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Beliebt Trigonometrie >

sin^4(x)+cos^4(x)= 5/8

Lösung

sin^{4} (x) + cos^{4} (x) = \frac{5}{8}

Lösung

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Grad

x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

Schritte zur Lösung

sin^{4} (x) + cos^{4} (x) = \frac{5}{8}

Subtrahiere

\frac{5}{8}

von beiden Seiten

sin^{4} (x) + cos^{4} (x) - \frac{5}{8} = 0

Vereinfache

sin^{4} (x) + cos^{4} (x) - \frac{5}{8} : \frac{8 sin ^{4} ( x ) + 8 cos ^{4} ( x ) - 5}{8}

sin^{4} (x) + cos^{4} (x) - \frac{5}{8}

Wandle das Element in einen Bruch um:

sin^{4} (x) = \frac{sin ^{4} ( x ) 8}{8}, cos^{4} (x) = \frac{cos ^{4} ( x ) 8}{8}

= \frac{sin ^{4} ( x ) \cdot 8}{8} + \frac{cos ^{4} ( x ) \cdot 8}{8} - \frac{5}{8}

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

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

= \frac{sin ^{4} ( x ) \cdot 8 + cos ^{4} ( x ) \cdot 8 - 5}{8}

\frac{8 sin ^{4} ( x ) + 8 cos ^{4} ( x ) - 5}{8} = 0

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

8 sin^{4} (x) + 8 cos^{4} (x) - 5 = 0

Wende Exponentenregel an:

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

- 5 + 8 cos^{4} (x) + 8 sin^{2} (x) sin^{2} (x) = 0

Umschreiben mit Hilfe von Trigonometrie-Identitäten

- 5 + 8 cos^{4} (x) + 8 sin^{2} (x) sin^{2} (x)

Verwende die Pythagoreische Identität:

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

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

= - 5 + 8 cos^{4} (x) + 8 (1 - cos^{2} (x)) (1 - cos^{2} (x))

Vereinfache

- 5 + 8 cos^{4} (x) + 8 (1 - cos^{2} (x)) (1 - cos^{2} (x)) : 16 cos^{4} (x) - 16 cos^{2} (x) + 3

- 5 + 8 cos^{4} (x) + 8 (1 - cos^{2} (x)) (1 - cos^{2} (x))

8 (1 - cos^{2} (x)) (1 - cos^{2} (x)) = 8 (1 - cos^{2} (x))^{2}

8 (1 - cos^{2} (x)) (1 - cos^{2} (x))

Wende Exponentenregel an:

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

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

= 8 (1 - cos^{2} (x))^{1 + 1}

Addiere die Zahlen:

1 + 1 = 2

= 8 (1 - cos^{2} (x))^{2}

= - 5 + 8 cos^{4} (x) + 8 (- cos^{2} (x) + 1)^{2}

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

Wende Formel für perfekte quadratische Gleichungen an:

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

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

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

Vereinfache

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

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

Wende Regel an

1^{a} = 1

1^{2} = 1

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

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

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

Multipliziere die Zahlen:

2 \cdot 1 = 2

= 2 cos^{2} (x)

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

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

Wende Exponentenregel an:

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

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

Multipliziere die Zahlen:

2 \cdot 2 = 4

= cos^{4} (x)

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

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

= - 5 + 8 cos^{4} (x) + 8 (1 - 2 cos^{2} (x) + cos^{4} (x))

Multipliziere aus

8 (1 - 2 cos^{2} (x) + cos^{4} (x)) : 8 - 16 cos^{2} (x) + 8 cos^{4} (x)

8 (1 - 2 cos^{2} (x) + cos^{4} (x))

Setze Klammern

= 8 \cdot 1 + 8 (- 2 cos^{2} (x)) + 8 cos^{4} (x)

Wende Minus-Plus Regeln an

+ (- a) = - a

= 8 \cdot 1 - 8 \cdot 2 cos^{2} (x) + 8 cos^{4} (x)

Vereinfache

8 \cdot 1 - 8 \cdot 2 cos^{2} (x) + 8 cos^{4} (x) : 8 - 16 cos^{2} (x) + 8 cos^{4} (x)

