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Beliebt Trigonometrie >

cos^4(x)-sin^4(x)=1

Lösung

cos^{4} (x) - sin^{4} (x) = 1

Lösung

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

cos^{4} (x) - sin^{4} (x) = 1

Subtrahiere

1

von beiden Seiten

cos^{4} (x) - sin^{4} (x) - 1 = 0

Wende Exponentenregel an:

a^{b} = a^{2} a^{b - 2}

- 1 - sin^{4} (x) + cos^{2} (x) cos^{2} (x) = 0

Umschreiben mit Hilfe von Trigonometrie-Identitäten

- 1 - sin^{4} (x) + cos^{2} (x) cos^{2} (x)

Verwende die Pythagoreische Identität:

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

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

= - 1 - sin^{4} (x) + (1 - sin^{2} (x)) (1 - sin^{2} (x))

Vereinfache

- 1 - sin^{4} (x) + (1 - sin^{2} (x)) (1 - sin^{2} (x)) : - 2 sin^{2} (x)

- 1 - sin^{4} (x) + (1 - sin^{2} (x)) (1 - sin^{2} (x))

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

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

Wende Exponentenregel an:

a^{b} \cdot a^{c} = a^{b + c}

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

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

Addiere die Zahlen:

1 + 1 = 2

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

= - 1 - sin^{4} (x) + (- sin^{2} (x) + 1)^{2}

(1 - sin^{2} (x))^{2} : 1 - 2 sin^{2} (x) + sin^{4} (x)

Wende Formel für perfekte quadratische Gleichungen an:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

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

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

Vereinfache

1^{2} - 2 \cdot 1 \cdot sin^{2} (x) + (sin^{2} (x))^{2} : 1 - 2 sin^{2} (x) + sin^{4} (x)

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

Wende Regel an

1^{a} = 1

1^{2} = 1

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

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

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

Multipliziere die Zahlen:

2 \cdot 1 = 2

= 2 sin^{2} (x)

(sin^{2} (x))^{2} = sin^{4} (x)

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

Wende Exponentenregel an:

(a^{b})^{c} = a^{b c}

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

Multipliziere die Zahlen:

2 \cdot 2 = 4

= sin^{4} (x)

= 1 - 2 sin^{2} (x) + sin^{4} (x)

= 1 - 2 sin^{2} (x) + sin^{4} (x)

= - 1 - sin^{4} (x) + 1 - 2 sin^{2} (x) + sin^{4} (x)

Vereinfache

- 1 - sin^{4} (x) + 1 - 2 sin^{2} (x) + sin^{4} (x) : - 2 sin^{2} (x)

- 1 - sin^{4} (x) + 1 - 2 sin^{2} (x) + sin^{4} (x)

Fasse gleiche Terme zusammen

= - sin^{4} (x) - 2 sin^{2} (x) + sin^{4} (x) - 1 + 1

Addiere gleiche Elemente:

- sin^{4} (x) + sin^{4} (x) = 0

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

- 1 + 1 = 0

= - 2 sin^{2} (x)

= - 2 sin^{2} (x)

= - 2 sin^{2} (x)

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

Teile beide Seiten durch

- 2

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

Teile beide Seiten durch

- 2

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

Teile beide Seiten durch

- 2

\frac{- 2 sin ^{2} ( x )}{- 2} = \frac{0}{- 2}

Vereinfache

sin^{2} (x) = 0

sin^{2} (x) = 0

Wende Regel an

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

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

Graph

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