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العربية

(4cos(x)-2sin(x))^2+12sin^2(x)=16

Lösung

(4 cos (x) - 2 sin (x))^{2} + 12 sin^{2} (x) = 16

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

Grad

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

Schritte zur Lösung

(4 cos (x) - 2 sin (x))^{2} + 12 sin^{2} (x) = 16

Subtrahiere

16

von beiden Seiten

16 cos^{2} (x) - 16 cos (x) sin (x) + 16 sin^{2} (x) - 16 = 0

Faktorisiere

16 cos^{2} (x) - 16 cos (x) sin (x) + 16 sin^{2} (x) - 16 : 16 (cos^{2} (x) - sin (x) cos (x) + sin^{2} (x) - 1)

16 cos^{2} (x) - 16 cos (x) sin (x) + 16 sin^{2} (x) - 16

Klammere gleiche Terme aus

16

= 16 (cos^{2} (x) - sin (x) cos (x) + sin^{2} (x) - 1)

16 (cos^{2} (x) - sin (x) cos (x) + sin^{2} (x) - 1) = 0

Umschreiben mit Hilfe von Trigonometrie-Identitäten

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

Verwende die Pythagoreische Identität:

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

= 16 (- 1 - cos (x) sin (x) + 1)

Vereinfache

16 (- 1 - cos (x) sin (x) + 1) : - 16 cos (x) sin (x)

16 (- 1 - cos (x) sin (x) + 1)

Vereinfache

- 1 - cos (x) sin (x) + 1 : - cos (x) sin (x)

- 1 - cos (x) sin (x) + 1

Fasse gleiche Terme zusammen

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

- 1 + 1 = 0

= - cos (x) sin (x)

= 16 (- cos (x) sin (x))

Entferne die Klammern:

(- a) = - a

= - 16 cos (x) sin (x)

= - 16 cos (x) sin (x)

- 16 cos (x) sin (x) = 0

Löse jeden Teil einzeln

cos (x) = 0 or sin (x) = 0

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

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

Kombiniere alle Lösungen

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