tag:blogger.com,1999:blog-70096806330457342052024-03-21T20:54:58.526-07:00Teorema del seno y el cosenoKaren Encisohttp://www.blogger.com/profile/08030437550934449063noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-7009680633045734205.post-15454036606253008242011-10-31T09:36:00.000-07:002011-10-31T09:36:42.330-07:00videos<div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/Wgmy9mzPEmg?feature=player_embedded' frameborder='0'></iframe></div><div class="separator" style="clear: both; text-align: center;"><br />
<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/hZGc-q715OA?feature=player_embedded' frameborder='0'></iframe></div>Karen Encisohttp://www.blogger.com/profile/08030437550934449063noreply@blogger.com0tag:blogger.com,1999:blog-7009680633045734205.post-47687380002734108002011-10-30T14:23:00.000-07:002011-10-30T14:32:23.530-07:00juegos*<a href="http://www.juegosdelogica.com/neuronas/acertijo.htm">http://www.juegosdelogica.com/neuronas/acertijo.htm</a><br />
*<a href="http://www.juegosdelogica.com/neuronas/matemagi.htm">http://www.juegosdelogica.com/neuronas/matemagi.htm</a><br />
*<a href="http://www.juegosdelogica.com/neuronas/ilusiones.htm">http://www.juegosdelogica.com/neuronas/ilusiones.htm</a><br />
*<a href="http://www.jugargratis.org/juego/1614/carrera-matematica">http://www.jugargratis.org/juego/1614/carrera-matematica</a><br />
*<a href="http://www.jugargratis.org/juego/1616/triangulos-matematicos">http://www.jugargratis.org/juego/1616/triangulos-matematicos</a><br />
*<a href="http://www.vedoque.com/juegos/granja-matematicas.html">http://www.vedoque.com/juegos/granja-matematicas.html</a>Karen Encisohttp://www.blogger.com/profile/08030437550934449063noreply@blogger.com3tag:blogger.com,1999:blog-7009680633045734205.post-82319083397477850872011-10-30T12:04:00.001-07:002011-10-30T14:34:25.331-07:00Webgrafia<ul><li><span class="Apple-style-span" style="color: #4d4d4d; font-family: Arial, Helvetica, sans-serif;"><span class="Apple-style-span" style="font-size: 14px; line-height: 18px;"><a href="http://espanol.answers.yahoo.com/question/index?qid=20090318041030AAwM1mQ">http://espanol.answers.yahoo.com/question/index?qid=20090318041030AAwM1mQ</a></span></span></li>
<li><span class="Apple-style-span" style="color: #4d4d4d; font-family: Arial, Helvetica, sans-serif;"><span class="Apple-style-span" style="line-height: 18px;"><a href="http://www.virtual.unal.edu.co/cursos/sedes/medellin/nivelacion/uv00004/lecciones/unidades/generalidades/vectores/concepto/index13.htm">http://www.virtual.unal.edu.co/cursos/sedes/medellin/nivelacion/uv00004/lecciones/unidades/generalidades/vectores/concepto/index13.htm</a></span></span></li>
<li><span class="Apple-style-span" style="background-color: white; color: #4d4d4d; font-family: Arial, Helvetica, sans-serif; line-height: 18px;"><a href="http://www.jugargratis.org/matematicos/">http://www.jugargratis.org/matematicos/</a> </span></li>
<li><a href="http://www.vedoque.com/juegos/granja-matematicas.html"><span class="Apple-style-span" style="font-family: Arial, Helvetica, sans-serif;">http://www.vedoque.com/juegos/granja-matematicas.html</span></a></li>
</ul>Karen Encisohttp://www.blogger.com/profile/08030437550934449063noreply@blogger.com0tag:blogger.com,1999:blog-7009680633045734205.post-17805771781079583522011-10-15T19:05:00.000-07:002011-10-15T19:05:03.760-07:00taller<m:smallfrac m:val="off"> <m:dispdef> <m:lmargin m:val="0"> <m:rmargin m:val="0"> <m:defjc m:val="centerGroup"> <m:wrapindent m:val="1440"> <m:intlim m:val="subSup"> <m:narylim m:val="undOvr"> </m:narylim></m:intlim> </m:wrapindent><span style="font-size: small;"><span style="line-height: 115%;"><span style="font-family: Arial,Helvetica,sans-serif;"> 1.Resuelve el triángulo ABC</span></span></span></m:defjc></m:rmargin></m:lmargin></m:dispdef></m:smallfrac><br />
<ul><li><span style="font-size: small;"><span style="line-height: 115%;"><span style="font-family: Arial,Helvetica,sans-serif;">α</span>= 60º </span><span style="font-family: Arial,Helvetica,sans-serif; line-height: 115%;">β=20 </span><span style="font-family: Arial,Helvetica,sans-serif; line-height: 115%;"> γ=30</span></span></li>
<li><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif; line-height: 115%;">β=150º a</span></span><span style="font-size: small;"><span style="line-height: 115%;">=150</span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif; line-height: 115%;"> c=30</span></span></li>
<li><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif; line-height: 115%;">a=10 b= 15 c=12</span></span></li>
<li><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif; line-height: 115%;">a=2.0 b=3.0 c=4.0</span></span></li>
<li><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif; line-height: 115%;">γ=45º</span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif; line-height: 115%;"> b=10.0 a=15.0 </span></span></li>
<li><span style="font-size: small;"><span style="line-height: 115%;"><span style="font-family: Arial,Helvetica,sans-serif;">α</span>=41º </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif; line-height: 115%;">γ=77º a=10.5</span></span></li>
<li><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif; line-height: 115%;"></span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif; line-height: 115%;">β=20º </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif; line-height: 115%;">γ=31º b=210</span></span></li>
<li><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif; line-height: 115%;">γ=81º c=11 b=12</span></span></li>
<li><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif; line-height: 115%;">γ=47.74º a=131.08 c=97.84</span></span></li>
<li><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif; line-height: 115%;">γ=73.01º a=17.31 c=</span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif; line-height: 115%;">20.24</span></span></li>
</ul><span style="font-size: small;">2.<span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;">Resolver los siguientes problemas:</span></span><br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj96Bhg091qY2BWnamn5cDyO3rvn8RqGy72ft68vgp3IrSQsVgjvQbVh0mhW95qKt1gm0H8EvfGOquBLcG7nJed_909ClR9UrfziYUvAsv9pHwChuprUiFvMMWQvR0IJZG49dLm8gGGaX2B/s1600/ejercicio+19.BMP" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="175" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj96Bhg091qY2BWnamn5cDyO3rvn8RqGy72ft68vgp3IrSQsVgjvQbVh0mhW95qKt1gm0H8EvfGOquBLcG7nJed_909ClR9UrfziYUvAsv9pHwChuprUiFvMMWQvR0IJZG49dLm8gGGaX2B/s320/ejercicio+19.BMP" width="320" /></a></div><br />
<ul><li><span style="font-size: small;"><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"> como se muestra en la figura, un teleférico transporta pasajeros dede el punto punto A, que está a 1.2 millas del punto B. que se halla en la base de una montaña, hasta un punto P de la cima de la montaña.Los ángulos de elevación de P desde A y B son 21º y 65º.respectivamente.</span></span></li>
