Trigonometrie Beispiele

\frac{\sin (x)}{\sqrt{1 - \sin^{2} (x)}}

$\frac{\sin (x)}{\sqrt{1 - \sin^{2} (x)}}$

Schritt 1

Vereinfache den Nenner.

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Schritt 1.1

Schreibe

1

$1$ als

1^{2}

$1^{2}$ um.

\frac{\sin (x)}{\sqrt{1^{2} - \sin^{2} (x)}}

$\frac{\sin (x)}{\sqrt{1^{2} - \sin^{2} (x)}}$

Schritt 1.2

Da beide Terme perfekte Quadrate sind, faktorisiere durch Anwendung der dritten binomischen Formel,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ , mit

a = 1

$a = 1$ und

b = \sin (x)

$b = \sin (x)$ .

\frac{\sin (x)}{\sqrt{(1 + \sin (x)) (1 - \sin (x))}}

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

\frac{\sin (x)}{\sqrt{(1 + \sin (x)) (1 - \sin (x))}}

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

Schritt 2

Mutltipliziere

\frac{\sin (x)}{\sqrt{(1 + \sin (x)) (1 - \sin (x))}}

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

\frac{\sqrt{(1 + \sin (x)) (1 - \sin (x))}}{\sqrt{(1 + \sin (x)) (1 - \sin (x))}}

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

\frac{\sin (x)}{\sqrt{(1 + \sin (x)) (1 - \sin (x))}} \cdot \frac{\sqrt{(1 + \sin (x)) (1 - \sin (x))}}{\sqrt{(1 + \sin (x)) (1 - \sin (x))}}

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

Schritt 3

Vereinige und vereinfache den Nenner.

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Schritt 3.1

Mutltipliziere

\frac{\sin (x)}{\sqrt{(1 + \sin (x)) (1 - \sin (x))}}

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

\frac{\sqrt{(1 + \sin (x)) (1 - \sin (x))}}{\sqrt{(1 + \sin (x)) (1 - \sin (x))}}

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

\frac{\sin (x) \sqrt{(1 + \sin (x)) (1 - \sin (x))}}{\sqrt{(1 + \sin (x)) (1 - \sin (x))} \sqrt{(1 + \sin (x)) (1 - \sin (x))}}

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

Schritt 3.2

Potenziere

\sqrt{(1 + \sin (x)) (1 - \sin (x))}

$\sqrt{(1 + \sin (x)) (1 - \sin (x))}$ mit

1

$1$ .

\frac{\sin (x) \sqrt{(1 + \sin (x)) (1 - \sin (x))}}{{\sqrt{(1 + \sin (x)) (1 - \sin (x))}}^{1} \sqrt{(1 + \sin (x)) (1 - \sin (x))}}

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

Schritt 3.3

Potenziere

\sqrt{(1 + \sin (x)) (1 - \sin (x))}

$\sqrt{(1 + \sin (x)) (1 - \sin (x))}$ mit

1

$1$ .

\frac{\sin (x) \sqrt{(1 + \sin (x)) (1 - \sin (x))}}{{\sqrt{(1 + \sin (x)) (1 - \sin (x))}}^{1} {\sqrt{(1 + \sin (x)) (1 - \sin (x))}}^{1}}

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

Schritt 3.4

Wende die Exponentenregel

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

\frac{\sin (x) \sqrt{(1 + \sin (x)) (1 - \sin (x))}}{{\sqrt{(1 + \sin (x)) (1 - \sin (x))}}^{1 + 1}}

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

Schritt 3.5

Addiere

1

$1$ und

1

$1$ .

