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Beliebt Trigonometrie >

sqrt(1-cos^2(x))=0.75sqrt(1-cos^2(y))

Lösung

1 - cos^{2} (x) = 0.75 1 - cos^{2} (y)

Lösung

x = arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn

Schritte zur Lösung

1 - cos^{2} (x) = 0.75 1 - cos^{2} (y)

Löse mit Substitution

1 - cos^{2} (x) = 0.75 1 - cos^{2} (y)

Angenommen:

cos (x) = u

1 - u^{2} = 0.75 1 - cos^{2} (y)

1 - u^{2} = 0.75 1 - cos^{2} (y) : u = 0.5625 cos^{2} (y) + 0.4375, u = - 0.5625 cos^{2} (y) + 0.4375

1 - u^{2} = 0.75 1 - cos^{2} (y)

Quadriere beide Seiten:

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y)

1 - u^{2} = 0.75 1 - cos^{2} (y)

(1 - u^{2})^{2} = (0.75 1 - cos^{2} (y))^{2}

Schreibe

(1 - u^{2})^{2}

um:

1 - u^{2}

(1 - u^{2})^{2}

Wende Radikal Regel an:

a = a^{\frac{1}{2}}

= ((1 - u^{2})^{\frac{1}{2}})^{2}

Wende Exponentenregel an:

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

= (1 - u^{2})^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multipliziere Brüche:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Streiche die gemeinsamen Faktoren:

2

= 1

= 1 - u^{2}

Schreibe

(0.75 1 - cos^{2} (y))^{2}

um:

0.5625 - 0.5625 cos^{2} (y)

(0.75 1 - cos^{2} (y))^{2}

Wende Exponentenregel an:

(a \cdot b)^{n} = a^{n} b^{n}

= 0.7 5^{2} (- cos^{2} (y) + 1)^{2}

(1 - cos^{2} (y))^{2} : 1 - cos^{2} (y)

Wende Radikal Regel an:

a = a^{\frac{1}{2}}

= ((1 - cos^{2} (y))^{\frac{1}{2}})^{2}

Wende Exponentenregel an:

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

= (1 - cos^{2} (y))^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multipliziere Brüche:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Streiche die gemeinsamen Faktoren:

2

= 1

= 1 - cos^{2} (y)

= 0.7 5^{2} (1 - cos^{2} (y))

0.7 5^{2} = 0.5625

= 0.5625 (- cos^{2} (y) + 1)

Wende das Distributivgesetz an:

a (b - c) = ab - a c

a = 0.5625, b = 1, c = cos^{2} (y)

= 0.5625 \cdot 1 - 0.5625 cos^{2} (y)

= 1 \cdot 0.5625 - 0.5625 cos^{2} (y)

Multipliziere die Zahlen:

1 \cdot 0.5625 = 0.5625

= 0.5625 - 0.5625 cos^{2} (y)

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y)

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y)

Löse

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y) : u = 0.5625 cos^{2} (y) + 0.4375, u = - 0.5625 cos^{2} (y) + 0.4375

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y)

Verschiebe

1

auf die rechte Seite

1 - u^{2} = 0.5625 - 0.5625 cos^{2} (y)

Subtrahiere

1

von beiden Seiten

1 - u^{2} - 1 = 0.5625 - 0.5625 cos^{2} (y) - 1

Vereinfache

- u^{2} = - 0.5625 cos^{2} (y) - 0.4375

- u^{2} = - 0.5625 cos^{2} (y) - 0.4375

Teile beide Seiten durch

- 1

- u^{2} = - 0.5625 cos^{2} (y) - 0.4375

Teile beide Seiten durch

- 1

\frac{- u ^{2}}{- 1} = - \frac{0.5625 cos ^{2} ( y )}{- 1} - \frac{0.4375}{- 1}

Vereinfache

\frac{- u ^{2}}{- 1} = - \frac{0.5625 cos ^{2} ( y )}{- 1} - \frac{0.4375}{- 1}

Vereinfache

\frac{- u ^{2}}{- 1} : u^{2}

\frac{- u ^{2}}{- 1}

Wende Bruchregel an:

\frac{- a}{- b} = \frac{a}{b}

= \frac{u ^{2}}{1}

Wende Regel an

\frac{a}{1} = a

= u^{2}

Vereinfache

- \frac{0.5625 cos ^{2} ( y )}{- 1} - \frac{0.4375}{- 1} : 0.5625 cos^{2} (y) + 0.4375

- \frac{0.5625 cos ^{2} ( y )}{- 1} - \frac{0.4375}{- 1}

Wende Regel an

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 0.5625 cos ^{2} ( y ) - 0.4375}{- 1}

Wende Bruchregel an:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{- 0.5625 cos ^{2} ( y ) - 0.4375}{1}

Wende Regel an

\frac{a}{1} = a

= - (- 0.5625 cos^{2} (y) - 0.4375)

Setze Klammern

= - (- 0.5625 cos^{2} (y)) - (- 0.4375)

Wende Minus-Plus Regeln an

- (- a) = a

= 0.5625 cos^{2} (y) + 0.4375

u^{2} = 0.5625 cos^{2} (y) + 0.4375

u^{2} = 0.5625 cos^{2} (y) + 0.4375

u^{2} = 0.5625 cos^{2} (y) + 0.4375

Für

x^{2} = f (a)

sind die Lösungen

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

u = 0.5625 cos^{2} (y) + 0.4375, u = - 0.5625 cos^{2} (y) + 0.4375

u = 0.5625 cos^{2} (y) + 0.4375, u = - 0.5625 cos^{2} (y) + 0.4375

Überprüfe die Lösungen:

u = 0.5625 cos^{2} (y) + 0.4375

Wahr

, u = - 0.5625 cos^{2} (y) + 0.4375

Wahr

Überprüfe die Lösungen, in dem die sie in

1 - u^{2} = 0.75 1 - cos^{2} (y)

einsetzt und entferne die Lösungen, die mit der Gleichung nicht übereinstimmen.

Setze ein

u = 0.5625 cos^{2} (y) + 0.4375 :

Wahr

1 - (0.5625 cos^{2} (y) + 0.4375)^{2} = 0.75 1 - cos^{2} (y)

Vereinfache

1 - (0.5625 cos^{2} (y) + 0.4375)^{2} : - 0.5625 cos^{2} (y) + 0.5625

1 - (0.5625 cos^{2} (y) + 0.4375)^{2}

(0.5625 cos^{2} (y) + 0.4375)^{2} = 0.5625 cos^{2} (y) + 0.4375

(0.5625 cos^{2} (y) + 0.4375)^{2}

Wende Radikal Regel an:

a = a^{\frac{1}{2}}

= ((0.5625 cos^{2} (y) + 0.4375)^{\frac{1}{2}})^{2}

Wende Exponentenregel an:

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

= (0.5625 cos^{2} (y) + 0.4375)^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multipliziere Brüche:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Streiche die gemeinsamen Faktoren:

2

= 1

= 0.5625 cos^{2} (y) + 0.4375

= 1 - (0.5625 cos^{2} (y) + 0.4375)

Multipliziere aus

1 - (0.5625 cos^{2} (y) + 0.4375) : - 0.5625 cos^{2} (y) + 0.5625

1 - (0.5625 cos^{2} (y) + 0.4375)

- (0.5625 cos^{2} (y) + 0.4375) : - 0.5625 cos^{2} (y) - 0.4375

- (0.5625 cos^{2} (y) + 0.4375)

Setze Klammern

= - (0.5625 cos^{2} (y)) - (0.4375)

Wende Minus-Plus Regeln an

+ (- a) = - a

= - 0.5625 cos^{2} (y) - 0.4375

= 1 - 0.5625 cos^{2} (y) - 0.4375

Subtrahiere die Zahlen:

1 - 0.4375 = 0.5625

= - 0.5625 cos^{2} (y) + 0.5625

= - 0.5625 cos^{2} (y) + 0.5625

- 0.5625 cos^{2} (y) + 0.5625 = 0.75 1 - cos^{2} (y)