</ul><span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;">a) Calcula la distancia qntre A y P</span></span><br />
<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;">b) Calcula la altura de la montaña</span></span><br />
<br />
<ul><li><span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;"> </span></span><span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;">Un camino recto hace un ángulo de 15º con la horizontal. Cuando el ángulo de elevación del sol es de 57º, un poste vertical que ésta a un lado del camino proyecta una sombra de 75 pies de largo directamente cuesta abajo, como se muestra en la figura.</span></span></li>
</ul><span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;">a) Calcula longitud del poste</span></span><br />
<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;"> </span></span><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9fDxfmWu2RpGt9QqkqgJlsvKVHViciMO8N7Z0y8fS-UBxehSPISc0p9cWpmllQFutJSwqlfSbwZmIoMizP_WclvLmyeWBbzfFxYnwUVQ9OL9yW7k0m1s3TwOikaMl_W6gj4Vsh1eLFkJ3/s1600/ejercicio+20.BMP" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="242" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9fDxfmWu2RpGt9QqkqgJlsvKVHViciMO8N7Z0y8fS-UBxehSPISc0p9cWpmllQFutJSwqlfSbwZmIoMizP_WclvLmyeWBbzfFxYnwUVQ9OL9yW7k0m1s3TwOikaMl_W6gj4Vsh1eLFkJ3/s320/ejercicio+20.BMP" width="320" /></a></div><span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;"> </span></span><br />
<ul><li><span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;"> Originalmente, esta torre estaba perpendicular al suelo y medía 179 pies de altura; debido al hundimiento del suelo, ahora se ha inclinado al cierto ángulo </span></span><!--[if gte mso 9]><xml> <w:WordDocument> <w:View>Normal</w:View> <w:Zoom>0</w:Zoom> <w:TrackMoves/> <w:TrackFormatting/> <w:HyphenationZone>21</w:HyphenationZone> <w:PunctuationKerning/> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:DoNotPromoteQF/> <w:LidThemeOther>ES</w:LidThemeOther> <w:LidThemeAsian>X-NONE</w:LidThemeAsian> <w:LidThemeComplexScript>X-NONE</w:LidThemeComplexScript> <w:Compatibility> <w:BreakWrappedTables/> <w:SnapToGridInCell/> <w:WrapTextWithPunct/> <w:UseAsianBreakRules/> <w:DontGrowAutofit/> <w:SplitPgBreakAndParaMark/> <w:DontVertAlignCellWithSp/> <w:DontBreakConstrainedForcedTables/> <w:DontVertAlignInTxbx/> <w:Word11KerningPairs/> <w:CachedColBalance/> </w:Compatibility> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> <m:mathPr> <m:mathFont m:val="Cambria Math"/> <m:brkBin m:val="before"/> <m:brkBinSub m:val="--> <m:smallfrac m:val="off"> <m:dispdef> <m:lmargin m:val="0"> <m:rmargin m:val="0"> <m:defjc m:val="centerGroup"> <m:wrapindent m:val="1440"> <m:intlim m:val="subSup"> <m:narylim m:val="undOvr"> </m:narylim></m:intlim> </m:wrapindent><!--[endif]--><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" DefUnhideWhenUsed="true"
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</style> <![endif]--><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;">Ѳ</span></m:defjc></m:rmargin></m:lmargin></m:dispdef></m:smallfrac></div><span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;">b) Calcula la distancia <span style="font-family: "Courier New",Courier,monospace;">d <span style="font-family: Arial,Helvetica,sans-serif;">que se ha movido el centro de la parte superior de la torre con respecto a la perpendicular.</span></span></span></span><br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlzCfFpdUVJT7-4bJg1A5dKzxv8h50FVCPRtUh5m42QrP9sW_nqkSVq4yjYQ6oBftvxgIZo6y3jTXsbsSbkM5cjlSxUPgGVAHL5d4GGyeUcCN3uy6WVJcJFxdfcY5-9m1z_y9MwZqBQQ3q/s1600/ejercicio26.BMP" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="199" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlzCfFpdUVJT7-4bJg1A5dKzxv8h50FVCPRtUh5m42QrP9sW_nqkSVq4yjYQ6oBftvxgIZo6y3jTXsbsSbkM5cjlSxUPgGVAHL5d4GGyeUcCN3uy6WVJcJFxdfcY5-9m1z_y9MwZqBQQ3q/s320/ejercicio26.BMP" width="320" /></a></div><ul><li><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">Una catedral se encuentra sobre una colina, como en la figura.Cuando se observa la parte superior del campanario desde la base de la colina, el ángulo de elevación es de 48º; cuando se ve a una distancia de 200 pies desde la base de la colina, es de 41º. La colina se eleva en un ángulo de 32º.</span></span></li>
</ul><div style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;">a) Calcula la altura de la catedral.</span></div><div style="font-family: Arial,Helvetica,sans-serif;"><br />
</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9u0fRW64RPy4P_qjil93A8aX97LnJWfWMGenkmcfcNWb3G-2JT1M5ROFouYnX6wUoYyrKiCgoIgLtBvshFXmM0SMIh4NFgNqL_zV1girZzXDns0l6oMoMU1pD1A8ZOelwdAbAkDiRo68Z/s1600/ejercicio27.BMP" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="216" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9u0fRW64RPy4P_qjil93A8aX97LnJWfWMGenkmcfcNWb3G-2JT1M5ROFouYnX6wUoYyrKiCgoIgLtBvshFXmM0SMIh4NFgNqL_zV1girZzXDns0l6oMoMU1pD1A8ZOelwdAbAkDiRo68Z/s320/ejercicio27.BMP" width="320" /></a></div><div style="font-family: Arial,Helvetica,sans-serif;"><br />
</div><ul style="font-family: Arial,Helvetica,sans-serif;"><li><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;">U</span><span style="font-size: small;">n helicóptero vuela a una actitud de 1000 pies sobre la cima de una montaña que mide 5210 pies, según se indica en la figura. Desde lo alto de esta montaña, más elevada que la primera. Desde el helicóptero, el ángulo de depresión de 43º, y desde la cima de la primera montaña, el ángulo de elevacion es de 18º.</span></li>
</ul><div style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;">a) Calcula la distancia de pico a pico.</span></div><div style="font-family: Arial,Helvetica,sans-serif;"><br />
</div><div style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;">b) Calcula la altitud de la montaña más alta.</span> </div><br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMAwjMup5EwvPKyeQ6Gi0BA-ZmHIFXCe4C14VaN_94p4p-KvJBH8H6pL19bagEyL2KTAVNb-Wg2R-JtxcHHHwLd6HyMuaH-dsj5aa2avooQaonC9OKoehe0Sb9GU-9e9fM-iogGxmUvcmK/s1600/ejercicio28.BMP" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="275" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMAwjMup5EwvPKyeQ6Gi0BA-ZmHIFXCe4C14VaN_94p4p-KvJBH8H6pL19bagEyL2KTAVNb-Wg2R-JtxcHHHwLd6HyMuaH-dsj5aa2avooQaonC9OKoehe0Sb9GU-9e9fM-iogGxmUvcmK/s320/ejercicio28.BMP" width="320" /></a></div><ul style="font-family: Arial,Helvetica,sans-serif;"><li><span style="font-size: small;">Los puntos P y Q ubicados a nivel del suelo están en ladosopuestos de un edificio. para hallar la distancia entre los puntos, un agrimensor escoge un punto R que está a 300 pies del punto P y a 328 pies del Q, y luego determina que el ángulo PRQ mide 37º40. </span></li>