\frac{\sin (x) \sqrt{(1 + \sin (x)) (1 - \sin (x))}}{{\sqrt{(1 + \sin (x)) (1 - \sin (x))}}^{2}}

$\frac{\sin (x) \sqrt{(1 + \sin (x)) (1 - \sin (x))}}{{\sqrt{(1 + \sin (x)) (1 - \sin (x))}}^{2}}$

Schritt 3.6

Schreibe

{\sqrt{(1 + \sin (x)) (1 - \sin (x))}}^{2}

${\sqrt{(1 + \sin (x)) (1 - \sin (x))}}^{2}$ als

(1 + \sin (x)) (1 - \sin (x))

$(1 + \sin (x)) (1 - \sin (x))$ um.

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Schritt 3.6.1

Benutze

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ , um

\sqrt{(1 + \sin (x)) (1 - \sin (x))}

$\sqrt{(1 + \sin (x)) (1 - \sin (x))}$ als

{((1 + \sin (x)) (1 - \sin (x)))}^{\frac{1}{2}}

${((1 + \sin (x)) (1 - \sin (x)))}^{\frac{1}{2}}$ neu zu schreiben.

\frac{\sin (x) \sqrt{(1 + \sin (x)) (1 - \sin (x))}}{{({((1 + \sin (x)) (1 - \sin (x)))}^{\frac{1}{2}})}^{2}}

$\frac{\sin (x) \sqrt{(1 + \sin (x)) (1 - \sin (x))}}{{({((1 + \sin (x)) (1 - \sin (x)))}^{\frac{1}{2}})}^{2}}$

Schritt 3.6.2

Wende die Potenzregel an und multipliziere die Exponenten,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\frac{\sin (x) \sqrt{(1 + \sin (x)) (1 - \sin (x))}}{{((1 + \sin (x)) (1 - \sin (x)))}^{\frac{1}{2} \cdot 2}}

$\frac{\sin (x) \sqrt{(1 + \sin (x)) (1 - \sin (x))}}{{((1 + \sin (x)) (1 - \sin (x)))}^{\frac{1}{2} \cdot 2}}$

Schritt 3.6.3

Kombiniere

\frac{1}{2}

$\frac{1}{2}$ und

2

$2$ .

\frac{\sin (x) \sqrt{(1 + \sin (x)) (1 - \sin (x))}}{{((1 + \sin (x)) (1 - \sin (x)))}^{\frac{2}{2}}}

$\frac{\sin (x) \sqrt{(1 + \sin (x)) (1 - \sin (x))}}{{((1 + \sin (x)) (1 - \sin (x)))}^{\frac{2}{2}}}$

Schritt 3.6.4

Kürze den gemeinsamen Faktor von

2

$2$ .

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Schritt 3.6.4.1

Kürze den gemeinsamen Faktor.

\frac{\sin (x) \sqrt{(1 + \sin (x)) (1 - \sin (x))}}{{((1 + \sin (x)) (1 - \sin (x)))}^{\frac{2}{2}}}

$\frac{\sin (x) \sqrt{(1 + \sin (x)) (1 - \sin (x))}}{{((1 + \sin (x)) (1 - \sin (x)))}^{\frac{2}{2}}}$

Schritt 3.6.4.2

Forme den Ausdruck um.

\frac{\sin (x) \sqrt{(1 + \sin (x)) (1 - \sin (x))}}{{((1 + \sin (x)) (1 - \sin (x)))}^{1}}

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

\frac{\sin (x) \sqrt{(1 + \sin (x)) (1 - \sin (x))}}{{((1 + \sin (x)) (1 - \sin (x)))}^{1}}

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

Schritt 3.6.5

Vereinfache.

\frac{\sin (x) \sqrt{(1 + \sin (x)) (1 - \sin (x))}}{(1 + \sin (x)) (1 - \sin (x))}

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

\frac{\sin (x) \sqrt{(1 + \sin (x)) (1 - \sin (x))}}{(1 + \sin (x)) (1 - \sin (x))}

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

\frac{\sin (x) \sqrt{(1 + \sin (x)) (1 - \sin (x))}}{(1 + \sin (x)) (1 - \sin (x))}

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