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Beliebt Trigonometrie >

sec^2(x)tan^2(x)+3sec^2(x)-2tan^2(x)=3

Lösung

sec^{2} (x) tan^{2} (x) + 3 sec^{2} (x) - 2 tan^{2} (x) = 3

Lösung

x = 2 πn, x = π + 2 πn

Grad

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

Schritte zur Lösung

sec^{2} (x) tan^{2} (x) + 3 sec^{2} (x) - 2 tan^{2} (x) = 3

Subtrahiere

3

von beiden Seiten

sec^{2} (x) tan^{2} (x) + 3 sec^{2} (x) - 2 tan^{2} (x) - 3 = 0

Umschreiben mit Hilfe von Trigonometrie-Identitäten

- 3 - 2 tan^{2} (x) + 3 sec^{2} (x) + sec^{2} (x) tan^{2} (x)

Verwende die Pythagoreische Identität:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

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

Vereinfache

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

- 3 - 2 (sec^{2} (x) - 1) + 3 sec^{2} (x) + sec^{2} (x) (sec^{2} (x) - 1)

Multipliziere aus

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

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

Wende das Distributivgesetz an:

a (b - c) = ab - a c

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

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

Wende Minus-Plus Regeln an

- (- a) = a

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

Multipliziere die Zahlen:

2 \cdot 1 = 2

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

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

Multipliziere aus

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

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

Wende das Distributivgesetz an:

a (b - c) = ab - a c

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

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

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

Vereinfache

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

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

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

sec^{2} (x) sec^{2} (x)

Wende Exponentenregel an:

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

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

= sec^{2 + 2} (x)

Addiere die Zahlen:

2 + 2 = 4

= sec^{4} (x)

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

1 \cdot sec^{2} (x)

Multipliziere:

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

= sec^{2} (x)

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

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

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

Vereinfache

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

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

Fasse gleiche Terme zusammen

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

Addiere gleiche Elemente:

- 2 sec^{2} (x) + 3 sec^{2} (x) - sec^{2} (x) = 0

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

Addiere/Subtrahiere die Zahlen:

- 3 + 2 = - 1

= sec^{4} (x) - 1

= sec^{4} (x) - 1

= sec^{4} (x) - 1

- 1 + sec^{4} (x) = 0

Löse mit Substitution

- 1 + sec^{4} (x) = 0

Angenommen:

sec (x) = u

- 1 + u^{4} = 0

- 1 + u^{4} = 0 : u = 1, u = - 1, u = i, u = - i

- 1 + u^{4} = 0

Verschiebe

1

auf die rechte Seite

- 1 + u^{4} = 0

Füge

1

zu beiden Seiten hinzu

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

Vereinfache

u^{4} = 1

u^{4} = 1

Schreibe die Gleichung um mit

v = u^{2}

und

v^{2} = u^{4}

v^{2} = 1

Löse

v^{2} = 1 : v = 1, v = - 1

v^{2} = 1

Für

(g (x))^{2} = f (a)

sind die Lösungen

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

v = 1, v = - 1

v = 1, v = - 1

Setze

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

löse für

u

Löse

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

u^{2} = 1

Wende Regel an

1 = 1

u^{2} = 1

Für

x^{2} = f (a)

sind die Lösungen

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

u = 1, u = - 1

1 = 1

1

Wende Regel an

1 = 1

= 1

- 1 = - 1

- 1

Wende Regel an

1 = 1

= - 1

u = 1, u = - 1

Löse

u^{2} = - 1 : u = i, u = - i

u^{2} = - 1

Wende Regel an

1 = 1

u^{2} = - 1

Für

x^{2} = f (a)

sind die Lösungen

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

u = - 1, u = - - 1

Vereinfache

- 1 : i

- 1

Wende imaginäre Zahlenregel an:

- 1 = i

= i

Vereinfache

- - 1 : - i

- - 1

Wende imaginäre Zahlenregel an:

- 1 = i

= - i

u = i, u = - i

Die Lösungen sind

u = 1, u = - 1, u = i, u = - i

Setze in

u = sec (x)

ein

sec (x) = 1, sec (x) = - 1, sec (x) = i, sec (x) = - i

sec (x) = 1, sec (x) = - 1, sec (x) = i, sec (x) = - i

sec (x) = 1 : x = 2 πn

sec (x) = 1

Allgemeine Lösung für

sec (x) = 1

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

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

sec (x) = - 1 : x = π + 2 πn

sec (x) = - 1

Allgemeine Lösung für

sec (x) = - 1

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = π + 2 πn

x = π + 2 πn

sec (x) = i :

Keine Lösung

sec (x) = i

Keine L \overset{o}{¨} sung

sec (x) = - i :

Keine Lösung

sec (x) = - i

Keine L \overset{o}{¨} sung

Kombiniere alle Lösungen

x = 2 πn, x = π + 2 πn

Graph

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