],{},[600,2455,2413],{},[600,2457,2413],{},[596,2459,624],{},[600,2461,627],{},[629,2463,2464],{"encoding":631},"\\cosh^2 x - \\sinh^2 x = \\frac{e^{2x} + 2 + e^{-2x}}{4} - \\frac{e^{2x} - 2 + e^{-2x}}{4} = \\frac{4}{4} = 1",[568,2466,2468,2518,2568,2741,2909,2988],{"className":2467,"ariaHidden":637},[636],[568,2469,2471,2474,2503,2506,2509,2512,2515],{"className":2470},[641],[568,2472],{"className":2473,"style":1761},[645],[568,2475,2477,2480],{"className":2476},[650],[568,2478,798],{"className":2479},[650],[568,2481,2483],{"className":2482},[657],[568,2484,2486],{"className":2485},[661],[568,2487,2489],{"className":2488},[665],[568,2490,2492],{"className":2491,"style":1078},[669],[568,2493,2494,2497],{"style":1099},[568,2495],{"className":2496,"style":678},[677],[568,2498,2500],{"className":2499},[682,683,684,685],[568,2501,602],{"className":2502},[689,685],[568,2504],{"className":2505,"style":694},[693],[568,2507,605],{"className":2508},[689,698],[568,2510],{"className":2511,"style":702},[693],[568,2513,914],{"className":2514},[706],[568,2516],{"className":2517,"style":702},[693],[568,2519,2521,2524,2553,2556,2559,2562,2565],{"className":2520},[641],[568,2522],{"className":2523,"style":1078},[645],[568,2525,2527,2530],{"className":2526},[650],[568,2528,808],{"className":2529},[650],[568,2531,2533],{"className":2532},[657],[568,2534,2536],{"className":2535},[661],[568,2537,2539],{"className":2538},[665],[568,2540,2542],{"className":2541,"style":1078},[669],[568,2543,2544,2547],{"style":1099},[568,2545],{"className":2546,"style":678},[677],[568,2548,2550],{"className":2549},[682,683,684,685],[568,2551,602],{"className":2552},[689,685],[568,2554],{"className":2555,"style":694},[693],[568,2557,605],{"className":2558},[689,698],[568,2560],{"className":2561,"style":756},[693],[568,2563,624],{"className":2564},[760],[568,2566],{"className":2567,"style":756},[693],[568,2569,2571,2575,2732,2735,2738],{"className":2570},[641],[568,2572],{"className":2573,"style":2574},[645],"height:2.1771em;vertical-align:-0.686em;",[568,2576,2578,2581,2729],{"className":2577},[689],[568,2579],{"className":2580},[832,1254],[568,2582,2584],{"className":2583},[1148],[568,2585,2587,2720],{"className":2586},[661,1261],[568,2588,2590,2717],{"className":2589},[665],[568,2591,2594,2606,2614],{"className":2592,"style":2593},[669],"height:1.4911em;",[568,2595,2597,2600],{"style":2596},"top:-2.314em;",[568,2598],{"className":2599,"style":1275},[677],[568,2601,2603],{"className":2602},[689],[568,2604,2413],{"className":2605},[689],[568,2607,2608,2611],{"style":1341},[568,2609],{"className":2610,"style":1275},[677],[568,2612],{"className":2613,"style":1349},[1348],[568,2615,2617,2620],{"style":2616},"top:-3.677em;",[568,2618],{"className":2619,"style":1275},[677],[568,2621,2623,2658,2661,2664,2667,2670,2673,2676,2679],{"className":2622},[689],[568,2624,2626,2629],{"className":2625},[689],[568,2627,2386],{"className":2628},[689,698],[568,2630,2632],{"className":2631},[657],[568,2633,2635],{"className":2634},[661],[568,2636,2638],{"className":2637},[665],[568,2639,2641],{"className":2640,"style":716},[669],[568,2642,2643,2646],{"style":737},[568,2644],{"className":2645,"style":678},[677],[568,2647,2649],{"className":2648},[682,683,684,685],[568,2650,2652,2655],{"className":2651},[689,685],[568,2653,602],{"className":2654},[689,685],[568,2656,605],{"className":2657},[689,698,685],[568,2659],{"className":2660,"style":702},[693],[568,2662,608],{"className":2663},[706],[568,2665],{"className":2666,"style":702},[693],[568,2668,602],{"className":2669},[689],[568,2671],{"className":2672,"style":702},[693],[568,2674,608],{"className":2675},[706],[568,2677],{"className":2678,"style":702},[693],[568,2680,2682,2685],{"className":2681},[689],[568,2683,2386],{"className":2684},[689,698],[568,2686,2688],{"className":2687},[657],[568,2689,2691],{"className":2690},[661],[568,2692,2694],{"className":2693},[665],[568,2695,2697],{"className":2696,"style":716},[669],[568,2698,2699,2702],{"style":737},[568,2700],{"className":2701,"style":678},[677],[568,2703,2705],{"className":2704},[682,683,684,685],[568,2706,2708,2711,2714],{"className":2707},[689,685],[568,2709,914],{"className":2710},[689,685],[568,2712,602],{"className":2713},[689,685],[568,2715,605],{"className":2716},[689,698,685],[568,2718,1369],{"className":2719},[1368],[568,2721,2723],{"className":2722},[665],[568,2724,2727],{"className":2725,"style":2726},[669],"height:0.686em;",[568,2728],{},[568,2730],{"className":2731},[861,1254],[568,2733],{"className":2734,"style":702},[693],[568,2736,914],{"className":2737},[706],[568,2739],{"className":2740,"style":702},[693],[568,2742,2744,2747,2900,2903,2906],{"className":2743},[641],[568,2745],{"className":2746,"style":2574},[645],[568,2748,2750,2753,2897],{"className":2749},[689],[568,2751],{"className":2752},[832,1254],[568,2754,2756],{"className":2755},[1148],[568,2757,2759,2889],{"className":2758},[661,1261],[568,2760,2762,2886],{"className":2761},[665],[568,2763,2765,2776,2784],{"className":2764,"style":2593},[669],[568,2766,2767,2770],{"style":2596},[568,2768],{"className":2769,"style":1275},[677],[568,2771,2773],{"className":2772},[689],[568,2774,2413],{"className":2775},[689],[568,2777,2778,2781],{"style":1341},[568,2779],{"className":2780,"style":1275},[677],[568,2782],{"className":2783,"style":1349},[1348],[568,2785,2786,2789],{"style":2616},[568,2787],{"className":2788,"style":1275},[677],[568,2790,2792,2827,2830,2833,2836,2839,2842,2845,2848],{"className":2791},[689],[568,2793,2795,2798],{"className":2794},[689],[568,2796,2386],{"className":2797},[689,698],[568,2799,2801],{"className":2800},[657],[568,2802,2804],{"className":2803},[661],[568,2805,2807],{"className":2806},[665],[568,2808,2810],{"className":2809,"style":716},[669],[568,2811,2812,2815],{"style":737},[568,2813],{"className":2814,"style":678},[677],[568,2816,2818],{"className":2817},[682,683,684,685],[568,2819,2821,2824],{"className":2820},[689,685],[568,2822,602],{"className":2823},[689,685],[568,2825,605],{"className":2826},[689,698,685],[568,2828],{"className":2829,"style":702},[693],[568,2831,914],{"className":2832},[706],[568,2834],{"className":2835,"style":702},[693],[568,2837,602],{"className":2838},[689],[568,2840],{"className":2841,"style":702},[693],[568,2843,608],{"className":2844},[706],[568,2846],{"className":2847,"style":702},[693],[568,2849,2851,2854],{"className":2850},[689],[568,2852,2386],{"className":2853},[689,698],[568,2855,2857],{"className":2856},[657],[568,2858,2860],{"className":2859},[661],[568,2861,2863],{"className":2862},[665],[568,2864,2866],{"className":2865,"style":716},[669],[568,2867,2868,2871],{"style":737},[568,2869],{"className":2870,"style":678},[677],[568,2872,2874],{"className":2873},[682,683,684,685],[568,2875,2877,2880,2883],{"className":2876},[689,685],[568,2878,914],{"className":2879},[689,685],[568,2881,602],{"className":2882},[689,685],[568,2884,605],{"className":2885},[689,698,685],[568,2887,1369],{"className":2888},[1368],[568,2890,2892],{"className":2891},[665],[568,2893,2895],{"className":2894,"style":2726},[669],[568,2896],{},[568,2898],{"className":2899},[861,1254],[568,2901],{"className":2902,"style":756},[693],[568,2904,624],{"className":2905},[760],[568,2907],{"className":2908,"style":756},[693],[568,2910,2912,2916,2979,2982,2985],{"className":2911},[641],[568,2913],{"className":2914,"style":2915},[645],"height:2.0074em;vertical-align:-0.686em;",[568,2917,2919,2922,2976],{"className":2918},[689],[568,2920],{"className":2921},[832,1254],[568,2923,2925],{"className":2924},[1148],[568,2926,2928,2968],{"className":2927},[661,1261],[568,2929,2931,2965],{"className":2930},[665],[568,2932,2935,2946,2954],{"className":2933,"style":2934},[669],"height:1.3214em;",[568,2936,2937,2940],{"style":2596},[568,2938],{"className":2939,"style":1275},[677],[568,2941,2943],{"className":2942},[689],[568,2944,2413],{"className":2945},[689],[568,2947,2948,2951],{"style":1341},[568,2949],{"className":2950,"style":1275},[677],[568,2952],{"className":2953,"style":1349},[1348],[568,2955,2956,2959],{"style":2616},[568,2957],{"className":2958,"style":1275},[677],[568,2960,2962],{"className":2961},[689],[568,2963,2413],{"className":2964},[689],[568,2966,1369],{"className":2967},[1368],[568,2969,2971],{"className":2970},[665],[568,2972,2974],{"className":2973,"style":2726},[669],[568,2975],{},[568,2977],{"className":2978},[861,1254],[568,2980],{"className":2981,"style":756},[693],[568,2983,624],{"className":2984},[760],[568,2986],{"className":2987,"style":756},[693],[568,2989,2991,2994],{"className":2990},[641],[568,2992],{"className":2993,"style":770},[645],[568,2995,627],{"className":2996},[689],[559,2998,3000],{"id":2999},"beispiele","Beispiele",[564,3002,3003,3004,3138],{},"Gegeben