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Beliebt Trigonometrie >

sin^3(x)=-2/3

Lösung

sin^{3} (x) = - \frac{2}{3}

Lösung

x = - 1.06251 \dots + 2 πn, x = π + 1.06251 \dots + 2 πn

Grad

x = - 60.87741 \dots^{\circ} + 36 0^{\circ} n, x = 240.87741 \dots^{\circ} + 36 0^{\circ} n

Schritte zur Lösung

sin^{3} (x) = - \frac{2}{3}

Löse mit Substitution

sin^{3} (x) = - \frac{2}{3}

Angenommen:

sin (x) = u

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

u^{3} = - \frac{2}{3} : u = - 3 \frac{2}{3}, u = \frac{3 ^{\frac{2}{3}} 3 2}{6} - i \frac{3 2 6 3}{2}, u = \frac{3 ^{\frac{2}{3}} 3 2}{6} + i \frac{3 2 6 3}{2}

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

Für

x^{3} = f (a)

sind die Lösungen

x = 3 f (a), 3 f (a) \frac{- 1 - 3 i}{2}, 3 f (a) \frac{- 1 + 3 i}{2}

u = 3 - \frac{2}{3}, u = 3 - \frac{2}{3} \frac{- 1 + 3 i}{2}, u = 3 - \frac{2}{3} \frac{- 1 - 3 i}{2}

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

3 - \frac{2}{3}

Wende Radikal Regel an:

n - a = - n a,

wenn

n

ungerade ist

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

= - 3 \frac{2}{3}

Vereinfache

3 - \frac{2}{3} \frac{- 1 + 3 i}{2} : \frac{3 ^{\frac{2}{3}} 3 2}{6} - i \frac{3 2 6 3}{2}

3 - \frac{2}{3} \frac{- 1 + 3 i}{2}

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

3 - \frac{2}{3}

Wende Radikal Regel an:

n - a = - n a,

wenn

n

ungerade ist

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

= - 3 \frac{2}{3}

= - \frac{- 1 + 3 i}{2} 3 \frac{2}{3}

Multipliziere Brüche:

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

= - \frac{( - 1 + 3 i ) 3 \frac{2}{3}}{2}

3 \frac{2}{3} = \frac{3 2}{3 3}

3 \frac{2}{3}

Wende Radikal Regel an:

n \frac{a}{b} = \frac{n a}{n b},

angenommen

a \geq 0, b \geq 0

= \frac{3 2}{3 3}

= - \frac{\frac{3 2}{3 3} ( - 1 + 3 i )}{2}

Multipliziere

(- 1 + 3 i) \frac{3 2}{3 3} : \frac{3 2 ( - 1 + 3 i )}{3 3}

(- 1 + 3 i) \frac{3 2}{3 3}

Multipliziere Brüche:

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

= \frac{3 2 ( - 1 + 3 i )}{3 3}

= - \frac{\frac{3 2 ( - 1 + 3 i )}{3 3}}{2}

Wende Bruchregel an:

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

= - \frac{3 2 ( - 1 + 3 i )}{2 3 3}

Wende Radikal Regel an:

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

32 = 2^{\frac{1}{3}}

= \frac{2 ^{\frac{1}{3}} ( - 1 + 3 i )}{2 3 3}

Wende Exponentenregel an:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{3}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{3}}}

= \frac{- 1 + 3 i}{3 3 \cdot 2 ^{- \frac{1}{3} + 1}}

Subtrahiere die Zahlen:

1 - \frac{1}{3} = \frac{2}{3}

= - \frac{- 1 + 3 i}{3 3 \cdot 2 ^{\frac{2}{3}}}

Rationalisiere

- \frac{- 1 + 3 i}{3 3 \cdot 2 ^{\frac{2}{3}}} : - \frac{3 ^{\frac{2}{3}} 3 2 ( - 1 + 3 i )}{6}

- \frac{- 1 + 3 i}{3 3 \cdot 2 ^{\frac{2}{3}}}

Multipliziere mit dem Konjugat

\frac{3 ^{\frac{2}{3}}}{3 ^{\frac{2}{3}}}

= - \frac{( - 1 + 3 i ) \cdot 3 ^{\frac{2}{3}}}{3 3 \cdot 2 ^{\frac{2}{3}} \cdot 3 ^{\frac{2}{3}}}