</ul><div style="text-align: left;"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">a) Calcula la distancia entre P y Q.</span></span></div><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"><br />
</span></span><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJ4Ks_8oUKAmzB-wx_AadQVRUXQoW2bZAaNCBfshUZed2U8F4QrHkVyGHpWK7-4F4x7ijea5m_RivBWKBoNqDQhbigfcrHPnAnY7jJCOySRYY2PGVA3pH3hfVyDDqN0W0TlV7Z7q3TYLmN/s1600/c-ejercicio+18.BMP" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="196" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJ4Ks_8oUKAmzB-wx_AadQVRUXQoW2bZAaNCBfshUZed2U8F4QrHkVyGHpWK7-4F4x7ijea5m_RivBWKBoNqDQhbigfcrHPnAnY7jJCOySRYY2PGVA3pH3hfVyDDqN0W0TlV7Z7q3TYLmN/s320/c-ejercicio+18.BMP" width="320" /></a></div><ul><li><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">Un crucero zarpa con rumbo N47ºE desde una isla a un puerto en la tierra firme, que está a 150 millas. Despues de navegar por aguas de fuertes corrientes, la nave está fuera de curso en una posición P ubicada a N33ºE y a 80 millas de la isla.</span></span></li>
</ul><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">a) ¿A qué distancia aproximada estará del punto de destino?</span></span><br />
<span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">b) ¿Que dirección debe tomar para corregir su curso?</span></span><br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjybW7UywZxsBEF-5HnA52ki6pvCNcv5TxPTrys98uzSbv8kojF2iiIu6xtRZ1H6XQfc27ZctEq4Y0I4gnnDCG_0YhlCamt4PYiJQqppcp5aRZ6MVJ0ei3EOStKm4zpOJ0B2a2wh8MLYfWH/s1600/c-ejercicio24.BMP" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="153" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjybW7UywZxsBEF-5HnA52ki6pvCNcv5TxPTrys98uzSbv8kojF2iiIu6xtRZ1H6XQfc27ZctEq4Y0I4gnnDCG_0YhlCamt4PYiJQqppcp5aRZ6MVJ0ei3EOStKm4zpOJ0B2a2wh8MLYfWH/s320/c-ejercicio24.BMP" width="320" /></a></div><br />
<span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"></span></span><br />
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<ul><li><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">Los sismólogos investigan la estructura del interior de la tierra analizando las ondas sísmeticas ocasionadas por terremotos. Si se supone que el interior del globo terráqueo </span></span><br />
<span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">es homogéneo, entonces estas ondas viajarán en línea recta a una velocidad constante v. Con la ley de los cosenos demuestra que el tiempo t para que una onda viaje por el interior de la tierra de E a S está dado por:</span></span><br />
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</style> <![endif]--><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;">Ѳ es el ángulo indicado por el vértice en el centro del planeta.</span></m:defjc></m:rmargin></m:lmargin></m:dispdef></m:smallfrac></div><div style="text-align: left;"><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;"> </span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhW1xkKJ_bk9JtS08PSEe4wfnDC6pwChd_Rrg0JM6rjWfDbNTCphv0Ylhnn4JuvJSC_oVY3Z10cMcTvAdvYXyhORzFU3uib2N2RGhIADP0z7mvKJYo2_4vGSls2G9snj7eYcVjT9a128qTO/s1600/c-ejercicio25.BMP" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="146" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhW1xkKJ_bk9JtS08PSEe4wfnDC6pwChd_Rrg0JM6rjWfDbNTCphv0Ylhnn4JuvJSC_oVY3Z10cMcTvAdvYXyhORzFU3uib2N2RGhIADP0z7mvKJYo2_4vGSls2G9snj7eYcVjT9a128qTO/s320/c-ejercicio25.BMP" width="320" /></a></div><div style="text-align: left;"><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;"> </span></div></li>
<li><div style="text-align: left;"><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;">La distancia de una margen a otra del río que se ve en la figura se puede encontrar sin medir ángulos. se seleccionan los puntos B y C de la orilla opuesta, y los segmentos de recta AB y AC se prolongan, según se observa. Se escogen los puntos D y E como se indica y se miden las distancias BC , BD, BE,CD y CE.</span></div><div style="text-align: left;"><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;">supón que BC=184 pies, BD= 102 pies, BE=218 pies, CD= 236 pies y CE=80 pies.</span></div><div style="text-align: left;"><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;"> </span></div><div style="text-align: left;"><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;">a) Calcula la distancias AB y AC</span></div><div style="text-align: left;"><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;">b) Calcula la distancia más corta a la otra orilla desde el punto A.</span></div><div style="text-align: left;"><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;"><br />
</span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6N025SM58xhaP1GjCuxouIDo47quph9k16igxkXmko4c3b2XG3BBx6d795iGsOKNC_ebo7OPz1m3yLDqldDGy8gCO5PjYCqlIWAYhelos-ElhMRbfakI-HUO-xS5HrFdbzMK9FA_m_lpk/s1600/c-ejercicio26.BMP" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="147" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6N025SM58xhaP1GjCuxouIDo47quph9k16igxkXmko4c3b2XG3BBx6d795iGsOKNC_ebo7OPz1m3yLDqldDGy8gCO5PjYCqlIWAYhelos-ElhMRbfakI-HUO-xS5HrFdbzMK9FA_m_lpk/s320/c-ejercicio26.BMP" width="320" /></a></div><div style="text-align: left;"><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;"></span></div></li>
<li><div style="text-align: left;"><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;">La portezuela trasera de un automide 42 pulgadas de largo. hay que fijar un soporte, que mide 24 pulgadas cuando está extendido por completo, tanto a la portezuela como a la carrocería, de modo que cuando la portezuela se abra del todo, el soporte quede en posición vertical y haya un espacio libre de 32 pulgadas.</span></div><div style="text-align: left;"><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;"><br />
</span></div><div style="text-align: left;"><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;">a) Calcula las longitudes de los segmentos TQ y TP.</span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbM5_pooGYj1ZUTbnzUV9XePi1-ek3NxicN197c8EL0UtdnWfuudZXfO1J81zFEYahTQSA4etrvvnKaVxou4oxOSLiGbeq8q3o6wM7sxn6n1Dn8y-ROSwePZbqk3Usun3TAyqsCrYuDMpE/s1600/c-ejercicio28.BMP" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="204" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbM5_pooGYj1ZUTbnzUV9XePi1-ek3NxicN197c8EL0UtdnWfuudZXfO1J81zFEYahTQSA4etrvvnKaVxou4oxOSLiGbeq8q3o6wM7sxn6n1Dn8y-ROSwePZbqk3Usun3TAyqsCrYuDMpE/s320/c-ejercicio28.BMP" width="320" /></a></div><div style="text-align: left;"><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;"><br />
</span></div></li>
