Algebra Beispiele

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

Schritt 1

Schreibe $\sec (x)$ mithilfe von Sinus und Kosinus um.

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

Schritt 2

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

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

Schritt 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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Schritt 3.1

Consider the corresponding sign chart.

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

Schritt 3.2

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

Schritt 3.3

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

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

Schritt 3.4

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

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

Schritt 3.5

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

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

Schritt 3.6

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

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

Schritt 3.7

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

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

Schritt 3.8

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

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

Schritt 3.9

Add the terms together.

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

Schritt 4

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

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

Schritt 5

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

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

Schritt 6

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

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Schritt 6.1

Die Determinante einer $2 \times 2$ -Matrix kann mithilfe der Formel $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ bestimmt werden.

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

Schritt 6.2

Mutltipliziere $\cos (x)$ mit $1$ .

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

Schritt 7

Vereinfache die Determinante.

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Schritt 7.1

Vereine die Terme mit entgegengesetztem Vorzeichen in $0 + 0 - \sec (x) (\cos (x) - \sin (x))$ .

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Schritt 7.1.1

Addiere $0$ und $0$ .

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

Schritt 7.1.2

Subtrahiere $\sec (x) (\cos (x) - \sin (x))$ von $0$ .

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

Schritt 7.2

Schreibe $\sec (x)$ mithilfe von Sinus und Kosinus um.

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

Schritt 7.3

Wende das Distributivgesetz an.

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

Schritt 7.4

Kürze den gemeinsamen Faktor von $\cos (x)$ .

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Schritt 7.4.1

Bringe das führende Minuszeichen in $- \frac{1}{\cos (x)}$ in den Zähler.

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

Schritt 7.4.2

Kürze den gemeinsamen Faktor.

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

Schritt 7.4.3

Forme den Ausdruck um.

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

Schritt 7.5

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

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Schritt 7.5.1

Mutltipliziere $- 1$ mit $- 1$ .

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

Schritt 7.5.2

Mutltipliziere $\frac{1}{\cos (x)}$ mit $1$ .

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

Schritt 7.5.3

Kombiniere $\frac{1}{\cos (x)}$ und $\sin (x)$ .

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

Schritt 7.6

Wandle von $\frac{\sin (x)}{\cos (x)}$ nach $\tan (x)$ um.

$- 1 + \tan (x)$