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    el uso de la ley de senos involucra tener como datos lados y ángulos de un triángulo

    Santiago

    Chicos, ¿alguien sabe la respuesta?

    obtenga el uso de la ley de senos involucra tener como datos lados y ángulos de un triángulo de este sitio.

    Ley de los senos

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    Ley de los senos

    La ley de los senos es la relación entre los lados y ángulos de triángulos no rectángulos (oblicuos). Simplemente, establece que la relación de la longitud de un lado de un triángulo al seno del ángulo opuesto a ese lado es igual para todos los lados y ángulos en un triángulo dado.

    En ∆ABC es un triángulo oblicuo con lados a, b y c , entonces .

    Para usar la ley de los senos necesita conocer ya sea dos ángulos y un lado del triángulo (AAL o ALA) o dos lados y un ángulo opuesto de uno de ellos (LLA). Dese cuenta que para el primero de los dos casos usamos las mismas partes que utilizó para probar la congruencia de triángulos en geometría pero en el segundo caso no podríamos probar los triángulos congruentes dadas esas partes. Esto es porque las partes faltantes podrían ser de diferentes tamaños. Esto es llamado el caso ambiguo y lo discutiremos más adelante.

    Ejemplo 1: Dado dos ángulos y un lado no incluído (AAL).

    Dado ∆ABC con A = 30°, B = 20° y a = 45 m. Encuentre el ángulo y los lados faltantes.

    El tercer ángulo del triángulo es

    C = 180° – A – B = 180° – 30° – 20 ° = 130°

    Por la ley de los senos,

    Por las propiedades de las proporciones

    Ejemplo 2: Dado dos ángulos y un lado incluído (ALA).

    Dado A = 42°, B = 75° y c = 22 cm. Encuentre el ángulo y los lados faltantes.

    El tercer ángulo del triángulo es:

    C = 180° – A – B = 180° – 42° – 75° = 63°

    Por la ley de los senos,

    Por las propiedades de las proporciones

    y

    El caso ambiguo

    Si dos lados y un ángulo opuesto a uno de ellos es dado, tres posibilidades pueden ocurrir.

    (1) No existe tal triángulo.

    (2) Dos triángulos diferentes existen.

    (3) Exactamente un triángulo existe.

    Considere un triángulo en el cual se le da a, b y A . (La altitud h del vértice B al lado , por la definición de los senos es igual a b sin A .)

    (1) No existe tal triángulo si A es agudo y a < h o A es obtuso y a ≤ b.

    (2) Dos triángulos diferentes existen si A es agudo y h < a < b.

    (3) En cualquier otro caso, exactamente un triángulo existe.

    Ejemplo 1: No existe solución

    Dado a = 15, b = 25 y A = 80°. Encuentre los otros ángulos y el lado.

    h = b sin A = 25 sin 80° ≈ 24.6

    Dese cuenta que a < h. Así parece que no hay solución. Verifique esto usando la ley de los senos.

    Esto contrae el hecho de que –1 ≤ sin B ≤ 1. Por lo tanto, no existe el triángulo.

    Ejemplo 2: Dos soluciones exist en

    Dado a = 6. b = 7 y A = 30°. Encuentre los otros ángulos y el lado.

    h = b sin A = 7 sin 30° = 3.5

    h < a < b por lo tanto, hay dos triángulos posibles.

    Por la ley de lo senos,

    Hay dos ángulos entre 0° y 180° cuyo seno es aproximadamente 0.5833, 35.69° y 144.31°.

    Si B ≈ 35.69°              Si B ≈ 144.31°

    C ≈180° – 30° – 35.69° ≈ 114.31°          C ≈ 180° – 30° – 144.31° ≈ 5.69°

    Ejemplo 3: Una solución existe

    Dado a = 22, b =12 y A = 40°. Encuentre los otros ángulos y el lado.

    a > b

    Por la ley de lo senos,

    B es agudo.

    C ≈ 180° – 40° – 20.52° ≈ 119.48°

    Por la ley de lo senos,

    Si se nos dan dos lados y un ángulo incluído de un triángulo o si se nos dan 3 lados de un triángulo, no podemos usar la ley de los senos porque no podemos establecer ninguna proporción donde información suficiente sea conocida. En estos dos casos debemos usar la ley de los cosenos .

    fuente : www.varsitytutors.com

    ::DIE::

    Se explicarán las leyes de senos y cosenos, a través de su definición, sus fórmulas, características y ejemplos, lo cual permitirá clarificar en qué casos o situaciones se aplica una ley u otra y cómo se resuelven problemas tanto simples como de un nivel más complejo; por ejemplo, un caso real.