ist ",[568,3005,3007,3034],{"className":3006},[571],[568,3008,3010],{"className":3009},[575],[577,3011,3012],{"xmlns":579},[581,3013,3014,3031],{},[584,3015,3016,3018,3020,3022,3024],{},[592,3017,808],{},[596,3019,598],{},[592,3021,605],{},[596,3023,624],{},[1148,3025,3026,3029],{},[600,3027,3028],{},"3",[600,3030,2413],{},[629,3032,3033],{"encoding":631},"\\sinh x = \\frac{3}{4}",[568,3035,3037,3061],{"className":3036,"ariaHidden":637},[636],[568,3038,3040,3043,3046,3049,3052,3055,3058],{"className":3039},[641],[568,3041],{"className":3042,"style":1528},[645],[568,3044,808],{"className":3045},[650],[568,3047],{"className":3048,"style":694},[693],[568,3050,605],{"className":3051},[689,698],[568,3053],{"className":3054,"style":756},[693],[568,3056,624],{"className":3057},[760],[568,3059],{"className":3060,"style":756},[693],[568,3062,3064,3068],{"className":3063},[641],[568,3065],{"className":3066,"style":3067},[645],"height:1.1901em;vertical-align:-0.345em;",[568,3069,3071,3074,3135],{"className":3070},[689],[568,3072],{"className":3073},[832,1254],[568,3075,3077],{"className":3076},[1148],[568,3078,3080,3126],{"className":3079},[661,1261],[568,3081,3083,3123],{"className":3082},[665],[568,3084,3086,3101,3109],{"className":3085,"style":1268},[669],[568,3087,3089,3092],{"style":3088},"top:-2.655em;",[568,3090],{"className":3091,"style":1275},[677],[568,3093,3095],{"className":3094},[682,683,684,685],[568,3096,3098],{"className":3097},[689,685],[568,3099,2413],{"className":3100},[689,685],[568,3102,3103,3106],{"style":1341},[568,3104],{"className":3105,"style":1275},[677],[568,3107],{"className":3108,"style":1349},[1348],[568,3110,3111,3114],{"style":1352},[568,3112],{"className":3113,"style":1275},[677],[568,3115,3117],{"className":3116},[682,683,684,685],[568,3118,3120],{"className":3119},[689,685],[568,3121,3028],{"className":3122},[689,685],[568,3124,1369],{"className":3125},[1368],[568,3127,3129],{"className":3128},[665],[568,3130,3133],{"className":3131,"style":3132},[669],"height:0.345em;",[568,3134],{},[568,3136],{"className":3137},[861,1254],". Dann ist",[568,3140,3142],{"className":3141},[2336],[568,3143,3145,3249],{"className":3144},[571],[568,3146,3148],{"className":3147},[575],[577,3149,3150],{"xmlns":579,"display":2345},[581,3151,3152,3246],{},[584,3153,3154,3156,3158,3160,3162,3181,3183,3194,3196,3205,3207,3210,3212,3214,3216,3218,3238,3240],{},[592,3155,798],{},[596,3157,598],{},[592,3159,605],{},[596,3161,624],{},[1496,3163,3164],{},[584,3165,3166,3168,3170],{},[600,3167,627],{},[596,3169,608],{},[3171,3172,3173],"mstyle",{"scriptlevel":1414,"displaystyle":794},[1148,3174,3175,3178],{},[600,3176,3177],{},"9",[600,3179,3180],{},"16",[596,3182,624],{},[1496,3184,3185],{},[3171,3186,3187],{"scriptlevel":1414,"displaystyle":794},[1148,3188,3189,3192],{},[600,3190,3191],{},"25",[600,3193,3180],{},[596,3195,624],{},[3171,3197,3198],{"scriptlevel":1414,"displaystyle":794},[1148,3199,3200,3203],{},[600,3201,3202],{},"5",[600,3204,2413],{},[596,3206,805],{"separator":637},[693,3208],{"width":3209},"2em",[592,3211,1138],{},[596,3213,598],{},[592,3215,605],{},[596,3217,624],{},[1148,3219,3220,3230],{},[584,3221,3222,3224,3228],{},[600,3223,3028],{},[592,3225,3227],{"mathvariant":3226},"normal","\u002F",[600,3229,2413],{},[584,3231,3232,3234,3236],{},[600,3233,3202],{},[592,3235,3227],{"mathvariant":3226},[600,3237,2413],{},[596,3239,624],{},[1148,3241,3242,3244],{},[600,3243,3028],{},[600,3245,3202],{},[629,3247,3248],{"encoding":631},"\\cosh x = \\sqrt{1 + \\tfrac{9}{16}} = \\sqrt{\\tfrac{25}{16}} = \\tfrac{5}{4},\\qquad \\tanh x = \\frac{3\u002F4}{5\u002F4} = \\frac{3}{5}",[568,3250,3252,3276,3425,3552,3654,3736],{"className":3251,"ariaHidden":637},[636],[568,3253,3255,3258,3261,3264,3267,3270,3273],{"className":3254},[641],[568,3256],{"className":3257,"style":1528},[645],[568,3259,798],{"className":3260},[650],[568,3262],{"className":3263,"style":694},[693],[568,3265,605],{"className":3266},[689,698],[568,3268],{"className":3269,"style":756},[693],[568,3271,624],{"className":3272},[760],[568,3274],{"className":3275,"style":756},[693],[568,3277,3279,3283,3416,3419,3422],{"className":3278},[641],[568,3280],{"className":3281,"style":3282},[645],"height:1.84em;vertical-align:-0.5561em;",[568,3284,3286],{"className":3285},[689,1557],[568,3287,3289,3407],{"className":3288},[661,1261],[568,3290,3292,3404],{"className":3291},[665],[568,3293,3296,3387],{"className":3294,"style":3295},[669],"height:1.2839em;",[568,3297,3300,3304],{"className":3298,"style":3299},[1571],"top:-3.8em;",[568,3301],{"className":3302,"style":3303},[677],"height:3.8em;",[568,3305,3307,3310,3313,3316,3319],{"className":3306,"style":1580},[689],[568,3308,627],{"className":3309},[689],[568,3311],{"className":3312,"style":702},[693],[568,3314,608],{"className":3315},[706],[568,3317],{"className":3318,"style":702},[693],[568,3320,3322,3325,3384],{"className":3321},[689],[568,3323],{"className":3324},[832,1254],[568,3326,3328],{"className":3327},[1148],[568,3329,3331,3376],{"className":3330},[661,1261],[568,3332,3334,3373],{"className":3333},[665],[568,3335,3337,3351,3359],{"className":3336,"style":1268},[669],[568,3338,3339,3342],{"style":3088},[568,3340],{"className":3341,"style":1275},[677],[568,3343,3345],{"className":3344},[682,683,684,685],[568,3346,3348],{"className":3347},[689,685],[568,3349,3180],{"className":3350},[689,685],[568,3352,3353,3356],{"style":1341},[568,3354],{"className":3355,"style":1275},[677],[568,3357],{"className":3358,"style":1349},[1348],[568,3360,3361,3364],{"style":1352},[568,3362],{"className":3363,"style":1275},[677],[568,3365,3367],{"className":3366},[682,683,684,685],[568,3368,3370],{"className":3369},[689,685],[568,3371,3177],{"className":3372},[689,685],[568,3374,1369],{"className":3375},[1368],[568,3377,3379],{"className":3378},[665],[568,3380,3382],{"className":3381,"style":3132},[669],[568,3383],{},[568,3385],{"className":3386},[861,1254],[568,3388,3390,3393],{"style":3389},"top:-3.2439em;",[568,3391],{"className":3392,"style":3303},[677],[568,3394,3397],{"className":3395,"style":3396},[1637],"min-width:1.02em;height:1.88em;",[1640,3398,3401],{"xmlns":1642,"width":1643,"height":3399,"viewBox":3400,"preserveAspectRatio":1646},"1.88em","0 0 400000 1944",[1648,3402],{"d":3403},"M983 90\nl0 -0\nc4,-6.7,10,-10,18,-10 H400000v40\nH1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7\ns-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744\nc-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30\nc26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722\nc56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5\nc53.7,-170.3,84.5,-266.8,92.5,-289.5z\nM1001 80h400000v40h-400000z",[568,3405,1369],{"className":3406},[1368],[568,3408,3410],{"className":3409},[665],[568,3411,3414],{"className":3412,"style":3413},[669],"height:0.5561em;",[568,3415],{},[568,3417],{"className":3418,"style":756},[693],[568,3420,624],{"className":3421},[760],[568,3423],{"className":3424,"style":756},[693],[568,3426,3428,3431,3543,3546,3549],{"className":3427},[641],[568,3429],{"className":3430,"style":3282},[645],[568,3432,3434],{"className":3433},[689,1557],[568,3435,3437,3535],{"className":3436},[661,1261],[568,3438,3440,3532],{"className":3439},[665],[568,3441,3443,3520],{"className":3442,"style":3295},[669],[568,3444,3446,3449],{"className":3445,"style":3299},[1571],[568,3447],{"className":3448,"style":3303},[677],[568,3450,3452],{"className":3451,"style":1580},[689],[568,3453,3455,3458,3517],{"className":3454},[689],[568,3456],{"className":3457},[832,1254],[568,3459,3461],{"className":3460},[1148],[568,3462,3464,3509],{"className":3463},[661,1261],[568,3465,3467,3506],{"className":3466},[665],[568,3468,3470,3484,3492],{"className":3469,"style":1268},[669],[568,3471,3472,3475],{"style":3088},[568,3473],{"className":3474,"style":1275},[677],[568,3476,3478],{"className":3477},[682,683,684,685],[568,3479,3481],{"className":3480},[689,685],[568,3482,3180],{"className":3483},[689,685],[568,3485,3486,3489],{"style":1341},[568,3487],{"className":3488,"style