41:@0.942772:0.974518:0.963274:0.974518:0.963274:0.956333:0.942772:0.956333:0.000000:0.000000 Pensamiento espacial, numérico y variacional:@0.525225:0.043660:0.930049:0.043660:0.930049:0.024915:0.525225:0.024915:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Solución:@0.073089:0.136836:0.140509:0.136836:0.140509:0.120167:0.073089:0.120167:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 : la relación entre , y está definida por . Reemplazamos los :@0.140588:0.136836:0.710399:0.136836:0.710399:0.121124:0.140588:0.121124:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a b:@0.274888:0.136836:0.301891:0.136836:0.301891:0.121401:0.274888:0.121401:0.000000:0.000000:0.000000 c:@0.320353:0.136836:0.328011:0.136836:0.328011:0.121401:0.320353:0.121401:0.000000 a:@0.464979:0.136836:0.474238:0.136836:0.474238:0.121401:0.464979:0.121401:0.000000 2:@0.474301:0.130772:0.479537:0.130772:0.479537:0.121612:0.474301:0.121612:0.000000 5:@0.484681:0.136281:0.500014:0.136281:0.500014:0.122566:0.484681:0.122566:0.000000 b:@0.505223:0.136836:0.514482:0.136836:0.514482:0.121401:0.505223:0.121401:0.000000 2:@0.514571:0.130772:0.519807:0.130772:0.519807:0.121612:0.514571:0.121612:0.000000 1:@0.524952:0.136281:0.540286:0.136281:0.540286:0.122566:0.524952:0.122566:0.000000 c:@0.545495:0.136836:0.553152:0.136836:0.553152:0.121401:0.545495:0.121401:0.000000 2:@0.553241:0.130772:0.558478:0.130772:0.558478:0.121612:0.553241:0.121612:0.000000 valores de y (2 5) 4:@0.073086:0.153478:0.267016:0.153478:0.267016:0.137765:0.073086:0.137765:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a b:@0.153655:0.153478:0.190018:0.153478:0.190018:0.138043:0.153655:0.138043:0.000000:0.000000:0.000000 2:@0.227705:0.147414:0.232942:0.147414:0.232942:0.138254:0.227705:0.138254:0.000000 5:@0.237804:0.152923:0.253137:0.152923:0.253137:0.139208:0.237804:0.139208:0.000000 2 :@0.267080:0.147414:0.274705:0.147414:0.274705:0.138254:0.267080:0.138254:0.000000:0.000000 1:@0.275166:0.152923:0.290499:0.152923:0.290499:0.139208:0.275166:0.139208:0.000000 . Por tanto, c (2 5):@0.290573:0.153478:0.464163:0.153478:0.464163:0.137765:0.290573:0.137765:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 c:@0.295395:0.153478:0.303053:0.153478:0.303053:0.138043:0.295395:0.138043:0.000000 2:@0.303105:0.147414:0.308342:0.147414:0.308342:0.138254:0.303105:0.138254:0.000000 2:@0.401108:0.147414:0.406345:0.147414:0.406345:0.138254:0.401108:0.138254:0.000000 5:@0.411205:0.152923:0.426539:0.152923:0.426539:0.139208:0.411205:0.139208:0.000000 2 :@0.464235:0.147414:0.471860:0.147414:0.471860:0.138254:0.464235:0.138254:0.000000:0.000000 2 5 :@0.472323:0.152923:0.530376:0.152923:0.530376:0.139208:0.472323:0.139208:0.000000:0.000000:0.000000:0.000000:0.000000 4:@0.493233:0.153478:0.502216:0.153478:0.502216:0.137765:0.493233:0.137765:0.000000 2 :@0.502281:0.147414:0.509906:0.147414:0.509906:0.138254:0.502281:0.138254:0.000000:0.000000 20:@0.531278:0.153478:0.549308:0.153478:0.549308:0.137765:0.531278:0.137765:0.000000:0.000000 2 :@0.549372:0.152923:0.574884:0.152923:0.574884:0.139208:0.549372:0.139208:0.000000:0.000000:0.000000 16 :@0.575786:0.153478:0.597912:0.153478:0.597912:0.137765:0.575786:0.137765:0.000000:0.000000:0.000000 5 :@0.598703:0.152923:0.618712:0.152923:0.618712:0.139208:0.598703:0.139208:0.000000:0.000000 4, :@0.619614:0.153478:0.635960:0.153478:0.635960:0.137765:0.619614:0.137765:0.000000:0.000000:0.000000 c:@0.636751:0.153478:0.644409:0.153478:0.644409:0.138043:0.636751:0.138043:0.000000 2 :@0.644467:0.147414:0.652092:0.147414:0.652092:0.138254:0.644467:0.138254:0.000000:0.000000 5 :@0.652553:0.152923:0.672562:0.152923:0.672562:0.139208:0.652553:0.139208:0.000000:0.000000 4 de :@0.673464:0.153478:0.710370:0.153478:0.710370:0.137765:0.673464:0.137765:0.000000:0.000000:0.000000:0.000000:0.000000 donde se concluye que :@0.073093:0.170120:0.247402:0.170120:0.247402:0.154407:0.073093:0.154407:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 c :@0.247189:0.170120:0.258086:0.170120:0.258086:0.154685:0.247189:0.154685:0.000000:0.000000 5 :@0.258030:0.169565:0.277873:0.169565:0.277873:0.155850:0.258030:0.155850:0.000000:0.000000 2:@0.277781:0.170120:0.286764:0.170120:0.286764:0.154407:0.277781:0.154407:0.000000 Para los triángulos oblicuángulos se cumplen dos leyes que se conocen como ley del :@0.073111:0.193904:0.710407:0.193904:0.710407:0.178191:0.073111:0.178191:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 seno y ley del coseno. Veamos qué dicen estas.:@0.073111:0.210546:0.412248:0.210546:0.412248:0.194833:0.073111:0.194833:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1. Ley del seno:@0.073089:0.234341:0.197713:0.234341:0.197713:0.217671:0.073089:0.217671:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 : en cualquier triángulo los lados son proporcionales a los senos de los:@0.197607:0.234341:0.706788:0.234341:0.706788:0.218628:0.197607:0.218628:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ángulos opuestos.:@0.103196:0.250983:0.235365:0.250983:0.235365:0.235270:0.103196:0.235270:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 !:@0.208629:0.150259:0.214691:0.150259:0.214691:0.137163:0.208629:0.137163:0.000000 !:@0.444404:0.150633:0.450466:0.150633:0.450466:0.137537:0.444404:0.137537:0.000000 A:@0.911101:0.227291:0.920840:0.227291:0.920840:0.212084:0.911101:0.212084:0.000000 B:@0.727082:0.227291:0.735787:0.227291:0.735787:0.212084:0.727082:0.212084:0.000000 :@0.908887:0.163813:0.909557:0.163813:0.909557:0.160954:0.908887:0.160954:0.000000 c:@0.823386:0.224243:0.830930:0.224243:0.830930:0.209036:0.823386:0.209036:0.000000 C:@0.903526:0.137385:0.913283:0.137385:0.913283:0.122178:0.903526:0.122178:0.000000 b:@0.912726:0.181721:0.921848:0.181721:0.921848:0.166513:0.912726:0.166513:0.000000 a:@0.821868:0.172020:0.830990:0.172020:0.830990:0.156812:0.821868:0.156812:0.000000 5:@0.115401:0.338346:0.130734:0.338346:0.130734:0.324630:0.115401:0.324630:0.000000 a:@0.087335:0.332813:0.096594:0.332813:0.096594:0.317377:0.087335:0.317377:0.000000 Sen A Sen B Sen C:@0.073507:0.345771:0.226740:0.345203:0.226740:0.329768:0.073507:0.330335:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.112919:0.345216:0.117598:0.345216:0.117598:0.331500:0.112919:0.331500:0.000000 b:@0.145117:0.332245:0.154376:0.332245:0.154376:0.316810:0.145117:0.316810:0.000000 :@0.169635:0.344649:0.174314:0.344649:0.174314:0.330933:0.169635:0.330933:0.000000 5:@0.171419:0.338346:0.186752:0.338346:0.186752:0.324630:0.171419:0.324630:0.000000 c:@0.201137:0.332245:0.208794:0.332245:0.208794:0.316810:0.201137:0.316810:0.000000 A:@0.270065:0.273565:0.279950:0.273565:0.279950:0.258130:0.270065:0.258130:0.000000 C:@0.308813:0.398379:0.318716:0.398379:0.318716:0.382944:0.308813:0.382944:0.000000 b:@0.277012:0.339119:0.286270:0.339119:0.286270:0.323684:0.277012:0.323684:0.000000 a:@0.393106:0.393760:0.402365:0.393760:0.402365:0.378324:0.393106:0.378324:0.000000 B:@0.512693:0.387209:0.521529:0.387209:0.521529:0.371774:0.512693:0.371774:0.000000 c:@0.393111:0.324702:0.400769:0.324702:0.400769:0.309267:0.393111:0.309267:0.000000 :@0.291763:0.287073:0.290540:0.287073:0.290540:0.281849:0.291763:0.281849:0.000000 Ejemplo 6:@0.073089:0.430759:0.148872:0.430759:0.148872:0.414089:0.073089:0.414089:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Encuentra el valor de en el triángulo :@0.073089:0.458121:0.350665:0.458121:0.350665:0.442408:0.073089:0.442408:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 c:@0.229716:0.458121:0.237374:0.458121:0.237374:0.442685:0.229716:0.442685:0.000000 ABC:@0.350486:0.458121:0.378962:0.458121:0.378962:0.442685:0.350486:0.442685:0.000000:0.000000:0.000000 , donde :@0.378907:0.458121:0.438270:0.458121:0.438270:0.442408:0.378907:0.442408:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 b :@0.438137:0.458703:0.450709:0.458703:0.450709:0.443268:0.438137:0.443268:0.000000:0.000000 5 :@0.450709:0.458148:0.470644:0.458148:0.470644:0.444433:0.450709:0.444433:0.000000:0.000000 7, :@0.470644:0.458703:0.486861:0.458703:0.486861:0.442990:0.470644:0.442990:0.000000:0.000000:0.000000 /:@0.486861:0.457843:0.502194:0.457843:0.502194:0.444946:0.486861:0.444946:0.000000 5 :@0.502194:0.458148:0.543480:0.458148:0.543480:0.444433:0.502194:0.444433:0.000000:0.000000:0.000000:0.000000 :@0.506796:0.458703:0.510109:0.458703:0.510109:0.443268:0.506796:0.443268:0.000000 C :@0.510108:0.458703:0.523545:0.458703:0.523545:0.443268:0.510108:0.443268:0.000000:0.000000 34°, :@0.543480:0.458703:0.574091:0.458703:0.574091:0.442990:0.543480:0.442990:0.000000:0.000000:0.000000:0.000000:0.000000 /:@0.574091:0.457843:0.589425:0.457843:0.589425:0.444946:0.574091:0.444946:0.000000 5 :@0.589423:0.458148:0.627625:0.458148:0.627625:0.444433:0.589423:0.444433:0.000000:0.000000:0.000000:0.000000 B:@0.594028:0.458703:0.603011:0.458703:0.603011:0.443268:0.594028:0.443268:0.000000 :@0.603011:0.458148:0.607690:0.458148:0.607690:0.444433:0.603011:0.444433:0.000000 30°.:@0.627625:0.458703:0.654205:0.458703:0.654205:0.442990:0.627625:0.442990:0.000000:0.000000:0.000000:0.000000 Solución: el primer paso es determinar el valor de :@0.073089:0.485485:0.434635:0.485485:0.434635:0.469773:0.073089:0.469773:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 /:@0.434233:0.484625:0.449566:0.484625:0.449566:0.471728:0.434233:0.471728:0.000000 :@0.449474:0.484930:0.454076:0.484930:0.454076:0.471215:0.449474:0.471215:0.000000 A:@0.453984:0.485485:0.463869:0.485485:0.463869:0.470050:0.453984:0.470050:0.000000 /:@0.073083:0.504970:0.088416:0.504970:0.088416:0.492073:0.073083:0.492073:0.000000 5:@0.088324:0.505275:0.121236:0.505275:0.121236:0.491559:0.088324:0.491559:0.000000:0.000000:0.000000 A :@0.092834:0.505830:0.124457:0.505830:0.124457:0.490395:0.092834:0.490395:0.000000:0.000000:0.000000:0.000000 180° :@0.124384:0.505830:0.160480:0.505830:0.160480:0.490117:0.124384:0.490117:0.000000:0.000000:0.000000:0.000000:0.000000 2 2 5 :@0.160370:0.505275:0.275305:0.505275:0.275305:0.491559:0.160370:0.491559:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 /:@0.175611:0.504970:0.190944:0.504970:0.190944:0.492073:0.175611:0.492073:0.000000 C :@0.195362:0.505830:0.208505:0.505830:0.208505:0.490395:0.195362:0.490395:0.000000:0.000000 /:@0.223691:0.504970:0.239024:0.504970:0.239024:0.492073:0.223691:0.492073:0.000000 B :@0.243442:0.505830:0.255517:0.505830:0.255517:0.490395:0.243442:0.490395:0.000000:0.000000 180° :@0.275213:0.505830:0.311309:0.505830:0.311309:0.490117:0.275213:0.490117:0.000000:0.000000:0.000000:0.000000:0.000000 2 :@0.311199:0.505275:0.331042:0.505275:0.331042:0.491559:0.311199:0.491559:0.000000:0.000000 34° :@0.330950:0.505830:0.358137:0.505830:0.358137:0.490117:0.330950:0.490117:0.000000:0.000000:0.000000:0.000000 2 :@0.358027:0.505275:0.377870:0.505275:0.377870:0.491559:0.358027:0.491559:0.000000:0.000000 30° 116°:@0.377778:0.505830:0.456193:0.505830:0.456193:0.490117:0.377778:0.490117:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.404855:0.505275:0.420188:0.505275:0.420188:0.491559:0.404855:0.491559:0.000000 .:@0.456082:0.505830:0.459212:0.505830:0.459212:0.489590:0.456082:0.489590:0.000000 Con el valor de :@0.073064:0.533192:0.187473:0.533192:0.187473:0.517479:0.073064:0.517479:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 /:@0.187944:0.532332:0.203277:0.532332:0.203277:0.519435:0.187944:0.519435:0.000000 :@0.203240:0.532637:0.207842:0.532637:0.207842:0.518921:0.203240:0.518921:0.000000 A,:@0.208266:0.533192:0.221464:0.533192:0.221464:0.517756:0.208266:0.517756:0.000000:0.000000 podemos usar una de las igualdades definidas por la ley del seno, :@0.221427:0.533192:0.709878:0.533192:0.709878:0.517479:0.221427:0.517479:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 que en este caso es :@0.073046:0.551401:0.218585:0.551401:0.218585:0.535688:0.073046:0.535688:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 b:@0.237833:0.546448:0.247091:0.546448:0.247091:0.530042:0.237833:0.530042:0.000000 Sen B:@0.218642:0.560682:0.256989:0.560682:0.256989:0.544276:0.218642:0.544276:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.263491:0.551404:0.286714:0.551404:0.286714:0.535692:0.263491:0.535692:0.000000:0.000000:0.000000 5:@0.267442:0.550850:0.282775:0.550850:0.282775:0.537134:0.267442:0.537134:0.000000 c:@0.306043:0.546448:0.313701:0.546448:0.313701:0.531013:0.306043:0.531013:0.000000 Sen C:@0.286853:0.560682:0.326285:0.560682:0.326285:0.545247:0.286853:0.545247:0.000000:0.000000:0.000000:0.000000:0.000000 y luego sustituimos los valores conocidos.:@0.331701:0.551404:0.638719:0.551404:0.638719:0.535692:0.331701:0.535692:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 :@0.073089:0.585756:0.077120:0.585756:0.077120:0.570043:0.073089:0.570043:0.000000 7:@0.098390:0.576507:0.107373:0.576507:0.107373:0.560073:0.098390:0.560073:0.000000 Sen:@0.077037:0.589765:0.102605:0.589765:0.102605:0.573359:0.077037:0.573359:0.000000:0.000000:0.000000 30˚:@0.102605:0.589765:0.126093:0.589765:0.126093:0.573331:0.102605:0.573331:0.000000:0.000000:0.000000 c:@0.169958:0.576646:0.177616:0.576646:0.177616:0.560240:0.169958:0.560240:0.000000 5:@0.130452:0.582682:0.145785:0.582682:0.145785:0.568967:0.130452:0.568967:0.000000 Sen:@0.148959:0.590823:0.174527:0.590823:0.174527:0.574417:0.148959:0.574417:0.000000:0.000000:0.000000 34˚:@0.174527:0.590823:0.198014:0.590823:0.198014:0.574389:0.174527:0.574389:0.000000:0.000000:0.000000 despejamos :@0.073089:0.615807:0.165309:0.615807:0.165309:0.600094:0.073089:0.600094:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 c :@0.165159:0.615807:0.179376:0.617086:0.179376:0.601650:0.165159:0.600372:0.000000:0.000000:0.000000 7:@0.198317:0.610859:0.207300:0.610859:0.207300:0.595146:0.198317:0.595146:0.000000 Sen:@0.207300:0.610859:0.232868:0.610859:0.232868:0.595423:0.207300:0.595423:0.000000:0.000000:0.000000 34˚ 7(0,55919):@0.232868:0.610859:0.352322:0.610859:0.352322:0.595146:0.232868:0.595146:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.179376:0.616530:0.194709:0.616530:0.194709:0.602814:0.179376:0.602814:0.000000 Sen:@0.202646:0.623664:0.228214:0.623664:0.228214:0.608229:0.202646:0.608229:0.000000:0.000000:0.000000 30˚:@0.228214:0.623664:0.253542:0.623664:0.253542:0.607951:0.228214:0.607951:0.000000:0.000000:0.000000 7,83 :@0.375367:0.615586:0.410145:0.615807:0.410145:0.600094:0.375367:0.599874:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.257577:0.616530:0.272910:0.616530:0.272910:0.602814:0.257577:0.602814:0.000000 ø:@0.356350:0.617667:0.371683:0.617667:0.371683:0.600013:0.356350:0.600013:0.000000 0,5:@0.302542:0.623817:0.323711:0.623817:0.323711:0.608104:0.302542:0.608104:0.000000:0.000000:0.000000 A:@0.786004:0.589576:0.795889:0.589576:0.795889:0.574141:0.786004:0.574141:0.000000 C:@0.747066:0.502383:0.756969:0.502383:0.756969:0.486948:0.747066:0.486948:0.000000 B:@0.904301:0.589576:0.913136:0.589576:0.913136:0.574141:0.904301:0.574141:0.000000 :@0.752591:0.526467:0.751912:0.526467:0.751912:0.523565:0.752591:0.523565:0.000000 Ejemplo 7:@0.073089:0.801759:0.148872:0.801759:0.148872:0.785090:0.073089:0.785090:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 En el triángulo dado aplica la ley del coseno para determinar el valor de .:@0.073089:0.829121:0.606052:0.829121:0.606052:0.813409:0.073089:0.813409:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 b:@0.593664:0.829121:0.602923:0.829121:0.602923:0.813686:0.593664:0.813686:0.000000 Solución:@0.073089:0.852905:0.139222:0.852905:0.139222:0.836236:0.073089:0.836236:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 : de acuerdo con la información dada, tenemos que aplicar la relación: :@0.139116:0.852905:0.653065:0.852905:0.653065:0.837193:0.139116:0.837193:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 b:@0.073089:0.869547:0.082348:0.869547:0.082348:0.854112:0.073089:0.854112:0.000000 