</ul><blockquote> </blockquote><div class="separator" style="clear: both; text-align: center;"></div>Karen Encisohttp://www.blogger.com/profile/08030437550934449063noreply@blogger.com4tag:blogger.com,1999:blog-7009680633045734205.post-15201648798838997182011-10-14T19:29:00.000-07:002011-10-14T19:29:10.791-07:00ley del coseno<span style="font-family: Arial,Helvetica,sans-serif; font-size: small;"><span class="notranslate" style="color: black;"><span class="a" style="left: 645px; letter-spacing: 2px; top: 1403px; word-spacing: 3px;">La ley del coseno es una expresión que te permite conocer un lado de un</span><span class="a" style="left: 645px; letter-spacing: 1px; top: 1507px; word-spacing: 6px;"> triángulo cualquiera, si conoces los otros dos lados y el ángulo opuesto al lado que quieres conocer.</span></span></span><br />
<span style="font-family: Arial,Helvetica,sans-serif; font-size: small;"><span class="notranslate" style="color: black;"><span class="a" style="left: 645px; letter-spacing: 1px; top: 1507px; word-spacing: 6px;">Cualquier lado de un </span></span></span><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;"><span class="notranslate" style="color: black;"><span class="a" style="left: 645px; letter-spacing: 1px; top: 1507px; word-spacing: 6px;">triángulo</span></span></span><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;"><span class="notranslate" style="color: black;"><span class="a" style="left: 645px; letter-spacing: 1px; top: 1507px; word-spacing: 6px;"> es igual a la suma de los cuadrados de las longitudes de los otros dos lados, menos el doble producto de las longitudes de los mismos lados por el coseno del </span></span></span><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;"><span class="notranslate" style="color: black;"><span class="a" style="left: 645px; letter-spacing: 1px; top: 1507px; word-spacing: 6px;">ángulo entre ellos.</span></span></span><br />
<br />
<span style="font-family: Arial,Helvetica,sans-serif; font-size: small;"><span class="notranslate" style="color: black;"><span class="a" style="left: 645px; letter-spacing: 1px; top: 1507px; word-spacing: 6px;">Dados dos lados y el </span></span></span><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;"><span class="notranslate" style="color: black;"><span class="a" style="left: 645px; letter-spacing: 1px; top: 1507px; word-spacing: 6px;">ángulo incluido de un </span></span></span><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;"><span class="notranslate" style="color: black;"><span class="a" style="left: 645px; letter-spacing: 1px; top: 1507px; word-spacing: 6px;">triángulo, podemos usar la ley de los cosenos para hallar el tercer lado;para eso recurrimos a la ley de los senos para encontrar otro </span></span></span><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;"><span class="notranslate" style="color: black;"><span class="a" style="left: 645px; letter-spacing: 1px; top: 1507px; word-spacing: 6px;">ángulo del </span></span></span><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;"><span class="notranslate" style="color: black;"><span class="a" style="left: 645px; letter-spacing: 1px; top: 1507px; word-spacing: 6px;">triángulo.siempre que se siga este procedimiento. es mejor determinar el </span></span></span><span style="font-family: Arial,Helvetica,sans-serif; font-size: small;"><span class="notranslate" style="color: black;"><span class="a" style="left: 645px; letter-spacing: 1px; top: 1507px; word-spacing: 6px;">ángulo opuesto al lado más corto. ya que siempre es agudo. De este modo evitamos la posibilidad de obtener dos soluciones cuando se resuelve una ecuación trigonométrica que comprende ese ángulo.</span></span></span><br />
<br />
<span style="font-family: Arial,Helvetica,sans-serif; font-size: small;"><span class="notranslate" style="color: black;"><span class="a" style="left: 645px; letter-spacing: 1px; top: 1507px; word-spacing: 6px;">EJEMPLO coseno (LAL)</span></span></span><br />
<span style="font-family: Arial,Helvetica,sans-serif; font-size: small;"><span class="notranslate" style="color: black;"><span class="a" style="left: 645px; letter-spacing: 1px; top: 1507px; word-spacing: 6px;">calcula las partes restantes del triángulo ABC si a=5.0 c=8.0 y </span></span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">β</span></span>=77º.<br />
<span style="font-family: Arial,Helvetica,sans-serif; font-size: small;">SOLUCIÓN </span><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfypGshfdG_CHL1sxLrqxwXUmhWDnuyBsrPuBY_9sQFFqiQw1gbn8FfnlFYmmzlIGqwOjUWK7s2fjjsgF_xAKeadUE0EaoHMYjdY9FCBNYLMkTt7dY5Ntg-kEoMPY12DwkELqN8RHb6XHx/s1600/Dibujo.bmp" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfypGshfdG_CHL1sxLrqxwXUmhWDnuyBsrPuBY_9sQFFqiQw1gbn8FfnlFYmmzlIGqwOjUWK7s2fjjsgF_xAKeadUE0EaoHMYjdY9FCBNYLMkTt7dY5Ntg-kEoMPY12DwkELqN8RHb6XHx/s1600/Dibujo.bmp" /></a></div><br />
<span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">como </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">β <span style="font-family: Arial,Helvetica,sans-serif;">es el ángulo entre los lados a y c. comenzamos por calcular b (lado opuesto a </span></span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">β):</span></span><br />
<br />
<span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">b^2= a^2+c^2- 2ac cos </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-family: Arial,Helvetica,sans-serif;"> </span></span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">β</span></span><br />
<span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> =(5.0) ^2 +(8.0) </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">^2 -2 (5.0)(8.0)cos 77º</span></span><br />
<span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> =89-80 cos77º =71.0</span></span><br />
<div class="actividadesv"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">β=</span></span> <m:smallfrac m:val="off"> <m:dispdef> <m:lmargin m:val="0"> <m:rmargin m:val="0"> <m:defjc m:val="centerGroup"> <m:wrapindent m:val="1440"> <m:intlim m:val="subSup"> <m:narylim m:val="undOvr"> </m:narylim></m:intlim> </m:wrapindent>√71.0 = 8.4</m:defjc></m:rmargin></m:lmargin></m:dispdef></m:smallfrac></div><div class="actividadesv"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">Ahora encontremos otro ángulo del triángulo mediante la ley de los senos. De acuerdo con las obsevarciones de este ejemplo, aplicaremos la ley de los senos y hallaremos </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">α</span></span>, <span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">puesto que es el ángulo opuesto al lado más corto,el cual es </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">α</span></span>.</div><div class="actividadesv"><br />