":1275},[677],[568,3490],{"className":3491,"style":1349},[1348],[568,3493,3494,3497],{"style":1352},[568,3495],{"className":3496,"style":1275},[677],[568,3498,3500],{"className":3499},[682,683,684,685],[568,3501,3503],{"className":3502},[689,685],[568,3504,3191],{"className":3505},[689,685],[568,3507,1369],{"className":3508},[1368],[568,3510,3512],{"className":3511},[665],[568,3513,3515],{"className":3514,"style":3132},[669],[568,3516],{},[568,3518],{"className":3519},[861,1254],[568,3521,3522,3525],{"style":3389},[568,3523],{"className":3524,"style":3303},[677],[568,3526,3528],{"className":3527,"style":3396},[1637],[1640,3529,3530],{"xmlns":1642,"width":1643,"height":3399,"viewBox":3400,"preserveAspectRatio":1646},[1648,3531],{"d":3403},[568,3533,1369],{"className":3534},[1368],[568,3536,3538],{"className":3537},[665],[568,3539,3541],{"className":3540,"style":3413},[669],[568,3542],{},[568,3544],{"className":3545,"style":756},[693],[568,3547,624],{"className":3548},[760],[568,3550],{"className":3551,"style":756},[693],[568,3553,3555,3558,3626,3629,3633,3636,3639,3642,3645,3648,3651],{"className":3554},[641],[568,3556],{"className":3557,"style":3067},[645],[568,3559,3561,3564,3623],{"className":3560},[689],[568,3562],{"className":3563},[832,1254],[568,3565,3567],{"className":3566},[1148],[568,3568,3570,3615],{"className":3569},[661,1261],[568,3571,3573,3612],{"className":3572},[665],[568,3574,3576,3590,3598],{"className":3575,"style":1268},[669],[568,3577,3578,3581],{"style":3088},[568,3579],{"className":3580,"style":1275},[677],[568,3582,3584],{"className":3583},[682,683,684,685],[568,3585,3587],{"className":3586},[689,685],[568,3588,2413],{"className":3589},[689,685],[568,3591,3592,3595],{"style":1341},[568,3593],{"className":3594,"style":1275},[677],[568,3596],{"className":3597,"style":1349},[1348],[568,3599,3600,3603],{"style":1352},[568,3601],{"className":3602,"style":1275},[677],[568,3604,3606],{"className":3605},[682,683,684,685],[568,3607,3609],{"className":3608},[689,685],[568,3610,3202],{"className":3611},[689,685],[568,3613,1369],{"className":3614},[1368],[568,3616,3618],{"className":3617},[665],[568,3619,3621],{"className":3620,"style":3132},[669],[568,3622],{},[568,3624],{"className":3625},[861,1254],[568,3627,805],{"className":3628},[845],[568,3630],{"className":3631,"style":3632},[693],"margin-right:2em;",[568,3634],{"className":3635,"style":694},[693],[568,3637,1138],{"className":3638},[650],[568,3640],{"className":3641,"style":694},[693],[568,3643,605],{"className":3644},[689,698],[568,3646],{"className":3647,"style":756},[693],[568,3649,624],{"className":3650},[760],[568,3652],{"className":3653,"style":756},[693],[568,3655,3657,3661,3727,3730,3733],{"className":3656},[641],[568,3658],{"className":3659,"style":3660},[645],"height:2.363em;vertical-align:-0.936em;",[568,3662,3664,3667,3724],{"className":3663},[689],[568,3665],{"className":3666},[832,1254],[568,3668,3670],{"className":3669},[1148],[568,3671,3673,3715],{"className":3672},[661,1261],[568,3674,3676,3712],{"className":3675},[665],[568,3677,3680,3692,3700],{"className":3678,"style":3679},[669],"height:1.427em;",[568,3681,3682,3685],{"style":2596},[568,3683],{"className":3684,"style":1275},[677],[568,3686,3688],{"className":3687},[689],[568,3689,3691],{"className":3690},[689],"5\u002F4",[568,3693,3694,3697],{"style":1341},[568,3695],{"className":3696,"style":1275},[677],[568,3698],{"className":3699,"style":1349},[1348],[568,3701,3702,3705],{"style":2616},[568,3703],{"className":3704,"style":1275},[677],[568,3706,3708],{"className":3707},[689],[568,3709,3711],{"className":3710},[689],"3\u002F4",[568,3713,1369],{"className":3714},[1368],[568,3716,3718],{"className":3717},[665],[568,3719,3722],{"className":3720,"style":3721},[669],"height:0.936em;",[568,3723],{},[568,3725],{"className":3726},[861,1254],[568,3728],{"className":3729,"style":756},[693],[568,3731,624],{"className":3732},[760],[568,3734],{"className":3735,"style":756},[693],[568,3737,3739,3742],{"className":3738},[641],[568,3740],{"className":3741,"style":2915},[645],[568,3743,3745,3748,3801],{"className":3744},[689],[568,3746],{"className":3747},[832,1254],[568,3749,3751],{"className":3750},[1148],[568,3752,3754,3793],{"className":3753},[661,1261],[568,3755,3757,3790],{"className":3756},[665],[568,3758,3760,3771,3779],{"className":3759,"style":2934},[669],[568,3761,3762,3765],{"style":2596},[568,3763],{"className":3764,"style":1275},[677],[568,3766,3768],{"className":3767},[689],[568,3769,3202],{"className":3770},[689],[568,3772,3773,3776],{"style":1341},[568,3774],{"className":3775,"style":1275},[677],[568,3777],{"className":3778,"style":1349},[1348],[568,3780,3781,3784],{"style":2616},[568,3782],{"className":3783,"style":1275},[677],[568,3785,3787],{"className":3786},[689],[568,3788,3028],{"className":3789},[689],[568,3791,1369],{"className":3792},[1368],[568,3794,3796],{"className":3795},[665],[568,3797,3799],{"className":3798,"style":2726},[669],[568,3800],{},[568,3802],{"className":3803},[861,1254],[564,3805,3806,3807,3835,3836,3876,3877,1847,4206,1382],{},"Zur Kontrolle: Das zugehörige ",[568,3808,3810,3823],{"className":3809},[571],[568,3811,3813],{"className":3812},[575],[577,3814,3815],{"xmlns":579},[581,3816,3817,3821],{},[584,3818,3819],{},[592,3820,605],{},[629,3822,605],{"encoding":631},[568,3824,3826],{"className":3825,"ariaHidden":637},[636],[568,3827,3829,3832],{"className":3828},[641],[568,3830],{"className":3831,"style":888},[645],[568,3833,605],{"className":3834},[689,698]," ist ",[568,3837,3839,3858],{"className":3838},[571],[568,3840,3842],{"className":3841},[575],[577,3843,3844],{"xmlns":579},[581,3845,3846,3855],{},[584,3847,3848,3851,3853],{},[592,3849,3850],{},"ln",[596,3852,598],{},[600,3854,602],{},[629,3856,3857],{"encoding":631},"\\ln 2",[568,3859,3861],{"className":3860,"ariaHidden":637},[636],[568,3862,3864,3867,3870,3873],{"className":3863},[641],[568,3865],{"className":3866,"style":1528},[645],[568,3868,3850],{"className":3869},[650],[568,3871],{"className":3872,"style":694},[693],[568,3874,602],{"className":3875},[689],", und tatsächlich ist ",[568,3878,3880,3932],{"className":3879},[571],[568,3881,3883],{"className":3882},[575],[577,3884,3885],{"xmlns":579},[581,3886,3887,3929],{},[584,3888,3889,3891,3893,3895,3897,3899,3901,3903,3905,3921,3923],{},[592,3890,808],{},[596,3892,598],{},[596,3894,795],{"stretchy":794},[592,3896,3850],{},[596,3898,598],{},[600,3900,602],{},[596,3902,815],{"stretchy":794},[596,3904,624],{},[1148,3906,3907,3919],{},[584,3908,3909,3911,3913],{},[600,3910,602],{},[596,3912,914],{},[1148,3914,3915,3917],{},[600,3916,627],{},[600,3918,602],{},[600,3920,602],{},[596,3922,624],{},[1148,3924,3925,3927],{},[600,3926,3028],{},[600,3928,2413],{},[629,3930,3931],{"encoding":631},"\\sinh(\\ln 2) = \\frac{2 - \\frac{1}{2}}{2} = \\frac{3}{4}",[568,3933,3935,3968,4132],{"className":3934,"ariaHidden":637},[636],[568,3936,3938,3941,3944,3947,3950,3953,3956,3959,3962,3965],{"className":3937},[641],[568,3939],{"className":3940,"style":828},[645],[568,3942,808],{"className":3943},[650],[568,3945,795],{"className":3946},[832],[568,3948,3850],{"className":3949},[650],[568,3951],{"className":3952,"style":694},[693],[568,3954,602],{"className":3955},[689],[568,3957,815],{"className":3958},[861],[568,3960],{"className":3961,"style":756},[693],[568,3963,624],{"className":3964},[760],[568,3966],{"className":3967,"style":756},[693],[568,3969,3971,3975,4123,4126,4129],{"className":3970},[641],[568,3972],{"className":3973,"style":3974},[645],"height:1.4868em;vertical-align:-0.345em;",[568,3976,3978,3981,4120],{"className":3977},[689],[568,3979],{"className":3980},[832,1254],[568,3982,3984],{"className":3983},[1148],[568,3985,3987,4112],{"className":3986},[661,1261],[568,3988,3990,4109],{"className":3989},[665],[568,3991,3994,4008,4016],{"className":3992,"style":3993},[669],"height:1.1418em;",[568,3995,3996,3999],{"style":3088},[568,3997],{"className":3998,"style":1275},[677],[568,4000,4002],{"className":4001},[682,683,684,685],[568,4003,4005],{"className":4004},[689,685],[568,4006,602],{"className":4007},[689,685],[568,4009,4010,4013],{"style":1341},[568,4011],{"className":4012,"style":1275},[677],[568,4014],{"className":4015,"style":1349},[1348],[568,4017,4019,4022],{"style":4018},"top:-3.5508em;",[568,4020],{"className":4021,"style":1275},[677],[568,4023,4025],{"className":4024},[682,683,684,685],[568,4026,4