Trigonometría Ejemplos

{(\sin (x) + \cos (x))}^{2} = 1 + \sin (2 x)

${(\sin (x) + \cos (x))}^{2} = 1 + \sin (2 x)$

Step 1

Simplifica el lado izquierdo.

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Simplifica

{(\sin (x) + \cos (x))}^{2}

${(\sin (x) + \cos (x))}^{2}$ .

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Reescribe.

0 + 0 + {(\sin (x) + \cos (x))}^{2} = 1 + \sin (2 x)

$0 + 0 + {(\sin (x) + \cos (x))}^{2} = 1 + \sin (2 x)$

Reescribe

{(\sin (x) + \cos (x))}^{2}

${(\sin (x) + \cos (x))}^{2}$ como

(\sin (x) + \cos (x)) (\sin (x) + \cos (x))

$(\sin (x) + \cos (x)) (\sin (x) + \cos (x))$ .

(\sin (x) + \cos (x)) (\sin (x) + \cos (x)) = 1 + \sin (2 x)

$(\sin (x) + \cos (x)) (\sin (x) + \cos (x)) = 1 + \sin (2 x)$

Expande

(\sin (x) + \cos (x)) (\sin (x) + \cos (x))

$(\sin (x) + \cos (x)) (\sin (x) + \cos (x))$ con el método PEIU (primero, exterior, interior, ultimo).

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Aplica la propiedad distributiva.

\sin (x) (\sin (x) + \cos (x)) + \cos (x) (\sin (x) + \cos (x)) = 1 + \sin (2 x)

$\sin (x) (\sin (x) + \cos (x)) + \cos (x) (\sin (x) + \cos (x)) = 1 + \sin (2 x)$

Aplica la propiedad distributiva.

\sin (x) \sin (x) + \sin (x) \cos (x) + \cos (x) (\sin (x) + \cos (x)) = 1 + \sin (2 x)

$\sin (x) \sin (x) + \sin (x) \cos (x) + \cos (x) (\sin (x) + \cos (x)) = 1 + \sin (2 x)$

Aplica la propiedad distributiva.

\sin (x) \sin (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos (x) \cos (x) = 1 + \sin (2 x)

$\sin (x) \sin (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos (x) \cos (x) = 1 + \sin (2 x)$

\sin (x) \sin (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos (x) \cos (x) = 1 + \sin (2 x)

$\sin (x) \sin (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos (x) \cos (x) = 1 + \sin (2 x)$

Simplifica y combina los términos similares.

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Simplifica cada término.

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Multiplica

\sin (x) \sin (x)

$\sin (x) \sin (x)$ .

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Eleva

\sin (x)

$\sin (x)$ a la potencia de

1

$1$ .

\sin^{1} (x) \sin (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos (x) \cos (x) = 1 + \sin (2 x)

$\sin^{1} (x) \sin (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos (x) \cos (x) = 1 + \sin (2 x)$

Eleva

\sin (x)

$\sin (x)$ a la potencia de

1

$1$ .

\sin^{1} (x) \sin^{1} (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos (x) \cos (x) = 1 + \sin (2 x)

$\sin^{1} (x) \sin^{1} (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos (x) \cos (x) = 1 + \sin (2 x)$

Usa la regla de la potencia

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

{\sin (x)}^{1 + 1} + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos (x) \cos (x) = 1 + \sin (2 x)

${\sin (x)}^{1 + 1} + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos (x) \cos (x) = 1 + \sin (2 x)$

Suma

1

$1$ y

1

$1$ .

\sin^{2} (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos (x) \cos (x) = 1 + \sin (2 x)

$\sin^{2} (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos (x) \cos (x) = 1 + \sin (2 x)$

\sin^{2} (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos (x) \cos (x) = 1 + \sin (2 x)

$\sin^{2} (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos (x) \cos (x) = 1 + \sin (2 x)$

Multiplica

\cos (x) \cos (x)

$\cos (x) \cos (x)$ .

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Eleva

\cos (x)

$\cos (x)$ a la potencia de

1

$1$ .

\sin^{2} (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos^{1} (x) \cos (x) = 1 + \sin (2 x)

$\sin^{2} (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos^{1} (x) \cos (x) = 1 + \sin (2 x)$

Eleva

\cos (x)

$\cos (x)$ a la potencia de

1

$1$ .

\sin^{2} (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos^{1} (x) \cos^{1} (x) = 1 + \sin (2 x)

$\sin^{2} (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos^{1} (x) \cos^{1} (x) = 1 + \sin (2 x)$

Usa la regla de la potencia

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

\sin^{2} (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + {\cos (x)}^{1 + 1} = 1 + \sin (2 x)

$\sin^{2} (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + {\cos (x)}^{1 + 1} = 1 + \sin (2 x)$

Suma

1

$1$ y

1

$1$ .