</div><div class="actividadesv"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">sen </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">α = sen</span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">β</span></span></div><div class="actividadesv"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> a b</span></span></div><div class="actividadesv"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">sen </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">α =a </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">sen</span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">β</span></span></div><div class="actividadesv"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> b</span></span></div><div class="actividadesv"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> =5.0 sen 77º = 0.5782</span></span></div><div class="MsoNormal"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> </span></span> <m:smallfrac m:val="off"> <m:dispdef> <m:lmargin m:val="0"> <m:rmargin m:val="0"> <m:defjc m:val="centerGroup"> <m:wrapindent m:val="1440"> <m:intlim m:val="subSup"> <m:narylim m:val="undOvr"> </m:narylim></m:intlim> </m:wrapindent> √71.0</m:defjc></m:rmargin></m:lmargin></m:dispdef></m:smallfrac></div><div class="MsoNormal"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">como </span></span> <span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">α es agudo,</span></span></div><div class="MsoNormal"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">α= sen ^ -1(0.5782)=35.3º=35º</span></span></div><div class="MsoNormal"><br />
</div><div class="MsoNormal"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">Dados los tres lados de un triángulo, se puede usar la ley de los cosenos para hallar cualquiera de los tres ángulos. siempre encontraremos primero el ángulo más grande;es decir, el ángulo opuesto al lado mas largo, ya que esta práctica garantizará que los ángulos restantes sean agudos. Luego se puede encontrar otro ángulo del triángulo aplicando la ley de los senos o la de los cosenos.</span></span></div><div class="MsoNormal"><br />
</div><div class="MsoNormal"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">EJEMPLO coseno(LLL)</span></span></div><div class="MsoNormal" style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">si el triángulo ABC tiene lados a=90, b=70 y c=40, calcula los ángulos </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">α,</span></span> <m:smallfrac m:val="off"> <m:dispdef> <m:lmargin m:val="0"> <m:rmargin m:val="0"> <m:defjc m:val="centerGroup"> <m:wrapindent m:val="1440"> <m:intlim m:val="subSup"> <m:narylim m:val="undOvr"> </m:narylim></m:intlim> </m:wrapindent><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">β y </span>γ al grado mas cercano.</span></m:defjc></m:rmargin></m:lmargin></m:dispdef></m:smallfrac></div><div class="MsoNormal" style="font-family: Arial,Helvetica,sans-serif;"><br />
</div><div class="MsoNormal" style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;">SOLUCIÓN </span></div><div class="MsoNormal" style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;">De acuerdo con la observación previa, primero se encuentra el ángulo opuesto al lado más largo a; por lo tanto, se escoge la forma de la ley de los cosenos donde aparece </span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">α; por lo tanto, se escoge la forma de la ley del coseno donde aparece </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">α:</span></span></div><div class="MsoNormal" style="font-family: Arial,Helvetica,sans-serif;"><br />
</div><div class="MsoNormal" style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> a^2=b^2 + c^2 -2bc cos </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">α</span></span></div><div class="MsoNormal" style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">cos </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">α= b^2 +c^2 -a^2</span></span></div><div class="MsoNormal" style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> 2bc</span></span></div><div class="MsoNormal" style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> =70^2+40^2-90^2 = -2</span></span></div><div class="MsoNormal" style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> 2(70)(40) 7</span></span></div><div class="MsoNormal" style="font-family: Arial,Helvetica,sans-serif;"><br />
</div><div class="MsoNormal" style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">α= cos ^-1 (-2/7)= 106.6º=107º</span></span></div><div class="MsoNormal" style="font-family: Arial,Helvetica,sans-serif;"><br />
<span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">En estas condiciones se puede usar la ley de los senos o la de los cosenos para hallar </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"></span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">β. Usemos la ley de los coseno:</span></span></div><div class="MsoNormal" style="font-family: Arial,Helvetica,sans-serif;"><br />
</div><div class="MsoNormal" style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> b^2 =a^2+c^52-2ac cos </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"></span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">β</span></span></div><div class="MsoNormal" style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">cos</span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">β=</span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">a^2+c^2-b^2</span></span></div><div class="MsoNormal" style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> 2ac<span style="background-color: white;"></span></span></span></div><div class="MsoNormal" style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">=90^2+40^2-70^2= 2 </span></span><br />
<span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> 2(90)(40) 3</span></span><br />
<div style="text-align: left;"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">β=cos^-1(2/3)=48.2º=48º</span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> </span></span></div><div style="text-align: left;"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"><br />
</span></span></div><div class="MsoNormal"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">En este punto de la solución podemos encontrar </span></span> <m:smallfrac m:val="off"> <m:dispdef> <m:lmargin m:val="0"> <m:rmargin m:val="0"> <m:defjc m:val="centerGroup"> <m:wrapindent m:val="1440"> <m:intlim m:val="subSup"> <m:narylim m:val="undOvr"> </m:narylim></m:intlim> </m:wrapindent> <m:smallfrac m:val="off"> <m:dispdef> <m:lmargin m:val="0"> <m:rmargin m:val="0"> <m:defjc m:val="centerGroup"> <m:wrapindent m:val="1440"> <m:intlim m:val="subSup"> <m:narylim m:val="undOvr"> </m:narylim></m:intlim> </m:wrapindent>γ usando la relación <span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">α</span></span> +<span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">β</span></span> + γ=18. Pero si alguno de <span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">α o </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;">βse calculó erróneamente, entonces </span></span>γ y luego comprobar que la suma de los tres ángulos es 180º. así:</m:defjc></m:rmargin></m:lmargin></m:dispdef></m:smallfrac></m:defjc></m:rmargin></m:lmargin></m:dispdef></m:smallfrac></div><div class="MsoNormal"><br />
