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cos^2(x)=(sqrt(3))/2

Lösung

cos^{2} (x) = \frac{3}{2}

x = 0.37473 \dots + 2 πn, x = 2 π - 0.37473 \dots + 2 πn, x = 2.76685 \dots + 2 πn, x = - 2.76685 \dots + 2 πn

Grad

x = 21.47070 \dots^{\circ} + 36 0^{\circ} n, x = 338.52929 \dots^{\circ} + 36 0^{\circ} n, x = 158.52929 \dots^{\circ} + 36 0^{\circ} n, x = - 158.52929 \dots^{\circ} + 36 0^{\circ} n

Schritte zur Lösung

cos^{2} (x) = \frac{3}{2}

Löse mit Substitution

cos^{2} (x) = \frac{3}{2}

Angenommen:

cos (x) = u

u^{2} = \frac{3}{2}

u^{2} = \frac{3}{2} : u = \frac{3}{2}, u = - \frac{3}{2}

u^{2} = \frac{3}{2}

Für

x^{2} = f (a)

sind die Lösungen

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

u = \frac{3}{2}, u = - \frac{3}{2}

Setze in

u = cos (x)

ein

cos (x) = \frac{3}{2}, cos (x) = - \frac{3}{2}

cos (x) = \frac{3}{2}, cos (x) = - \frac{3}{2}

cos (x) = \frac{3}{2} : x = arccos \frac{3}{2} + 2 πn, x = 2 π - arccos \frac{3}{2} + 2 πn

cos (x) = \frac{3}{2}

Wende die Eigenschaften der Trigonometrie an

cos (x) = \frac{3}{2}

Allgemeine Lösung für

cos (x) = \frac{3}{2}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos \frac{3}{2} + 2 πn, x = 2 π - arccos \frac{3}{2} + 2 πn

x = arccos \frac{3}{2} + 2 πn, x = 2 π - arccos \frac{3}{2} + 2 πn

cos (x) = - \frac{3}{2} : x = arccos - \frac{3}{2} + 2 πn, x = - arccos - \frac{3}{2} + 2 πn

cos (x) = - \frac{3}{2}

Wende die Eigenschaften der Trigonometrie an

cos (x) = - \frac{3}{2}

Allgemeine Lösung für

cos (x) = - \frac{3}{2}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos - \frac{3}{2} + 2 πn, x = - arccos - \frac{3}{2} + 2 πn

x = arccos - \frac{3}{2} + 2 πn, x = - arccos - \frac{3}{2} + 2 πn

Kombiniere alle Lösungen

x = arccos \frac{3}{2} + 2 πn, x = 2 π - arccos \frac{3}{2} + 2 πn, x = arccos - \frac{3}{2} + 2 πn, x = - arccos - \frac{3}{2} + 2 πn

Zeige Lösungen in Dezimalform

x = 0.37473 \dots + 2 πn, x = 2 π - 0.37473 \dots + 2 πn, x = 2.76685 \dots + 2 πn, x = - 2.76685 \dots + 2 πn

sin (\frac{x}{2}) = \frac{3}{5}

16 cos^{2} (x) - 8 = 0

2 sin (4 θ) = 0

so l v e f or θ, 4 sin^{2} (\frac{θ}{2}) + 8 cos (\frac{θ}{2}) - 7 = 0

2 sin^{2} (x) - sin (x) = 0, 0 \leq x \leq 2 π