代数示例

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

解题步骤 1

将 $\sec (x)$ 重写为正弦和余弦形式。

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

解题步骤 2

将 $\frac{1}{\cos (x)}$ 转换成 $\sec (x)$ 。

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

解题步骤 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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解题步骤 3.1

Consider the corresponding sign chart.

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

解题步骤 3.2

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

解题步骤 3.3

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

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

解题步骤 3.4

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

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

解题步骤 3.5

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

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

解题步骤 3.6

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

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

解题步骤 3.7

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

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

解题步骤 3.8

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

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

解题步骤 3.9

Add the terms together.

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

解题步骤 4

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

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

解题步骤 5

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

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

解题步骤 6

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

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解题步骤 6.1

可以使用公式 $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ 求 $2 \times 2$ 矩阵的行列式。

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

解题步骤 6.2

将 $\cos (x)$ 乘以 $1$ 。

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

解题步骤 7

化简行列式。

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解题步骤 7.1

合并 $0 + 0 - \sec (x) (\cos (x) - \sin (x))$ 中相反的项。

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解题步骤 7.1.1

将 $0$ 和 $0$ 相加。

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

解题步骤 7.1.2

从 $0$ 中减去 $\sec (x) (\cos (x) - \sin (x))$ 。

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

解题步骤 7.2

将 $\sec (x)$ 重写为正弦和余弦形式。

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

解题步骤 7.3

运用分配律。

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

解题步骤 7.4

约去 $\cos (x)$ 的公因数。

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解题步骤 7.4.1

将 $- \frac{1}{\cos (x)}$ 中前置负号移到分子中。

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

解题步骤 7.4.2

约去公因数。

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

解题步骤 7.4.3

重写表达式。

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

解题步骤 7.5

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

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解题步骤 7.5.1

将 $- 1$ 乘以 $- 1$ 。

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

解题步骤 7.5.2

将 $\frac{1}{\cos (x)}$ 乘以 $1$ 。

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

解题步骤 7.5.3

组合 $\frac{1}{\cos (x)}$ 和 $\sin (x)$ 。

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

解题步骤 7.6

将 $\frac{\sin (x)}{\cos (x)}$ 转换成 $\tan (x)$ 。

$- 1 + \tan (x)$

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