Elementarmathematik Beispiele

${(4 \cos (x) - 7 \sin (x))}^{2} + {(7 \cos (x) + 4 \sin (x))}^{2} = 65$

Schritt 1

Quadriere beide Seiten der Gleichung.

${({(4 \cos (x) - 7 \sin (x))}^{2} + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Schritt 2

Vereinfache ${({(4 \cos (x) - 7 \sin (x))}^{2} + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2}$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.1

Vereinfache jeden Term.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.1.1

Schreibe ${(4 \cos (x) - 7 \sin (x))}^{2}$ als $(4 \cos (x) - 7 \sin (x)) (4 \cos (x) - 7 \sin (x))$ um.

${((4 \cos (x) - 7 \sin (x)) (4 \cos (x) - 7 \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Schritt 2.1.2

Multipliziere $(4 \cos (x) - 7 \sin (x)) (4 \cos (x) - 7 \sin (x))$ aus unter Verwendung der FOIL-Methode.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.1.2.1

Wende das Distributivgesetz an.

${(4 \cos (x) (4 \cos (x) - 7 \sin (x)) - 7 \sin (x) (4 \cos (x) - 7 \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Schritt 2.1.2.2

Wende das Distributivgesetz an.

${(4 \cos (x) (4 \cos (x)) + 4 \cos (x) (- 7 \sin (x)) - 7 \sin (x) (4 \cos (x) - 7 \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Schritt 2.1.2.3

Wende das Distributivgesetz an.

${(4 \cos (x) (4 \cos (x)) + 4 \cos (x) (- 7 \sin (x)) - 7 \sin (x) (4 \cos (x)) - 7 \sin (x) (- 7 \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Schritt 2.1.3

Vereinfache und fasse gleichartige Terme zusammen.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.1.3.1

Vereinfache jeden Term.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.1.3.1.1

Multipliziere $4 \cos (x) (4 \cos (x))$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.1.3.1.1.1

Mutltipliziere $4$ mit $4$ .

${(16 \cos (x) \cos (x) + 4 \cos (x) (- 7 \sin (x)) - 7 \sin (x) (4 \cos (x)) - 7 \sin (x) (- 7 \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Schritt 2.1.3.1.1.2

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

${(16 (\cos^{1} (x) \cos (x)) + 4 \cos (x) (- 7 \sin (x)) - 7 \sin (x) (4 \cos (x)) - 7 \sin (x) (- 7 \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Schritt 2.1.3.1.1.3

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

${(16 (\cos^{1} (x) \cos^{1} (x)) + 4 \cos (x) (- 7 \sin (x)) - 7 \sin (x) (4 \cos (x)) - 7 \sin (x) (- 7 \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Schritt 2.1.3.1.1.4

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

${(16 {\cos (x)}^{1 + 1} + 4 \cos (x) (- 7 \sin (x)) - 7 \sin (x) (4 \cos (x)) - 7 \sin (x) (- 7 \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Schritt 2.1.3.1.1.5

Addiere $1$ und $1$ .

${(16 \cos^{2} (x) + 4 \cos (x) (- 7 \sin (x)) - 7 \sin (x) (4 \cos (x)) - 7 \sin (x) (- 7 \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Schritt 2.1.3.1.2

Mutltipliziere $- 7$ mit $4$ .

${(16 \cos^{2} (x) - 28 \cos (x) \sin (x) - 7 \sin (x) (4 \cos (x)) - 7 \sin (x) (- 7 \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Schritt 2.1.3.1.3

Mutltipliziere $4$ mit $- 7$ .

${(16 \cos^{2} (x) - 28 \cos (x) \sin (x) - 28 \sin (x) \cos (x) - 7 \sin (x) (- 7 \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Schritt 2.1.3.1.4

Multipliziere $- 7 \sin (x) (- 7 \sin (x))$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.1.3.1.4.1

Mutltipliziere $- 7$ mit $- 7$ .