    -

    LEY DE SENOS Y COSENOS

    Álgebra

    Departamento de Ingenería en Computación

    Introducción

    La trigonometría es una rama de las matemáticas que está presente en la vida cotidiana, pues nos ayuda a resolver diversas situaciones. Por ejemplo, el siguiente problema: Una fábrica de papel ha sufrido un incendio, cerca de la fábrica hay dos estaciones de bomberos, ¿cuál es la estación más cercana a la fábrica?

    Se puede utilizar la ley de senos o la ley de cosenos para conocer la distancia a la estación de bomberos más cercana a la fábrica, y así poder calcular el menor de los tiempos. Esta unidad te permitirá recordar por medio de preguntas y ejercicios estas leyes.

    Objetivo

    Identificarás la ley de senos y de cosenos, a través de su concepto, características y postulados, con la finalidad de resolver problemas físicos y geométricos necesarios en tu formación académica.

    Ley de senos

    La ley de senos es una herramienta básica para resolver triángulos de cualquier tipo y establece lo siguiente:

    a senA = b senB = c senC

    Esta ley se utiliza cuando se conocen:

    1) Dos ángulos interiores del triángulo y uno de sus lados;

    2) Dos lados del triángulo y el ángulo opuesto a cualquiera de estos lados.

    A continuación revisa los siguientes ejemplos:

    Ejemplo 1

    Resolución de ejemplo 1

    Ejemplo 2

    Resolución de ejemplo 2

    Si A=40º y B=60º determinar la longitud de los lados b y c y el valor del ángulo C para el siguiente triángulo.

    Ley de cosenos

    Al igual que la ley de senos, la ley de cosenos también es una herramienta básica para resolver triángulos de cualquier tipo y establece que:

    a2 = b2 + c2 - 2bc cosA

    b2 = a2 + c2 - 2ac cosB

    c2 = a2 + b2 - 2ab cosC

    Esta ley se utiliza para lo siguiente:

    1) Determinar la longitud de un lado del triángulo cuando se conocen los otros dos lados y el ángulo opuesto al lado que se desea calcular;

    2) Determinar un ángulo cuando se conocen los tres lados del triángulo.

    Ejemplo

    Para el triángulo que se muestra en la siguiente figura, calcular la longitud del lado a si A=35°, b=11 cm y c=15 cm

    a2 = b2 + c2 - 2bc cosA

    a2= (11)2 + (15)2 - 2(11)(15)cos(35º)

    a2= 121 + 225 - 330cos(35º)

    a2= 121 + 225 - 330(0.8191)

    a2= 75.679 a= 8.699 cm

    Actividad. Identificando las características de las leyes de seno y coseno

    Las leyes de seno y coseno resultan útiles para aplicarlas en la vida cotidiana, ya que pueden implementarse adecuadamente en problemas o situaciones específicas y también pueden explotarse en otros conocimientos, igual de necesarios para tu formación académica o laboral; por ello es importante que las identifiques y practiques.

    Con base en las características y ejemplos que revisaste en este tema, realiza la siguiente actividad.

    Autoevaluación. Resolución de problemas con ayuda de las leyes seno y coseno.

    Saber cómo y cuándo aplicar la ley de senos y cosenos resulta de suma importancia, ya que dichas leyes permiten resolver diversas situaciones de la vida cotidiana, por ejemplo, calcular la distancia de un lugar a otro.

    Fuentes de información

    Bibliografía

    Barrera, F., De la Cruz, B. y Soriano, L. H. (2006). Trigonometría, teoría y ejercicios. México: UNAM, pp. 87.

    Sitios electrónicos

    División de Ciencias Básicas. (2008a). Sección: Coordinaciones Académicas. Consultado el 31 de agosto de 2016 de http://www.dcb.unam.mx/CoordinacionesAcademicas/Matematicas/CapsulasAntecedentes/Ley_senos_Ley_Cosenos.pdf

    División de Ciencias Básicas. (2008b). Sección: Coordinaciones Académicas. Consultado el 31 de agosto de 2016 de http://www.dcb.unam.mx/CoordinacionesAcademicas/Matematicas/CapsulasAntecedentes/ley_de_los_senos.html

    fuente : uapa.cuaieed.unam.mx

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    fuente : brainly.lat

    ¿Quieres ver la respuesta o más?
    Santiago 13 day ago
    4

    Chicos, ¿alguien sabe la respuesta?

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