Precálculo Ejemplos

$[\begin{array}{c} - 1 & \cos (c) & \cos (b) \\ \cos (c) & - 1 & \cos (a) \\ \cos (b) & \cos (a) & - 1 \end{array}]$

Paso 1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Paso 1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Paso 1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Paso 1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} - 1 & \cos (a) \\ \cos (a) & - 1 \end{array} |$

Paso 1.4

Multiply element $a_{11}$ by its cofactor.

$- 1 | \begin{array}{c} - 1 & \cos (a) \\ \cos (a) & - 1 \end{array} |$

Paso 1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} \cos (c) & \cos (a) \\ \cos (b) & - 1 \end{array} |$

Paso 1.6

Multiply element $a_{12}$ by its cofactor.

$- \cos (c) | \begin{array}{c} \cos (c) & \cos (a) \\ \cos (b) & - 1 \end{array} |$

Paso 1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} \cos (c) & - 1 \\ \cos (b) & \cos (a) \end{array} |$

Paso 1.8

Multiply element $a_{13}$ by its cofactor.

$\cos (b) | \begin{array}{c} \cos (c) & - 1 \\ \cos (b) & \cos (a) \end{array} |$

Paso 1.9

Add the terms together.

$- 1 | \begin{array}{c} - 1 & \cos (a) \\ \cos (a) & - 1 \end{array} | - \cos (c) | \begin{array}{c} \cos (c) & \cos (a) \\ \cos (b) & - 1 \end{array} | + \cos (b) | \begin{array}{c} \cos (c) & - 1 \\ \cos (b) & \cos (a) \end{array} |$

Paso 2

Evalúa $| \begin{array}{c} - 1 & \cos (a) \\ \cos (a) & - 1 \end{array} |$ .

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Paso 2.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- 1 (- - 1 - \cos (a) \cos (a)) - \cos (c) | \begin{array}{c} \cos (c) & \cos (a) \\ \cos (b) & - 1 \end{array} | + \cos (b) | \begin{array}{c} \cos (c) & - 1 \\ \cos (b) & \cos (a) \end{array} |$

Paso 2.2

Simplifica el determinante.

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Paso 2.2.1

Simplifica cada término.

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Paso 2.2.1.1

Multiplica $- 1$ por $- 1$ .

$- 1 (1 - \cos (a) \cos (a)) - \cos (c) | \begin{array}{c} \cos (c) & \cos (a) \\ \cos (b) & - 1 \end{array} | + \cos (b) | \begin{array}{c} \cos (c) & - 1 \\ \cos (b) & \cos (a) \end{array} |$

Paso 2.2.1.2

Multiplica $- \cos (a) \cos (a)$ .

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Paso 2.2.1.2.1

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

$- 1 (1 - (\cos^{1} (a) \cos (a))) - \cos (c) | \begin{array}{c} \cos (c) & \cos (a) \\ \cos (b) & - 1 \end{array} | + \cos (b) | \begin{array}{c} \cos (c) & - 1 \\ \cos (b) & \cos (a) \end{array} |$

Paso 2.2.1.2.2

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

$- 1 (1 - (\cos^{1} (a) \cos^{1} (a))) - \cos (c) | \begin{array}{c} \cos (c) & \cos (a) \\ \cos (b) & - 1 \end{array} | + \cos (b) | \begin{array}{c} \cos (c) & - 1 \\ \cos (b) & \cos (a) \end{array} |$

Paso 2.2.1.2.3

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

$- 1 (1 - {\cos (a)}^{1 + 1}) - \cos (c) | \begin{array}{c} \cos (c) & \cos (a) \\ \cos (b) & - 1 \end{array} | + \cos (b) | \begin{array}{c} \cos (c) & - 1 \\ \cos (b) & \cos (a) \end{array} |$

Paso 2.2.1.2.4

Suma $1$ y $1$ .

$- 1 (1 - \cos^{2} (a)) - \cos (c) | \begin{array}{c} \cos (c) & \cos (a) \\ \cos (b) & - 1 \end{array} | + \cos (b) | \begin{array}{c} \cos (c) & - 1 \\ \cos (b) & \cos (a) \end{array} |$

Paso 2.2.2

Aplica la identidad pitagórica.

$- 1 \sin^{2} (a) - \cos (c) | \begin{array}{c} \cos (c) & \cos (a) \\ \cos (b) & - 1 \end{array} | + \cos (b) | \begin{array}{c} \cos (c) & - 1 \\ \cos (b) & \cos (a) \end{array} |$

Paso 3

Evalúa $| \begin{array}{c} \cos (c) & \cos (a) \\ \cos (b) & - 1 \end{array} |$ .

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Paso 3.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- 1 \sin^{2} (a) - \cos (c) (\cos (c) \cdot - 1 - \cos (b) \cos (a)) + \cos (b) | \begin{array}{c} \cos (c) & - 1 \\ \cos (b) & \cos (a) \end{array} |$

Paso 3.2

Simplifica cada término.

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Paso 3.2.1

Mueve $- 1$ a la izquierda de $\cos (c)$ .

$- 1 \sin^{2} (a) - \cos (c) (- 1 \cdot \cos (c) - \cos (b) \cos (a)) + \cos (b) | \begin{array}{c} \cos (c) & - 1 \\ \cos (b) & \cos (a) \end{array} |$

Paso 3.2.2

Reescribe $- 1 \cos (c)$ como $- \cos (c)$ .

$- 1 \sin^{2} (a) - \cos (c) (- \cos (c) - \cos (b) \cos (a)) + \cos (b) | \begin{array}{c} \cos (c) & - 1 \\ \cos (b) & \cos (a) \end{array} |$

Paso 4

Evalúa $| \begin{array}{c} \cos (c) & - 1 \\ \cos (b) & \cos (a) \end{array} |$ .

