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Beliebt Trigonometrie >

sin^2(x)-2cos^4(x)=0

Lösung

sin^{2} (x) - 2 cos^{4} (x) = 0

Lösung

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Grad

x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n

Schritte zur Lösung

sin^{2} (x) - 2 cos^{4} (x) = 0

Faktorisiere

sin^{2} (x) - 2 cos^{4} (x) : (sin (x) + 2 cos^{2} (x)) (sin (x) - 2 cos^{2} (x))

sin^{2} (x) - 2 cos^{4} (x)

Schreibe

sin^{2} (x) - 2 cos^{4} (x)

um:

sin^{2} (x) - (2 cos^{2} (x))^{2}

sin^{2} (x) - 2 cos^{4} (x)

Wende Radikal Regel an:

a = (a)^{2}

2 = (2)^{2}

= sin^{2} (x) - (2)^{2} cos^{4} (x)

Wende Exponentenregel an:

a^{b c} = (a^{b})^{c}

cos^{4} (x) = (cos^{2} (x))^{2}

= sin^{2} (x) - (2)^{2} (cos^{2} (x))^{2}

Wende Exponentenregel an:

a^{m} b^{m} = (ab)^{m}

(2)^{2} (cos^{2} (x))^{2} = (2 cos^{2} (x))^{2}

= sin^{2} (x) - (2 cos^{2} (x))^{2}

= sin^{2} (x) - (2 cos^{2} (x))^{2}

Wende Formel zur Differenz von zwei Quadraten an:

x^{2} - y^{2} = (x + y) (x - y)

sin^{2} (x) - (2 cos^{2} (x))^{2} = (sin (x) + 2 cos^{2} (x)) (sin (x) - 2 cos^{2} (x))

= (sin (x) + 2 cos^{2} (x)) (sin (x) - 2 cos^{2} (x))

(sin (x) + 2 cos^{2} (x)) (sin (x) - 2 cos^{2} (x)) = 0

Löse jeden Teil einzeln

sin (x) + 2 cos^{2} (x) = 0 or sin (x) - 2 cos^{2} (x) = 0

sin (x) + 2 cos^{2} (x) = 0 : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sin (x) + 2 cos^{2} (x) = 0

Umschreiben mit Hilfe von Trigonometrie-Identitäten

sin (x) + cos^{2} (x) 2

Verwende die Pythagoreische Identität:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= sin (x) + (1 - sin^{2} (x)) 2

sin (x) + (1 - sin^{2} (x)) 2 = 0

Löse mit Substitution

sin (x) + (1 - sin^{2} (x)) 2 = 0

Angenommen:

sin (x) = u

u + (1 - u^{2}) 2 = 0

u + (1 - u^{2}) 2 = 0 : u = - \frac{2}{2}, u = 2

u + (1 - u^{2}) 2 = 0

Schreibe

u + (1 - u^{2}) 2

um:

u + 2 - 2 u^{2}

u + (1 - u^{2}) 2

= u + 2 (1 - u^{2})

Multipliziere aus

2 (1 - u^{2}) : 2 - 2 u^{2}

2 (1 - u^{2})

Wende das Distributivgesetz an:

a (b - c) = ab - a c

a = 2, b = 1, c = u^{2}

= 2 \cdot 1 - 2 u^{2}

= 1 \cdot 2 - 2 u^{2}

Multipliziere:

1 \cdot 2 = 2

= 2 - 2 u^{2}

= u + 2 - 2 u^{2}

u + 2 - 2 u^{2} = 0

Schreibe in der Standard Form

a x^{2} + b x + c = 0

- 2 u^{2} + u + 2 = 0

Löse mit der quadratischen Formel

- 2 u^{2} + u + 2 = 0

Quadratische Formel für Gliechungen:

Für

a = - 2, b = 1, c = 2

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 2 ) 2}{2 ( - 2 )}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 2 ) 2}{2 ( - 2 )}

1^{2} - 4 (- 2) 2 = 3

1^{2} - 4 (- 2) 2

Wende Regel an

1^{a} = 1

1^{2} = 1

= 1 - 42 (- 2)

