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Beliebt Trigonometrie >

cos^4(x)-sin^4(x)= 1/2

Lösung

cos^{4} (x) - sin^{4} (x) = \frac{1}{2}

Lösung

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Grad

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Schritte zur Lösung

cos^{4} (x) - sin^{4} (x) = \frac{1}{2}

Subtrahiere

\frac{1}{2}

von beiden Seiten

cos^{4} (x) - sin^{4} (x) - \frac{1}{2} = 0

Vereinfache

cos^{4} (x) - sin^{4} (x) - \frac{1}{2} : \frac{2 cos ^{4} ( x ) - 2 sin ^{4} ( x ) - 1}{2}

cos^{4} (x) - sin^{4} (x) - \frac{1}{2}

Wandle das Element in einen Bruch um:

cos^{4} (x) = \frac{cos ^{4} ( x ) 2}{2}, sin^{4} (x) = \frac{sin ^{4} ( x ) 2}{2}

= \frac{cos ^{4} ( x ) \cdot 2}{2} - \frac{sin ^{4} ( x ) \cdot 2}{2} - \frac{1}{2}

Da die Nenner gleich sind, fasse die Brüche zusammen.:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{cos ^{4} ( x ) \cdot 2 - sin ^{4} ( x ) \cdot 2 - 1}{2}

\frac{2 cos ^{4} ( x ) - 2 sin ^{4} ( x ) - 1}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 cos^{4} (x) - 2 sin^{4} (x) - 1 = 0

Wende Exponentenregel an:

a^{b} = a^{2} a^{b - 2}

- 1 - 2 sin^{4} (x) + 2 cos^{2} (x) cos^{2} (x) = 0

Umschreiben mit Hilfe von Trigonometrie-Identitäten

- 1 - 2 sin^{4} (x) + 2 cos^{2} (x) cos^{2} (x)

Verwende die Pythagoreische Identität:

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

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

= - 1 - 2 sin^{4} (x) + 2 (1 - sin^{2} (x)) (1 - sin^{2} (x))

Vereinfache

- 1 - 2 sin^{4} (x) + 2 (1 - sin^{2} (x)) (1 - sin^{2} (x)) : - 4 sin^{2} (x) + 1

- 1 - 2 sin^{4} (x) + 2 (1 - sin^{2} (x)) (1 - sin^{2} (x))

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

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

Wende Exponentenregel an:

a^{b} \cdot a^{c} = a^{b + c}

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

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

Addiere die Zahlen:

1 + 1 = 2

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

= - 1 - 2 sin^{4} (x) + 2 (- sin^{2} (x) + 1)^{2}

(1 - sin^{2} (x))^{2} : 1 - 2 sin^{2} (x) + sin^{4} (x)

Wende Formel für perfekte quadratische Gleichungen an:

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

a = 1, b = sin^{2} (x)

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

Vereinfache

1^{2} - 2 \cdot 1 \cdot sin^{2} (x) + (sin^{2} (x))^{2} : 1 - 2 sin^{2} (x) + sin^{4} (x)

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

Wende Regel an

1^{a} = 1

1^{2} = 1

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

2 \cdot 1 \cdot sin^{2} (x) = 2 sin^{2} (x)

2 \cdot 1 \cdot sin^{2} (x)

Multipliziere die Zahlen:

2 \cdot 1 = 2

= 2 sin^{2} (x)

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

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

Wende Exponentenregel an:

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

= sin^{2 \cdot 2} (x)

Multipliziere die Zahlen:

2 \cdot 2 = 4

= sin^{4} (x)

= 1 - 2 sin^{2} (x) + sin^{4} (x)

= 1 - 2 sin^{2} (x) + sin^{4} (x)

= - 1 - 2 sin^{4} (x) + 2 (1 - 2 sin^{2} (x) + sin^{4} (x))

Multipliziere aus

2 (1 - 2 sin^{2} (x) + sin^{4} (x)) : 2 - 4 sin^{2} (x) + 2 sin^{4} (x)

2 (1 - 2 sin^{2} (x) + sin^{4} (x))

Setze Klammern

= 2 \cdot 1 + 2 (- 2 sin^{2} (x)) + 2 sin^{4} (x)