${(16 \cos^{2} (x) - 28 \cos (x) \sin (x) - 28 \sin (x) \cos (x) + 49 \sin (x) \sin (x) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Schritt 2.1.3.1.4.2

Potenziere $\sin (x)$ mit $1$ .

${(16 \cos^{2} (x) - 28 \cos (x) \sin (x) - 28 \sin (x) \cos (x) + 49 (\sin^{1} (x) \sin (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Schritt 2.1.3.1.4.3

Potenziere $\sin (x)$ mit $1$ .

${(16 \cos^{2} (x) - 28 \cos (x) \sin (x) - 28 \sin (x) \cos (x) + 49 (\sin^{1} (x) \sin^{1} (x)) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Schritt 2.1.3.1.4.4

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

${(16 \cos^{2} (x) - 28 \cos (x) \sin (x) - 28 \sin (x) \cos (x) + 49 {\sin (x)}^{1 + 1} + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Schritt 2.1.3.1.4.5

Addiere $1$ und $1$ .

${(16 \cos^{2} (x) - 28 \cos (x) \sin (x) - 28 \sin (x) \cos (x) + 49 \sin^{2} (x) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Schritt 2.1.3.2

Stelle die Faktoren von $- 28 \sin (x) \cos (x)$ um.

${(16 \cos^{2} (x) - 28 \cos (x) \sin (x) - 28 \cos (x) \sin (x) + 49 \sin^{2} (x) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Schritt 2.1.3.3

Subtrahiere $28 \cos (x) \sin (x)$ von $- 28 \cos (x) \sin (x)$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + {(7 \cos (x) + 4 \sin (x))}^{2})}^{2} = {(65)}^{2}$

Schritt 2.1.4

Schreibe ${(7 \cos (x) + 4 \sin (x))}^{2}$ als $(7 \cos (x) + 4 \sin (x)) (7 \cos (x) + 4 \sin (x))$ um.

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + (7 \cos (x) + 4 \sin (x)) (7 \cos (x) + 4 \sin (x)))}^{2} = {(65)}^{2}$

Schritt 2.1.5

Multipliziere $(7 \cos (x) + 4 \sin (x)) (7 \cos (x) + 4 \sin (x))$ aus unter Verwendung der FOIL-Methode.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.1.5.1

Wende das Distributivgesetz an.

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 7 \cos (x) (7 \cos (x) + 4 \sin (x)) + 4 \sin (x) (7 \cos (x) + 4 \sin (x)))}^{2} = {(65)}^{2}$

Schritt 2.1.5.2

Wende das Distributivgesetz an.

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 7 \cos (x) (7 \cos (x)) + 7 \cos (x) (4 \sin (x)) + 4 \sin (x) (7 \cos (x) + 4 \sin (x)))}^{2} = {(65)}^{2}$

Schritt 2.1.5.3

Wende das Distributivgesetz an.

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 7 \cos (x) (7 \cos (x)) + 7 \cos (x) (4 \sin (x)) + 4 \sin (x) (7 \cos (x)) + 4 \sin (x) (4 \sin (x)))}^{2} = {(65)}^{2}$

Schritt 2.1.6

Vereinfache und fasse gleichartige Terme zusammen.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.1.6.1

Vereinfache jeden Term.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.1.6.1.1

Multipliziere $7 \cos (x) (7 \cos (x))$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.1.6.1.1.1

Mutltipliziere $7$ mit $7$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos (x) \cos (x) + 7 \cos (x) (4 \sin (x)) + 4 \sin (x) (7 \cos (x)) + 4 \sin (x) (4 \sin (x)))}^{2} = {(65)}^{2}$

Schritt 2.1.6.1.1.2

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

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 (\cos^{1} (x) \cos (x)) + 7 \cos (x) (4 \sin (x)) + 4 \sin (x) (7 \cos (x)) + 4 \sin (x) (4 \sin (x)))}^{2} = {(65)}^{2}$

