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Beliebt >

2cos(x)=2cos(3x)

Lösung

2 cos (x) = 2 cos (3 x)

Lösung

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

Grad

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n

Schritte zur Lösung

2 cos (x) = 2 cos (3 x)

Subtrahiere

2 cos (3 x)

von beiden Seiten

2 cos (x) - 2 cos (3 x) = 0

Umschreiben mit Hilfe von Trigonometrie-Identitäten

- 2 cos (3 x) + 2 cos (x)

cos (3 x) = 4 cos^{3} (x) - 3 cos (x)

cos (3 x)

Umschreiben mit Hilfe von Trigonometrie-Identitäten

cos (3 x)

Schreibe um

= cos (2 x + x)

Benutze die Identität der Winkelsumme:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (2 x) cos (x) - sin (2 x) sin (x)

Verwende die Doppelwinkelidentität:

sin (2 x) = 2 sin (x) cos (x)

= cos (2 x) cos (x) - 2 sin (x) cos (x) sin (x)

Vereinfache

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

cos (2 x) cos (x) - 2 sin (x) cos (x) sin (x)

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

2 sin (x) cos (x) sin (x)

Wende Exponentenregel an:

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

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

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

Addiere die Zahlen:

1 + 1 = 2

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

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

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

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

Verwende die Doppelwinkelidentität:

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

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

Verwende die Pythagoreische Identität:

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

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

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

Multipliziere aus

(2 cos^{2} (x) - 1) cos (x) - 2 (1 - cos^{2} (x)) cos (x) : 4 cos^{3} (x) - 3 cos (x)

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

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

Multipliziere aus

cos (x) (2 cos^{2} (x) - 1) : 2 cos^{3} (x) - cos (x)

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

Wende das Distributivgesetz an:

a (b - c) = ab - a c

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

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

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

Vereinfache

2 cos^{2} (x) cos (x) - 1 \cdot cos (x) : 2 cos^{3} (x) - cos (x)

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

2 cos^{2} (x) cos (x) = 2 cos^{3} (x)

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

Wende Exponentenregel an:

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

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

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

Addiere die Zahlen:

2 + 1 = 3

= 2 cos^{3} (x)

1 \cdot cos (x) = cos (x)

1 cos (x)

Multipliziere:

1 \cdot cos (x) = cos (x)

= cos (x)

= 2 cos^{3} (x) - cos (x)

= 2 cos^{3} (x) - cos (x)

= 2 cos^{3} (x) - cos (x) - 2 (1 - cos^{2} (x)) cos (x)

Multipliziere aus

- 2 cos (x) (1 - cos^{2} (x)) : - 2 cos (x) + 2 cos^{3} (x)

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

Wende das Distributivgesetz an:

a (b - c) = ab - a c

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

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

Wende Minus-Plus Regeln an

- (- a) = a

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

Vereinfache

- 2 \cdot 1 \cdot cos (x) + 2 cos^{2} (x) cos (x) : - 2 cos (x) + 2 cos^{3} (x)

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

2 \cdot 1 \cdot cos (x) = 2 cos (x)

2 \cdot 1 cos (x)

Multipliziere die Zahlen:

2 \cdot 1 = 2

= 2 cos (x)

2 cos^{2} (x) cos (x) = 2 cos^{3} (x)

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

Wende Exponentenregel an:

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

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

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

Addiere die Zahlen:

2 + 1 = 3

= 2 cos^{3} (x)

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

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

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

Vereinfache

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

2 cos^{3} (x) - cos (x) - 2 cos (x) + 2 cos^{3} (x)

Fasse gleiche Terme zusammen

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

Addiere gleiche Elemente:

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

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

Addiere gleiche Elemente:

- cos (x) - 2 cos (x) = - 3 cos (x)

= 4 cos^{3} (x) - 3 cos (x)

= 4 cos^{3} (x) - 3 cos (x)

= 4 cos^{3} (x) - 3 cos (x)

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

Vereinfache

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

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

Multipliziere aus

- 2 (4 cos^{3} (x) - 3 cos (x)) : - 8 cos^{3} (x) + 6 cos (x)

- 2 (4 cos^{3} (x) - 3 cos (x))