Wende Regel an

- (- a) = a

= 1 + 422

422 = 8

422

Wende Radikal Regel an:

a a = a

22 = 2

= 4 \cdot 2

Multipliziere die Zahlen:

4 \cdot 2 = 8

= 8

= 1 + 8

Addiere die Zahlen:

1 + 8 = 9

= 9

Faktorisiere die Zahl:

9 = 3^{2}

= 3^{2}

Wende Radikal Regel an:

n a^{n} = a

3^{2} = 3

= 3

u_{1, 2} = \frac{- 1 \pm 3}{2 ( - 2 )}

Trenne die Lösungen

u_{1} = \frac{- 1 + 3}{2 ( - 2 )}, u_{2} = \frac{- 1 - 3}{2 ( - 2 )}

u = \frac{- 1 + 3}{2 ( - 2 )} : - \frac{2}{2}

\frac{- 1 + 3}{2 ( - 2 )}

Entferne die Klammern:

(- a) = - a

= \frac{- 1 + 3}{- 2 2}

Addiere/Subtrahiere die Zahlen:

- 1 + 3 = 2

= \frac{2}{- 2 2}

Wende Bruchregel an:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{2}{2 2}

Teile die Zahlen:

\frac{2}{2} = 1

= - \frac{1}{2}

Rationalisiere

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multipliziere mit dem Konjugat

\frac{2}{2}

= - \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Wende Radikal Regel an:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

u = \frac{- 1 - 3}{2 ( - 2 )} : 2

\frac{- 1 - 3}{2 ( - 2 )}

Entferne die Klammern:

(- a) = - a

= \frac{- 1 - 3}{- 2 2}

Subtrahiere die Zahlen:

- 1 - 3 = - 4

= \frac{- 4}{- 2 2}

Wende Bruchregel an:

\frac{- a}{- b} = \frac{a}{b}

= \frac{4}{2 2}

Teile die Zahlen:

\frac{4}{2} = 2

= \frac{2}{2}

Wende Radikal Regel an:

n a = a^{\frac{1}{n}}

2 = 2^{\frac{1}{2}}

= \frac{2}{2 ^{\frac{1}{2}}}

Wende Exponentenregel an:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{1}}{2 ^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}}

= 2^{1 - \frac{1}{2}}

Subtrahiere die Zahlen:

1 - \frac{1}{2} = \frac{1}{2}

= 2^{\frac{1}{2}}

Wende Radikal Regel an:

a^{\frac{1}{n}} = n a

2^{\frac{1}{2}} = 2

= 2

Die Lösungen für die quadratische Gleichung sind:

u = - \frac{2}{2}, u = 2

Setze in

u = sin (x)

ein

sin (x) = - \frac{2}{2}, sin (x) = 2

sin (x) = - \frac{2}{2}, sin (x) = 2

sin (x) = - \frac{2}{2} : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sin (x) = - \frac{2}{2}

Allgemeine Lösung für

sin (x) = - \frac{2}{2}

sin (x)

Periodizitätstabelle mit

2 πn

Zyklus:

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sin (x) = 2 :

Keine Lösung

sin (x) = 2

- 1 \leq sin (x) \leq 1

Keine L \overset{o}{¨} sung

Kombiniere alle Lösungen

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sin (x) - 2 cos^{2} (x) = 0 : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sin (x) - 2 cos^{2} (x) = 0

Umschreiben mit Hilfe von Trigonometrie-Identitäten

sin (x) - cos^{2} (x) 2

Verwende die Pythagoreische Identität:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= sin (x) - (1 - sin^{2} (x)) 2

sin (x) - (1 - sin^{2} (x)) 2 = 0

Löse mit Substitution

sin (x) - (1 - sin^{2} (x)) 2 = 0

Angenommen:

sin (x) = u

u - (1 - u^{2}) 2 = 0

u - (1 - u^{2}) 2 = 0 : u = \frac{2}{2}, u = - 2

u - (1 - u^{2}) 2 = 0

Schreibe

u - (1 - u^{2}) 2

um:

u - 2 + 2 u^{2}

u - (1 - u^{2}) 2

= u - 2 (1 - u^{2})

