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العربية

(-5cos(x)-6sin(x))^2-11sin^2(x)=25

Lösung

(- 5 cos (x) - 6 sin (x))^{2} - 11 sin^{2} (x) = 25

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

(- 5 cos (x) - 6 sin (x))^{2} - 11 sin^{2} (x) = 25

Subtrahiere

25

von beiden Seiten

25 cos^{2} (x) + 60 cos (x) sin (x) + 25 sin^{2} (x) - 25 = 0

Umschreiben mit Hilfe von Trigonometrie-Identitäten

- 25 + 25 cos^{2} (x) + 25 sin^{2} (x) + 60 cos (x) sin (x)

Verwende die Pythagoreische Identität:

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

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

= - 25 + 25 (1 - sin^{2} (x)) + 25 sin^{2} (x) + 60 cos (x) sin (x)

Vereinfache

- 25 + 25 (1 - sin^{2} (x)) + 25 sin^{2} (x) + 60 cos (x) sin (x) : 60 cos (x) sin (x)

- 25 + 25 (1 - sin^{2} (x)) + 25 sin^{2} (x) + 60 cos (x) sin (x)

Multipliziere aus

25 (1 - sin^{2} (x)) : 25 - 25 sin^{2} (x)

25 (1 - sin^{2} (x))

Wende das Distributivgesetz an:

a (b - c) = ab - a c

a = 25, b = 1, c = sin^{2} (x)

= 25 \cdot 1 - 25 sin^{2} (x)

Multipliziere die Zahlen:

25 \cdot 1 = 25

= 25 - 25 sin^{2} (x)

= - 25 + 25 - 25 sin^{2} (x) + 25 sin^{2} (x) + 60 cos (x) sin (x)

Vereinfache

- 25 + 25 - 25 sin^{2} (x) + 25 sin^{2} (x) + 60 cos (x) sin (x) : 60 cos (x) sin (x)

- 25 + 25 - 25 sin^{2} (x) + 25 sin^{2} (x) + 60 cos (x) sin (x)

Addiere gleiche Elemente:

- 25 sin^{2} (x) + 25 sin^{2} (x) = 0

= - 25 + 25 + 60 cos (x) sin (x)

- 25 + 25 = 0

= 60 cos (x) sin (x)

= 60 cos (x) sin (x)

= 60 cos (x) sin (x)

60 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

2 sin^{2} (x) - 9 sin (x) - 5 = 0

2 sin^{2} (x) = 5 cos (x) - 1

so l v e f or x, sec (x + 1 0^{\circ}) = 5.759

tan (x) cot (x) + 2 cot (x) = 0

3 cos (x) = 3 sin (x)