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Beliebt Trigonometrie >

6tan^2(x)-2cos^2(x)=cos(2x)

Lösung

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

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

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

Schritte zur Lösung

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

Subtrahiere

cos (2 x)

von beiden Seiten

6 tan^{2} (x) - 2 cos^{2} (x) - cos (2 x) = 0

Drücke mit sin, cos aus

- cos (2 x) - 2 cos^{2} (x) + 6 tan^{2} (x)

Verwende die grundlegende trigonometrische Identität:

tan (x) = \frac{sin ( x )}{cos ( x )}

= - cos (2 x) - 2 cos^{2} (x) + 6 (\frac{sin ( x )}{cos ( x )})^{2}

Vereinfache

- cos (2 x) - 2 cos^{2} (x) + 6 (\frac{sin ( x )}{cos ( x )})^{2} : \frac{- cos ^{2} ( x ) cos ( 2 x ) - 2 cos ^{4} ( x ) + 6 sin ^{2} ( x )}{cos ^{2} ( x )}

- cos (2 x) - 2 cos^{2} (x) + 6 (\frac{sin ( x )}{cos ( x )})^{2}

6 (\frac{sin ( x )}{cos ( x )})^{2} = \frac{6 sin ^{2} ( x )}{cos ^{2} ( x )}

6 (\frac{sin ( x )}{cos ( x )})^{2}

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

(\frac{sin ( x )}{cos ( x )})^{2}

Wende Exponentenregel an:

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

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

= 6 \cdot \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

Multipliziere Brüche:

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

= \frac{sin ^{2} ( x ) \cdot 6}{cos ^{2} ( x )}

= - cos (2 x) - 2 cos^{2} (x) + \frac{6 sin ^{2} ( x )}{cos ^{2} ( x )}

Wandle das Element in einen Bruch um:

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

= - \frac{cos ( 2 x ) cos ^{2} ( x )}{cos ^{2} ( x )} - \frac{2 cos ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )} + \frac{sin ^{2} ( x ) \cdot 6}{cos ^{2} ( x )}

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

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

= \frac{- cos ( 2 x ) cos ^{2} ( x ) - 2 cos ^{2} ( x ) cos ^{2} ( x ) + sin ^{2} ( x ) \cdot 6}{cos ^{2} ( x )}

- cos (2 x) cos^{2} (x) - 2 cos^{2} (x) cos^{2} (x) + sin^{2} (x) \cdot 6 = - cos^{2} (x) cos (2 x) - 2 cos^{4} (x) + 6 sin^{2} (x)

- cos (2 x) cos^{2} (x) - 2 cos^{2} (x) cos^{2} (x) + sin^{2} (x) \cdot 6

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

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

Wende Exponentenregel an:

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

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

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

Addiere die Zahlen:

2 + 2 = 4

= 2 cos^{4} (x)

= - cos^{2} (x) cos (2 x) - 2 cos^{4} (x) + 6 sin^{2} (x)

= \frac{- cos ^{2} ( x ) cos ( 2 x ) - 2 cos ^{4} ( x ) + 6 sin ^{2} ( x )}{cos ^{2} ( x )}

= \frac{- cos ^{2} ( x ) cos ( 2 x ) - 2 cos ^{4} ( x ) + 6 sin ^{2} ( x )}{cos ^{2} ( x )}

\frac{- 2 cos ^{4} ( x ) + 6 sin ^{2} ( x ) - cos ( 2 x ) cos ^{2} ( x )}{cos ^{2} ( x )} = 0

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

- 2 cos^{4} (x) + 6 sin^{2} (x) - cos (2 x) cos^{2} (x) = 0

Umschreiben mit Hilfe von Trigonometrie-Identitäten

- 2 cos^{4} (x) + 6 sin^{2} (x) - cos (2 x) cos^{2} (x)

Verwende die Doppelwinkelidentität:

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

= - 2 cos^{4} (x) + 6 sin^{2} (x) - cos^{2} (x) (cos^{2} (x) - sin^{2} (x))