Multipliziere aus

- 2 (1 - u^{2}) : - 2 + 2 u^{2}

- 2 (1 - u^{2})

Wende das Distributivgesetz an:

a (b - c) = ab - a c

a = - 2, b = 1, c = u^{2}

= - 2 \cdot 1 - (- 2) u^{2}

Wende Minus-Plus Regeln an

- (- a) = a

= - 1 \cdot 2 + 2 u^{2}

Multipliziere:

1 \cdot 2 = 2

= - 2 + 2 u^{2}

= u - 2 + 2 u^{2}

u - 2 + 2 u^{2} = 0

Schreibe in der Standard Form

a x^{2} + b x + c = 0

2 u^{2} + u - 2 = 0

Löse mit der quadratischen Formel

2 u^{2} + u - 2 = 0

Quadratische Formel für Gliechungen:

Für

a = 2, b = 1, c = - 2

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 2 ( - 2 )}{2 2}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 2 ( - 2 )}{2 2}

1^{2} - 42 (- 2) = 3

1^{2} - 42 (- 2)

Wende Regel an

1^{a} = 1

1^{2} = 1

= 1 - 42 (- 2)

Wende Regel an

- (- a) = a

= 1 + 422

422 = 8

422

Wende Radikal Regel an:

a a = a

22 = 2

= 4 \cdot 2

Multipliziere die Zahlen:

4 \cdot 2 = 8

= 8

= 1 + 8

Addiere die Zahlen:

1 + 8 = 9

= 9

Faktorisiere die Zahl:

9 = 3^{2}

= 3^{2}

Wende Radikal Regel an:

n a^{n} = a

3^{2} = 3

= 3

u_{1, 2} = \frac{- 1 \pm 3}{2 2}

Trenne die Lösungen

u_{1} = \frac{- 1 + 3}{2 2}, u_{2} = \frac{- 1 - 3}{2 2}

u = \frac{- 1 + 3}{2 2} : \frac{2}{2}

\frac{- 1 + 3}{2 2}

Addiere/Subtrahiere die Zahlen:

- 1 + 3 = 2

= \frac{2}{2 2}

Teile die Zahlen:

\frac{2}{2} = 1

= \frac{1}{2}

Rationalisiere

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multipliziere mit dem Konjugat

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Wende Radikal Regel an:

a a = a

22 = 2

= 2

= \frac{2}{2}

= \frac{2}{2}

u = \frac{- 1 - 3}{2 2} : - 2

\frac{- 1 - 3}{2 2}

Subtrahiere die Zahlen:

- 1 - 3 = - 4

= \frac{- 4}{2 2}

Wende Bruchregel an:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{4}{2 2}

Teile die Zahlen:

\frac{4}{2} = 2

= \frac{2}{2}

Wende Radikal Regel an:

n a = a^{\frac{1}{n}}

2 = 2^{\frac{1}{2}}

= \frac{2}{2 ^{\frac{1}{2}}}

Wende Exponentenregel an:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{1}}{2 ^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}}

= 2^{1 - \frac{1}{2}}

Subtrahiere die Zahlen:

1 - \frac{1}{2} = \frac{1}{2}

= 2^{\frac{1}{2}}

Wende Radikal Regel an:

a^{\frac{1}{n}} = n a

2^{\frac{1}{2}} = 2

= - 2

Die Lösungen für die quadratische Gleichung sind:

u = \frac{2}{2}, u = - 2

Setze in

u = sin (x)

ein

sin (x) = \frac{2}{2}, sin (x) = - 2

sin (x) = \frac{2}{2}, sin (x) = - 2

sin (x) = \frac{2}{2} : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sin (x) = \frac{2}{2}

Allgemeine Lösung für

sin (x) = \frac{2}{2}

sin (x)

Periodizitätstabelle mit

2 πn

Zyklus:

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sin (x) = - 2 :

Keine Lösung

sin (x) = - 2

- 1 \leq sin (x) \leq 1

Keine L \overset{o}{¨} sung

Kombiniere alle Lösungen

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Kombiniere alle Lösungen

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Graph

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