Schritt 2.1.6.1.1.3

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

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 (\cos^{1} (x) \cos^{1} (x)) + 7 \cos (x) (4 \sin (x)) + 4 \sin (x) (7 \cos (x)) + 4 \sin (x) (4 \sin (x)))}^{2} = {(65)}^{2}$

Schritt 2.1.6.1.1.4

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

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 {\cos (x)}^{1 + 1} + 7 \cos (x) (4 \sin (x)) + 4 \sin (x) (7 \cos (x)) + 4 \sin (x) (4 \sin (x)))}^{2} = {(65)}^{2}$

Schritt 2.1.6.1.1.5

Addiere $1$ und $1$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 7 \cos (x) (4 \sin (x)) + 4 \sin (x) (7 \cos (x)) + 4 \sin (x) (4 \sin (x)))}^{2} = {(65)}^{2}$

Schritt 2.1.6.1.2

Mutltipliziere $4$ mit $7$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 28 \cos (x) \sin (x) + 4 \sin (x) (7 \cos (x)) + 4 \sin (x) (4 \sin (x)))}^{2} = {(65)}^{2}$

Schritt 2.1.6.1.3

Mutltipliziere $7$ mit $4$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 28 \cos (x) \sin (x) + 28 \sin (x) \cos (x) + 4 \sin (x) (4 \sin (x)))}^{2} = {(65)}^{2}$

Schritt 2.1.6.1.4

Multipliziere $4 \sin (x) (4 \sin (x))$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.1.6.1.4.1

Mutltipliziere $4$ mit $4$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 28 \cos (x) \sin (x) + 28 \sin (x) \cos (x) + 16 \sin (x) \sin (x))}^{2} = {(65)}^{2}$

Schritt 2.1.6.1.4.2

Potenziere $\sin (x)$ mit $1$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 28 \cos (x) \sin (x) + 28 \sin (x) \cos (x) + 16 (\sin^{1} (x) \sin (x)))}^{2} = {(65)}^{2}$

Schritt 2.1.6.1.4.3

Potenziere $\sin (x)$ mit $1$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 28 \cos (x) \sin (x) + 28 \sin (x) \cos (x) + 16 (\sin^{1} (x) \sin^{1} (x)))}^{2} = {(65)}^{2}$

Schritt 2.1.6.1.4.4

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

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 28 \cos (x) \sin (x) + 28 \sin (x) \cos (x) + 16 {\sin (x)}^{1 + 1})}^{2} = {(65)}^{2}$

Schritt 2.1.6.1.4.5

Addiere $1$ und $1$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 28 \cos (x) \sin (x) + 28 \sin (x) \cos (x) + 16 \sin^{2} (x))}^{2} = {(65)}^{2}$

Schritt 2.1.6.2

Stelle die Faktoren von $28 \sin (x) \cos (x)$ um.

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 28 \cos (x) \sin (x) + 28 \cos (x) \sin (x) + 16 \sin^{2} (x))}^{2} = {(65)}^{2}$

Schritt 2.1.6.3

Addiere $28 \cos (x) \sin (x)$ und $28 \cos (x) \sin (x)$ .

${(16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 56 \cos (x) \sin (x) + 16 \sin^{2} (x))}^{2} = {(65)}^{2}$

Schritt 2.2

Vereinfache Terme.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.2.1

Vereine die Terme mit entgegengesetztem Vorzeichen in $16 \cos^{2} (x) - 56 \cos (x) \sin (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 56 \cos (x) \sin (x) + 16 \sin^{2} (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 2.2.1.1

Addiere $- 56 \cos (x) \sin (x)$ und $56 \cos (x) \sin (x)$ .

${(16 \cos^{2} (x) + 0 + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 16 \sin^{2} (x))}^{2} = {(65)}^{2}$

Schritt 2.2.1.2

Addiere $16 \cos^{2} (x)$ und $0$ .

