Trigonometrie Beispiele

\frac{\sin (x) + \tan (x)}{1 + \cos (x)}

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

Schritt 1

Vereinfache den Zähler.

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Schritt 1.1

Schreibe

\tan (x)

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

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

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

Schritt 1.2

Faktorisiere

\sin (x)

$\sin (x)$ aus

\sin (x) + \frac{\sin (x)}{\cos (x)}

$\sin (x) + \frac{\sin (x)}{\cos (x)}$ heraus.

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Schritt 1.2.1

Multipliziere mit

1

$1$ .

\frac{\sin (x) \cdot 1 + \frac{\sin (x)}{\cos (x)}}{1 + \cos (x)}

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

Schritt 1.2.2

Faktorisiere

\sin (x)

$\sin (x)$ aus

\frac{\sin (x)}{\cos (x)}

$\frac{\sin (x)}{\cos (x)}$ heraus.

\frac{\sin (x) \cdot 1 + \sin (x) \frac{1}{\cos (x)}}{1 + \cos (x)}

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

Schritt 1.2.3

Faktorisiere

\sin (x)

$\sin (x)$ aus

\sin (x) \cdot 1 + \sin (x) \frac{1}{\cos (x)}

$\sin (x) \cdot 1 + \sin (x) \frac{1}{\cos (x)}$ heraus.

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

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

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

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

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

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

Schritt 2

Wandle von

\frac{1}{\cos (x)}

$\frac{1}{\cos (x)}$ nach

\sec (x)

$\sec (x)$ um.

\frac{\sin (x) (1 + \sec (x))}{1 + \cos (x)}

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