Trigonometrie Beispiele

{sin}^{3} (t) + {cos}^{3} (t) + sin (t) {cos}^{2} (t) + {sin}^{2} (t) cos (t) = sin (t) + cos (t)

$\sin^{3} (t) + \cos^{3} (t) + \sin (t) \cos^{2} (t) + \sin^{2} (t) \cos (t) = \sin (t) + \cos (t)$

Schritt 1

Beginne auf der linken Seite.

{sin}^{3} (t) + {cos}^{3} (t) + sin (t) {cos}^{2} (t) + {sin}^{2} (t) cos (t)

$\sin^{3} (t) + \cos^{3} (t) + \sin (t) \cos^{2} (t) + \sin^{2} (t) \cos (t)$

Schritt 2

Faktorisiere.

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Schritt 2.1

Da beide Terme perfekte Terme zur dritten Potenz sind, faktorisiere mithilfe der Formel für die Summe kubischer Terme,

a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})

$a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ , wobei

a = sin (t)

$a = \sin (t)$ und

b = cos (t)

$b = \cos (t)$ .

(sin (t) + cos (t)) ({sin}^{2} (t) - sin (t) cos (t) + {cos}^{2} (t)) + sin (t) {cos}^{2} (t) + {sin}^{2} (t) cos (t)

$(\sin (t) + \cos (t)) (\sin^{2} (t) - \sin (t) \cos (t) + \cos^{2} (t)) + \sin (t) \cos^{2} (t) + \sin^{2} (t) \cos (t)$

Schritt 2.2

Vereinfache.

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Schritt 2.2.1

Ordne Terme um.

(sin (t) + cos (t)) (- sin (t) cos (t) + {sin}^{2} (t) + {cos}^{2} (t)) + sin (t) {cos}^{2} (t) + {sin}^{2} (t) cos (t)

$(\sin (t) + \cos (t)) (- \sin (t) \cos (t) + \sin^{2} (t) + \cos^{2} (t)) + \sin (t) \cos^{2} (t) + \sin^{2} (t) \cos (t)$

Schritt 2.2.2

Wende den trigonometrischen Pythagoras an.

(sin (t) + cos (t)) (- sin (t) cos (t) + 1) + sin (t) {cos}^{2} (t) + {sin}^{2} (t) cos (t)

$(\sin (t) + \cos (t)) (- \sin (t) \cos (t) + 1) + \sin (t) \cos^{2} (t) + \sin^{2} (t) \cos (t)$

(sin (t) + cos (t)) (- sin (t) cos (t) + 1) + sin (t) {cos}^{2} (t) + {sin}^{2} (t) cos (t)

$(\sin (t) + \cos (t)) (- \sin (t) \cos (t) + 1) + \sin (t) \cos^{2} (t) + \sin^{2} (t) \cos (t)$

(sin (t) + cos (t)) (- sin (t) cos (t) + 1) + sin (t) {cos}^{2} (t) + {sin}^{2} (t) cos (t)

$(\sin (t) + \cos (t)) (- \sin (t) \cos (t) + 1) + \sin (t) \cos^{2} (t) + \sin^{2} (t) \cos (t)$

Schritt 3

Vereinfache.

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Schritt 3.1

Vereinfache jeden Term.

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Schritt 3.1.1

Multipliziere

(sin (t) + cos (t)) (- sin (t) cos (t) + 1)

$(\sin (t) + \cos (t)) (- \sin (t) \cos (t) + 1)$ aus unter Verwendung der FOIL-Methode.

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Schritt 3.1.1.1

Wende das Distributivgesetz an.

sin (t) (- sin (t) cos (t) + 1) + cos (t) (- sin (t) cos (t) + 1) + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$\sin (t) (- \sin (t) \cos (t) + 1) + \cos (t) (- \sin (t) \cos (t) + 1) + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

Schritt 3.1.1.2

Wende das Distributivgesetz an.

sin (t) (- sin (t) cos (t)) + sin (t) \cdot 1 + cos (t) (- sin (t) cos (t) + 1) + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$\sin (t) (- \sin (t) \cos (t)) + \sin (t) \cdot 1 + \cos (t) (- \sin (t) \cos (t) + 1) + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

Schritt 3.1.1.3

Wende das Distributivgesetz an.

sin (t) (- sin (t) cos (t)) + sin (t) \cdot 1 + cos (t) (- sin (t) cos (t)) + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$\sin (t) (- \sin (t) \cos (t)) + \sin (t) \cdot 1 + \cos (t) (- \sin (t) \cos (t)) + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

sin (t) (- sin (t) cos (t)) + sin (t) \cdot 1 + cos (t) (- sin (t) cos (t)) + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$\sin (t) (- \sin (t) \cos (t)) + \sin (t) \cdot 1 + \cos (t) (- \sin (t) \cos (t)) + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

Schritt 3.1.2

Vereinfache jeden Term.

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Schritt 3.1.2.1

Multipliziere

sin (t) (- sin (t) cos (t))

$\sin (t) (- \sin (t) \cos (t))$ .

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Schritt 3.1.2.1.1

Potenziere

sin (t)

$\sin (t)$ mit

1

$1$ .

- ({sin (t)}^{1} sin (t)) cos (t) + sin (t) \cdot 1 + cos (t) (- sin (t) cos (t)) + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$- ({\sin (t)}^{1} \sin (t)) \cos (t) + \sin (t) \cdot 1 + \cos (t) (- \sin (t) \cos (t)) + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

Schritt 3.1.2.1.2

Potenziere

sin (t)

$\sin (t)$ mit

1

$1$ .