</div><div class="MsoNormal">cosγ=a^2+b^2-c^2 , de modo que γ=107º+48º+25º=180º</div><div class="MsoNormal" style="text-align: left;"> 2ab</div><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"><br />
</span></span></div><div class="MsoNormal" style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"><span style="background-color: black;"> </span></span></span></div><div class="MsoNormal" style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> </span></span><br />
<span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> </span></span></div><div class="MsoNormal" style="font-family: Arial,Helvetica,sans-serif;"><br />
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</span></span></div><div class="MsoNormal"></div><div class="actividadesv"><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> </span></span></div><m:smallfrac m:val="off"> <m:dispdef> <m:lmargin m:val="0"> <m:rmargin m:val="0"> <m:defjc m:val="centerGroup"> <m:wrapindent m:val="1440"> <m:intlim m:val="subSup"> <m:narylim m:val="undOvr"> </m:narylim></m:intlim> </m:wrapindent> </m:defjc></m:rmargin></m:lmargin></m:dispdef></m:smallfrac><br />
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<div class="MsoNormal"><m:smallfrac m:val="off"> <m:dispdef> <m:lmargin m:val="0"> <m:rmargin m:val="0"> <m:defjc m:val="centerGroup"> <m:wrapindent m:val="1440"> <m:intlim m:val="subSup"> <m:narylim m:val="undOvr"> </m:narylim></m:intlim> </m:wrapindent> </m:defjc></m:rmargin></m:lmargin></m:dispdef></m:smallfrac></div><div class="MsoNormal"><span style="font-family: "Calibri","sans-serif"; font-size: 11pt; line-height: 115%;"></span></div><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"> </span></span><span style="font-size: small;"><span style="font-family: Arial,Helvetica,sans-serif;"></span></span>Karen Encisohttp://www.blogger.com/profile/08030437550934449063noreply@blogger.com1tag:blogger.com,1999:blog-7009680633045734205.post-8241344605607597102011-10-10T11:07:00.001-07:002011-10-25T14:34:52.256-07:00Teorema del seno y coseno en nuestras vidas<b><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;">Teorema del seno y coseno en nuestras vidas</span><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;">:</span></b><br />
<span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;">En nuestros tiempos de avances tecnológicos es necesario y casi prioritario el uso de cálculos y funciones que a pesar que fueron creadas hace mucho tiempo siempre van a ser información y material de importancia en el moderno mundo de hoy</span><br />
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<span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;"><b>Historia del teorema del seno y el coseno:</b></span><br />
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<span class="Apple-style-span" style="background-color: white; line-height: 16px;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;">Los Elementos de Euclides, que datan del siglo III a. C., contienen ya una aproximación geométrica de la generalización del teorema de Pitágoras .</span></span><span class="Apple-style-span" style="background-color: white;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;"><span class="Apple-style-span" style="line-height: 16px;">La formulación de la época es arcaica ya que la ausencia de funciones trigonométricas y del álgebra obligó a razonar en términos de diferencias de áreas. Por esto se comienzan a utilizar los siguientes términos:</span></span></span><br />
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<span class="Apple-style-span" style="background-color: white; line-height: 16px;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;">«En los triángulos obtusángulos, el cuadrado del lado opuesto al ángulo obtuso es mayor que los cuadrados de los lados que comprenden el ángulo obtuso en dos veces el rectángulo comprendido por un lado de los del ángulo obtuso sobre el que cae la perpendicular y la recta exterior cortada por la perpendicular, hasta el ángulo obtuso.»</span></span><br />
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<span class="Apple-style-span" style="background-color: white; line-height: 16px;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;">Faltaba esperar la trigonometría árabe-musulmana de la Edad Media para ver al teorema evolucionar a su forma y en su alcance: el astrónomo y matemático al-Battani generalizó el resultado de Euclides en la geometría esférica a principios del siglo X, lo que permitió efectuar los cálculos de la distancia angular entre el Sol y la Tierra. Fue durante el mismo período cuando se establecieron las primeras tablas trigonométricas, para las funciones seno y coseno. Eso permitió a Ghiyath al-Kashi, matemático de la escuela de Samarcanda, de poner el teorema bajo una forma utilizable para la triangulación durante el siglo XV. La propiedad fue popularizada en occidente por François Viète quien, al parecer, lo redescubrió independientemente.</span></span><br />
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<span class="Apple-style-span" style="background-color: white; line-height: 16px;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;">Fue a finales del siglo XVII cuando la notación algebraica moderna, aunada a la notación moderna de las funciones trigonométricas introducida por Euler en su libro Introductio in analysin infinitorum, permitieron escribir el teorema bajo su forma actual, extendiéndose el nombre de teorema (o ley) del coseno.</span></span><br />
<div class="separator" style="clear: both; text-align: left;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhugTkGujUP9SFllnUi-COkp28eHtqXI2aoldZrdQzuky5yPm5tmslfr1ExntHTDkjU9M95BzMddbxwGoOu_pC6DggeVPfdQxjWt63J5Iq-aqpen8yWh7zaa6nSKp073S0ILgbGCMPux88Q/s1600/descarga" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhugTkGujUP9SFllnUi-COkp28eHtqXI2aoldZrdQzuky5yPm5tmslfr1ExntHTDkjU9M95BzMddbxwGoOu_pC6DggeVPfdQxjWt63J5Iq-aqpen8yWh7zaa6nSKp073S0ILgbGCMPux88Q/s1600/descarga" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqjBqM3tMHI6ncLphVljmkfMEhx-zFS55zDl8kS4SM45Dt1GGv5RilezH3ovnPmJexqknDuicwBzRfu-xbEECv5bOm69s5Lir8ZhkBoV1HW_hjBp5h9cYWwAzpnIpvbBuhK33wAYba0fgT/s1600/descarga+%25281%2529" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqjBqM3tMHI6ncLphVljmkfMEhx-zFS55zDl8kS4SM45Dt1GGv5RilezH3ovnPmJexqknDuicwBzRfu-xbEECv5bOm69s5Lir8ZhkBoV1HW_hjBp5h9cYWwAzpnIpvbBuhK33wAYba0fgT/s200/descarga+%25281%2529" width="148" /></a></div><br />
