Precálculo Ejemplos

${(4 \cos (x) - 7 \sin (x))}^{2} + {(7 \cos (x) + 4 \sin (x))}^{2} = 65$

Paso 1

Eleva al cuadrado ambos lados de la ecuación.

${({(4 \cos (x) - 7 \sin (x))}^{2} + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Paso 2

Simplifica ${({(4 \cos (x) - 7 \sin (x))}^{2} + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2}$ .

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Paso 2.1

Simplifica cada término.

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Paso 2.1.1

Reescribe ${(4 \cos (x) - 7 \sin (x))}^{2}$ como $(4 \cos (x) - 7 \sin (x)) (4 \cos (x) - 7 \sin (x))$ .

${((4 \cos (x) - 7 \sin (x)) (4 \cos (x) - 7 \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Paso 2.1.2

Expande $(4 \cos (x) - 7 \sin (x)) (4 \cos (x) - 7 \sin (x))$ con el método PEIU (primero, exterior, interior, ultimo).

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Paso 2.1.2.1

Aplica la propiedad distributiva.

${(4 \cos (x) (4 \cos (x) - 7 \sin (x)) - 7 \sin (x) (4 \cos (x) - 7 \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Paso 2.1.2.2

Aplica la propiedad distributiva.

${(4 \cos (x) (4 \cos (x)) + 4 \cos (x) (- 7 \sin (x)) - 7 \sin (x) (4 \cos (x) - 7 \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Paso 2.1.2.3

Aplica la propiedad distributiva.

${(4 \cos (x) (4 \cos (x)) + 4 \cos (x) (- 7 \sin (x)) - 7 \sin (x) (4 \cos (x)) - 7 \sin (x) (- 7 \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Paso 2.1.3

Simplifica y combina los términos similares.

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Paso 2.1.3.1

Simplifica cada término.

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Paso 2.1.3.1.1

Multiplica $4 \cos (x) (4 \cos (x))$ .

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Paso 2.1.3.1.1.1

Multiplica $4$ por $4$ .

${(16 \cos (x) \cos (x) + 4 \cos (x) (- 7 \sin (x)) - 7 \sin (x) (4 \cos (x)) - 7 \sin (x) (- 7 \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Paso 2.1.3.1.1.2

Eleva $\cos (x)$ a la potencia de $1$ .

${(16 (\cos^{1} (x) \cos (x)) + 4 \cos (x) (- 7 \sin (x)) - 7 \sin (x) (4 \cos (x)) - 7 \sin (x) (- 7 \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Paso 2.1.3.1.1.3

Eleva $\cos (x)$ a la potencia de $1$ .

${(16 (\cos^{1} (x) \cos^{1} (x)) + 4 \cos (x) (- 7 \sin (x)) - 7 \sin (x) (4 \cos (x)) - 7 \sin (x) (- 7 \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Paso 2.1.3.1.1.4

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

${(16 {\cos (x)}^{1 + 1} + 4 \cos (x) (- 7 \sin (x)) - 7 \sin (x) (4 \cos (x)) - 7 \sin (x) (- 7 \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Paso 2.1.3.1.1.5

Suma $1$ y $1$ .

${(16 \cos^{2} (x) + 4 \cos (x) (- 7 \sin (x)) - 7 \sin (x) (4 \cos (x)) - 7 \sin (x) (- 7 \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Paso 2.1.3.1.2

Multiplica $- 7$ por $4$ .

${(16 \cos^{2} (x) - 28 \cos (x) \sin (x) - 7 \sin (x) (4 \cos (x)) - 7 \sin (x) (- 7 \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Paso 2.1.3.1.3

Multiplica $4$ por $- 7$ .

${(16 \cos^{2} (x) - 28 \cos (x) \sin (x) - 28 \sin (x) \cos (x) - 7 \sin (x) (- 7 \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Paso 2.1.3.1.4

Multiplica $- 7 \sin (x) (- 7 \sin (x))$ .

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Paso 2.1.3.1.4.1

Multiplica $- 7$ por $- 7$ .

${(16 \cos^{2} (x) - 28 \cos (x) \sin (x) - 28 \sin (x) \cos (x) + 49 \sin (x) \sin (x) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Paso 2.1.3.1.4.2

Eleva $\sin (x)$ a la potencia de $1$ .

${(16 \cos^{2} (x) - 28 \cos (x) \sin (x) - 28 \sin (x) \cos (x) + 49 (\sin^{1} (x) \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Paso 2.1.3.1.4.3

Eleva $\sin (x)$ a la potencia de $1$ .

${(16 \cos^{2} (x) - 28 \cos (x) \sin (x) - 28 \sin (x) \cos (x) + 49 (\sin^{1} (x) \sin^{1} (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Paso 2.1.3.1.4.4

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

${(16 \cos^{2} (x) - 28 \cos (x) \sin (x) - 28 \sin (x) \cos (x) + 49 {\sin (x)}^{1 + 1} + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Paso 2.1.3.1.4.5

Suma $1$ y $1$ .

