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Beliebt Trigonometrie >

3sin^2(x)+8cos^2(x)=6

Lösung

3 sin^{2} (x) + 8 cos^{2} (x) = 6

Lösung

x = 0.68471 \dots + 2 πn, x = π - 0.68471 \dots + 2 πn, x = - 0.68471 \dots + 2 πn, x = π + 0.68471 \dots + 2 πn

Grad

x = 39.23152 \dots^{\circ} + 36 0^{\circ} n, x = 140.76847 \dots^{\circ} + 36 0^{\circ} n, x = - 39.23152 \dots^{\circ} + 36 0^{\circ} n, x = 219.23152 \dots^{\circ} + 36 0^{\circ} n

Schritte zur Lösung

3 sin^{2} (x) + 8 cos^{2} (x) = 6

Subtrahiere

6

von beiden Seiten

3 sin^{2} (x) + 8 cos^{2} (x) - 6 = 0

Umschreiben mit Hilfe von Trigonometrie-Identitäten

- 6 + 3 sin^{2} (x) + 8 cos^{2} (x)

Verwende die Pythagoreische Identität:

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

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

= - 6 + 3 sin^{2} (x) + 8 (1 - sin^{2} (x))

Vereinfache

- 6 + 3 sin^{2} (x) + 8 (1 - sin^{2} (x)) : - 5 sin^{2} (x) + 2

- 6 + 3 sin^{2} (x) + 8 (1 - sin^{2} (x))

Multipliziere aus

8 (1 - sin^{2} (x)) : 8 - 8 sin^{2} (x)

8 (1 - sin^{2} (x))

Wende das Distributivgesetz an:

a (b - c) = ab - a c

a = 8, b = 1, c = sin^{2} (x)

= 8 \cdot 1 - 8 sin^{2} (x)

Multipliziere die Zahlen:

8 \cdot 1 = 8

= 8 - 8 sin^{2} (x)

= - 6 + 3 sin^{2} (x) + 8 - 8 sin^{2} (x)

Vereinfache

- 6 + 3 sin^{2} (x) + 8 - 8 sin^{2} (x) : - 5 sin^{2} (x) + 2

- 6 + 3 sin^{2} (x) + 8 - 8 sin^{2} (x)

Fasse gleiche Terme zusammen

= 3 sin^{2} (x) - 8 sin^{2} (x) - 6 + 8

Addiere gleiche Elemente:

3 sin^{2} (x) - 8 sin^{2} (x) = - 5 sin^{2} (x)

= - 5 sin^{2} (x) - 6 + 8

Addiere/Subtrahiere die Zahlen:

- 6 + 8 = 2

= - 5 sin^{2} (x) + 2

= - 5 sin^{2} (x) + 2

= - 5 sin^{2} (x) + 2

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

Löse mit Substitution

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

Angenommen:

sin (x) = u

2 - 5 u^{2} = 0

2 - 5 u^{2} = 0 : u = \frac{2}{5}, u = - \frac{2}{5}

2 - 5 u^{2} = 0

Verschiebe

2

auf die rechte Seite

2 - 5 u^{2} = 0

Subtrahiere

2

von beiden Seiten

2 - 5 u^{2} - 2 = 0 - 2

Vereinfache

- 5 u^{2} = - 2

- 5 u^{2} = - 2

Teile beide Seiten durch

- 5

- 5 u^{2} = - 2

Teile beide Seiten durch

- 5

\frac{- 5 u ^{2}}{- 5} = \frac{- 2}{- 5}

Vereinfache

u^{2} = \frac{2}{5}

u^{2} = \frac{2}{5}

Für

x^{2} = f (a)

sind die Lösungen

x = f (a), - f (a)

u = \frac{2}{5}, u = - \frac{2}{5}

Setze in

u = sin (x)

ein

sin (x) = \frac{2}{5}, sin (x) = - \frac{2}{5}

sin (x) = \frac{2}{5}, sin (x) = - \frac{2}{5}

sin (x) = \frac{2}{5} : x = arcsin (\frac{2}{5}) + 2 πn, x = π - arcsin (\frac{2}{5}) + 2 πn

sin (x) = \frac{2}{5}

Wende die Eigenschaften der Trigonometrie an

sin (x) = \frac{2}{5}

Allgemeine Lösung für

sin (x) = \frac{2}{5}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{2}{5}) + 2 πn, x = π - arcsin (\frac{2}{5}) + 2 πn

x = arcsin (\frac{2}{5}) + 2 πn, x = π - arcsin (\frac{2}{5}) + 2 πn

sin (x) = - \frac{2}{5} : x = arcsin (- \frac{2}{5}) + 2 πn, x = π + arcsin (\frac{2}{5}) + 2 πn

sin (x) = - \frac{2}{5}

Wende die Eigenschaften der Trigonometrie an

sin (x) = - \frac{2}{5}

Allgemeine Lösung für

sin (x) = - \frac{2}{5}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- \frac{2}{5}) + 2 πn, x = π + arcsin (\frac{2}{5}) + 2 πn

x = arcsin (- \frac{2}{5}) + 2 πn, x = π + arcsin (\frac{2}{5}) + 2 πn

Kombiniere alle Lösungen

x = arcsin (\frac{2}{5}) + 2 πn, x = π - arcsin (\frac{2}{5}) + 2 πn, x = arcsin (- \frac{2}{5}) + 2 πn, x = π + arcsin (\frac{2}{5}) + 2 πn

Zeige Lösungen in Dezimalform

x = 0.68471 \dots + 2 πn, x = π - 0.68471 \dots + 2 πn, x = - 0.68471 \dots + 2 πn, x = π + 0.68471 \dots + 2 πn

Graph

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