028,4031,4034],{"className":4027},[689,685],[568,4029,602],{"className":4030},[689,685],[568,4032,914],{"className":4033},[706,685],[568,4035,4037,4041,4106],{"className":4036},[689,685],[568,4038],{"className":4039},[832,1254,682,1327,4040],"size6",[568,4042,4044],{"className":4043},[1148],[568,4045,4047,4097],{"className":4046},[661,1261],[568,4048,4050,4094],{"className":4049},[665],[568,4051,4054,4069,4079],{"className":4052,"style":4053},[669],"height:0.8443em;",[568,4055,4057,4060],{"style":4056},"top:-2.656em;",[568,4058],{"className":4059,"style":1275},[677],[568,4061,4063],{"className":4062},[682,1327,1328,685],[568,4064,4066],{"className":4065},[689,685],[568,4067,602],{"className":4068},[689,685],[568,4070,4072,4075],{"style":4071},"top:-3.2255em;",[568,4073],{"className":4074,"style":1275},[677],[568,4076],{"className":4077,"style":4078},[1348,685],"border-bottom-width:0.049em;",[568,4080,4082,4085],{"style":4081},"top:-3.384em;",[568,4083],{"className":4084,"style":1275},[677],[568,4086,4088],{"className":4087},[682,1327,1328,685],[568,4089,4091],{"className":4090},[689,685],[568,4092,627],{"className":4093},[689,685],[568,4095,1369],{"className":4096},[1368],[568,4098,4100],{"className":4099},[665],[568,4101,4104],{"className":4102,"style":4103},[669],"height:0.344em;",[568,4105],{},[568,4107],{"className":4108},[861,1254,682,1327,4040],[568,4110,1369],{"className":4111},[1368],[568,4113,4115],{"className":4114},[665],[568,4116,4118],{"className":4117,"style":3132},[669],[568,4119],{},[568,4121],{"className":4122},[861,1254],[568,4124],{"className":4125,"style":756},[693],[568,4127,624],{"className":4128},[760],[568,4130],{"className":4131,"style":756},[693],[568,4133,4135,4138],{"className":4134},[641],[568,4136],{"className":4137,"style":3067},[645],[568,4139,4141,4144,4203],{"className":4140},[689],[568,4142],{"className":4143},[832,1254],[568,4145,4147],{"className":4146},[1148],[568,4148,4150,4195],{"className":4149},[661,1261],[568,4151,4153,4192],{"className":4152},[665],[568,4154,4156,4170,4178],{"className":4155,"style":1268},[669],[568,4157,4158,4161],{"style":3088},[568,4159],{"className":4160,"style":1275},[677],[568,4162,4164],{"className":4163},[682,683,684,685],[568,4165,4167],{"className":4166},[689,685],[568,4168,2413],{"className":4169},[689,685],[568,4171,4172,4175],{"style":1341},[568,4173],{"className":4174,"style":1275},[677],[568,4176],{"className":4177,"style":1349},[1348],[568,4179,4180,4183],{"style":1352},[568,4181],{"className":4182,"style":1275},[677],[568,4184,4186],{"className":4185},[682,683,684,685],[568,4187,4189],{"className":4188},[689,685],[568,4190,3028],{"className":4191},[689,685],[568,4193,1369],{"className":4194},[1368],[568,4196,4198],{"className":4197},[665],[568,4199,4201],{"className":4200,"style":3132},[669],[568,4202],{},[568,4204],{"className":4205},[861,1254],[568,4207,4209,4261],{"className":4208},[571],[568,4210,4212],{"className":4211},[575],[577,4213,4214],{"xmlns":579},[581,4215,4216,4258],{},[584,4217,4218,4220,4222,4224,4226,4228,4230,4232,4234,4250,4252],{},[592,4219,798],{},[596,4221,598],{},[596,4223,795],{"stretchy":794},[592,4225,3850],{},[596,4227,598],{},[600,4229,602],{},[596,4231,815],{"stretchy":794},[596,4233,624],{},[1148,4235,4236,4248],{},[584,4237,4238,4240,4242],{},[600,4239,602],{},[596,4241,608],{},[1148,4243,4244,4246],{},[600,4245,627],{},[600,4247,602],{},[600,4249,602],{},[596,4251,624],{},[1148,4253,4254,4256],{},[600,4255,3202],{},[600,4257,2413],{},[629,4259,4260],{"encoding":631},"\\cosh(\\ln 2) = \\frac{2 + \\frac{1}{2}}{2} = \\frac{5}{4}",[568,4262,4264,4297,4451],{"className":4263,"ariaHidden":637},[636],[568,4265,4267,4270,4273,4276,4279,4282,4285,4288,4291,4294],{"className":4266},[641],[568,4268],{"className":4269,"style":828},[645],[568,4271,798],{"className":4272},[650],[568,4274,795],{"className":4275},[832],[568,4277,3850],{"className":4278},[650],[568,4280],{"className":4281,"style":694},[693],[568,4283,602],{"className":4284},[689],[568,4286,815],{"className":4287},[861],[568,4289],{"className":4290,"style":756},[693],[568,4292,624],{"className":4293},[760],[568,4295],{"className":4296,"style":756},[693],[568,4298,4300,4303,4442,4445,4448],{"className":4299},[641],[568,4301],{"className":4302,"style":3974},[645],[568,4304,4306,4309,4439],{"className":4305},[689],[568,4307],{"className":4308},[832,1254],[568,4310,4312],{"className":4311},[1148],[568,4313,4315,4431],{"className":4314},[661,1261],[568,4316,4318,4428],{"className":4317},[665],[568,4319,4321,4335,4343],{"className":4320,"style":3993},[669],[568,4322,4323,4326],{"style":3088},[568,4324],{"className":4325,"style":1275},[677],[568,4327,4329],{"className":4328},[682,683,684,685],[568,4330,4332],{"className":4331},[689,685],[568,4333,602],{"className":4334},[689,685],[568,4336,4337,4340],{"style":1341},[568,4338],{"className":4339,"style":1275},[677],[568,4341],{"className":4342,"style":1349},[1348],[568,4344,4345,4348],{"style":4018},[568,4346],{"className":4347,"style":1275},[677],[568,4349,4351],{"className":4350},[682,683,684,685],[568,4352,4354,4357,4360],{"className":4353},[689,685],[568,4355,602],{"className":4356},[689,685],[568,4358,608],{"className":4359},[706,685],[568,4361,4363,4366,4425],{"className":4362},[689,685],[568,4364],{"className":4365},[832,1254,682,1327,4040],[568,4367,4369],{"className":4368},[1148],[568,4370,4372,4417],{"className":4371},[661,1261],[568,4373,4375,4414],{"className":4374},[665],[568,4376,4378,4392,4400],{"className":4377,"style":4053},[669],[568,4379,4380,4383],{"style":4056},[568,4381],{"className":4382,"style":1275},[677],[568,4384,4386],{"className":4385},[682,1327,1328,685],[568,4387,4389],{"className":4388},[689,685],[568,4390,602],{"className":4391},[689,685],[568,4393,4394,4397],{"style":4071},[568,4395],{"className":4396,"style":1275},[677],[568,4398],{"className":4399,"style":4078},[1348,685],[568,4401,4402,4405],{"style":4081},[568,4403],{"className":4404,"style":1275},[677],[568,4406,4408],{"className":4407},[682,1327,1328,685],[568,4409,4411],{"className":4410},[689,685],[568,4412,627],{"className":4413},[689,685],[568,4415,1369],{"className":4416},[1368],[568,4418,4420],{"className":4419},[665],[568,4421,4423],{"className":4422,"style":4103},[669],[568,4424],{},[568,4426],{"className":4427},[861,1254,682,1327,4040],[568,4429,1369],{"className":4430},[1368],[568,4432,4434],{"className":4433},[665],[568,4435,4437],{"className":4436,"style":3132},[669],[568,4438],{},[568,4440],{"className":4441},[861,1254],[568,4443],{"className":4444,"style":756},[693],[568,4446,624],{"className":4447},[760],[568,4449],{"className":4450,"style":756},[693],[568,4452,4454,4457],{"className":4453},[641],[568,4455],{"className":4456,"style":3067},[645],[568,4458,4460,4463,4522],{"className":4459},[689],[568,4461],{"className":4462},[832,1254],[568,4464,4466],{"className":4465},[1148],[568,4467,4469,4514],{"className":4468},[661,1261],[568,4470,4472,4511],{"className":4471},[665],[568,4473,4475,4489,4497],{"className":4474,"style":1268},[669],[568,4476,4477,4480],{"style":3088},[568,4478],{"className":4479,"style":1275},[677],[568,4481,4483],{"className":4482},[682,683,684,685],[568,4484,4486],{"className":4485},[689,685],[568,4487,2413],{"className":4488},[689,685],[568,4490,4491,4494],{"style":1341},[568,4492],{"className":4493,"style":1275},[677],[568,4495],{"className":4496,"style":1349},[1348],[568,4498,4499,4502],{"style":1352},[568,4500],{"className":4501,"style":1275},[677],[568,4503,4505],{"className":4504},[682,683,684,685],[568,4506,4508],{"className":4507},[689,685],[568,4509,3202],{"className":4510},[689,685],[568,4512,1369],{"className":4513},[1368],[568,4515,4517],{"className":4516},[665],[568,4518,4520],{"className":4519,"style":3132},[669],[568,4521],{},[568,4523],{"className":4524},[861,1254],[559,4526,4528],{"id":4527},"typische-fehler","Typische Fehler",[1036,4530,4531,4750,4959],{},[1039,4532,4533,4534,4669,4670,4698,4699,1382],{},"Das Pluszeichen aus der Trigonometrie übernehmen: ",[568,4535,4537,4575],{"className":4536},[571],[568,4538,4540],{"className":4539},[575],[577,4541,4542],{"xmlns":579},[581,4543,4544,4572],{},[584,4545,4546,4556,4558,4560,4570],{},[587,4547,4548,4554],{},[584,4549,4550,4552],{},[592,4551,798],{},[596,4553,598],{},[600,4555,602],{},[592,4557,605],{},[596,4559,608],{},[587,4561,4562,4568],{},[584,4563,4564,4566],{},[592,4565,808],{},[596,4567,598],{},[600,4569,602],{},[592,4571,605],{},[629,4573,4574],{"encoding":631},"\\cosh^2 