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Paso 4.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- 1 \sin^{2} (a) - \cos (c) (- \cos (c) - \cos (b) \cos (a)) + \cos (b) (\cos (c) \cos (a) - \cos (b) \cdot - 1)$

Paso 4.2

Multiplica $- \cos (b) \cdot - 1$ .

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Paso 4.2.1

Multiplica $- 1$ por $- 1$ .

$- 1 \sin^{2} (a) - \cos (c) (- \cos (c) - \cos (b) \cos (a)) + \cos (b) (\cos (c) \cos (a) + 1 \cos (b))$

Paso 4.2.2

Multiplica $\cos (b)$ por $1$ .

$- 1 \sin^{2} (a) - \cos (c) (- \cos (c) - \cos (b) \cos (a)) + \cos (b) (\cos (c) \cos (a) + \cos (b))$

Paso 5

Simplifica el determinante.

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Paso 5.1

Simplifica cada término.

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Paso 5.1.1

Reescribe $- 1 \sin^{2} (a)$ como $- \sin^{2} (a)$ .

$- \sin^{2} (a) - \cos (c) (- \cos (c) - \cos (b) \cos (a)) + \cos (b) (\cos (c) \cos (a) + \cos (b))$

Paso 5.1.2

Aplica la propiedad distributiva.

$- \sin^{2} (a) - \cos (c) (- \cos (c)) - \cos (c) (- \cos (b) \cos (a)) + \cos (b) (\cos (c) \cos (a) + \cos (b))$

Paso 5.1.3

Multiplica $- \cos (c) (- \cos (c))$ .

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Paso 5.1.3.1

Multiplica $- 1$ por $- 1$ .

$- \sin^{2} (a) + 1 \cos (c) \cos (c) - \cos (c) (- \cos (b) \cos (a)) + \cos (b) (\cos (c) \cos (a) + \cos (b))$

Paso 5.1.3.2

Multiplica $\cos (c)$ por $1$ .

$- \sin^{2} (a) + \cos (c) \cos (c) - \cos (c) (- \cos (b) \cos (a)) + \cos (b) (\cos (c) \cos (a) + \cos (b))$

Paso 5.1.3.3

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

$- \sin^{2} (a) + \cos^{1} (c) \cos (c) - \cos (c) (- \cos (b) \cos (a)) + \cos (b) (\cos (c) \cos (a) + \cos (b))$

Paso 5.1.3.4

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

$- \sin^{2} (a) + \cos^{1} (c) \cos^{1} (c) - \cos (c) (- \cos (b) \cos (a)) + \cos (b) (\cos (c) \cos (a) + \cos (b))$

Paso 5.1.3.5

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

$- \sin^{2} (a) + {\cos (c)}^{1 + 1} - \cos (c) (- \cos (b) \cos (a)) + \cos (b) (\cos (c) \cos (a) + \cos (b))$

Paso 5.1.3.6

Suma $1$ y $1$ .

$- \sin^{2} (a) + \cos^{2} (c) - \cos (c) (- \cos (b) \cos (a)) + \cos (b) (\cos (c) \cos (a) + \cos (b))$

Paso 5.1.4

Multiplica $- \cos (c) (- \cos (b) \cos (a))$ .

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Paso 5.1.4.1

Multiplica $- 1$ por $- 1$ .

$- \sin^{2} (a) + \cos^{2} (c) + 1 \cos (c) (\cos (b) \cos (a)) + \cos (b) (\cos (c) \cos (a) + \cos (b))$

Paso 5.1.4.2

Multiplica $\cos (c)$ por $1$ .

$- \sin^{2} (a) + \cos^{2} (c) + \cos (c) (\cos (b) \cos (a)) + \cos (b) (\cos (c) \cos (a) + \cos (b))$

Paso 5.1.5

Aplica la propiedad distributiva.

$- \sin^{2} (a) + \cos^{2} (c) + \cos (c) \cos (b) \cos (a) + \cos (b) (\cos (c) \cos (a)) + \cos (b) \cos (b)$

Paso 5.1.6

Multiplica $\cos (b) \cos (b)$ .

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Paso 5.1.6.1

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

$- \sin^{2} (a) + \cos^{2} (c) + \cos (c) \cos (b) \cos (a) + \cos (b) \cos (c) \cos (a) + \cos^{1} (b) \cos (b)$

Paso 5.1.6.2

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

$- \sin^{2} (a) + \cos^{2} (c) + \cos (c) \cos (b) \cos (a) + \cos (b) \cos (c) \cos (a) + \cos^{1} (b) \cos^{1} (b)$

Paso 5.1.6.3

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

$- \sin^{2} (a) + \cos^{2} (c) + \cos (c) \cos (b) \cos (a) + \cos (b) \cos (c) \cos (a) + {\cos (b)}^{1 + 1}$

Paso 5.1.6.4

Suma $1$ y $1$ .

$- \sin^{2} (a) + \cos^{2} (c) + \cos (c) \cos (b) \cos (a) + \cos (b) \cos (c) \cos (a) + \cos^{2} (b)$

Paso 5.2

Reordena los factores en los términos $\cos (c) \cos (b) \cos (a)$ y $\cos (b) \cos (c) \cos (a)$ .

$- \sin^{2} (a) + \cos^{2} (c) + \cos (a) \cos (b) \cos (c) + \cos (a) \cos (b) \cos (c) + \cos^{2} (b)$

Paso 5.3

Suma $\cos (a) \cos (b) \cos (c)$ y $\cos (a) \cos (b) \cos (c)$ .

$- \sin^{2} (a) + \cos^{2} (c) + 2 \cos (a) \cos (b) \cos (c) + \cos^{2} (b)$