- ({sin (t)}^{1} {sin (t)}^{1}) cos (t) + sin (t) \cdot 1 + cos (t) (- sin (t) cos (t)) + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$- ({\sin (t)}^{1} {\sin (t)}^{1}) \cos (t) + \sin (t) \cdot 1 + \cos (t) (- \sin (t) \cos (t)) + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

Schritt 3.1.2.1.3

Wende die Exponentenregel

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

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

- {sin (t)}^{1 + 1} cos (t) + sin (t) \cdot 1 + cos (t) (- sin (t) cos (t)) + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$- {\sin (t)}^{1 + 1} \cos (t) + \sin (t) \cdot 1 + \cos (t) (- \sin (t) \cos (t)) + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

Schritt 3.1.2.1.4

Addiere

1

$1$ und

1

$1$ .

- {sin (t)}^{2} cos (t) + sin (t) \cdot 1 + cos (t) (- sin (t) cos (t)) + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$- {\sin (t)}^{2} \cos (t) + \sin (t) \cdot 1 + \cos (t) (- \sin (t) \cos (t)) + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

- {sin (t)}^{2} cos (t) + sin (t) \cdot 1 + cos (t) (- sin (t) cos (t)) + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$- {\sin (t)}^{2} \cos (t) + \sin (t) \cdot 1 + \cos (t) (- \sin (t) \cos (t)) + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

Schritt 3.1.2.2

Mutltipliziere

sin (t)

$\sin (t)$ mit

1

$1$ .

- {sin (t)}^{2} cos (t) + sin (t) + cos (t) (- sin (t) cos (t)) + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$- {\sin (t)}^{2} \cos (t) + \sin (t) + \cos (t) (- \sin (t) \cos (t)) + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

Schritt 3.1.2.3

Multipliziere

cos (t) (- sin (t) cos (t))

$\cos (t) (- \sin (t) \cos (t))$ .

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Schritt 3.1.2.3.1

Potenziere

cos (t)

$\cos (t)$ mit

1

$1$ .

- {sin (t)}^{2} cos (t) + sin (t) - sin (t) ({cos (t)}^{1} cos (t)) + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$- {\sin (t)}^{2} \cos (t) + \sin (t) - \sin (t) ({\cos (t)}^{1} \cos (t)) + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

Schritt 3.1.2.3.2

Potenziere

cos (t)

$\cos (t)$ mit

1

$1$ .

- {sin (t)}^{2} cos (t) + sin (t) - sin (t) ({cos (t)}^{1} {cos (t)}^{1}) + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$- {\sin (t)}^{2} \cos (t) + \sin (t) - \sin (t) ({\cos (t)}^{1} {\cos (t)}^{1}) + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

Schritt 3.1.2.3.3

Wende die Exponentenregel

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

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

- {sin (t)}^{2} cos (t) + sin (t) - sin (t) {cos (t)}^{1 + 1} + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$- {\sin (t)}^{2} \cos (t) + \sin (t) - \sin (t) {\cos (t)}^{1 + 1} + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

Schritt 3.1.2.3.4

Addiere

1

$1$ und

1

$1$ .

- {sin (t)}^{2} cos (t) + sin (t) - sin (t) {cos (t)}^{2} + cos (t) \cdot 1 + sin (t) {cos (t)}^{2} + {sin (t)}^{2} cos (t)

$- {\sin (t)}^{2} \cos (t) + \sin (t) - \sin (t) {\cos (t)}^{2} + \cos (t) \cdot 1 + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

Schritt 3.1.2.4

Mutltipliziere

$\cos (t)$ mit

$1$ .

$- {\sin (t)}^{2} \cos (t) + \sin (t) - \sin (t) {\cos (t)}^{2} + \cos (t) + \sin (t) {\cos (t)}^{2} + {\sin (t)}^{2} \cos (t)$

Schritt 3.2

Addiere

$- {\sin (t)}^{2} \cos (t)$ und

${\sin (t)}^{2} \cos (t)$ .

$\sin (t) - \sin (t) {\cos (t)}^{2} + \cos (t) + \sin (t) {\cos (t)}^{2} + 0$

Schritt 3.3

Addiere

$\sin (t)$ und

$0$ .

$- \sin (t) {\cos (t)}^{2} + \cos (t) + \sin (t) {\cos (t)}^{2} + \sin (t)$

Schritt 3.4

Addiere

$- \sin (t) {\cos (t)}^{2}$ und

$\sin (t) {\cos (t)}^{2}$ .

$0 + \cos (t) + \sin (t)$

Schritt 3.5

Addiere

$0$ und

$\cos (t)$ .

$\cos (t) + \sin (t)$

Schritt 4

Schreibe

$\cos (t) + \sin (t)$ als

$\sin (t) + \cos (t)$ um.

$\sin (t) + \cos (t)$

Schritt 5

Da gezeigt wurde, dass die beiden Seiten äquivalent sind, ist die Gleichung eine Identitätsgleichung.

$\sin^{3} (t) + \cos^{3} (t) + \sin (t) \cos^{2} (t) + \sin^{2} (t) \cos (t) = \sin (t) + \cos (t)$ ist eine Identitätsgleichung