Wende das Distributivgesetz an:

a (b - c) = ab - a c

a = - 2, b = 4 cos^{3} (x), c = 3 cos (x)

= - 2 \cdot 4 cos^{3} (x) - (- 2) \cdot 3 cos (x)

Wende Minus-Plus Regeln an

- (- a) = a

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

Vereinfache

- 2 \cdot 4 cos^{3} (x) + 2 \cdot 3 cos (x) : - 8 cos^{3} (x) + 6 cos (x)

- 2 \cdot 4 cos^{3} (x) + 2 \cdot 3 cos (x)

Multipliziere die Zahlen:

2 \cdot 4 = 8

= - 8 cos^{3} (x) + 2 \cdot 3 cos (x)

Multipliziere die Zahlen:

2 \cdot 3 = 6

= - 8 cos^{3} (x) + 6 cos (x)

= - 8 cos^{3} (x) + 6 cos (x)

= - 8 cos^{3} (x) + 6 cos (x) + 2 cos (x)

Addiere gleiche Elemente:

6 cos (x) + 2 cos (x) = 8 cos (x)

= - 8 cos^{3} (x) + 8 cos (x)

= - 8 cos^{3} (x) + 8 cos (x)

8 cos (x) - 8 cos^{3} (x) = 0

Löse mit Substitution

8 cos (x) - 8 cos^{3} (x) = 0

Angenommen:

cos (x) = u

8 u - 8 u^{3} = 0

8 u - 8 u^{3} = 0 : u = 0, u = - 1, u = 1

8 u - 8 u^{3} = 0

Faktorisiere

8 u - 8 u^{3} : - 8 u (u + 1) (u - 1)

8 u - 8 u^{3}

Klammere gleiche Terme aus

- 8 u : - 8 u (u^{2} - 1)

- 8 u^{3} + 8 u

Wende Exponentenregel an:

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

u^{3} = u^{2} u

= - 8 u^{2} u + 8 u

Klammere gleiche Terme aus

- 8 u

= - 8 u (u^{2} - 1)

= - 8 u (u^{2} - 1)

Faktorisiere

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

u^{2} - 1

Schreibe

1

um:

1^{2}

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

Wende Formel zur Differenz von zwei Quadraten an:

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

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

= (u + 1) (u - 1)

= - 8 u (u + 1) (u - 1)

- 8 u (u + 1) (u - 1) = 0

Anwendung des Nullfaktorprinzips: Wenn

ab = 0

dann

a = 0

oder

b = 0

u = 0 or u + 1 = 0 or u - 1 = 0

Löse

u + 1 = 0 : u = - 1

u + 1 = 0

Verschiebe

1

auf die rechte Seite

u + 1 = 0

Subtrahiere

1

von beiden Seiten

u + 1 - 1 = 0 - 1

Vereinfache

u = - 1

u = - 1

Löse

u - 1 = 0 : u = 1

u - 1 = 0

Verschiebe

1

auf die rechte Seite

u - 1 = 0

Füge

1

zu beiden Seiten hinzu

u - 1 + 1 = 0 + 1

Vereinfache

u = 1

u = 1

Die Lösungen sind

u = 0, u = - 1, u = 1

Setze in

u = cos (x)

ein

cos (x) = 0, cos (x) = - 1, cos (x) = 1

cos (x) = 0, cos (x) = - 1, cos (x) = 1

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

cos (x) = 0

Allgemeine Lösung für

cos (x) = 0

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}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

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

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

Allgemeine Lösung für

cos (x) = - 1

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 = π + 2 πn

x = π + 2 πn

cos (x) = 1 : x = 2 πn

cos (x) = 1

Allgemeine Lösung für

cos (x) = 1

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 = 0 + 2 πn

x = 0 + 2 πn

Löse

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

Kombiniere alle Lösungen

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

Graph

Beliebte Beispiele

2sin^2(x)+3cos(x)-3=0,0<= x<= 2pi

2 sin^{2} (x) + 3 cos (x) - 3 = 0, 0 \leq x \leq 2 π

2+2cos(θ)=0

2 + 2 cos (θ) = 0

2=4cos(θ)

2 = 4 cos (θ)

8cos^2(x)-2cos(x)-1=0

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

sin(θ)= 19/18

sin (θ) = \frac{19}{18}