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Identidades Trigonométricas

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Muestra de las principales Identidades Trigonométricas aplicadas para un mismo arco o ángulo.

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  • @Kamilo Stifler cierto,esta mal,casi la utilizo incorrectamente.
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  • @miryamparedes90 son derivadas?
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Identidades Trigonométricas

  1. 1. Curso: Trigonometría Cód.: Trig 05 Tema: Identidades Trigonométricas para un mismo Arco - Identidades Fundamentales - Deducción de Fórmulas - Ejercicios aplicativos
  2. 2.  Identidades Fundamentales: Identidades Pitagóricas: Se denominan de esa manera por que son producto de la aplicación del Teorema de Pitágoras con las razones trigonométricas C Sen 2α + Cos 2α = 1 A α Tg α + 1 = Sec α 2 2 B A2 + B 2 = C 2 Ctg α + 1 = Csc α 2 2 (Teorema de Pitágoras)
  3. 3.  Identidades Fundamentales: Identidades Recíprocas: Se denominan de esa manera por que son obtenidas al efectuar el producto entre dos razones recíprocas. Ejm: “Seno y Cosecante” Senα .Cscα = 1 C A Cosα .Secα = 1 α B Tgα .Ctgα = 1 C.O A C. A B C.O A No olvides que: Senα = = ; Cosα = = ; Tgα = = Hip C Hip C C. A B
  4. 4.  Identidades Fundamentales: Identidades por Cociente: Denominadas así por que cada una de ellas representa la división o cociente entre otras dos razones trigonométricas. Senα Tgα = C Cosα A α Cosα Ctgα = B Senα OK… pero… ¿de donde salen esas fórmulas?
  5. 5.  Deducción de Fórmulas Veamos este ejemplo: Hipótesis : Sen 2α + Cos 2α = 1 Como ésta es una “Identidad Pitagórica”, usaremos el “Teorema de Pitágoras” para su demostración … listos? Del triángulo trigonométrico sabemos que: A B Senα = y Cosα = … entonces: C C C 2 2 A  A B Sen α =   y Cos 2α =   2 α C  C B A2 + B 2 Sen α + Cos α = 2 2 A2 + B = C 2 2 C2 (Teorema de Pitágoras) C2 Por lo tanto: Sen α + Cos α = 2 = 1 2 2 C
  6. 6.  Deducción de Fórmulas Una deducción más para que quede clara la idea ok? Senα Hipótesis : Tgα = Cosα Como ésta es una “Identidad por Cociente“, vamos a dividir las razones Seno y Coseno para la deducción. Del triángulo trigonométrico sabemos que: A B Senα = y Cosα = C C C A A Senα C A A α Dividiendo : = / = Pero : = Tgα Cosα B B B B C/ Senα Por lo tanto: Tgα = Cosα
  7. 7.  Ejercicios aplicativos Ahora veamos cómo se resuelven algunos ejercicios: 1. Simplifica: E = Cosx.Ctgx − Cscx 1 − Sen 2 x ( ) Solución: Por lo general, es conveniente convertir todo a Senos y Cosenos. Entonces Cosx 1 i ) Ctgx = ...Id . por cociente ii ) Cscx = ... Id . Recíproca Senx Senx iii )1 − Sen 2 x = Cos 2 x... Id . Pitagórica Cosx 1 Reemplazando las identidades tenemos: E = Cosx. − .Cos 2 x Senx Senx Cos 2 x Cos 2 x Cos 2 x − Cos 2 x Multiplicando y agrupando: E = − = Senx Senx Senx 0 Y llegamos a la respuesta: E= =0 Senx
  8. 8.  Ejercicios aplicativos 1. Simplifica: M = 1 + 2 Senα .Cosα − Senα Solución: Recordemos que una de las identidades Pitagóricas es ( Sen α + Cos α ) = 1 Reemplazando tenemos: 2 2 M = ( Sen α + Cos α ) + 2 Senα .Cosα − Senα 2 2 ¿Esto no es un producto notable?... Sí: M= ( Senα + Cosα ) 2 − Senα M = Senα + Cosα − Senα Y llegamos a la respuesta: M = Cosα
  9. 9.  Resumen de Fórmulas Identidades Fundamentales Pitagóricas : Por Cociente : Recíprocas : Sen x + Cos x = 1 2 2 Senx Senx.Cscx = 1 Tgx = Cosx Cosx.Secx = 1 Tg 2 x + 1 = Sec 2 x Cosx Ctg 2 x + 1 = Ctg 2 x Ctgx = Tgx.Ctgx = 1 Senx Ahora a seguir practicando …

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