${(16 \cos^{2} (x) - 28 \cos (x) \sin (x) - 28 \sin (x) \cos (x) + 49 \sin^{2} (x) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Paso 2.1.3.2

Reordena los factores de $- 28 \sin (x) \cos (x)$ .

${(16 \cos^{2} (x) - 28 \cos (x) \sin (x) - 28 \cos (x) \sin (x) + 49 \sin^{2} (x) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Paso 2.1.3.3

Resta $28 \cos (x) \sin (x)$ de $- 28 \cos (x) \sin (x)$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Paso 2.1.4

Reescribe ${(7 \cos (x) + 4 \sin (x))}^{2}$ como $(7 \cos (x) + 4 \sin (x)) (7 \cos (x) + 4 \sin (x))$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + (7 \cos (x) + 4 \sin (x)) (7 \cos (x) + 4 \sin (x)))}^{2} = {(65)}^{2}$

Paso 2.1.5

Expande $(7 \cos (x) + 4 \sin (x)) (7 \cos (x) + 4 \sin (x))$ con el método PEIU (primero, exterior, interior, ultimo).

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Paso 2.1.5.1

Aplica la propiedad distributiva.

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 7 \cos (x) (7 \cos (x) + 4 \sin (x)) + 4 \sin (x) (7 \cos (x) + 4 \sin (x)))}^{2} = {(65)}^{2}$

Paso 2.1.5.2

Aplica la propiedad distributiva.

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 7 \cos (x) (7 \cos (x)) + 7 \cos (x) (4 \sin (x)) + 4 \sin (x) (7 \cos (x) + 4 \sin (x)))}^{2} = {(65)}^{2}$

Paso 2.1.5.3

Aplica la propiedad distributiva.

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 7 \cos (x) (7 \cos (x)) + 7 \cos (x) (4 \sin (x)) + 4 \sin (x) (7 \cos (x)) + 4 \sin (x) (4 \sin (x)))}^{2} = {(65)}^{2}$

Paso 2.1.6

Simplifica y combina los términos similares.

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Paso 2.1.6.1

Simplifica cada término.

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Paso 2.1.6.1.1

Multiplica $7 \cos (x) (7 \cos (x))$ .

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Paso 2.1.6.1.1.1

Multiplica $7$ por $7$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos (x) \cos (x) + 7 \cos (x) (4 \sin (x)) + 4 \sin (x) (7 \cos (x)) + 4 \sin (x) (4 \sin (x)))}^{2} = {(65)}^{2}$

Paso 2.1.6.1.1.2

Eleva $\cos (x)$ a la potencia de $1$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 (\cos^{1} (x) \cos (x)) + 7 \cos (x) (4 \sin (x)) + 4 \sin (x) (7 \cos (x)) + 4 \sin (x) (4 \sin (x)))}^{2} = {(65)}^{2}$

Paso 2.1.6.1.1.3

Eleva $\cos (x)$ a la potencia de $1$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 (\cos^{1} (x) \cos^{1} (x)) + 7 \cos (x) (4 \sin (x)) + 4 \sin (x) (7 \cos (x)) + 4 \sin (x) (4 \sin (x)))}^{2} = {(65)}^{2}$

Paso 2.1.6.1.1.4

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 {\cos (x)}^{1 + 1} + 7 \cos (x) (4 \sin (x)) + 4 \sin (x) (7 \cos (x)) + 4 \sin (x) (4 \sin (x)))}^{2} = {(65)}^{2}$

Paso 2.1.6.1.1.5

Suma $1$ y $1$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 7 \cos (x) (4 \sin (x)) + 4 \sin (x) (7 \cos (x)) + 4 \sin (x) (4 \sin (x)))}^{2} = {(65)}^{2}$

Paso 2.1.6.1.2

Multiplica $4$ por $7$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 28 \cos (x) \sin (x) + 4 \sin (x) (7 \cos (x)) + 4 \sin (x) (4 \sin (x)))}^{2} = {(65)}^{2}$

Paso 2.1.6.1.3

Multiplica $7$ por $4$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 28 \cos (x) \sin (x) + 28 \sin (x) \cos (x) + 4 \sin (x) (4 \sin (x)))}^{2} = {(65)}^{2}$

Paso 2.1.6.1.4

Multiplica $4 \sin (x) (4 \sin (x))$ .

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Paso 2.1.6.1.4.1

Multiplica $4$ por $4$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 28 \cos (x) \sin (x) + 28 \sin (x) \cos (x) + 16 \sin (x) \sin (x))}^{2} = {(65)}^{2}$

Paso 2.1.6.1.4.2

Eleva $\sin (x)$ a la potencia de $1$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 28 \cos (x) \sin (x) + 28 \sin (x) \cos (x) + 16 (\sin^{1} (x) \sin (x)))}^{2} = {(65)}^{2}$

Paso 2.1.6.1.4.3

Eleva $\sin (x)$ a la potencia de $1$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 28 \cos (x) \sin (x) + 28 \sin (x) \cos (x) + 16 (\sin^{1} (x) \sin^{1} (x)))}^{2} = {(65)}^{2}$

