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Beliebt Trigonometrie >

sin(x)-sin(2x)+sin(3x)=0

Lösung

sin (x) - sin (2 x) + sin (3 x) = 0

Lösung

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

Grad

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Schritte zur Lösung

sin (x) - sin (2 x) + sin (3 x) = 0

Umschreiben mit Hilfe von Trigonometrie-Identitäten

- sin (2 x) + sin (3 x) + sin (x)

Verwende die Doppelwinkelidentität:

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

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

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

sin (3 x)

Umschreiben mit Hilfe von Trigonometrie-Identitäten

sin (3 x)

Schreibe um

= sin (2 x + x)

Benutze die Identität der Winkelsumme:

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

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

Verwende die Doppelwinkelidentität:

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

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

Vereinfache

cos (2 x) sin (x) + cos (x) \cdot 2 sin (x) cos (x) : sin (x) cos (2 x) + 2 cos^{2} (x) sin (x)

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

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

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

Wende Exponentenregel an:

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

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

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

Addiere die Zahlen:

1 + 1 = 2

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

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

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

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

Verwende die Doppelwinkelidentität:

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

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

Verwende die Pythagoreische Identität:

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

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

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

Multipliziere aus

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

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

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

Multipliziere aus

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

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

Wende das Distributivgesetz an:

a (b - c) = ab - a c

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

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

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

Vereinfache

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

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

1 \cdot sin (x) = sin (x)

1 sin (x)

Multipliziere:

1 \cdot sin (x) = sin (x)

= sin (x)

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

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

Wende Exponentenregel an:

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

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

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

Addiere die Zahlen:

2 + 1 = 3

= 2 sin^{3} (x)

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

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

= sin (x) - 2 sin^{3} (x) + 2 (1 - sin^{2} (x)) sin (x)

Multipliziere aus

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

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

Wende das Distributivgesetz an:

a (b - c) = ab - a c

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

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

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

Vereinfache

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

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

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

2 \cdot 1 sin (x)

Multipliziere die Zahlen:

2 \cdot 1 = 2

= 2 sin (x)

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

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

Wende Exponentenregel an:

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

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

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

Addiere die Zahlen:

2 + 1 = 3

= 2 sin^{3} (x)

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

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

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

Vereinfache

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

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

Fasse gleiche Terme zusammen

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

Addiere gleiche Elemente:

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

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

Addiere gleiche Elemente:

sin (x) + 2 sin (x) = 3 sin (x)

= - 4 sin^{3} (x) + 3 sin (x)

= - 4 sin^{3} (x) + 3 sin (x)

= - 4 sin^{3} (x) + 3 sin (x)

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

Vereinfache

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

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

Faktorisiere

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

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

Wende Exponentenregel an:

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

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

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

Schreibe

- 4

um:

2 \cdot 2

Schreibe

4

um:

2 \cdot 2

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

Klammere gleiche Terme aus

2 sin (x)

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

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

Löse jeden Teil einzeln

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

Allgemeine Lösung für

sin (x) = 0

sin (x)

Periodizitätstabelle mit

2 πn

Zyklus:

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 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, x = π + 2 πn

2 - 2 sin^{2} (x) - cos (x) = 0 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Verwende die Pythagoreische Identität:

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

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

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

Vereinfache

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

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

Multipliziere aus

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

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

Wende das Distributivgesetz an:

a (b - c) = ab - a c

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

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

Wende Minus-Plus Regeln an

- (- a) = a

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

Multipliziere die Zahlen:

2 \cdot 1 = 2

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

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

Vereinfache

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

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

Fasse gleiche Terme zusammen

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

2 - 2 = 0

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

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

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

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

Löse mit Substitution

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

Angenommen:

cos (x) = u

- u + 2 u^{2} = 0

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

- u + 2 u^{2} = 0

Schreibe in der Standard Form

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

2 u^{2} - u = 0

Löse mit der quadratischen Formel

2 u^{2} - u = 0

Quadratische Formel für Gliechungen:

Für

a = 2, b = - 1, c = 0

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 2 \cdot 0}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 2 \cdot 0}{2 \cdot 2}

(- 1)^{2} - 4 \cdot 2 \cdot 0 = 1

(- 1)^{2} - 4 \cdot 2 \cdot 0

(- 1)^{2} = 1

(- 1)^{2}

Wende Exponentenregel an:

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

wenn

n

gerade ist

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

= 1^{2}

Wende Regel an

1^{a} = 1

= 1

4 \cdot 2 \cdot 0 = 0

4 \cdot 2 \cdot 0

Wende Regel an

0 \cdot a = 0

= 0

= 1 - 0

Subtrahiere die Zahlen:

1 - 0 = 1

= 1

Wende Regel an

1 = 1

= 1

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

Trenne die Lösungen

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

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

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

Wende Regel an

- (- a) = a

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

Addiere die Zahlen:

1 + 1 = 2

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

Multipliziere die Zahlen:

2 \cdot 2 = 4

= \frac{2}{4}

Streiche die gemeinsamen Faktoren:

2

= \frac{1}{2}

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

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

Wende Regel an

- (- a) = a

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

Subtrahiere die Zahlen:

1 - 1 = 0

= \frac{0}{2 \cdot 2}

Multipliziere die Zahlen:

2 \cdot 2 = 4

= \frac{0}{4}

Wende Regel an

\frac{0}{a} = 0, a \neq = 0

= 0

Die Lösungen für die quadratische Gleichung sind:

u = \frac{1}{2}, u = 0

Setze in

u = cos (x)

ein

cos (x) = \frac{1}{2}, cos (x) = 0

cos (x) = \frac{1}{2}, cos (x) = 0

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) = 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

Kombiniere alle Lösungen

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

Kombiniere alle Lösungen

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

Graph

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