Wende Minus-Plus Regeln an

+ (- a) = - a

= 2 \cdot 1 - 2 \cdot 2 sin^{2} (x) + 2 sin^{4} (x)

Vereinfache

2 \cdot 1 - 2 \cdot 2 sin^{2} (x) + 2 sin^{4} (x) : 2 - 4 sin^{2} (x) + 2 sin^{4} (x)

2 \cdot 1 - 2 \cdot 2 sin^{2} (x) + 2 sin^{4} (x)

Multipliziere die Zahlen:

2 \cdot 1 = 2

= 2 - 2 \cdot 2 sin^{2} (x) + 2 sin^{4} (x)

Multipliziere die Zahlen:

2 \cdot 2 = 4

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

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

= - 1 - 2 sin^{4} (x) + 2 - 4 sin^{2} (x) + 2 sin^{4} (x)

Vereinfache

- 1 - 2 sin^{4} (x) + 2 - 4 sin^{2} (x) + 2 sin^{4} (x) : - 4 sin^{2} (x) + 1

- 1 - 2 sin^{4} (x) + 2 - 4 sin^{2} (x) + 2 sin^{4} (x)

Fasse gleiche Terme zusammen

= - 2 sin^{4} (x) - 4 sin^{2} (x) + 2 sin^{4} (x) - 1 + 2

Addiere gleiche Elemente:

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

= - 4 sin^{2} (x) - 1 + 2

Addiere/Subtrahiere die Zahlen:

- 1 + 2 = 1

= - 4 sin^{2} (x) + 1

= - 4 sin^{2} (x) + 1

= - 4 sin^{2} (x) + 1

1 - 4 sin^{2} (x) = 0

Löse mit Substitution

1 - 4 sin^{2} (x) = 0

Angenommen:

sin (x) = u

1 - 4 u^{2} = 0

1 - 4 u^{2} = 0 : u = \frac{1}{2}, u = - \frac{1}{2}

1 - 4 u^{2} = 0

Verschiebe

1

auf die rechte Seite

1 - 4 u^{2} = 0

Subtrahiere

1

von beiden Seiten

1 - 4 u^{2} - 1 = 0 - 1

Vereinfache

- 4 u^{2} = - 1

- 4 u^{2} = - 1

Teile beide Seiten durch

- 4

- 4 u^{2} = - 1

Teile beide Seiten durch

- 4

\frac{- 4 u ^{2}}{- 4} = \frac{- 1}{- 4}

Vereinfache

u^{2} = \frac{1}{4}

u^{2} = \frac{1}{4}

Für

x^{2} = f (a)

sind die Lösungen

x = f (a), - f (a)

u = \frac{1}{4}, u = - \frac{1}{4}

\frac{1}{4} = \frac{1}{2}

\frac{1}{4}

Wende Radikal Regel an:

angenommen

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Faktorisiere die Zahl:

4 = 2^{2}

= 2^{2}

Wende Radikal Regel an:

2^{2} = 2

= 2

= \frac{1}{2}

Wende Regel an

1 = 1

= \frac{1}{2}

- \frac{1}{4} = - \frac{1}{2}

- \frac{1}{4}

Vereinfache

\frac{1}{4} : \frac{1}{2}

\frac{1}{4}

Wende Radikal Regel an:

angenommen

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Faktorisiere die Zahl:

4 = 2^{2}

= 2^{2}

Wende Radikal Regel an:

2^{2} = 2

= 2

= \frac{1}{2}

= - \frac{1}{2}

Wende Regel an

1 = 1

= - \frac{1}{2}

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

Setze in

u = sin (x)

ein

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = \frac{1}{2}

Allgemeine Lösung für

sin (x) = \frac{1}{2}

sin (x)

Periodizitätstabelle mit

2 πn

Zyklus:

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = - \frac{1}{2} : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

sin (x) = - \frac{1}{2}

Allgemeine Lösung für

sin (x) = - \frac{1}{2}

sin (x)

Periodizitätstabelle mit

2 πn

Zyklus:

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Kombiniere alle Lösungen

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Graph

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