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العربية

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

Lösung

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

x = π + 2 πn

Grad

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

Schritte zur Lösung

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

Subtrahiere

cos (2 x)

von beiden Seiten

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Verwende die Doppelwinkelidentität:

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

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

Vereinfache

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

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

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

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

Setze Klammern

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

Wende Minus-Plus Regeln an

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

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

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

Addiere gleiche Elemente:

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

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

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

Verwende die Pythagoreische Identität:

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

= cos (x) + 1

cos (x) + 1 = 0

Verschiebe

1

auf die rechte Seite

cos (x) + 1 = 0

Subtrahiere

1

von beiden Seiten

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

Vereinfache

cos (x) = - 1

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 (θ) = \frac{2}{5}, cot (θ) < 0

so l v e f or b, s (t) = tan (b t^{2})

tan (x) = \frac{80}{90}

4 cos^{2} (x) - 11 cos (x) + 6 = 0

tanh (x) = 0.99