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<span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;"><span class="Apple-style-span" style="line-height: 16px;"><b>Ley del seno:</b></span></span><br />
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<span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;"><span class="Apple-style-span" style="line-height: 16px;">Un triángulo oblicuo es aquel que no tiene un ángulo recto. Usaremos la literales <i>A, B, C, a, b, c ,</i></span></span><span class="Apple-style-span" style="font-family: Calibri, sans-serif; line-height: 17px;"><i>α</i>,</span><span class="Apple-style-span" style="background-color: white; font-family: arial, sans-serif; line-height: 14px;"><i>β, </i></span><span style="font-family: Calibri, sans-serif; font-style: italic; line-height: 115%;"> </span><span style="line-height: 115%;"><i style="font-family: Calibri, sans-serif;">λ, </i><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;">para distinguir</span></span><span style="font-family: Calibri, sans-serif; line-height: 115%;"> </span><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;"><span class="Apple-style-span" style="line-height: 115%;">partes de los </span><span class="Apple-style-span" style="line-height: 18px;">triángulos.</span></span><br />
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<span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif; line-height: 15px;">Supongamos que <i>ABC </i>es un triangulo oblicuo con los angulos y lados marcados en la forma acostumbrada: </span><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZPXFyuWLGr5szek_e5nR3H6QXLhfhyphenhyphenGweSL8Lqx84hc-wx91rpu2VjFrDjzAL5-2zvwsyOItdqWwaZ287J_78wn-z9K1IpJxAiKzDK3fCe1JUGG6SXs-ksV71QCp6lT-RS8Sc7QOkQVcu/s1600/Sin+5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="178" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZPXFyuWLGr5szek_e5nR3H6QXLhfhyphenhyphenGweSL8Lqx84hc-wx91rpu2VjFrDjzAL5-2zvwsyOItdqWwaZ287J_78wn-z9K1IpJxAiKzDK3fCe1JUGG6SXs-ksV71QCp6lT-RS8Sc7QOkQVcu/s200/Sin+5.png" width="200" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
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</div><div class="separator" style="clear: both; text-align: left;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;">Entonces: </span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxpdBKesO-QUCGKfEnZkaWEXwvQKCre7SisYT2XdwYMgG_UJJCJ-uCS8G8CTP0b_zorTyD6mE25RHey6FpeJKjSFpTmd4DLrQbDHN4G5siVbe-fBWPVxk7ZW5AjP2WmpB1O42MrPX4CcrB/s1600/Sin+5.png5.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="49" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxpdBKesO-QUCGKfEnZkaWEXwvQKCre7SisYT2XdwYMgG_UJJCJ-uCS8G8CTP0b_zorTyD6mE25RHey6FpeJKjSFpTmd4DLrQbDHN4G5siVbe-fBWPVxk7ZW5AjP2WmpB1O42MrPX4CcrB/s200/Sin+5.png5.png" width="200" /><span class="Apple-style-span" style="color: black; font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;"> </span></a></div><div class="separator" style="clear: both; text-align: left;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;">observa que la ley de los senos consta de los senos consta de las siguientes tres formulas: </span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuT8meEebXHd6YMc_Ee33XczGI76rtWr2nD0NmDFp1S5V8sa10ahzN5sUwXVdkR1Ar1d8R6IZ6BVltzp5jiL_ZdnK6OlPmvzgNkqWOUff8BspMg-HTR9Gki6zGQrMSHqEalFm-hILQkvOa/s1600/Sin+5.png4.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="41" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuT8meEebXHd6YMc_Ee33XczGI76rtWr2nD0NmDFp1S5V8sa10ahzN5sUwXVdkR1Ar1d8R6IZ6BVltzp5jiL_ZdnK6OlPmvzgNkqWOUff8BspMg-HTR9Gki6zGQrMSHqEalFm-hILQkvOa/s320/Sin+5.png4.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;"><span class="Apple-style-span" style="line-height: 18px;">para aplicar cualquiera de estas fórmulas a un triangulo especifico debemos conocer los valores de tres de las cuatro variables. Si sustituyes estos tres valores en la forma apropiada, podrás despejar el valor de la cuarta variable.</span></span></div><div class="separator" style="clear: both; text-align: left;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;"><span class="Apple-style-span" style="line-height: 18px;">Se deduce que la ley de los senos se puede hallar las partes restantes de un triangulo oblicuo, siempre que se conozcan cualquiera de las siguientes dos condiciones (las tres letras dentro del paréntesis significan las partes conocidas con "L" representando un lado y "A" ángulo) :</span></span></div><div class="separator" style="clear: both; text-align: left;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;"><span class="Apple-style-span" style="line-height: 18px;"><br />
</span></span></div><div class="separator" style="clear: both; text-align: left;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;"><span class="Apple-style-span" style="line-height: 18px;">1) dos lados y un ángulo <i>opuesto </i>a uno de ellos (LLA) </span></span></div><span class="Apple-style-span" style="background-color: white; line-height: 16px;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;">2) dos ángulos y cualquier lado (AAL) o (ALA) </span></span><br />
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<span class="Apple-style-span" style="background-color: white;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;"><span class="Apple-style-span" style="line-height: 16px;">en la sección siguiente estudiaremos la ley de los cosenos y y demostraremos como se usa para hallar las partes restantes del triángulo oblicuo cuando se conoce los siguiente:</span></span></span><br />
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<span class="Apple-style-span" style="background-color: white;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;"><span class="Apple-style-span" style="line-height: 16px;">1) dos lados y el triangulo <i>entre</i> ellos (LAL)</span></span></span><br />
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<span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;"><span class="Apple-style-span" style="line-height: 16px;">La ley de los senos no se puede aplicar directamente a los dos últimos casos. Esta ley también se puede escribir en la forma.</span></span><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEu0DwT1x_MivenO3Oj3tLI8jyap27_8UiSYZb4fsUDFM6F3HJgLOAAeFzAGVKVQE02C7Pciyrlv8Fc-1fRMPUpyA1ZyrGrcYEzd-J006-i4ncmzkYd14AwwLKJbKgbhHqTB8otuzwMPK1/s1600/Sin+5.png3.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="47" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEu0DwT1x_MivenO3Oj3tLI8jyap27_8UiSYZb4fsUDFM6F3HJgLOAAeFzAGVKVQE02C7Pciyrlv8Fc-1fRMPUpyA1ZyrGrcYEzd-J006-i4ncmzkYd14AwwLKJbKgbhHqTB8otuzwMPK1/s200/Sin+5.png3.png" width="200" /></a></div><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;"><span class="Apple-style-span" style="line-height: 16px;"><br />