Wahr

Setze ein

u = - 0.5625 cos^{2} (y) + 0.4375 :

Wahr

1 - (- 0.5625 cos^{2} (y) + 0.4375)^{2} = 0.75 1 - cos^{2} (y)

Vereinfache

1 - (- 0.5625 cos^{2} (y) + 0.4375)^{2} : - 0.5625 cos^{2} (y) + 0.5625

1 - (- 0.5625 cos^{2} (y) + 0.4375)^{2}

(- 0.5625 cos^{2} (y) + 0.4375)^{2} = 0.5625 cos^{2} (y) + 0.4375

(- 0.5625 cos^{2} (y) + 0.4375)^{2}

Wende Exponentenregel an:

(- a)^{n} = a^{n},

wenn

n

gerade ist

(- 0.5625 cos^{2} (y) + 0.4375)^{2} = (0.5625 cos^{2} (y) + 0.4375)^{2}

= (0.5625 cos^{2} (y) + 0.4375)^{2}

Wende Radikal Regel an:

a = a^{\frac{1}{2}}

= ((0.5625 cos^{2} (y) + 0.4375)^{\frac{1}{2}})^{2}

Wende Exponentenregel an:

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

= (0.5625 cos^{2} (y) + 0.4375)^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multipliziere Brüche:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Streiche die gemeinsamen Faktoren:

2

= 1

= 0.5625 cos^{2} (y) + 0.4375

= 1 - (0.5625 cos^{2} (y) + 0.4375)

Multipliziere aus

1 - (0.5625 cos^{2} (y) + 0.4375) : - 0.5625 cos^{2} (y) + 0.5625

1 - (0.5625 cos^{2} (y) + 0.4375)

- (0.5625 cos^{2} (y) + 0.4375) : - 0.5625 cos^{2} (y) - 0.4375

- (0.5625 cos^{2} (y) + 0.4375)

Setze Klammern

= - (0.5625 cos^{2} (y)) - (0.4375)

Wende Minus-Plus Regeln an

+ (- a) = - a

= - 0.5625 cos^{2} (y) - 0.4375

= 1 - 0.5625 cos^{2} (y) - 0.4375

Subtrahiere die Zahlen:

1 - 0.4375 = 0.5625

= - 0.5625 cos^{2} (y) + 0.5625

= - 0.5625 cos^{2} (y) + 0.5625

- 0.5625 cos^{2} (y) + 0.5625 = 0.75 1 - cos^{2} (y)

Wahr

Die Lösungen sind

u = 0.5625 cos^{2} (y) + 0.4375, u = - 0.5625 cos^{2} (y) + 0.4375

Setze in

u = cos (x)

ein

cos (x) = 0.5625 cos^{2} (y) + 0.4375, cos (x) = - 0.5625 cos^{2} (y) + 0.4375

cos (x) = 0.5625 cos^{2} (y) + 0.4375, cos (x) = - 0.5625 cos^{2} (y) + 0.4375

cos (x) = 0.5625 cos^{2} (y) + 0.4375 : x = arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn

cos (x) = 0.5625 cos^{2} (y) + 0.4375

Wende die Eigenschaften der Trigonometrie an

cos (x) = 0.5625 cos^{2} (y) + 0.4375

Allgemeine Lösung für

cos (x) = 0.5625 cos^{2} (y) + 0.4375

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

x = arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn

x = arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn

cos (x) = - 0.5625 cos^{2} (y) + 0.4375 : x = arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn

cos (x) = - 0.5625 cos^{2} (y) + 0.4375

Wende die Eigenschaften der Trigonometrie an

cos (x) = - 0.5625 cos^{2} (y) + 0.4375

Allgemeine Lösung für

cos (x) = - 0.5625 cos^{2} (y) + 0.4375

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

x = arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn

x = arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn

Kombiniere alle Lösungen

x = arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn, x = - arccos (- 0.5625 cos^{2} (y) + 0.4375) + 2 πn

Graph

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