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العربية

sin(2x)+cos(x)=0,x<= 2pi,0

Lösung

sin (2 x) + cos (x) = 0, x \leq 2 π, 0

x = \frac{π}{2}, x = \frac{3 π}{2}, x = - \frac{3 π}{2}, x = - \frac{π}{2}, x = - \frac{7 π}{2}, x = - \frac{5 π}{2}, x = - \frac{11 π}{2}, x = - \frac{9 π}{2}, x = - \frac{15 π}{2}, x = - \frac{13 π}{2}, x = \frac{7 π}{6}, x = \frac{11 π}{6}, x = - \frac{5 π}{6}, x = - \frac{π}{6}, x = - \frac{17 π}{6}, x = - \frac{13 π}{6}, x = - \frac{29 π}{6}, x = - \frac{25 π}{6}, x = - \frac{41 π}{6}, x = - \frac{37 π}{6}

Grad

x = 9 0^{\circ}, x = 27 0^{\circ}, x = - 27 0^{\circ}, x = - 9 0^{\circ}, x = - 63 0^{\circ}, x = - 45 0^{\circ}, x = - 99 0^{\circ}, x = - 81 0^{\circ}, x = - 135 0^{\circ}, x = - 117 0^{\circ}, x = 21 0^{\circ}, x = 33 0^{\circ}, x = - 15 0^{\circ}, x = - 3 0^{\circ}, x = - 51 0^{\circ}, x = - 39 0^{\circ}, x = - 87 0^{\circ}, x = - 75 0^{\circ}, x = - 123 0^{\circ}, x = - 111 0^{\circ}

Schritte zur Lösung

sin (2 x) + cos (x) = 0, x \leq 2 π, 0

Umschreiben mit Hilfe von Trigonometrie-Identitäten

cos (x) + sin (2 x)

Verwende die Doppelwinkelidentität:

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

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

cos (x) + 2 cos (x) sin (x) = 0

Faktorisiere

cos (x) + 2 cos (x) sin (x) : cos (x) (2 sin (x) + 1)

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

Klammere gleiche Terme aus

cos (x)

= cos (x) (1 + 2 sin (x))

cos (x) (2 sin (x) + 1) = 0

Löse jeden Teil einzeln

cos (x) = 0 or 2 sin (x) + 1 = 0

cos (x) = 0, x \leq 2 π : x = \frac{π}{2}, x = \frac{3 π}{2}, x = - \frac{3 π}{2}, x = - \frac{π}{2}, x = - \frac{7 π}{2}, x = - \frac{5 π}{2}, x = - \frac{11 π}{2}, x = - \frac{9 π}{2}, x = - \frac{15 π}{2}, x = - \frac{13 π}{2}

cos (x) = 0, x \leq 2 π

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

Lösungen für den Bereich

x \leq 2 π

x = \frac{π}{2}, x = \frac{3 π}{2}, x = - \frac{3 π}{2}, x = - \frac{π}{2}, x = - \frac{7 π}{2}, x = - \frac{5 π}{2}, x = - \frac{11 π}{2}, x = - \frac{9 π}{2}, x = - \frac{15 π}{2}, x = - \frac{13 π}{2}

2 sin (x) + 1 = 0, x \leq 2 π : x = \frac{7 π}{6}, x = \frac{11 π}{6}, x = - \frac{5 π}{6}, x = - \frac{π}{6}, x = - \frac{17 π}{6}, x = - \frac{13 π}{6}, x = - \frac{29 π}{6}, x = - \frac{25 π}{6}, x = - \frac{41 π}{6}, x = - \frac{37 π}{6}

2 sin (x) + 1 = 0, x \leq 2 π

Verschiebe

1

auf die rechte Seite

2 sin (x) + 1 = 0

Subtrahiere

1

von beiden Seiten

2 sin (x) + 1 - 1 = 0 - 1

Vereinfache

2 sin (x) = - 1

2 sin (x) = - 1

Teile beide Seiten durch

2

2 sin (x) = - 1