Vereinfache

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

- 2 cos^{4} (x) + 6 sin^{2} (x) - cos^{2} (x) (cos^{2} (x) - sin^{2} (x))

Multipliziere aus

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

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

Wende das Distributivgesetz an:

a (b - c) = ab - a c

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

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

Wende Minus-Plus Regeln an

- (- a) = a

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

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

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

Wende Exponentenregel an:

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

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

= cos^{2 + 2} (x)

Addiere die Zahlen:

2 + 2 = 4

= cos^{4} (x)

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

= - 2 cos^{4} (x) + 6 sin^{2} (x) - cos^{4} (x) + cos^{2} (x) sin^{2} (x)

Addiere gleiche Elemente:

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

= - 3 cos^{4} (x) + 6 sin^{2} (x) + cos^{2} (x) sin^{2} (x)

= - 3 cos^{4} (x) + 6 sin^{2} (x) + cos^{2} (x) sin^{2} (x)

Verwende die Pythagoreische Identität:

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

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

= - 3 cos^{4} (x) + 6 (1 - cos^{2} (x)) + cos^{2} (x) (1 - cos^{2} (x))

Vereinfache

- 3 cos^{4} (x) + 6 (1 - cos^{2} (x)) + cos^{2} (x) (1 - cos^{2} (x)) : - 4 cos^{4} (x) - 5 cos^{2} (x) + 6

- 3 cos^{4} (x) + 6 (1 - cos^{2} (x)) + cos^{2} (x) (1 - cos^{2} (x))

Multipliziere aus

6 (1 - cos^{2} (x)) : 6 - 6 cos^{2} (x)

6 (1 - cos^{2} (x))

Wende das Distributivgesetz an:

a (b - c) = ab - a c

a = 6, b = 1, c = cos^{2} (x)

= 6 \cdot 1 - 6 cos^{2} (x)

Multipliziere die Zahlen:

6 \cdot 1 = 6

= 6 - 6 cos^{2} (x)

= - 3 cos^{4} (x) + 6 - 6 cos^{2} (x) + cos^{2} (x) (1 - cos^{2} (x))

Multipliziere aus

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

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

Wende das Distributivgesetz an:

a (b - c) = ab - a c

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

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

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

Vereinfache

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

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

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

1 \cdot cos^{2} (x)

Multipliziere:

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

= cos^{2} (x)

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

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

Wende Exponentenregel an:

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

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

= cos^{2 + 2} (x)

Addiere die Zahlen:

2 + 2 = 4

= cos^{4} (x)

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

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

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

Vereinfache

- 3 cos^{4} (x) + 6 - 6 cos^{2} (x) + cos^{2} (x) - cos^{4} (x) : - 4 cos^{4} (x) - 5 cos^{2} (x) + 6

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

Fasse gleiche Terme zusammen

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

Addiere gleiche Elemente:

- 6 cos^{2} (x) + cos^{2} (x) = - 5 cos^{2} (x)

= - 3 cos^{4} (x) - 5 cos^{2} (x) - cos^{4} (x) + 6

Addiere gleiche Elemente:

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

= - 4 cos^{4} (x) - 5 cos^{2} (x) + 6

= - 4 cos^{4} (x) - 5 cos^{2} (x) + 6

= - 4 cos^{4} (x) - 5 cos^{2} (x) + 6

6 - 4 cos^{4} (x) - 5 cos^{2} (x) = 0

Löse mit Substitution

6 - 4 cos^{4} (x) - 5 cos^{2} (x) = 0

Angenommen:

cos (x) = u

6 - 4 u^{4} - 5 u^{2} = 0

6 - 4 u^{4} - 5 u^{2} = 0 : u = 2 i, u = - 2 i, u = \frac{3}{2}, u = - \frac{3}{2}

6 - 4 u^{4} - 5 u^{2} = 0

Schreibe in der Standard Form

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

- 4 u^{4} - 5 u^{2} + 6 = 0

Schreibe die Gleichung um mit

v = u^{2}

und

v^{2} = u^{4}

- 4 v^{2} - 5 v + 6 = 0

Löse

- 4 v^{2} - 5 v + 6 = 0 : v = - 2, v = \frac{3}{4}

- 4 v^{2} - 5 v + 6 = 0

Löse mit der quadratischen Formel

- 4 v^{2} - 5 v + 6 = 0

Quadratische Formel für Gliechungen:

Für

a = - 4, b = - 5, c = 6

v_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 4 ) \cdot 6}{2 ( - 4 )}

v_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 4 ) \cdot 6}{2 ( - 4 )}

(- 5)^{2} - 4 (- 4) \cdot 6 = 11

(- 5)^{2} - 4 (- 4) \cdot 6

Wende Regel an

- (- a) = a

= (- 5)^{2} + 4 \cdot 4 \cdot 6

Wende Exponentenregel an:

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

wenn

n

gerade ist

(- 5)^{2} = 5^{2}

= 5^{2} + 4 \cdot 4 \cdot 6

Multipliziere die Zahlen:

4 \cdot 4 \cdot 6 = 96

= 5^{2} + 96

5^{2} = 25

= 25 + 96

Addiere die Zahlen:

25 + 96 = 121

= 121

Faktorisiere die Zahl:

121 = 1 1^{2}

= 1 1^{2}

Wende Radikal Regel an:

n a^{n} = a

1 1^{2} = 11

= 11

v_{1, 2} = \frac{- ( - 5 ) \pm 11}{2 ( - 4 )}

Trenne die Lösungen

v_{1} = \frac{- ( - 5 ) + 11}{2 ( - 4 )}, v_{2} = \frac{- ( - 5 ) - 11}{2 ( - 4 )}

v = \frac{- ( - 5 ) + 11}{2 ( - 4 )} : - 2

\frac{- ( - 5 ) + 11}{2 ( - 4 )}

Entferne die Klammern:

(- a) = - a, - (- a) = a

= \frac{5 + 11}{- 2 \cdot 4}

Addiere die Zahlen:

5 + 11 = 16

= \frac{16}{- 2 \cdot 4}

Multipliziere die Zahlen:

2 \cdot 4 = 8

= \frac{16}{- 8}

Wende Bruchregel an:

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

= - \frac{16}{8}

Teile die Zahlen:

\frac{16}{8} = 2

= - 2

v = \frac{- ( - 5 ) - 11}{2 ( - 4 )} : \frac{3}{4}

\frac{- ( - 5 ) - 11}{2 ( - 4 )}

Entferne die Klammern:

(- a) = - a, - (- a) = a

= \frac{5 - 11}{- 2 \cdot 4}

Subtrahiere die Zahlen:

5 - 11 = - 6

= \frac{- 6}{- 2 \cdot 4}

Multipliziere die Zahlen:

2 \cdot 4 = 8

= \frac{- 6}{- 8}

Wende Bruchregel an:

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

= \frac{6}{8}

Streiche die gemeinsamen Faktoren:

2

= \frac{3}{4}

Die Lösungen für die quadratische Gleichung sind:

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

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

Setze

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

löse für

u

Löse

u^{2} = - 2 : u = 2 i, u = - 2 i

u^{2} = - 2

Für

x^{2} = f (a)

sind die Lösungen

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

u = - 2, u = - - 2

Vereinfache

- 2 : 2 i

- 2

Wende Radikal Regel an:

- a = - 1 a

- 2 = - 1 2

= - 1 2

Wende imaginäre Zahlenregel an:

- 1 = i

= 2 i

Vereinfache

- - 2 : - 2 i

- - 2

Vereinfache

- 2 : 2 i

- 2

Wende Radikal Regel an:

- a = - 1 a

- 2 = - 1 2

= - 1 2

Wende imaginäre Zahlenregel an:

- 1 = i

= 2 i

= - 2 i

u = 2 i, u = - 2 i

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}

Die Lösungen sind

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

Setze in

u = cos (x)

ein

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

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

cos (x) = 2 i :

Keine Lösung

cos (x) = 2 i

Keine L \overset{o}{¨} sung

cos (x) = - 2 i :

Keine Lösung

cos (x) = - 2 i

Keine L \overset{o}{¨} sung

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

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

Graph

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