x + \\sinh^2 x",[568,4576,4578,4628],{"className":4577,"ariaHidden":637},[636],[568,4579,4581,4584,4613,4616,4619,4622,4625],{"className":4580},[641],[568,4582],{"className":4583,"style":1761},[645],[568,4585,4587,4590],{"className":4586},[650],[568,4588,798],{"className":4589},[650],[568,4591,4593],{"className":4592},[657],[568,4594,4596],{"className":4595},[661],[568,4597,4599],{"className":4598},[665],[568,4600,4602],{"className":4601,"style":1078},[669],[568,4603,4604,4607],{"style":1099},[568,4605],{"className":4606,"style":678},[677],[568,4608,4610],{"className":4609},[682,683,684,685],[568,4611,602],{"className":4612},[689,685],[568,4614],{"className":4615,"style":694},[693],[568,4617,605],{"className":4618},[689,698],[568,4620],{"className":4621,"style":702},[693],[568,4623,608],{"className":4624},[706],[568,4626],{"className":4627,"style":702},[693],[568,4629,4631,4634,4663,4666],{"className":4630},[641],[568,4632],{"className":4633,"style":1078},[645],[568,4635,4637,4640],{"className":4636},[650],[568,4638,808],{"className":4639},[650],[568,4641,4643],{"className":4642},[657],[568,4644,4646],{"className":4645},[661],[568,4647,4649],{"className":4648},[665],[568,4650,4652],{"className":4651,"style":1078},[669],[568,4653,4654,4657],{"style":1099},[568,4655],{"className":4656,"style":678},[677],[568,4658,4660],{"className":4659},[682,683,684,685],[568,4661,602],{"className":4662},[689,685],[568,4664],{"className":4665,"style":694},[693],[568,4667,605],{"className":4668},[689,698]," ist nicht ",[568,4671,4673,4686],{"className":4672},[571],[568,4674,4676],{"className":4675},[575],[577,4677,4678],{"xmlns":579},[581,4679,4680,4684],{},[584,4681,4682],{},[600,4683,627],{},[629,4685,627],{"encoding":631},[568,4687,4689],{"className":4688,"ariaHidden":637},[636],[568,4690,4692,4695],{"className":4691},[641],[568,4693],{"className":4694,"style":770},[645],[568,4696,627],{"className":4697},[689],", sondern ",[568,4700,4702,4726],{"className":4701},[571],[568,4703,4705],{"className":4704},[575],[577,4706,4707],{"xmlns":579},[581,4708,4709,4723],{},[584,4710,4711,4713,4715,4717,4719,4721],{},[592,4712,798],{},[596,4714,598],{},[596,4716,795],{"stretchy":794},[600,4718,602],{},[592,4720,605],{},[596,4722,815],{"stretchy":794},[629,4724,4725],{"encoding":631},"\\cosh(2x)",[568,4727,4729],{"className":4728,"ariaHidden":637},[636],[568,4730,4732,4735,4738,4741,4744,4747],{"className":4731},[641],[568,4733],{"className":4734,"style":828},[645],[568,4736,798],{"className":4737},[650],[568,4739,795],{"className":4740},[832],[568,4742,602],{"className":4743},[689],[568,4745,605],{"className":4746},[689,698],[568,4748,815],{"className":4749},[861],[1039,4751,4752,4753,4926,4927,4958],{},"Bei ",[568,4754,4756,4799],{"className":4755},[571],[568,4757,4759],{"className":4758},[575],[577,4760,4761],{"xmlns":579},[581,4762,4763,4796],{},[584,4764,4765,4767,4769,4771,4773,4776],{},[592,4766,798],{},[596,4768,598],{},[592,4770,605],{},[596,4772,624],{},[596,4774,4775],{},"±",[1496,4777,4778],{},[584,4779,4780,4782,4784,4794],{},[600,4781,627],{},[596,4783,608],{},[587,4785,4786,4792],{},[584,4787,4788,4790],{},[592,4789,808],{},[596,4791,598],{},[600,4793,602],{},[592,4795,605],{},[629,4797,4798],{"encoding":631},"\\cosh x = \\pm\\sqrt{1 + \\sinh^2 x}",[568,4800,4802,4826],{"className":4801,"ariaHidden":637},[636],[568,4803,4805,4808,4811,4814,4817,4820,4823],{"className":4804},[641],[568,4806],{"className":4807,"style":1528},[645],[568,4809,798],{"className":4810},[650],[568,4812],{"className":4813,"style":694},[693],[568,4815,605],{"className":4816},[689,698],[568,4818],{"className":4819,"style":756},[693],[568,4821,624],{"className":4822},[760],[568,4824],{"className":4825,"style":756},[693],[568,4827,4829,4832,4835],{"className":4828},[641],[568,4830],{"className":4831,"style":1553},[645],[568,4833,4775],{"className":4834},[689],[568,4836,4838],{"className":4837},[689,1557],[568,4839,4841,4918],{"className":4840},[661,1261],[568,4842,4844,4915],{"className":4843},[665],[568,4845,4847,4903],{"className":4846,"style":1567},[669],[568,4848,4850,4853],{"className":4849,"style":1572},[1571],[568,4851],{"className":4852,"style":1576},[677],[568,4854,4856,4859,4862,4865,4868,4897,4900],{"className":4855,"style":1580},[689],[568,4857,627],{"className":4858},[689],[568,4860],{"className":4861,"style":702},[693],[568,4863,608],{"className":4864},[706],[568,4866],{"className":4867,"style":702},[693],[568,4869,4871,4874],{"className":4870},[650],[568,4872,808],{"className":4873},[650],[568,4875,4877],{"className":4876},[657],[568,4878,4880],{"className":4879},[661],[568,4881,4883],{"className":4882},[665],[568,4884,4886],{"className":4885,"style":1078},[669],[568,4887,4888,4891],{"style":1099},[568,4889],{"className":4890,"style":678},[677],[568,4892,4894],{"className":4893},[682,683,684,685],[568,4895,602],{"className":4896},[689,685],[568,4898],{"className":4899,"style":694},[693],[568,4901,605],{"className":4902},[689,698],[568,4904,4905,4908],{"style":1630},[568,4906],{"className":4907,"style":1576},[677],[568,4909,4911],{"className":4910,"style":1638},[1637],[1640,4912,4913],{"xmlns":1642,"width":1643,"height":1644,"viewBox":1645,"preserveAspectRatio":1646},[1648,4914],{"d":1650},[568,4916,1369],{"className":4917},[1368],[568,4919,4921],{"className":4920},[665],[568,4922,4924],{"className":4923,"style":1660},[669],[568,4925],{}," die negative Lösung mitnehmen. ",[568,4928,4930,4946],{"className":4929},[571],[568,4931,4933],{"className":4932},[575],[577,4934,4935],{"xmlns":579},[581,4936,4937,4943],{},[584,4938,4939,4941],{},[592,4940,798],{},[596,4942,598],{},[629,4944,4945],{"encoding":631},"\\cosh",[568,4947,4949],{"className":4948,"ariaHidden":637},[636],[568,4950,4952,4955],{"className":4951},[641],[568,4953],{"className":4954,"style":1528},[645],[568,4956,798],{"className":4957},[650]," ist immer positiv.",[1039,4960,4961,5201,5202,1382],{},[568,4962,4964,5012],{"className":4963},[571],[568,4965,4967],{"className":4966},[575],[577,4968,4969],{"xmlns":579},[581,4970,4971,5009],{},[584,4972,4973,4975,4977,4987,4989,4991],{},[600,4974,627],{},[596,4976,608],{},[587,4978,4979,4985],{},[584,4980,4981,4983],{},[592,4982,1138],{},[596,4984,598],{},[600,4986,602],{},[592,4988,605],{},[596,4990,624],{},[1148,4992,4993,4995],{},[600,4994,627],{},[584,4996,4997,5007],{},[587,4998,4999,5005],{},[584,5000,5001,5003],{},[592,5002,798],{},[596,5004,598],{},[600,5006,602],{},[592,5008,605],{},[629,5010,5011],{"encoding":631},"1 + \\tanh^2 x = \\frac{1}{\\cosh^2 x}",[568,5013,5015,5033,5083],{"className":5014,"ariaHidden":637},[636],[568,5016,5018,5021,5024,5027,5030],{"className":5017},[641],[568,5019],{"className":5020,"style":1178},[645],[568,5022,627],{"className":5023},[689],[568,5025],{"className":5026,"style":702},[693],[568,5028,608],{"className":5029},[706],[568,5031],{"className":5032,"style":702},[693],[568,5034,5036,5039,5068,5071,5074,5077,5080],{"className":5035},[641],[568,5037],{"className":5038,"style":1078},[645],[568,5040,5042,5045],{"className":5041},[650],[568,5043,1138],{"className":5044},[650],[568,5046,5048],{"className":5047},[657],[568,5049,5051],{"className":5050},[661],[568,5052,5054],{"className":5053},[665],[568,5055,5057],{"className":5056,"style":1078},[669],[568,5058,5059,5062],{"style":1099},[568,5060],{"className":5061,"style":678},[677],[568,5063,5065],{"className":5064},[682,683,684,685],[568,5066,602],{"className":5067},[689,685],[568,5069],{"className":5070,"style":694},[693],[568,5072,605],{"className":5073},[689,698],[568,5075],{"className":5076,"style":756},[693],[568,5078,624],{"className":5079},[760],[568,5081],{"className":5082,"style":756},[693],[568,5084,5086,5089],{"className":5085},[641],[568,5087],{"className":5088,"style":1247},[645],[568,5090,5092,5095,5198],{"className":5091},[689],[568,5093],{"className":5094},[832,1254],[568,5096,5098],{"className":5097},[1148],[568,5099,5101,5190],{"className":5100},[661,1261],[568,5102,5104,5187],{"className":5103},[665],[568,5105,5107,5165,5173],{"className":5106,"style":1268},[669],[568,5108,5109,5112],{"style":1271},[568,5110],{"className":5111,"style":1275},[677],[568,5113,5115],{"className":5114},[682,683,684,685],[568,5116,5118,5159,5162],{"className":5117},[689,685],[568,5119,5121,5136],{"className":5120},[650,685],[568,5122,5124,5127,5130,5133],{"className":5123},[650,685],[568,5125,1291],{"className":5126},[685],[568,5128,1295],{"className":5129},[685],[568,5131,1299],{"className":5132},[685],[568,5134,1303],{"className":5135},[685],[568,5137,5139],{"className":5138},[657],[568,5140,5142],{"className":5141},[661],[568,5143,5145],{"className":5144},[665],[568,5146,5148],{"className":5147,"style":1316},[669],[568,5149,5150,5153],{"style":1319},[568,5151],{"className":5152,"style":1323},[677],[568,5154,5156],{"className":5155},[682,1327,1328,685],[568,5157,602],{"className":5158},[689,685],[568,5160],{"className":5161,"style":1335},[693,685],[568,5163,605],{"className":5164},[689,698,685],[568,5166,5167,5170],{"style":1341},[568,5168],{"className":5169,"style":1275},[677],[568,5171],{"className":5172,"style":1349},[1348],[568,5174,5175,5178],{"style":1352},[568,5176],{"className":5177,"style":1275},[677],[568,5179,5181],{"className":5180},[682,683,684,685],[568,5182,5184],{"className":5183},[689,685],[568,5185,627],{"className":5186},[689,685],[568,5188,1369],{"className":5189},[1368],[568,5191,5193],{"className":5192},[665],[568,5194,5196],{"className":5195,"style":1376},[669],[568,5197],{},[568,5199],{"className":5200},[861,1254]," schreiben. Richtig