\sin^{2} (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos^{2} (x) = 1 + \sin (2 x)

$\sin^{2} (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos^{2} (x) = 1 + \sin (2 x)$

\sin^{2} (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos^{2} (x) = 1 + \sin (2 x)

$\sin^{2} (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos^{2} (x) = 1 + \sin (2 x)$

\sin^{2} (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos^{2} (x) = 1 + \sin (2 x)

$\sin^{2} (x) + \sin (x) \cos (x) + \cos (x) \sin (x) + \cos^{2} (x) = 1 + \sin (2 x)$

Reordena los factores de

\sin (x) \cos (x)

$\sin (x) \cos (x)$ .

\sin^{2} (x) + \cos (x) \sin (x) + \cos (x) \sin (x) + \cos^{2} (x) = 1 + \sin (2 x)

$\sin^{2} (x) + \cos (x) \sin (x) + \cos (x) \sin (x) + \cos^{2} (x) = 1 + \sin (2 x)$

Suma

\cos (x) \sin (x)

$\cos (x) \sin (x)$ y

\cos (x) \sin (x)

$\cos (x) \sin (x)$ .

\sin^{2} (x) + 2 \cos (x) \sin (x) + \cos^{2} (x) = 1 + \sin (2 x)

$\sin^{2} (x) + 2 \cos (x) \sin (x) + \cos^{2} (x) = 1 + \sin (2 x)$

\sin^{2} (x) + 2 \cos (x) \sin (x) + \cos^{2} (x) = 1 + \sin (2 x)

$\sin^{2} (x) + 2 \cos (x) \sin (x) + \cos^{2} (x) = 1 + \sin (2 x)$

Mueve

\cos^{2} (x)

$\cos^{2} (x)$ .

\sin^{2} (x) + \cos^{2} (x) + 2 \cos (x) \sin (x) = 1 + \sin (2 x)

$\sin^{2} (x) + \cos^{2} (x) + 2 \cos (x) \sin (x) = 1 + \sin (2 x)$

Aplica la identidad pitagórica.

1 + 2 \cos (x) \sin (x) = 1 + \sin (2 x)

$1 + 2 \cos (x) \sin (x) = 1 + \sin (2 x)$

Simplifica cada término.

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Reordena

2 \cos (x)

$2 \cos (x)$ y

\sin (x)

$\sin (x)$ .

1 + \sin (x) (2 \cos (x)) = 1 + \sin (2 x)

$1 + \sin (x) (2 \cos (x)) = 1 + \sin (2 x)$

Reordena

\sin (x)

$\sin (x)$ y

2

$2$ .

1 + 2 \cdot \sin (x) \cos (x) = 1 + \sin (2 x)

$1 + 2 \cdot \sin (x) \cos (x) = 1 + \sin (2 x)$

Aplica la razón del ángulo doble sinusoidal.

1 + \sin (2 x) = 1 + \sin (2 x)

$1 + \sin (2 x) = 1 + \sin (2 x)$

1 + \sin (2 x) = 1 + \sin (2 x)

$1 + \sin (2 x) = 1 + \sin (2 x)$

1 + \sin (2 x) = 1 + \sin (2 x)

$1 + \sin (2 x) = 1 + \sin (2 x)$

1 + \sin (2 x) = 1 + \sin (2 x)

$1 + \sin (2 x) = 1 + \sin (2 x)$

Step 2

Mueve todos los términos que contengan

\sin (2 x)

$\sin (2 x)$ al lado izquierdo de la ecuación.

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Resta

\sin (2 x)

$\sin (2 x)$ de ambos lados de la ecuación.

1 + \sin (2 x) - \sin (2 x) = 1

$1 + \sin (2 x) - \sin (2 x) = 1$

Combina los términos opuestos en

1 + \sin (2 x) - \sin (2 x)

$1 + \sin (2 x) - \sin (2 x)$ .

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Resta

\sin (2 x)

$\sin (2 x)$ de

\sin (2 x)

$\sin (2 x)$ .

1 + 0 = 1

$1 + 0 = 1$

Suma

1

$1$ y

0

$0$ .

1 = 1

$1 = 1$

1 = 1

$1 = 1$

1 = 1

$1 = 1$

Step 3

Como

1 = 1

$1 = 1$ , la ecuación siempre será verdadera para cualquier valor de

x

$x$ .

Todos los números reales

Step 4

El resultado puede mostrarse de distintas formas.

Todos los números reales

Notación de intervalo:

(- \infty, \infty)

$(- \infty, \infty)$