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2cos^2(x)-sqrt(3)*sin^2(x)-2=0

Lösung

2 cos^{2} (x) - 3 \cdot sin^{2} (x) - 2 = 0

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

2 cos^{2} (x) - 3 sin^{2} (x) - 2 = 0

Umschreiben mit Hilfe von Trigonometrie-Identitäten

- 2 + 2 cos^{2} (x) - sin^{2} (x) 3

Verwende die Pythagoreische Identität:

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

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

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

Vereinfache

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

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

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

Multipliziere aus

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

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

Wende das Distributivgesetz an:

a (b - c) = ab - a c

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

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

Multipliziere die Zahlen:

2 \cdot 1 = 2

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

= - 2 + 2 - 2 sin^{2} (x) - sin^{2} (x) 3

- 2 + 2 = 0

= - 2 sin^{2} (x) - 3 sin^{2} (x)

= - 2 sin^{2} (x) - 3 sin^{2} (x)

- 2 sin^{2} (x) - sin^{2} (x) 3 = 0

Löse mit Substitution

- 2 sin^{2} (x) - sin^{2} (x) 3 = 0

Angenommen:

sin (x) = u

- 2 u^{2} - u^{2} 3 = 0

- 2 u^{2} - u^{2} 3 = 0 : u = 0

- 2 u^{2} - u^{2} 3 = 0

Faktorisiere

- 2 u^{2} - u^{2} 3 : - (2 + 3) u^{2}

- 2 u^{2} - u^{2} 3

Klammere gleiche Terme aus

u^{2}

= - u^{2} (2 + 3)

- (2 + 3) u^{2} = 0

Teile beide Seiten durch

- 2 - 3

- (2 + 3) u^{2} = 0

Teile beide Seiten durch

- 2 - 3

- (2 + 3) u^{2} = 0

Teile beide Seiten durch

- 2 - 3

\frac{- ( 2 + 3 ) u ^{2}}{- 2 - 3} = \frac{0}{- 2 - 3}

Vereinfache

u^{2} = 0

u^{2} = 0

Wende Regel an

x^{n} = 0 \Rightarrow x = 0

u = 0

Setze in

u = sin (x)

ein

sin (x) = 0

sin (x) = 0

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 = 2 πn, x = π + 2 πn

so l v e f or x, lo g_{10} (y) = arctan (x) + c

2 sin (x) = \frac{( 4 sin ( x ) - cos ( x ) )}{2}

cos (x + 60) \cdot cos (x - 60) = \frac{1}{2}

\frac{cos ( 2 x + 9 )}{3} = \frac{1}{2}

sin^{6} (x) - cos^{6} (x) = 0