ist ",[568,5203,5205,5252],{"className":5204},[571],[568,5206,5208],{"className":5207},[575],[577,5209,5210],{"xmlns":579},[581,5211,5212,5250],{},[584,5213,5214,5216,5218,5228,5230,5232],{},[600,5215,627],{},[596,5217,914],{},[587,5219,5220,5226],{},[584,5221,5222,5224],{},[592,5223,1138],{},[596,5225,598],{},[600,5227,602],{},[592,5229,605],{},[596,5231,624],{},[1148,5233,5234,5236],{},[600,5235,627],{},[584,5237,5238,5248],{},[587,5239,5240,5246],{},[584,5241,5242,5244],{},[592,5243,798],{},[596,5245,598],{},[600,5247,602],{},[592,5249,605],{},[629,5251,1168],{"encoding":631},[568,5253,5255,5273,5323],{"className":5254,"ariaHidden":637},[636],[568,5256,5258,5261,5264,5267,5270],{"className":5257},[641],[568,5259],{"className":5260,"style":1178},[645],[568,5262,627],{"className":5263},[689],[568,5265],{"className":5266,"style":702},[693],[568,5268,914],{"className":5269},[706],[568,5271],{"className":5272,"style":702},[693],[568,5274,5276,5279,5308,5311,5314,5317,5320],{"className":5275},[641],[568,5277],{"className":5278,"style":1078},[645],[568,5280,5282,5285],{"className":5281},[650],[568,5283,1138],{"className":5284},[650],[568,5286,5288],{"className":5287},[657],[568,5289,5291],{"className":5290},[661],[568,5292,5294],{"className":5293},[665],[568,5295,5297],{"className":5296,"style":1078},[669],[568,5298,5299,5302],{"style":1099},[568,5300],{"className":5301,"style":678},[677],[568,5303,5305],{"className":5304},[682,683,684,685],[568,5306,602],{"className":5307},[689,685],[568,5309],{"className":5310,"style":694},[693],[568,5312,605],{"className":5313},[689,698],[568,5315],{"className":5316,"style":756},[693],[568,5318,624],{"className":5319},[760],[568,5321],{"className":5322,"style":756},[693],[568,5324,5326,5329],{"className":5325},[641],[568,5327],{"className":5328,"style":1247},[645],[568,5330,5332,5335,5438],{"className":5331},[689],[568,5333],{"className":5334},[832,1254],[568,5336,5338],{"className":5337},[1148],[568,5339,5341,5430],{"className":5340},[661,1261],[568,5342,5344,5427],{"className":5343},[665],[568,5345,5347,5405,5413],{"className":5346,"style":1268},[669],[568,5348,5349,5352],{"style":1271},[568,5350],{"className":5351,"style":1275},[677],[568,5353,5355],{"className":5354},[682,683,684,685],[568,5356,5358,5399,5402],{"className":5357},[689,685],[568,5359,5361,5376],{"className":5360},[650,685],[568,5362,5364,5367,5370,5373],{"className":5363},[650,685],[568,5365,1291],{"className":5366},[685],[568,5368,1295],{"className":5369},[685],[568,5371,1299],{"className":5372},[685],[568,5374,1303],{"className":5375},[685],[568,5377,5379],{"className":5378},[657],[568,5380,5382],{"className":5381},[661],[568,5383,5385],{"className":5384},[665],[568,5386,5388],{"className":5387,"style":1316},[669],[568,5389,5390,5393],{"style":1319},[568,5391],{"className":5392,"style":1323},[677],[568,5394,5396],{"className":5395},[682,1327,1328,685],[568,5397,602],{"className":5398},[689,685],[568,5400],{"className":5401,"style":1335},[693,685],[568,5403,605],{"className":5404},[689,698,685],[568,5406,5407,5410],{"style":1341},[568,5408],{"className":5409,"style":1275},[677],[568,5411],{"className":5412,"style":1349},[1348],[568,5414,5415,5418],{"style":1352},[568,5416],{"className":5417,"style":1275},[677],[568,5419,5421],{"className":5420},[682,683,684,685],[568,5422,5424],{"className":5423},[689,685],[568,5425,627],{"className":5426},[689,685],[568,5428,1369],{"className":5429},[1368],[568,5431,5433],{"className":5432},[665],[568,5434,5436],{"className":5435,"style":1376},[669],[568,5437],{},[568,5439],{"className":5440},[861,1254],{"title":5442,"searchDepth":5443,"depth":5443,"links":5444},"",2,[5445,5446,5447,5448],{"id":561,"depth":5443,"text":562},{"id":2328,"depth":5443,"text":2329},{"id":2999,"depth":5443,"text":3000},{"id":4527,"depth":5443,"text":4528},[5450,5452,5454,5457,5460,5463,5466,5469,5471],{"lhs":5451,"rhs":627},"cosh(x)^2 - sinh(x)^2",{"lhs":5453,"rhs":627},"((exp(x) + exp(-x))\u002F2)^2 - ((exp(x) - exp(-x))\u002F2)^2",{"lhs":5455,"rhs":5456},"1 - tanh(x)^2","1\u002Fcosh(x)^2",{"lhs":5458,"rhs":5459},"cosh(2*x)","cosh(x)^2 + sinh(x)^2",{"lhs":5461,"rhs":5462},"sinh(2*x)","2*sinh(x)*cosh(x)",{"lhs":5464,"rhs":5465},"sqrt(1 + 0.75^2)","1.25",{"lhs":5467,"rhs":5468},"sinh(log(2))","0.75",{"lhs":5470,"rhs":5465},"cosh(log(2))",{"lhs":5472,"rhs":5473},"0.75\u002F1.25","0.6",null,"md",{"caption":5477,"x":5478,"y":5481,"unit":5484,"items":5485},"Einheitshyperbel X² − Y² = 1 mit ihren Asymptoten; der Punkt (cosh x, sinh x) liegt auf ihr, für x = ln 2 ist er (5\u002F4, 3\u002F4)",[5479,5480],-2,3.8,[5482,5483],-2.3,2.3,48,[5486,5492,5495,5499,5502,5507,5510,5515,5518,5521,5523,5530,5535],{"type":5487,"from":5488,"to":5490,"style":5491},"segment",[5479,5489],0,[5480,5489],"muted",{"type":5487,"from":5493,"to":5494,"style":5491},[5489,5482],[5489,5483],{"type":5487,"from":5496,"to":5497,"style":5498},[5479,5479],[5483,5483],"dashed",{"type":5487,"from":5500,"to":5501,"style":5498},[5479,5443],[5483,5482],{"type":5487,"from":5503,"to":5505,"style":5498},[5504,5489],1.25,[5504,5506],0.75,{"type":5487,"from":5508,"to":5509,"style":5498},[5489,5506],[5504,5506],{"type":5511,"expr":5512,"domain":5513},"function","sqrt(x^2 - 1)",[5514,5480],1,{"type":5511,"expr":5516,"domain":5517},"-sqrt(x^2 - 1)",[5514,5480],{"type":5511,"expr":5512,"domain":5519},[5479,5520],-1,{"type":5511,"expr":5516,"domain":5522},[5479,5520],{"type":5524,"at":5525,"label":5526,"offset":5527},"point",[5504,5506],"(cosh x, sinh x)",[5528,5529],6,16,{"type":5531,"at":5532,"text":909},"label",[5533,5534],3.6,-0.3,{"type":5531,"at":5536,"text":919},[5537,5538],0.3,2.1,"\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"0 0 278.4 220.8\" width=\"278.4\" height=\"220.8\" role=\"img\" aria-label=\"Einheitshyperbel X² − Y² = 1 mit ihren Asymptoten; der Punkt (cosh x, sinh x) liegt auf ihr, für x = ln 2 ist er (5\u002F4, 3\u002F4)\" class=\"formula-figure-svg\">\u003Cdefs>\u003Cmarker id=\"fig-hyperbolic-identity-arrow\" viewBox=\"0 0 10 10\" refX=\"9\" refY=\"5\" markerWidth=\"7\" markerHeight=\"7\" orient=\"auto-start-reverse\">\u003Cpath d=\"M 0 0 L 10 5 L 0 10 z\" class=\"fig-arrowhead\"\u002F>\u003C\u002Fmarker>\u003CclipPath id=\"fig-hyperbolic-identity-clip\">\u003Crect x=\"0\" y=\"0\" width=\"278.4\" height=\"220.8\"\u002F>\u003C\u002FclipPath>\u003C\u002Fdefs>\u003Cline x1=\"0\" y1=\"110.4\" x2=\"278.4\" y2=\"110.4\" class=\"fig-muted\"\u002F>\u003Cline x1=\"96\" y1=\"220.8\" x2=\"96\" y2=\"0\" class=\"fig-muted\"\u002F>\u003Cline x1=\"0\" y1=\"206.4\" x2=\"206.4\" y2=\"0\" class=\"fig-dashed\"\u002F>\u003Cline x1=\"0\" y1=\"14.4\" x2=\"206.4\" y2=\"220.8\" class=\"fig-dashed\"\u002F>\u003Cline x1=\"156\" y1=\"110.4\" x2=\"156\" y2=\"74.4\" class=\"fig-dashed\"\u002F>\u003Cline x1=\"96\" y1=\"74.4\" x2=\"156\" y2=\"74.4\" class=\"fig-dashed\"\u002F>\u003Cpath d=\"M 144 110.4 L 144.56 103.05 L 145.12 99.97 L 145.68 97.59 L 146.24 95.57 L 146.8 93.77 L 147.36 92.13 L 147.92 90.61 L 148.48 89.18 L 149.04 87.83 L 149.6 86.55 L 150.16 85.31 L 150.72 84.13 L 151.28 82.98 L 151.84 81.87 L 152.4 80.79 L 152.96 79.73 L 153.52 78.71 L 154.08 77.7 L 154.64 76.72 L 155.2 75.75 L 155.76 