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Clase 71. Identidades trigonométricas. sen 2 x + cos 2 x = 1. cos(x + y) = cosx cosy – senx seny. sen 2x = 2 senx cos x. Igualdades donde al menos aparece una variable. Ecuaciones. Identidades. Solo se satisfacen para algunos valores del dominio de la varible.

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Clase 71

Identidades

trigonométricas

sen2x + cos2x = 1

cos(x + y) = cosx cosy – senx seny

sen 2x = 2 senx cos x


Igualdades donde al menos aparece una variable.

Ecuaciones

Identidades

Solo se satisfacen para algunos valores del dominio de la varible.

Se satisfacen para todos los valores del dominio de la varible.


sen x

tan x =

cos x

cos x

cot x =

sen x

1

1

1 + tan2x =

1 + cot2x =

cos2 x

sen2 x

Identidades fundamentales

sen2x + cos2x = 1

sen2x = 1 – cos2x

cos2x = 1 – sen2x


cos(x y) = cos x cos y sen x  sen y

tan x  tan y

tan(x  y) =

1 tan x tan y

Fórmulas de adición

sen(x y) = sen x cos y  cos x sen y


2 tan x

tan 2x =

1 – tan2x

Fórmulas del ángulo duplo

sen 2x = 2 senx cosx

cos 2x = cos2x – sen2x

= 1 – 2 sen2x

= 2 cos2x –1


Ejercicio 1

Demuestra las siguientes identidades.

a) (sen x + cos x)2 = 1 + sen 2x

b) sen 3x = 3 sen x – 4 sen3x

a) (sen x + cos x)2

= sen2x + 2 sen x cos x + cos2x

= 1 + sen 2x

se cumple


b) sen 3x = 3 sen x – 4 sen3x

sen 3x

= sen (x + 2x)

= sen x cos 2x + cos x sen 2x

+ 2 sen x cos2x

= sen x (1–2 sen2x)

= sen x –2 sen3x

+2 sen x (1–sen2x)

= sen x –2 sen3x

+2 sen x – 2 sen3x

– 4 sen3x

= 3 sen x

se cumple


2 cos2 x

2

c) – tan x =

sen 2x

sen 2x

Ejercicio 2

Demuestra las siguientes identidades para los valores admisibles de la variable.

Para el estudio individual

a) cos4y – sen4y = cos 2y

4 cot 2x

b) cot2x – tan2x =

d) cos 3x = 4 cos3x – 3 senx

sen 2x


a) cos4y – sen4y = cos 2y

cos4y – sen4y

1

= (cos2y + sen2y)(cos2y – sen2y)

cos2y – sen2y

=

= cos 2y

se cumple


cos2x

sen2x

sen2x

cos2x

4 cot 2x

b) cot2x – tan2x =

sen 2x

cot2x – tan2x

cos4x

–sen4x

=

=

sen2x cos2x

4 cos 2x

cos 2x

=

=

sen2x cos2x

4 sen2x cos2x

4 cos 2x

4 cot 2x

=

=

sen2 2x

sen 2x

se cumple


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