Paso 2.1.6.1.4.4

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 28 \cos (x) \sin (x) + 28 \sin (x) \cos (x) + 16 {\sin (x)}^{1 + 1})}^{2} = {(65)}^{2}$

Paso 2.1.6.1.4.5

Suma $1$ y $1$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 28 \cos (x) \sin (x) + 28 \sin (x) \cos (x) + 16 \sin^{2} (x))}^{2} = {(65)}^{2}$

Paso 2.1.6.2

Reordena los factores de $28 \sin (x) \cos (x)$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 28 \cos (x) \sin (x) + 28 \cos (x) \sin (x) + 16 \sin^{2} (x))}^{2} = {(65)}^{2}$

Paso 2.1.6.3

Suma $28 \cos (x) \sin (x)$ y $28 \cos (x) \sin (x)$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 56 \cos (x) \sin (x) + 16 \sin^{2} (x))}^{2} = {(65)}^{2}$

Paso 2.2

Simplifica los términos.

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Paso 2.2.1

Combina los términos opuestos en $16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 56 \cos (x) \sin (x) + 16 \sin^{2} (x)$ .

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Paso 2.2.1.1

Suma $- 56 \cos (x) \sin (x)$ y $56 \cos (x) \sin (x)$ .

${(16 \cos^{2} (x) + 0 + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 16 \sin^{2} (x))}^{2} = {(65)}^{2}$

Paso 2.2.1.2

Suma $16 \cos^{2} (x)$ y $0$ .

${(16 \cos^{2} (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 16 \sin^{2} (x))}^{2} = {(65)}^{2}$

Paso 2.2.2

Mueve $16 \sin^{2} (x)$ .

${(16 \cos^{2} (x) + 16 \sin^{2} (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x))}^{2} = {(65)}^{2}$

Paso 2.2.3

Factoriza $16$ de $16 \cos^{2} (x)$ .

${(16 (\cos^{2} (x)) + 16 \sin^{2} (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x))}^{2} = {(65)}^{2}$

Paso 2.2.4

Factoriza $16$ de $16 \sin^{2} (x)$ .

${(16 (\cos^{2} (x)) + 16 (\sin^{2} (x)) + 49 \sin^{2} (x) + 49 \cos^{2} (x))}^{2} = {(65)}^{2}$

Paso 2.2.5

Factoriza $16$ de $16 (\cos^{2} (x)) + 16 (\sin^{2} (x))$ .

${(16 (\cos^{2} (x) + \sin^{2} (x)) + 49 \sin^{2} (x) + 49 \cos^{2} (x))}^{2} = {(65)}^{2}$

Paso 2.3

Reorganiza los términos.

${(16 (\sin^{2} (x) + \cos^{2} (x)) + 49 \sin^{2} (x) + 49 \cos^{2} (x))}^{2} = {(65)}^{2}$

Paso 2.4

Aplica la identidad pitagórica.

${(16 \cdot 1 + 49 \sin^{2} (x) + 49 \cos^{2} (x))}^{2} = {(65)}^{2}$

Paso 2.5

Simplifica con la obtención del factor común.

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Paso 2.5.1

Factoriza $49$ de $49 \sin^{2} (x)$ .

${(16 \cdot 1 + 49 (\sin^{2} (x)) + 49 \cos^{2} (x))}^{2} = {(65)}^{2}$

Paso 2.5.2

Factoriza $49$ de $49 \cos^{2} (x)$ .

${(16 \cdot 1 + 49 (\sin^{2} (x)) + 49 (\cos^{2} (x)))}^{2} = {(65)}^{2}$

Paso 2.5.3

Factoriza $49$ de $49 (\sin^{2} (x)) + 49 (\cos^{2} (x))$ .

${(16 \cdot 1 + 49 (\sin^{2} (x) + \cos^{2} (x)))}^{2} = {(65)}^{2}$

Paso 2.6

Aplica la identidad pitagórica.

${(16 \cdot 1 + 49 \cdot 1)}^{2} = {(65)}^{2}$

Paso 2.7

Simplifica los términos.

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Paso 2.7.1

Simplifica cada término.

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Paso 2.7.1.1

Multiplica $16$ por $1$ .

${(16 + 49 \cdot 1)}^{2} = {(65)}^{2}$

Paso 2.7.1.2

Multiplica $49$ por $1$ .

${(16 + 49)}^{2} = {(65)}^{2}$

Paso 2.7.2

Simplifica la expresión.

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Paso 2.7.2.1

Suma $16$ y $49$ .

$65^{2} = {(65)}^{2}$

Paso 2.7.2.2

Eleva $65$ a la potencia de $2$ .

$4225 = {(65)}^{2}$

Paso 3

Eleva $65$ a la potencia de $2$ .

$4225 = 4225$

Paso 4

Como $4225 = 4225$ , la ecuación siempre será verdadera para cualquier valor de $x$ .

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Paso 5

El resultado puede mostrarse de distintas formas.

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Notación de intervalo:

$(- \infty, \infty)$