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cos^2(111)+cos^2(69.3)+cos^2(x)=1

Lösung

cos^{2} (11 1^{\circ}) + cos^{2} (69. 3^{\circ}) + cos^{2} (x) = 1

x = 0.52748 \dots + 36 0^{\circ} n, x = 36 0^{\circ} - 0.52748 \dots + 36 0^{\circ} n, x = 2.61410 \dots + 36 0^{\circ} n, x = - 2.61410 \dots + 36 0^{\circ} n

Radianten

x = 0.52748 \dots + 2 πn, x = 2 π - 0.52748 \dots + 2 πn, x = 2.61410 \dots + 2 πn, x = - 2.61410 \dots + 2 πn

Schritte zur Lösung

cos^{2} (11 1^{\circ}) + cos^{2} (69. 3^{\circ}) + cos^{2} (x) = 1

Löse mit Substitution

cos^{2} (11 1^{\circ}) + cos^{2} (69. 3^{\circ}) + cos^{2} (x) = 1

Angenommen:

cos (x) = u

cos^{2} (11 1^{\circ}) + cos^{2} (69. 3^{\circ}) + u^{2} = 1

cos^{2} (11 1^{\circ}) + cos^{2} (69. 3^{\circ}) + u^{2} = 1 : u = 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ}), u = - 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

cos^{2} (11 1^{\circ}) + cos^{2} (69. 3^{\circ}) + u^{2} = 1

Verschiebe

cos^{2} (11 1^{\circ})

auf die rechte Seite

cos^{2} (11 1^{\circ}) + cos^{2} (69. 3^{\circ}) + u^{2} = 1

Subtrahiere

cos^{2} (11 1^{\circ})

von beiden Seiten

cos^{2} (11 1^{\circ}) + cos^{2} (69. 3^{\circ}) + u^{2} - cos^{2} (11 1^{\circ}) = 1 - cos^{2} (11 1^{\circ})

Vereinfache

cos^{2} (69. 3^{\circ}) + u^{2} = 1 - cos^{2} (11 1^{\circ})

cos^{2} (69. 3^{\circ}) + u^{2} = 1 - cos^{2} (11 1^{\circ})

Verschiebe

cos^{2} (69. 3^{\circ})

auf die rechte Seite

cos^{2} (69. 3^{\circ}) + u^{2} = 1 - cos^{2} (11 1^{\circ})

Subtrahiere

cos^{2} (69. 3^{\circ})

von beiden Seiten

cos^{2} (69. 3^{\circ}) + u^{2} - cos^{2} (69. 3^{\circ}) = 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

Vereinfache

u^{2} = 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

u^{2} = 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

Für

x^{2} = f (a)

sind die Lösungen

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

u = 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ}), u = - 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

Setze in

u = cos (x)

ein

cos (x) = 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ}), cos (x) = - 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

cos (x) = 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ}), cos (x) = - 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

cos (x) = 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ}) : x = arccos (1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n

cos (x) = 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

Wende die Eigenschaften der Trigonometrie an

cos (x) = 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

Allgemeine Lösung für

cos (x) = 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

cos (x) = a \Rightarrow x = arccos (a) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (a) + 36 0^{\circ} n

x = arccos (1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n

x = arccos (1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n

cos (x) = - 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ}) : x = arccos (- 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n, x = - arccos (- 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n

cos (x) = - 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

Wende die Eigenschaften der Trigonometrie an

cos (x) = - 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

Allgemeine Lösung für

cos (x) = - 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})

cos (x) = - a \Rightarrow x = arccos (- a) + 36 0^{\circ} n, x = - arccos (- a) + 36 0^{\circ} n

x = arccos (- 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n, x = - arccos (- 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n

x = arccos (- 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n, x = - arccos (- 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n

Kombiniere alle Lösungen

x = arccos (1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n, x = arccos (- 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n, x = - arccos (- 1 - cos^{2} (11 1^{\circ}) - cos^{2} (69. 3^{\circ})) + 36 0^{\circ} n

Zeige Lösungen in Dezimalform

x = 0.52748 \dots + 36 0^{\circ} n, x = 36 0^{\circ} - 0.52748 \dots + 36 0^{\circ} n, x = 2.61410 \dots + 36 0^{\circ} n, x = - 2.61410 \dots + 36 0^{\circ} n

4 + 2 cos (2 x) = 6 cos (x)

cos (x) - tan (x) cos (x) = 0

cot (2 θ) = \frac{3}{4}

tan (x) = - \frac{4}{33}

2 cot^{2} (x) - 11 csc (x) = - 14