33 \cdot 2^{\frac{2}{3}} \cdot 3^{\frac{2}{3}} = 3 \cdot 2^{\frac{2}{3}}

33 \cdot 2^{\frac{2}{3}} \cdot 3^{\frac{2}{3}}

Wende Exponentenregel an:

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

3^{\frac{2}{3}} 33 = 3^{\frac{2}{3}} \cdot 3^{\frac{1}{3}} = 3^{\frac{2}{3} + \frac{1}{3}}

= 3^{\frac{2}{3} + \frac{1}{3}} \cdot 2^{\frac{2}{3}}

3^{\frac{2}{3} + \frac{1}{3}} = 3

3^{\frac{2}{3} + \frac{1}{3}}

Ziehe Brüche zusammen

\frac{2}{3} + \frac{1}{3} : 1

Wende Regel an

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

= \frac{2 + 1}{3}

Addiere die Zahlen:

2 + 1 = 3

= \frac{3}{3}

Wende Regel an

\frac{a}{a} = 1

= 1

= 3^{1}

Wende Regel an

a^{1} = a

= 3

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

= - \frac{3 ^{\frac{2}{3}} ( - 1 + 3 i )}{3 \cdot 2 ^{\frac{2}{3}}}

Multipliziere mit dem Konjugat

\frac{3 2}{3 2}

= - \frac{3 ^{\frac{2}{3}} ( - 1 + 3 i ) 3 2}{3 \cdot 2 ^{\frac{2}{3}} 3 2}

3 \cdot 2^{\frac{2}{3}} 32 = 6

3 \cdot 2^{\frac{2}{3}} 32

Wende Exponentenregel an:

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

2^{\frac{2}{3}} 32 = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{2}{3} + \frac{1}{3}}

= 3 \cdot 2^{\frac{2}{3} + \frac{1}{3}}

2^{\frac{2}{3} + \frac{1}{3}} = 2

2^{\frac{2}{3} + \frac{1}{3}}

Ziehe Brüche zusammen

\frac{2}{3} + \frac{1}{3} : 1

Wende Regel an

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

= \frac{2 + 1}{3}

Addiere die Zahlen:

2 + 1 = 3

= \frac{3}{3}

Wende Regel an

\frac{a}{a} = 1

= 1

= 2^{1}

Wende Regel an

a^{1} = a

= 2

= 3 \cdot 2

Multipliziere die Zahlen:

3 \cdot 2 = 6

= 6

= - \frac{3 ^{\frac{2}{3}} 3 2 ( - 1 + 3 i )}{6}

= - \frac{3 ^{\frac{2}{3}} 3 2 ( - 1 + 3 i )}{6}

Schreibe

- \frac{3 ^{\frac{2}{3}} 3 2 ( - 1 + 3 i )}{6}

in der Standard komplexen Form um:

\frac{3 2 \cdot 3 ^{\frac{2}{3}}}{6} - \frac{6 3 3 2}{2} i

- \frac{3 ^{\frac{2}{3}} 3 2 ( - 1 + 3 i )}{6}

Streiche

\frac{3 ^{\frac{2}{3}} 3 2 ( - 1 + 3 i )}{6} : \frac{- 1 + 3 i}{2 ^{\frac{2}{3}} 3 3}

\frac{3 ^{\frac{2}{3}} 3 2 ( - 1 + 3 i )}{6}

Faktorisiere

6 : 2 \cdot 3

Faktorisiere

6 = 2 \cdot 3

= \frac{3 ^{\frac{2}{3}} 3 2 ( - 1 + 3 i )}{2 \cdot 3}

Streiche

\frac{3 ^{\frac{2}{3}} 3 2 ( - 1 + 3 i )}{2 \cdot 3} : \frac{- 1 + 3 i}{2 ^{\frac{2}{3}} \cdot 3 ^{\frac{1}{3}}}