8 \cdot 1 - 8 \cdot 2 cos^{2} (x) + 8 cos^{4} (x)

Multipliziere die Zahlen:

8 \cdot 1 = 8

= 8 - 8 \cdot 2 cos^{2} (x) + 8 cos^{4} (x)

Multipliziere die Zahlen:

8 \cdot 2 = 16

= 8 - 16 cos^{2} (x) + 8 cos^{4} (x)

= 8 - 16 cos^{2} (x) + 8 cos^{4} (x)

= - 5 + 8 cos^{4} (x) + 8 - 16 cos^{2} (x) + 8 cos^{4} (x)

Vereinfache

- 5 + 8 cos^{4} (x) + 8 - 16 cos^{2} (x) + 8 cos^{4} (x) : 16 cos^{4} (x) - 16 cos^{2} (x) + 3

- 5 + 8 cos^{4} (x) + 8 - 16 cos^{2} (x) + 8 cos^{4} (x)

Fasse gleiche Terme zusammen

= 8 cos^{4} (x) - 16 cos^{2} (x) + 8 cos^{4} (x) - 5 + 8

Addiere gleiche Elemente:

8 cos^{4} (x) + 8 cos^{4} (x) = 16 cos^{4} (x)

= 16 cos^{4} (x) - 16 cos^{2} (x) - 5 + 8

Addiere/Subtrahiere die Zahlen:

- 5 + 8 = 3

= 16 cos^{4} (x) - 16 cos^{2} (x) + 3

= 16 cos^{4} (x) - 16 cos^{2} (x) + 3

= 16 cos^{4} (x) - 16 cos^{2} (x) + 3

3 - 16 cos^{2} (x) + 16 cos^{4} (x) = 0

Löse mit Substitution

3 - 16 cos^{2} (x) + 16 cos^{4} (x) = 0

Angenommen:

cos (x) = u

3 - 16 u^{2} + 16 u^{4} = 0

3 - 16 u^{2} + 16 u^{4} = 0 : u = \frac{3}{2}, u = - \frac{3}{2}, u = \frac{1}{2}, u = - \frac{1}{2}

3 - 16 u^{2} + 16 u^{4} = 0

Schreibe in der Standard Form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

16 u^{4} - 16 u^{2} + 3 = 0

Schreibe die Gleichung um mit

v = u^{2}

und

v^{2} = u^{4}

16 v^{2} - 16 v + 3 = 0

Löse

16 v^{2} - 16 v + 3 = 0 : v = \frac{3}{4}, v = \frac{1}{4}

16 v^{2} - 16 v + 3 = 0

Löse mit der quadratischen Formel

16 v^{2} - 16 v + 3 = 0

Quadratische Formel für Gliechungen:

Für

a = 16, b = - 16, c = 3

v_{1, 2} = \frac{- ( - 16 ) \pm ( - 16 ) ^{2} - 4 \cdot 16 \cdot 3}{2 \cdot 16}

v_{1, 2} = \frac{- ( - 16 ) \pm ( - 16 ) ^{2} - 4 \cdot 16 \cdot 3}{2 \cdot 16}

(- 16)^{2} - 4 \cdot 16 \cdot 3 = 8

(- 16)^{2} - 4 \cdot 16 \cdot 3

Wende Exponentenregel an:

(- a)^{n} = a^{n},

wenn

n

gerade ist

(- 16)^{2} = 1 6^{2}

= 1 6^{2} - 4 \cdot 16 \cdot 3

Multipliziere die Zahlen:

4 \cdot 16 \cdot 3 = 192

= 1 6^{2} - 192

1 6^{2} = 256

= 256 - 192

Subtrahiere die Zahlen:

256 - 192 = 64

= 64

Faktorisiere die Zahl:

64 = 8^{2}

= 8^{2}

Wende Radikal Regel an:

n a^{n} = a

8^{2} = 8

= 8

v_{1, 2} = \frac{- ( - 16 ) \pm 8}{2 \cdot 16}

Trenne die Lösungen

v_{1} = \frac{- ( - 16 ) + 8}{2 \cdot 16}, v_{2} = \frac{- ( - 16 ) - 8}{2 \cdot 16}

v = \frac{- ( - 16 ) + 8}{2 \cdot 16} : \frac{3}{4}

\frac{- ( - 16 ) + 8}{2 \cdot 16}

Wende Regel an

- (- a) = a

= \frac{16 + 8}{2 \cdot 16}

Addiere die Zahlen:

16 + 8 = 24

= \frac{24}{2 \cdot 16}

Multipliziere die Zahlen:

2 \cdot 16 = 32

= \frac{24}{32}

Streiche die gemeinsamen Faktoren:

8

= \frac{3}{4}

v = \frac{- ( - 16 ) - 8}{2 \cdot 16} : \frac{1}{4}

\frac{- ( - 16 ) - 8}{2 \cdot 16}

Wende Regel an

- (- a) = a

= \frac{16 - 8}{2 \cdot 16}

Subtrahiere die Zahlen:

16 - 8 = 8

= \frac{8}{2 \cdot 16}

Multipliziere die Zahlen:

2 \cdot 16 = 32

= \frac{8}{32}

Streiche die gemeinsamen Faktoren:

8

= \frac{1}{4}

Die Lösungen für die quadratische Gleichung sind:

v = \frac{3}{4}, v = \frac{1}{4}

v = \frac{3}{4}, v = \frac{1}{4}

Setze

v = u^{2} w i e d er e in,

löse für

u

Löse

u^{2} = \frac{3}{4} : u = \frac{3}{2}, u = - \frac{3}{2}

u^{2} = \frac{3}{4}

Für

x^{2} = f (a)

sind die Lösungen

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

u = \frac{3}{4}, u = - \frac{3}{4}

\frac{3}{4} = \frac{3}{2}

\frac{3}{4}

Wende Radikal Regel an:

n \frac{a}{b} = \frac{n a}{n b},

angenommen

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Faktorisiere die Zahl:

4 = 2^{2}

= 2^{2}

Wende Radikal Regel an:

n a^{n} = a

2^{2} = 2

= 2

= \frac{3}{2}

- \frac{3}{4} = - \frac{3}{2}

- \frac{3}{4}

Vereinfache

\frac{3}{4} : \frac{3}{2}

\frac{3}{4}

Wende Radikal Regel an:

n \frac{a}{b} = \frac{n a}{n b},

angenommen

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Faktorisiere die Zahl:

4 = 2^{2}

= 2^{2}

Wende Radikal Regel an:

n a^{n} = a

2^{2} = 2

= 2

= \frac{3}{2}

= - \frac{3}{2}

u = \frac{3}{2}, u = - \frac{3}{2}

Löse

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

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:

n \frac{a}{b} = \frac{n a}{n b},

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:

n a^{n} = a

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:

n \frac{a}{b} = \frac{n a}{n b},

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:

n a^{n} = a

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}

Die Lösungen sind

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

Setze in

u = cos (x)

ein

cos (x) = \frac{3}{2}, cos (x) = - \frac{3}{2}, cos (x) = \frac{1}{2}, cos (x) = - \frac{1}{2}

cos (x) = \frac{3}{2}, cos (x) = - \frac{3}{2}, cos (x) = \frac{1}{2}, cos (x) = - \frac{1}{2}

cos (x) = \frac{3}{2} : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

cos (x) = \frac{3}{2}

Allgemeine Lösung für

cos (x) = \frac{3}{2}

cos (x)

Periodizitätstabelle mit

2 πn

Zyklus:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

cos (x) = - \frac{3}{2} : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

cos (x) = - \frac{3}{2}

Allgemeine Lösung für

cos (x) = - \frac{3}{2}

cos (x)

Periodizitätstabelle mit

2 πn

Zyklus:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

Allgemeine Lösung für

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

cos (x)

Periodizitätstabelle mit

2 πn

Zyklus:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

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

Allgemeine Lösung für

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

cos (x)

Periodizitätstabelle mit

2 πn

Zyklus:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

Kombiniere alle Lösungen

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Graph

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