2:@0.082281:0.863490:0.087518:0.863490:0.087518:0.854329:0.082281:0.854329:0.000000 2 :@0.087471:0.869553:0.209847:0.869553:0.209847:0.853841:0.087471:0.853841:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.091429:0.868999:0.106762:0.868999:0.106762:0.855283:0.091429:0.855283:0.000000 a:@0.110628:0.869553:0.119886:0.869553:0.119886:0.854118:0.110628:0.854118:0.000000 2:@0.119806:0.863490:0.125043:0.863490:0.125043:0.854329:0.119806:0.854329:0.000000 1:@0.128954:0.868999:0.144287:0.868999:0.144287:0.855283:0.128954:0.855283:0.000000 c:@0.148153:0.869553:0.155810:0.869553:0.155810:0.854118:0.148153:0.854118:0.000000 2:@0.155730:0.863490:0.160966:0.863490:0.160966:0.854329:0.155730:0.854329:0.000000 2:@0.164878:0.868999:0.180211:0.868999:0.180211:0.855283:0.164878:0.855283:0.000000 ac CosB:@0.189028:0.869553:0.243724:0.869553:0.243724:0.854118:0.189028:0.854118:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 . Sustituyendo los valores conocidos, obtenemos::@0.243679:0.869553:0.599830:0.869553:0.599830:0.853841:0.243679:0.853841:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 b:@0.073118:0.893337:0.082377:0.893337:0.082377:0.877902:0.073118:0.877902:0.000000 2:@0.082281:0.887280:0.087518:0.887280:0.087518:0.878119:0.082281:0.878119:0.000000 10 8 2(10)(8) :@0.087471:0.893344:0.248925:0.893344:0.248925:0.877631:0.087471:0.877631:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.091429:0.892789:0.106762:0.892789:0.106762:0.879073:0.091429:0.879073:0.000000 2:@0.128418:0.887280:0.133655:0.887280:0.133655:0.878119:0.128418:0.878119:0.000000 1:@0.137565:0.892789:0.152899:0.892789:0.152899:0.879073:0.137565:0.879073:0.000000 2:@0.165653:0.887280:0.170890:0.887280:0.170890:0.878119:0.165653:0.878119:0.000000 2:@0.174799:0.892789:0.190133:0.892789:0.190133:0.879073:0.174799:0.879073:0.000000 Cos:@0.248797:0.893344:0.273985:0.893344:0.273985:0.877908:0.248797:0.877908:0.000000:0.000000:0.000000 45°.:@0.273923:0.893344:0.300282:0.893344:0.300282:0.877631:0.273923:0.877631:0.000000:0.000000:0.000000:0.000000 b:@0.073099:0.917128:0.082358:0.917128:0.082358:0.901692:0.073099:0.901692:0.000000 2:@0.082281:0.911069:0.087518:0.911069:0.087518:0.901908:0.082281:0.901908:0.000000 100 64 160(0,707) :@0.087471:0.917133:0.276588:0.917133:0.276588:0.901420:0.087471:0.901420:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.091429:0.916578:0.106762:0.916578:0.106762:0.902862:0.091429:0.902862:0.000000 1:@0.141276:0.916578:0.156609:0.916578:0.156609:0.902862:0.141276:0.902862:0.000000 2:@0.182214:0.916578:0.197547:0.916578:0.197547:0.902862:0.182214:0.902862:0.000000 5 :@0.276440:0.916578:0.296283:0.916578:0.296283:0.902862:0.276440:0.902862:0.000000:0.000000 50,88:@0.296191:0.917133:0.335031:0.917133:0.335031:0.901420:0.296191:0.901420:0.000000:0.000000:0.000000:0.000000:0.000000 b :@0.075818:0.939149:0.088316:0.939149:0.088316:0.923714:0.075818:0.923714:0.000000:0.000000 5 :@0.088261:0.938927:0.112706:0.938927:0.112706:0.925211:0.088261:0.925211:0.000000:0.000000:0.000000 50,88 7,13:@0.112522:0.939149:0.204337:0.939149:0.204337:0.923436:0.112522:0.923436:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 <:@0.155208:0.939537:0.170542:0.939537:0.170542:0.921883:0.155208:0.921883:0.000000 !:@0.107107:0.937787:0.113170:0.937787:0.113170:0.924690:0.107107:0.924690:0.000000 A:@0.736184:0.911836:0.746069:0.911836:0.746069:0.896401:0.736184:0.896401:0.000000 C:@0.765635:0.831692:0.775539:0.831692:0.775539:0.816256:0.765635:0.816256:0.000000 a :@0.815556:0.870536:0.828128:0.870536:0.828128:0.855101:0.815556:0.855101:0.000000:0.000000 5:@0.828128:0.869982:0.843461:0.869982:0.843461:0.856266:0.828128:0.856266:0.000000 10:@0.843461:0.870536:0.865458:0.870536:0.865458:0.854824:0.843461:0.854824:0.000000:0.000000:0.000000 c :@0.782108:0.919134:0.793079:0.919134:0.793079:0.903698:0.782108:0.903698:0.000000:0.000000 5:@0.793079:0.918579:0.808412:0.918579:0.808412:0.904863:0.793079:0.904863:0.000000 8:@0.808412:0.919134:0.821426:0.919134:0.821426:0.903421:0.808412:0.903421:0.000000:0.000000 B :@0.782808:0.904176:0.794957:0.904176:0.794957:0.888741:0.782808:0.888741:0.000000:0.000000 5:@0.794957:0.903621:0.810290:0.903621:0.810290:0.889906:0.794957:0.889906:0.000000 45˚:@0.810290:0.904176:0.839649:0.904176:0.839649:0.888463:0.810290:0.888463:0.000000:0.000000:0.000000:0.000000 b:@0.748202:0.874484:0.757461:0.874484:0.757461:0.859049:0.748202:0.859049:0.000000 B:@0.861167:0.912885:0.870003:0.912885:0.870003:0.897450:0.861167:0.897450:0.000000 :@0.749901:0.855353:0.749221:0.855353:0.749221:0.852451:0.749901:0.852451:0.000000 2. Ley del coseno::@0.073089:0.659053:0.218329:0.659053:0.218329:0.642384:0.073089:0.642384:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 en todo triángulo, el cuadrado de unos de los lados es igual:@0.218205:0.659053:0.648920:0.659053:0.648920:0.643341:0.218205:0.643341:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a la suma de los cuadrados de los otros dos lados, menos dos veces el:@0.103215:0.675695:0.648900:0.675695:0.648900:0.659982:0.103215:0.659982:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 producto de estos por el coseno del ángulo determinado por estos.:@0.103215:0.692337:0.593069:0.692337:0.593069:0.676624:0.103215:0.676624:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a:@0.103210:0.712552:0.112469:0.712552:0.112469:0.697116:0.103210:0.697116:0.000000 2:@0.112402:0.706488:0.117639:0.706488:0.117639:0.697327:0.112402:0.697327:0.000000 2 :@0.117592:0.712552:0.239968:0.712552:0.239968:0.696839:0.117592:0.696839:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.121550:0.711997:0.136883:0.711997:0.136883:0.698281:0.121550:0.698281:0.000000 b:@0.140749:0.712552:0.150007:0.712552:0.150007:0.697116:0.140749:0.697116:0.000000 2:@0.149928:0.706488:0.155165:0.706488:0.155165:0.697327:0.149928:0.697327:0.000000 1:@0.159075:0.711997:0.174408:0.711997:0.174408:0.698281:0.159075:0.698281:0.000000 c:@0.178273:0.712552:0.185931:0.712552:0.185931:0.697116:0.178273:0.697116:0.000000 2:@0.185851:0.706488:0.191087:0.706488:0.191087:0.697327:0.185851:0.697327:0.000000 2:@0.194999:0.711997:0.210332:0.711997:0.210332:0.698281:0.194999:0.698281:0.000000 bc CosA:@0.219149:0.712552:0.274894:0.712552:0.274894:0.697116:0.219149:0.697116:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 .:@0.274850:0.712552:0.278052:0.712552:0.278052:0.696839:0.274850:0.696839:0.000000 b:@0.103210:0.734554:0.112469:0.734554:0.112469:0.719119:0.103210:0.719119:0.000000 2:@0.112402:0.728490:0.117639:0.728490:0.117639:0.719330:0.112402:0.719330:0.000000 2 :@0.117592:0.734554:0.239968:0.734554:0.239968:0.718842:0.117592:0.718842:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.121550:0.733999:0.136883:0.733999:0.136883:0.720284:0.121550:0.720284:0.000000 a:@0.140749:0.734554:0.150008:0.734554:0.150008:0.719119:0.140749:0.719119:0.000000 2:@0.149928:0.728490:0.155165:0.728490:0.155165:0.719330:0.149928:0.719330:0.000000 1:@0.159075:0.733999:0.174408:0.733999:0.174408:0.720284:0.159075:0.720284:0.000000 c:@0.178273:0.734554:0.185931:0.734554:0.185931:0.719119:0.178273:0.719119:0.000000 2:@0.185851:0.728490:0.191087:0.728490:0.191087:0.719330:0.185851:0.719330:0.000000 2:@0.194999:0.733999:0.210332:0.733999:0.210332:0.720284:0.194999:0.720284:0.000000 ac CosB:@0.219149:0.734554:0.273845:0.734554:0.273845:0.719119:0.219149:0.719119:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 .:@0.273800:0.734554:0.277003:0.734554:0.277003:0.718842:0.273800:0.718842:0.000000 c:@0.103210:0.756557:0.110867:0.756557:0.110867:0.741121:0.103210:0.741121:0.000000 2:@0.110801:0.750493:0.116038:0.750493:0.116038:0.741332:0.110801:0.741332:0.000000 2 :@0.115991:0.756557:0.241569:0.756557:0.241569:0.740844:0.115991:0.740844:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 5:@0.119949:0.756002:0.135282:0.756002:0.135282:0.742286:0.119949:0.742286:0.000000 a:@0.139147:0.756557:0.148406:0.756557:0.148406:0.741121:0.139147:0.741121:0.000000 2:@0.148326:0.750493:0.153563:0.750493:0.153563:0.741332:0.148326:0.741332:0.000000 1:@0.157473:0.756002:0.172806:0.756002:0.172806:0.742286:0.157473:0.742286:0.000000 b:@0.176672:0.756557:0.185931:0.756557:0.185931:0.741121:0.176672:0.741121:0.000000 2:@0.185851:0.750493:0.191087:0.750493:0.191087:0.741332:0.185851:0.741332:0.000000 2:@0.194999:0.756002:0.210332:0.756002:0.210332:0.742286:0.194999:0.742286:0.000000 ab CosC:@0.219149:0.756557:0.276514:0.756557:0.276514:0.741121:0.219149:0.741121:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 .:@0.276469:0.756557:0.279672:0.756557:0.279672:0.740844:0.276469:0.740844:0.000000 A:@0.792137:0.659326:0.802022:0.659326:0.802022:0.643890:0.792137:0.643890:0.000000 C:@0.675361:0.764238:0.685264:0.764238:0.685264:0.748803:0.675361:0.748803:0.000000 b:@0.732663:0.704772:0.741922:0.704772:0.741922:0.689336:0.732663:0.689336:0.000000 a:@0.799261:0.762630:0.808519:0.762630:0.808519:0.747194:0.799261:0.747194:0.000000 B:@0.914159:0.762782:0.922994:0.762782:0.922994:0.747347:0.914159:0.747347:0.000000 c:@0.853047:0.703177:0.860704:0.703177:0.860704:0.687741:0.853047:0.687741:0.000000 :@0.696725:0.662414:0.695503:0.662414:0.695503:0.657190:0.696725:0.657190:0.000000 MUESTRA EDITORIAL:@0.248902:0.947499:0.888171:0.113285:0.743922:0.050540:0.104653:0.884754:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Muestra editorial solo para fines didácticos – Prohibida su venta:@0.095585:0.659700:0.095585:0.341447:0.077558:0.341447:0.077558:0.659700:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000