74.8 L 156.32 73.87 L 156.88 72.95 L 157.44 72.05 L 158 71.16 L 158.56 70.28 L 159.12 69.41 L 159.68 68.55 L 160.24 67.71 L 160.8 66.87 L 161.36 66.04 L 161.92 65.22 L 162.48 64.4 L 163.04 63.6 L 163.6 62.8 L 164.16 62.01 L 164.72 61.22 L 165.28 60.44 L 165.84 59.67 L 166.4 58.9 L 166.96 58.14 L 167.52 57.38 L 168.08 56.63 L 168.64 55.88 L 169.2 55.13 L 169.76 54.4 L 170.32 53.66 L 170.88 52.93 L 171.44 52.2 L 172 51.48 L 172.56 50.76 L 173.12 50.04 L 173.68 49.32 L 174.24 48.61 L 174.8 47.91 L 175.36 47.2 L 175.92 46.5 L 176.48 45.8 L 177.04 45.1 L 177.6 44.41 L 178.16 43.72 L 178.72 43.03 L 179.28 42.34 L 179.84 41.66 L 180.4 40.98 L 180.96 40.3 L 181.52 39.62 L 182.08 38.95 L 182.64 38.27 L 183.2 37.6 L 183.76 36.93 L 184.32 36.26 L 184.88 35.6 L 185.44 34.93 L 186 34.27 L 186.56 33.61 L 187.12 32.95 L 187.68 32.29 L 188.24 31.63 L 188.8 30.98 L 189.36 30.32 L 189.92 29.67 L 190.48 29.02 L 191.04 28.37 L 191.6 27.72 L 192.16 27.08 L 192.72 26.43 L 193.28 25.79 L 193.84 25.14 L 194.4 24.5 L 194.96 23.86 L 195.52 23.22 L 196.08 22.58 L 196.64 21.94 L 197.2 21.31 L 197.76 20.67 L 198.32 20.04 L 198.88 19.4 L 199.44 18.77 L 200 18.14 L 200.56 17.51 L 201.12 16.88 L 201.68 16.25 L 202.24 15.62 L 202.8 14.99 L 203.36 14.37 L 203.92 13.74 L 204.48 13.12 L 205.04 12.49 L 205.6 11.87 L 206.16 11.25 L 206.72 10.63 L 207.28 10 L 207.84 9.38 L 208.4 8.76 L 208.96 8.15 L 209.52 7.53 L 210.08 6.91 L 210.64 6.29 L 211.2 5.68 L 211.76 5.06 L 212.32 4.45 L 212.88 3.83 L 213.44 3.22 L 214 2.6 L 214.56 1.99 L 215.12 1.38 L 215.68 0.77 L 216.24 0.16 L 216.8 -0.45 L 217.36 -1.06 L 217.92 -1.67 L 218.48 -2.28 L 219.04 -2.89 L 219.6 -3.5 L 220.16 -4.11 L 220.72 -4.71 L 221.28 -5.32 L 221.84 -5.93 L 222.4 -6.53 L 222.96 -7.14 L 223.52 -7.74 L 224.08 -8.35 L 224.64 -8.95 L 225.2 -9.55 L 225.76 -10.16 L 226.32 -10.76 L 226.88 -11.36 L 227.44 -11.96 L 228 -12.56 L 228.56 -13.16 L 229.12 -13.76 L 229.68 -14.37 L 230.24 -14.96 L 230.8 -15.56 L 231.36 -16.16 L 231.92 -16.76 L 232.48 -17.36 L 233.04 -17.96 L 233.6 -18.56 L 234.16 -19.15 L 234.72 -19.75 L 235.28 -20.35 L 235.84 -20.94 L 236.4 -21.54 L 236.96 -22.14 L 237.52 -22.73 L 238.08 -23.33 L 238.64 -23.92 L 239.2 -24.52 L 239.76 -25.11 L 240.32 -25.7 L 240.88 -26.3 L 241.44 -26.89 L 242 -27.48 L 242.56 -28.08 L 243.12 -28.67 L 243.68 -29.26 L 244.24 -29.85 L 244.8 -30.45 L 245.36 -31.04 L 245.92 -31.63 L 246.48 -32.22 L 247.04 -32.81 L 247.6 -33.4 L 248.16 -33.99 L 248.72 -34.58 L 249.28 -35.17 L 249.84 -35.76 L 250.4 -36.35 L 250.96 -36.94 L 251.52 -37.53 L 252.08 -38.12 L 252.64 -38.7 L 253.2 -39.29 L 253.76 -39.88 L 254.32 -40.47 L 254.88 -41.06 L 255.44 -41.64 L 256 -42.23 L 256.56 -42.82 L 257.12 -43.4 L 257.68 -43.99 L 258.24 -44.58 L 258.8 -45.16 L 259.36 -45.75 L 259.92 -46.33 L 260.48 -46.92 L 261.04 -47.51 L 261.6 -48.09 L 262.16 -48.68 L 262.72 -49.26 L 263.28 -49.85 L 263.84 -50.43 L 264.4 -51.01 L 264.96 -51.6 L 265.52 -52.18 L 266.08 -52.77 L 266.64 -53.35 L 267.2 -53.93 L 267.76 -54.52 L 268.32 -55.1 L 268.88 -55.68 L 269.44 -56.27 L 270 -56.85 L 270.56 -57.43 L 271.12 -58.01 L 271.68 -58.6 L 272.24 -59.18 L 272.8 -59.76 L 273.36 -60.34 L 273.92 -60.92 L 274.48 -61.5 L 275.04 -62.09 L 275.6 -62.67 L 276.16 -63.25 L 276.72 -63.83 L 277.28 -64.41 L 277.84 -64.99 L 278.4 -65.57\" class=\"fig-main fig-graph\" clip-path=\"url(#fig-hyperbolic-identity-clip)\"\u002F>\u003Cpath d=\"M 144 110.4 L 144.56 117.75 L 145.12 120.83 L 145.68 123.21 L 146.24 125.23 L 146.8 127.03 L 147.36 128.67 L 147.92 130.19 L 148.48 131.62 L 149.04 132.97 L 149.6 134.25 L 150.16 135.49 L 150.72 136.67 L 151.28 137.82 L 151.84 138.93 L 152.4 140.01 L 152.96 141.07 L 153.52 142.09 L 154.08 143.1 L 154.64 144.08 L 155.2 145.05 L 155.76 146 L 156.32 146.93 L 156.88 147.85 L 157.44 148.75 L 158 149.64 L 158.56 150.52 L 159.12 151.39 L 159.68 152.25 L 160.24 153.09 L 160.8 153.93 L 161.36 154.76 L 161.92 155.58 L 162.48 156.4 L 163.04 157.2 L 163.6 158 L 164.16 158.79 L 164.72 159.58 L 165.28 160.36 L 165.84 161.13 L 166.4 161.9 L 166.96 162.66 L 167.52 163.42 L 168.08 164.17 L 168.64 164.92 L 169.2 165.67 L 169.76 166.4 L 170.32 167.14 L 170.88 167.87 L 171.44 168.6 L 172 169.32 L 172.56 170.04 L 173.12 170.76 L 173.68 171.48 L 174.24 172.19 L 174.8 172.89 L 175.36 173.6 L 175.92 174.3 L 176.48 175 L 177.04 175.7 L 177.6 176.39 L 178.16 177.08 L 178.72 177.77 L 179.28 178.46 L 179.84 179.14 L 180.4 179.82 L 180.96 180.5 L 181.52 181.18 L 182.08 181.85 L 182.64 182.53 L 183.2 183.2 L 183.76 183.87 L 184.32 184.54 L 184.88 185.2 L 185.44 185.87 L 186 186.53 L 186.56 187.19 L 187.12 187.85 L 187.68 188.51 L 188.24 189.17 L 188.8 189.82 L 189.36 190.48 L 189.92 191.13 L 190.48 191.78 L 191.04 192.43 L 191.6 193.08 L 192.16 193.72 L 192.72 194.37 L 193.28 195.01 L 193.84 195.66 L 194.4 196.3 L 194.96 196.94 L 195.52 197.58 L 196.08 198.22 L 196.64 198.86 L 197.2 199.49 L 197.76 200.13 L 198.32 200.76 L 198.88 201.4 L 199.44 202.03 L 200 202.66 L 200.56 203.29 L 201.12 203.92 L 201.68 204.55 L 202.24 205.18 L 202.8 205.81 L 203.36 206.43 L 203.92 207.06 L 204.48 207.68 L 205.04 208.31 L 205.6 208.93 L 206.16 209.55 L 206.72 210.17 L 207.28 210.8 L 207.84 211.42 L 208.4 212.04 L 208.96 212.65 L 209.52 213.27 L 210.08 213.89 L 210.64 214.51 L 211.2 215.12 L 211.76 215.74 L 212.32 216.35 L 212.88 216.97 L 213.44 217.58 L 214 218.2 L 214.56 218.81 L 215.12 219.42 L 215.68 220.03 L 216.24 220.64 L 216.8 221.25 L 217.36 221.86 L 217.92 222.47 L 218.48 223.08 L 219.04 223.69 L 219.6 224.3 L 220.16 224.91 L 220.72 225.51 L 221.28 226.12 L 221.84 226.73 L 222.4 227.33 L 222.96 227.94 L 223.52 228.54 L 224.08 229.15 L 224.64 229.75 L 225.2 230.35 L 225.76 230.96 L 226.32 231.56 L 226.88 232.16 L 227.44 232.76 L 228 233.36 L 228.56 233.96 L 229.12 234.56 L 229.68 235.17 L 230.24 235.76 L 230.8 236.36 L 231.36 236.96 L 231.92 237.56 L 232.48 238.16 L 233.04 238.76 L 233.6 239.36 L 234.16 239.95 L 234.72 240.55 L 235.28 241.15 L 235.84 241.74 L 236.4 242.34 L 236.96 242.94 L 237.52 243.53 L 238.08 244.13 L 238.64 244.72 L 239.2 245.32 L 239.76 245.91 L 240.32 246.5 L 240.88 247.1 L 241.44 247.69 L 242 248.28 L 242.56 248.88 L 243.12 249.47 L 243.68 250.06 L 244.24 250.65 L 244.8 251.25 L 245.36 251.84 L 245.92 252.43 L 246.48 253.02 L 247.04 253.61 L 247.6 254.2 L 248.16 254.79 L 248.72 255.38 L 249.28 255.97 L 249.84 256.56 L 250.4 257.15 L 250.96 257.74 L 251.52 258.33 L 252.08 258.92 L 252.64 259.5 L 253.2 260.09 L 253.76 260.68 L 254.32 261.27 L 254.88 261.86 L 255.44 262.44 L 256 263.03 L 256.56 263.62 L 257.12 264.2 L 257.68 264.79 L 258.24 265.38 L 258.8 265.96 L 259.36 266.55 L 259.92 267.13 L 260.48 267.72 L 261.04 268.31 L 261.6 268.89 L 262.16 269.48 L 262.72 270.06 L 263.28 270.65 L 263.84 271.23 L 264.4 271.81 L 264.96 272.4 L 265.52 272.98 L 266.08 273.57 L 266.64 274.15 L 267.2 274.73 L 267.76 275.32 L 268.32 275.9 L 268.88 276.48 L 269.44 277.07 L 270 277.65 L 270.56 278.23 L 271.12 278.81 L 271.68 279.4 L 272.24 279.98 L 272.8 280.56 L 273.36 281.14 L 273.92 281.72 L 274.48 282.3 L 275.04 282.89 L 275.6 283.47 L 276.16 284.05 L 276.72 284.63 L 277.28 285.21 L 277.84 285.79 L 278.4 286.37\" class=\"fig-main fig-graph\" clip-path=\"url(#fig-hyperbolic-identity-clip)\"\u002F>\u003Cpath d=\"M 0 27.26 L 0.2 27.49 L 0.4 27.72 L 0.6 27.96 L 0.8 28.19 L 1 28.42 L 1.2 28.65 L 1.4 28.88 L 1.6 29.11 L 1.8 29.35 L 2 29.58 L 2.2 29.81 L 2.4 30.04 L 2.6 30.28 L 2.8 30.51 L 3 30.74 L 3.2 30.98 L 3.4 31.21 L 3.6 31.45 L 3.8 31.68 L 4 31.91 L 4.2 32.15 L 4.4 32.38 L 4.6 32.62 L 4.8 32.85 L 5 33.09 L 5.2 33.32 L 5.4 33.56 L 5.6 33.8 L 5.8 34.03 L 6 34.27 L 6.2 34.51 L 6.4 34.74 L 6.6 34.98 L 6.8 35.22 L 7 35.45 L 7.2 35.69 L 7.4 35.93 L 7.6 36.17 L 7.8 36.41 L 8 36.64 L 8.2 36.88 L 8.4 37.12 L 8.6 37.36 L 8.8 37.6 L 9 37.84 L 9.2 38.08 L 9.4 38.32 L 9.6 38.56 L 9.8 38.8 L 10 39.04 L 10.2 39.28 L 10.4 39.52 L 10.6 39.77 L 10.8 40.01 L 11 40.25 L 11.2 40.49 L 11.4 40.74 L 11.6 40.98 L 11.8 41.22 L 12 41.47 L 12.2 41.71 L 12.4 41.95 L 12.6 42.2 L 12.8 42.44 L 13 42.69 L 13.2 42.93 L 13.4 43.18 L 13.6 43.42 L 13.8 43.67 L 14 43.92 L 14.2 44.16 L 14.4 44.41 L 14.6 44.66 L 14.8 44.91 L 15 45.15 L 15.2 45.4 L 15.4 45.65 L 15.6 45.9 L 15.8 46.15 L 16 46.4 L 16.2 46.65 L 16.4 46.9 L 16.6 47.15 L 16.8 47.4 L 17 47.65 L 17.2 47.91 L 17.4 48.16 L 17.6 48.41 L 17.8 48.66 L 18 48.92 L 18.2 49.17 L 18.4 49.43 L 18.6 49.68 L 18.8 49.94 L 19 50.19 L 19.2 50.45 L 19.4 50.7 L 19.6 50.96 L 19.8 51.22 L 20 