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<span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;"><span class="Apple-style-span" style="line-height: 16px;">en lugar de memorizar las tres fórmulas relacionadas con la ley de los senos. Pueden ser mas conveniente recordar el siguiente enunciado, que toma en cuenta a todas ellas.</span></span><br />
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<span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;"><span class="Apple-style-span" style="line-height: 16px;"><b>Ley de los senos: </b>en cualquier triángulo, la razón entre el seno de un ángulo y el lado opuesto a ese ángulo es igual a la razón entre el seno de otro ángulo y el lado opuesto a ese ángulo.</span></span><br />
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<span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;"><span class="Apple-style-span"><b style="line-height: 16px;">Ejemplo: </b><span class="Apple-style-span" style="line-height: 16px;">Cuando el ángulo de elevación del sol es de 64</span></span><span class="Apple-style-span" style="background-color: white; line-height: 16px;">°, un poste de teléfonos inclinado a un ángulo de de 9</span><span class="Apple-style-span" style="background-color: white; line-height: 16px;">° en direccion opuesta al sol arroja una sombra de 21 pies de largo a nivel del suelo. Calcula la longitud del poste</span></span><span class="Apple-style-span" style="background-color: white; font-family: arial, sans-serif; line-height: 16px;">. </span><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCkusUla-fG1nf_J50zn1u112I_emTJflmxjicSnMJeI0u6k7X6vJjkoUqYHmPAxo8FrzepbC7NEyi9kmNWf3FKhOHww5GDNPP3hQTTVUVXdS_CXyBB0jFUbdbvqOMbv1M8BV2LKkYcNkW/s1600/Sin+5..png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCkusUla-fG1nf_J50zn1u112I_emTJflmxjicSnMJeI0u6k7X6vJjkoUqYHmPAxo8FrzepbC7NEyi9kmNWf3FKhOHww5GDNPP3hQTTVUVXdS_CXyBB0jFUbdbvqOMbv1M8BV2LKkYcNkW/s200/Sin+5..png" width="156" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><span class="Apple-style-span" style="background-color: white; line-height: 16px;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;">en la siguiente imagen se ilustra el triangulo <i>ABC </i> con los datos dados. Observamos que en la figura anterior hemos calculado los siguientes ángulos.</span></span><br />
<span class="Apple-style-span" style="line-height: 16px;"><span style="line-height: 115%;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;">β</span></span><span style="font-family: Calibri, sans-serif; line-height: 115%;"> </span><span style="line-height: 115%;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;">= 90</span></span></span><span class="Apple-style-span" style="background-color: white; color: #222222; font-family: Arial, sans-serif; line-height: 17px;">°</span><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;"><span class="Apple-style-span" style="background-color: white; line-height: 17px;">- 9</span><span class="Apple-style-span" style="background-color: white; color: #222222; line-height: 17px;">°</span><span class="Apple-style-span" style="line-height: 15px;"> = 81</span></span><span class="Apple-style-span" style="background-color: white; color: #222222; font-family: Arial, sans-serif; line-height: 17px;">°</span><br />
<span class="Apple-style-span" style="background-color: white; color: #222222; font-family: Arial, sans-serif; line-height: 17px;"><br />
</span><br />
<span class="Apple-style-span" style="background-color: white; line-height: 17px;"></span><br />
<div class="MsoNormal"><span style="line-height: 115%;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;">λ = 180</span><span class="Apple-style-span" style="color: #222222; font-family: Arial, sans-serif;"><o:p></o:p></span></span><span class="Apple-style-span" style="background-color: white; color: #222222; font-family: Arial, sans-serif; line-height: 17px;">°</span><span class="Apple-style-span" style="background-color: white; line-height: 17px;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;">- 64</span></span><span class="Apple-style-span" style="background-color: white; color: #222222; font-family: Arial, sans-serif; line-height: 17px;">° </span><span class="Apple-style-span" style="background-color: white; line-height: 17px;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;">- 81 = 35</span></span><span class="Apple-style-span" style="background-color: white; color: #222222; font-family: Arial, sans-serif; line-height: 17px;">° </span></div><div class="MsoNormal"><span class="Apple-style-span" style="background-color: white; color: #222222; font-family: Arial, sans-serif; line-height: 17px;"><br />
</span></div><div class="MsoNormal"><span class="Apple-style-span" style="line-height: 17px;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;">Para hallar la longitud del poste; es decir , el lado de </span></span><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;">α del triagulo <i>ABC </i> se procede como sigue:</span></div><div class="MsoNormal"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;"><br />
</span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEubMWV40HLMJapIk1fyKfW8JU3z8FhfsAvr-dkmAnMzrfYVqYgFzd8Zs9tprVaWqkB8RZepCwMt5PSKZuWN7srFAsZ-5hFn6uds264OGFiHmiZf9Kv3ybr6BrYZMFbrHZbJcdQV7JEMTJ/s1600/Sin+5.png3.png5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="99" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEubMWV40HLMJapIk1fyKfW8JU3z8FhfsAvr-dkmAnMzrfYVqYgFzd8Zs9tprVaWqkB8RZepCwMt5PSKZuWN7srFAsZ-5hFn6uds264OGFiHmiZf9Kv3ybr6BrYZMFbrHZbJcdQV7JEMTJ/s320/Sin+5.png3.png5.png" width="320" /></a><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;">por tanto, el poste de teléfono mide 33pies de largo</span></div><div class="separator" style="clear: both; text-align: center;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, Helvetica, sans-serif;"><br />
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</span></div>Karen Encisohttp://www.blogger.com/profile/08030437550934449063noreply@blogger.com3tag:blogger.com,1999:blog-7009680633045734205.post-11782008638685506332011-10-10T10:58:00.000-07:002011-10-15T19:06:11.201-07:00Importancia del teorema del seno y coseno<div style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;">En nuestros tiempos de avances tecnológicos es necesario y casi prioritario el uso de cálculos y funciones que a pesar que fueron creadas hace mucho tiempo siempre van a ser información y material de importancia en el moderno mundo de hoy</span></div>Karen Encisohttp://www.blogger.com/profile/08030437550934449063noreply@blogger.com1