Álgebra Ejemplos

$[\begin{array}{c} 1 & 0 & \sin (x) \\ 1 & 0 & \cos (x) \\ 1 & \sec (x) & - \sin (x) \end{array}]$

Paso 1

Reescribe $\sec (x)$ en términos de senos y cosenos.

$[\begin{array}{c} 1 & 0 & \sin (x) \\ 1 & 0 & \cos (x) \\ 1 & \frac{1}{\cos (x)} & - \sin (x) \end{array}]$

Paso 2

Convierte de $\frac{1}{\cos (x)}$ a $\sec (x)$ .

$[\begin{array}{c} 1 & 0 & \sin (x) \\ 1 & 0 & \cos (x) \\ 1 & \sec (x) & - \sin (x) \end{array}]$

Paso 3

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 column $2$ by its cofactor and add.

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Paso 3.1

Consider the corresponding sign chart.

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

Paso 3.2

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

Paso 3.3

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

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

Paso 3.4

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

$0 | \begin{array}{c} 1 & \cos (x) \\ 1 & - \sin (x) \end{array} |$

Paso 3.5

The minor for $a_{22}$ is the determinant with row $2$ and column $2$ deleted.

$| \begin{array}{c} 1 & \sin (x) \\ 1 & - \sin (x) \end{array} |$

Paso 3.6

Multiply element $a_{22}$ by its cofactor.

$0 | \begin{array}{c} 1 & \sin (x) \\ 1 & - \sin (x) \end{array} |$

Paso 3.7

The minor for $a_{32}$ is the determinant with row $3$ and column $2$ deleted.

$| \begin{array}{c} 1 & \sin (x) \\ 1 & \cos (x) \end{array} |$

Paso 3.8

Multiply element $a_{32}$ by its cofactor.

$- \sec (x) | \begin{array}{c} 1 & \sin (x) \\ 1 & \cos (x) \end{array} |$

Paso 3.9

Add the terms together.

$0 | \begin{array}{c} 1 & \cos (x) \\ 1 & - \sin (x) \end{array} | + 0 | \begin{array}{c} 1 & \sin (x) \\ 1 & - \sin (x) \end{array} | - \sec (x) | \begin{array}{c} 1 & \sin (x) \\ 1 & \cos (x) \end{array} |$

Paso 4

Multiplica $0$ por $| \begin{array}{c} 1 & \cos (x) \\ 1 & - \sin (x) \end{array} |$ .

$0 + 0 | \begin{array}{c} 1 & \sin (x) \\ 1 & - \sin (x) \end{array} | - \sec (x) | \begin{array}{c} 1 & \sin (x) \\ 1 & \cos (x) \end{array} |$

Paso 5

Multiplica $0$ por $| \begin{array}{c} 1 & \sin (x) \\ 1 & - \sin (x) \end{array} |$ .

$0 + 0 - \sec (x) | \begin{array}{c} 1 & \sin (x) \\ 1 & \cos (x) \end{array} |$

Paso 6

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

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Paso 6.1

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

$0 + 0 - \sec (x) (1 \cos (x) - \sin (x))$

Paso 6.2

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

$0 + 0 - \sec (x) (\cos (x) - \sin (x))$

Paso 7

Simplifica el determinante.

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Paso 7.1

Combina los términos opuestos en $0 + 0 - \sec (x) (\cos (x) - \sin (x))$ .

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Paso 7.1.1

Suma $0$ y $0$ .

$0 - \sec (x) (\cos (x) - \sin (x))$

Paso 7.1.2

Resta $\sec (x) (\cos (x) - \sin (x))$ de $0$ .

$- \sec (x) (\cos (x) - \sin (x))$

Paso 7.2

Reescribe $\sec (x)$ en términos de senos y cosenos.

$- \frac{1}{\cos (x)} (\cos (x) - \sin (x))$

Paso 7.3

Aplica la propiedad distributiva.

$- \frac{1}{\cos (x)} \cos (x) - \frac{1}{\cos (x)} (- \sin (x))$

Paso 7.4

Cancela el factor común de $\cos (x)$ .

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Paso 7.4.1

Mueve el signo menos inicial en $- \frac{1}{\cos (x)}$ al numerador.

$\frac{- 1}{\cos (x)} \cos (x) - \frac{1}{\cos (x)} (- \sin (x))$

Paso 7.4.2

Cancela el factor común.

$\frac{- 1}{\cos (x)} \cos (x) - \frac{1}{\cos (x)} (- \sin (x))$

Paso 7.4.3

Reescribe la expresión.

$- 1 - \frac{1}{\cos (x)} (- \sin (x))$

Paso 7.5

Multiplica $- \frac{1}{\cos (x)} (- \sin (x))$ .

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Paso 7.5.1

Multiplica $- 1$ por $- 1$ .

$- 1 + 1 \frac{1}{\cos (x)} \sin (x)$

Paso 7.5.2

Multiplica $\frac{1}{\cos (x)}$ por $1$ .

$- 1 + \frac{1}{\cos (x)} \sin (x)$

Paso 7.5.3

Combina $\frac{1}{\cos (x)}$ y $\sin (x)$ .

$- 1 + \frac{\sin (x)}{\cos (x)}$

Paso 7.6

Convierte de $\frac{\sin (x)}{\cos (x)}$ a $\tan (x)$ .

$- 1 + \tan (x)$