51.48 L 20.2 51.73 L 20.4 51.99 L 20.6 52.25 L 20.8 52.51 L 21 52.77 L 21.2 53.03 L 21.4 53.29 L 21.6 53.55 L 21.8 53.82 L 22 54.08 L 22.2 54.34 L 22.4 54.61 L 22.6 54.87 L 22.8 55.13 L 23 55.4 L 23.2 55.67 L 23.4 55.93 L 23.6 56.2 L 23.8 56.47 L 24 56.73 L 24.2 57 L 24.4 57.27 L 24.6 57.54 L 24.8 57.81 L 25 58.08 L 25.2 58.36 L 25.4 58.63 L 25.6 58.9 L 25.8 59.17 L 26 59.45 L 26.2 59.72 L 26.4 60 L 26.6 60.28 L 26.8 60.55 L 27 60.83 L 27.2 61.11 L 27.4 61.39 L 27.6 61.67 L 27.8 61.95 L 28 62.23 L 28.2 62.52 L 28.4 62.8 L 28.6 63.08 L 28.8 63.37 L 29 63.66 L 29.2 63.94 L 29.4 64.23 L 29.6 64.52 L 29.8 64.81 L 30 65.1 L 30.2 65.39 L 30.4 65.69 L 30.6 65.98 L 30.8 66.27 L 31 66.57 L 31.2 66.87 L 31.4 67.17 L 31.6 67.47 L 31.8 67.77 L 32 68.07 L 32.2 68.37 L 32.4 68.68 L 32.6 68.98 L 32.8 69.29 L 33 69.6 L 33.2 69.91 L 33.4 70.22 L 33.6 70.53 L 33.8 70.84 L 34 71.16 L 34.2 71.47 L 34.4 71.79 L 34.6 72.11 L 34.8 72.43 L 35 72.76 L 35.2 73.08 L 35.4 73.41 L 35.6 73.74 L 35.8 74.07 L 36 74.4 L 36.2 74.73 L 36.4 75.07 L 36.6 75.41 L 36.8 75.75 L 37 76.09 L 37.2 76.44 L 37.4 76.79 L 37.6 77.14 L 37.8 77.49 L 38 77.84 L 38.2 78.2 L 38.4 78.56 L 38.6 78.92 L 38.8 79.29 L 39 79.66 L 39.2 80.03 L 39.4 80.41 L 39.6 80.79 L 39.8 81.17 L 40 81.56 L 40.2 81.95 L 40.4 82.34 L 40.6 82.74 L 40.8 83.14 L 41 83.55 L 41.2 83.96 L 41.4 84.38 L 41.6 84.8 L 41.8 85.23 L 42 85.66 L 42.2 86.1 L 42.4 86.55 L 42.6 87 L 42.8 87.46 L 43 87.93 L 43.2 88.4 L 43.4 88.89 L 43.6 89.38 L 43.8 89.89 L 44 90.4 L 44.2 90.93 L 44.4 91.46 L 44.6 92.02 L 44.8 92.58 L 45 93.17 L 45.2 93.77 L 45.4 94.39 L 45.6 95.03 L 45.8 95.7 L 46 96.4 L 46.2 97.13 L 46.4 97.9 L 46.6 98.72 L 46.8 99.6 L 47 100.55 L 47.2 101.6 L 47.4 102.79 L 47.6 104.19 L 47.8 106.01 L 48 110.4\" class=\"fig-main fig-graph\" clip-path=\"url(#fig-hyperbolic-identity-clip)\"\u002F>\u003Cpath d=\"M 0 193.54 L 0.2 193.31 L 0.4 193.08 L 0.6 192.84 L 0.8 192.61 L 1 192.38 L 1.2 192.15 L 1.4 191.92 L 1.6 191.69 L 1.8 191.45 L 2 191.22 L 2.2 190.99 L 2.4 190.76 L 2.6 190.52 L 2.8 190.29 L 3 190.06 L 3.2 189.82 L 3.4 189.59 L 3.6 189.35 L 3.8 189.12 L 4 188.89 L 4.2 188.65 L 4.4 188.42 L 4.6 188.18 L 4.8 187.95 L 5 187.71 L 5.2 187.48 L 5.4 187.24 L 5.6 187 L 5.8 186.77 L 6 186.53 L 6.2 186.29 L 6.4 186.06 L 6.6 185.82 L 6.8 185.58 L 7 185.35 L 7.2 185.11 L 7.4 184.87 L 7.6 184.63 L 7.8 184.39 L 8 184.16 L 8.2 183.92 L 8.4 183.68 L 8.6 183.44 L 8.8 183.2 L 9 182.96 L 9.2 182.72 L 9.4 182.48 L 9.6 182.24 L 9.8 182 L 10 181.76 L 10.2 181.52 L 10.4 181.28 L 10.6 181.03 L 10.8 180.79 L 11 180.55 L 11.2 180.31 L 11.4 180.06 L 11.6 179.82 L 11.8 179.58 L 12 179.33 L 12.2 179.09 L 12.4 178.85 L 12.6 178.6 L 12.8 178.36 L 13 178.11 L 13.2 177.87 L 13.4 177.62 L 13.6 177.38 L 13.8 177.13 L 14 176.88 L 14.2 176.64 L 14.4 176.39 L 14.6 176.14 L 14.8 175.89 L 15 175.65 L 15.2 175.4 L 15.4 175.15 L 15.6 174.9 L 15.8 174.65 L 16 174.4 L 16.2 174.15 L 16.4 173.9 L 16.6 173.65 L 16.8 173.4 L 17 173.15 L 17.2 172.89 L 17.4 172.64 L 17.6 172.39 L 17.8 172.14 L 18 171.88 L 18.2 171.63 L 18.4 171.37 L 18.6 171.12 L 18.8 170.86 L 19 170.61 L 19.2 170.35 L 19.4 170.1 L 19.6 169.84 L 19.8 169.58 L 20 169.32 L 20.2 169.07 L 20.4 168.81 L 20.6 168.55 L 20.8 168.29 L 21 168.03 L 21.2 167.77 L 21.4 167.51 L 21.6 167.25 L 21.8 166.98 L 22 166.72 L 22.2 166.46 L 22.4 166.19 L 22.6 165.93 L 22.8 165.67 L 23 165.4 L 23.2 165.13 L 23.4 164.87 L 23.6 164.6 L 23.8 164.33 L 24 164.07 L 24.2 163.8 L 24.4 163.53 L 24.6 163.26 L 24.8 162.99 L 25 162.72 L 25.2 162.44 L 25.4 162.17 L 25.6 161.9 L 25.8 161.63 L 26 161.35 L 26.2 161.08 L 26.4 160.8 L 26.6 160.52 L 26.8 160.25 L 27 159.97 L 27.2 159.69 L 27.4 159.41 L 27.6 159.13 L 27.8 158.85 L 28 158.57 L 28.2 158.28 L 28.4 158 L 28.6 157.72 L 28.8 157.43 L 29 157.14 L 29.2 156.86 L 29.4 156.57 L 29.6 156.28 L 29.8 155.99 L 30 155.7 L 30.2 155.41 L 30.4 155.11 L 30.6 154.82 L 30.8 154.53 L 31 154.23 L 31.2 153.93 L 31.4 153.63 L 31.6 153.33 L 31.8 153.03 L 32 152.73 L 32.2 152.43 L 32.4 152.12 L 32.6 151.82 L 32.8 151.51 L 33 151.2 L 33.2 150.89 L 33.4 150.58 L 33.6 150.27 L 33.8 149.96 L 34 149.64 L 34.2 149.33 L 34.4 149.01 L 34.6 148.69 L 34.8 148.37 L 35 148.04 L 35.2 147.72 L 35.4 147.39 L 35.6 147.06 L 35.8 146.73 L 36 146.4 L 36.2 146.07 L 36.4 145.73 L 36.6 145.39 L 36.8 145.05 L 37 144.71 L 37.2 144.36 L 37.4 144.01 L 37.6 143.66 L 37.8 143.31 L 38 142.96 L 38.2 142.6 L 38.4 142.24 L 38.6 141.88 L 38.8 141.51 L 39 141.14 L 39.2 140.77 L 39.4 140.39 L 39.6 140.01 L 39.8 139.63 L 40 139.24 L 40.2 138.85 L 40.4 138.46 L 40.6 138.06 L 40.8 137.66 L 41 137.25 L 41.2 136.84 L 41.4 136.42 L 41.6 136 L 41.8 135.57 L 42 135.14 L 42.2 134.7 L 42.4 134.25 L 42.6 133.8 L 42.8 133.34 L 43 132.87 L 43.2 132.4 L 43.4 131.91 L 43.6 131.42 L 43.8 130.91 L 44 130.4 L 44.2 129.87 L 44.4 129.34 L 44.6 128.78 L 44.8 128.22 L 45 127.63 L 45.2 127.03 L 45.4 126.41 L 45.6 125.77 L 45.8 125.1 L 46 124.4 L 46.2 123.67 L 46.4 122.9 L 46.6 122.08 L 46.8 121.2 L 47 120.25 L 47.2 119.2 L 47.4 118.01 L 47.6 116.61 L 47.8 114.79 L 48 110.4\" class=\"fig-main fig-graph\" clip-path=\"url(#fig-hyperbolic-identity-clip)\"\u002F>\u003Ccircle cx=\"156\" cy=\"74.4\" r=\"3\" class=\"fig-dot\"\u002F>\u003Ctext x=\"162\" y=\"90.4\" text-anchor=\"start\" dominant-baseline=\"middle\" class=\"fig-label fig-upright\">(cosh x, sinh x)\u003C\u002Ftext>\u003Ctext x=\"268.8\" y=\"124.8\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">X\u003C\u002Ftext>\u003Ctext x=\"110.4\" y=\"9.6\" text-anchor=\"middle\" dominant-baseline=\"middle\" class=\"fig-label\">Y\u003C\u002Ftext>\u003C\u002Fsvg>","hyperbolic-identity","\\cosh^2 x - \\sinh^2 x = 1","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmrow>\u003Cmi>cosh\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsup>\u003Cmi>x\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmsup>\u003Cmrow>\u003Cmi>sinh\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsup>\u003Cmi>x\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\cosh^2 x - \\sinh^2 x = 1\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.9818em;vertical-align:-0.0833em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mop\">cosh\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8984em;\">\u003Cspan style=\"top:-3.1473em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8984em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mop\">sinh\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8984em;\">\u003Cspan style=\"top:-3.1473em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","university",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Fcalculus\u002Fgrundidentitaet-hyperbelfunktionen",[],[5549,5550,5551],"hyperbolic-functions","derivative-hyperbolic","pythagorean-identity",{"title":548,"description":5442},"de\u002Fformulas\u002Fmath\u002Fcalculus\u002Fgrundidentitaet-hyperbelfunktionen","Für jedes reelle x ist cosh² x − sinh² x = 1, das hyperbolische Gegenstück zu sin² x + cos² x = 1.",[5556,5557,5558,5559],"hyperbelfunktionen","identität","sinus hyperbolicus","kosinus hyperbolicus",[5561,5564,5568],{"symbol":605,"meaning":5562,"symbolHtml":5563},"reelle Zahl","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>x\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">x\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":5565,"meaning":5566,"symbolHtml":5567},"\\cosh x","Kosinus hyperbolicus von x","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>cosh\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\cosh x\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">cosh\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":5569,"meaning":5570,"symbolHtml":5571},"\\sinh x","Sinus hyperbolicus von x","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>sinh\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\sinh x\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">sinh\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","0SsrV2UVNmSzNoYtmnPTXav_RPj7K9MI1E3Wt7NSkX0",[5574,5577,5580],{"stem":5575,"title":5576,"category":101},"de\u002Fformulas\u002Fmath\u002Fcalculus\u002Fableitung-hyperbelfunktionen","Ableitungen der Hyperbelfunktionen",{"stem":5578,"title":5579,"category":101},"de\u002Fformulas\u002Fmath\u002Fcalculus\u002Fhyperbelfunktionen","Hyperbelfunktionen",{"stem":5581,"title":5582,"category":518},"de\u002Fformulas\u002Fmath\u002Ftrigonometry\u002Ftrigonometrischer-pythagoras","Trigonometrischer Pythagoras",1793824281898]