${(16 \cos^{2} (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x) + 16 \sin^{2} (x))}^{2} = {(65)}^{2}$

Schritt 2.2.2

Bewege $16 \sin^{2} (x)$ .

${(16 \cos^{2} (x) + 16 \sin^{2} (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x))}^{2} = {(65)}^{2}$

Schritt 2.2.3

Faktorisiere $16$ aus $16 \cos^{2} (x)$ heraus.

${(16 (\cos^{2} (x)) + 16 \sin^{2} (x) + 49 \sin^{2} (x) + 49 \cos^{2} (x))}^{2} = {(65)}^{2}$

Schritt 2.2.4

Faktorisiere $16$ aus $16 \sin^{2} (x)$ heraus.

${(16 (\cos^{2} (x)) + 16 (\sin^{2} (x)) + 49 \sin^{2} (x) + 49 \cos^{2} (x))}^{2} = {(65)}^{2}$

Schritt 2.2.5

Faktorisiere $16$ aus $16 (\cos^{2} (x)) + 16 (\sin^{2} (x))$ heraus.

${(16 (\cos^{2} (x) + \sin^{2} (x)) + 49 \sin^{2} (x) + 49 \cos^{2} (x))}^{2} = {(65)}^{2}$

Schritt 2.3

Ordne Terme um.

${(16 (\sin^{2} (x) + \cos^{2} (x)) + 49 \sin^{2} (x) + 49 \cos^{2} (x))}^{2} = {(65)}^{2}$

Schritt 2.4

Wende den trigonometrischen Pythagoras an.

${(16 \cdot 1 + 49 \sin^{2} (x) + 49 \cos^{2} (x))}^{2} = {(65)}^{2}$

Schritt 2.5

Vereinfache durch Herausfaktorisieren.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.5.1

Faktorisiere $49$ aus $49 \sin^{2} (x)$ heraus.

${(16 \cdot 1 + 49 (\sin^{2} (x)) + 49 \cos^{2} (x))}^{2} = {(65)}^{2}$

Schritt 2.5.2

Faktorisiere $49$ aus $49 \cos^{2} (x)$ heraus.

${(16 \cdot 1 + 49 (\sin^{2} (x)) + 49 (\cos^{2} (x)))}^{2} = {(65)}^{2}$

Schritt 2.5.3

Faktorisiere $49$ aus $49 (\sin^{2} (x)) + 49 (\cos^{2} (x))$ heraus.

${(16 \cdot 1 + 49 (\sin^{2} (x) + \cos^{2} (x)))}^{2} = {(65)}^{2}$

Schritt 2.6

Wende den trigonometrischen Pythagoras an.

${(16 \cdot 1 + 49 \cdot 1)}^{2} = {(65)}^{2}$

Schritt 2.7

Vereinfache Terme.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.7.1

Vereinfache jeden Term.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.7.1.1

Mutltipliziere $16$ mit $1$ .

${(16 + 49 \cdot 1)}^{2} = {(65)}^{2}$

Schritt 2.7.1.2

Mutltipliziere $49$ mit $1$ .

${(16 + 49)}^{2} = {(65)}^{2}$

Schritt 2.7.2

Vereinfache den Ausdruck.

Tippen, um mehr Schritte zu sehen ...

Schritt 2.7.2.1

Addiere $16$ und $49$ .

$65^{2} = {(65)}^{2}$

Schritt 2.7.2.2

Potenziere $65$ mit $2$ .

$4225 = {(65)}^{2}$

Schritt 3

Potenziere $65$ mit $2$ .

$4225 = 4225$

Schritt 4

Da $4225 = 4225$ , wird die Gleichung immer erfüllt sein für jeden Wert von $x$ .

Alle reellen Zahlen

Schritt 5

Das Ergebnis kann in mehreren Formen wiedergegeben werden.

Alle reellen Zahlen

Intervallschreibweise:

$(- \infty, \infty)$