\frac{3 ^{\frac{2}{3}} 3 2 ( - 1 + 3 i )}{2 \cdot 3}

Wende Exponentenregel an:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{3 ^{\frac{2}{3}}}{3} = \frac{1}{3 ^{1 - \frac{2}{3}}}

= \frac{3 2 ( - 1 + 3 i )}{2 \cdot 3 ^{- \frac{2}{3} + 1}}

Subtrahiere die Zahlen:

1 - \frac{2}{3} = \frac{1}{3}

= \frac{3 2 ( - 1 + 3 i )}{2 \cdot 3 ^{\frac{1}{3}}}

Wende Radikal Regel an:

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

32 = 2^{\frac{1}{3}}

= \frac{2 ^{\frac{1}{3}} ( - 1 + 3 i )}{2 \cdot 3 ^{\frac{1}{3}}}

Wende Exponentenregel an:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{3}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{3}}}

= \frac{- 1 + 3 i}{3 ^{\frac{1}{3}} \cdot 2 ^{- \frac{1}{3} + 1}}

Subtrahiere die Zahlen:

1 - \frac{1}{3} = \frac{2}{3}

= \frac{- 1 + 3 i}{2 ^{\frac{2}{3}} \cdot 3 ^{\frac{1}{3}}}

= \frac{- 1 + 3 i}{2 ^{\frac{2}{3}} \cdot 3 ^{\frac{1}{3}}}

3^{\frac{1}{3}} = 33

Wende Radikal Regel an:

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

3^{\frac{1}{3}} = 33

= \frac{- 1 + 3 i}{2 ^{\frac{2}{3}} 3 3}

= - \frac{- 1 + 3 i}{2 ^{\frac{2}{3}} 3 3}

= - \frac{- 1 + 3 i}{2 ^{\frac{2}{3}} 3 3}

Wende Bruchregel an:

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

\frac{- 1 + 3 i}{2 ^{\frac{2}{3}} 3 3} = - (- \frac{1}{2 ^{\frac{2}{3}} 3 3}) - (\frac{3 i}{2 ^{\frac{2}{3}} 3 3})

= - (- \frac{1}{2 ^{\frac{2}{3}} 3 3}) - (\frac{3 i}{2 ^{\frac{2}{3}} 3 3})

Entferne die Klammern:

(a) = a, - (- a) = a

= \frac{1}{2 ^{\frac{2}{3}} 3 3} - \frac{3 i}{2 ^{\frac{2}{3}} 3 3}

Streiche

\frac{3 i}{2 ^{\frac{2}{3}} 3 3} : \frac{6 3 i}{2 ^{\frac{2}{3}}}

\frac{3 i}{2 ^{\frac{2}{3}} 3 3}

Streiche

\frac{3 i}{2 ^{\frac{2}{3}} 3 3} : \frac{6 3 i}{2 ^{\frac{2}{3}}}

\frac{3 i}{2 ^{\frac{2}{3}} 3 3}

Wende Radikal Regel an:

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

33 = 3^{\frac{1}{3}}, 3 = 3^{\frac{1}{2}}

= \frac{3 ^{\frac{1}{2}} i}{2 ^{\frac{2}{3}} \cdot 3 ^{\frac{1}{3}}}

Wende Exponentenregel an:

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

\frac{3 ^{\frac{1}{2}}}{3 ^{\frac{1}{3}}} = 3^{\frac{1}{2} - \frac{1}{3}}

= \frac{3 ^{\frac{1}{2} - \frac{1}{3}} i}{2 ^{\frac{2}{3}}}

Subtrahiere die Zahlen:

\frac{1}{2} - \frac{1}{3} = \frac{1}{6}

= \frac{3 ^{\frac{1}{6}} i}{2 ^{\frac{2}{3}}}

Wende Radikal Regel an:

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

3^{\frac{1}{6}} = 63

= \frac{6 3 i}{2 ^{\frac{2}{3}}}

= \frac{6 3 i}{2 ^{\frac{2}{3}}}

= \frac{1}{2 ^{\frac{2}{3}} 3 3} - \frac{6 3 i}{2 ^{\frac{2}{3}}}

- \frac{6 3}{2 ^{\frac{2}{3}}} = - \frac{6 3 3 2}{2}

- \frac{6 3}{2 ^{\frac{2}{3}}}

Multipliziere mit dem Konjugat

\frac{3 2}{3 2}

= - \frac{6 3 3 2}{2 ^{\frac{2}{3}} 3 2}

2^{\frac{2}{3}} 32 = 2

2^{\frac{2}{3}} 32

Wende Exponentenregel an:

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

2^{\frac{2}{3}} 32 = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{2}{3} + \frac{1}{3}}

= 2^{\frac{2}{3} + \frac{1}{3}}

Füge

\frac{2}{3} + \frac{1}{3}

zusammen:

1

\frac{2}{3} + \frac{1}{3}

Da die Nenner gleich sind, fasse die Brüche zusammen.:

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

= \frac{2 + 1}{3}

Addiere die Zahlen:

2 + 1 = 3

= \frac{3}{3}

Wende Regel an

\frac{a}{a} = 1

= 1

= 2^{1}

Wende Regel an

a^{1} = a

= 2

= - \frac{6 3 3 2}{2}

= \frac{1}{2 ^{\frac{2}{3}} 3 3} - \frac{6 3 3 2}{2} i

\frac{1}{2 ^{\frac{2}{3}} 3 3} = \frac{3 2 \cdot 3 ^{\frac{2}{3}}}{6}

\frac{1}{2 ^{\frac{2}{3}} 3 3}

Multipliziere mit dem Konjugat

\frac{3 2}{3 2}

= \frac{1 \cdot 3 2}{2 ^{\frac{2}{3}} 3 3 3 2}

1 \cdot 32 = 32

2^{\frac{2}{3}} 3332 = 233

2^{\frac{2}{3}} 3332

Wende Exponentenregel an:

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

2^{\frac{2}{3}} 32 = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{2}{3} + \frac{1}{3}}

= 33 \cdot 2^{\frac{2}{3} + \frac{1}{3}}

2^{\frac{2}{3} + \frac{1}{3}} = 2

2^{\frac{2}{3} + \frac{1}{3}}

Ziehe Brüche zusammen

\frac{2}{3} + \frac{1}{3} : 1

Wende Regel an

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

= \frac{2 + 1}{3}

Addiere die Zahlen:

2 + 1 = 3

= \frac{3}{3}

Wende Regel an

\frac{a}{a} = 1

= 1

= 2^{1}

Wende Regel an

a^{1} = a

= 2

= 233

= \frac{3 2}{2 3 3}

Multipliziere mit dem Konjugat

\frac{3 ^{\frac{2}{3}}}{3 ^{\frac{2}{3}}}

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

233 \cdot 3^{\frac{2}{3}} = 6

233 \cdot 3^{\frac{2}{3}}

Wende Exponentenregel an:

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

3^{\frac{2}{3}} 33 = 3^{\frac{2}{3}} \cdot 3^{\frac{1}{3}} = 3^{\frac{2}{3} + \frac{1}{3}}

= 2 \cdot 3^{\frac{2}{3} + \frac{1}{3}}

3^{\frac{2}{3} + \frac{1}{3}} = 3

3^{\frac{2}{3} + \frac{1}{3}}

Ziehe Brüche zusammen

\frac{2}{3} + \frac{1}{3} : 1

Wende Regel an

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

= \frac{2 + 1}{3}

Addiere die Zahlen:

2 + 1 = 3

= \frac{3}{3}

Wende Regel an

\frac{a}{a} = 1

= 1

= 3^{1}

Wende Regel an

a^{1} = a

= 3

= 2 \cdot 3

Multipliziere die Zahlen:

2 \cdot 3 = 6

= 6

= \frac{3 2 \cdot 3 ^{\frac{2}{3}}}{6}

= \frac{3 2 \cdot 3 ^{\frac{2}{3}}}{6} - \frac{6 3 3 2}{2} i

= \frac{3 2 \cdot 3 ^{\frac{2}{3}}}{6} - \frac{6 3 3 2}{2} i

Vereinfache

3 - \frac{2}{3} \frac{- 1 - 3 i}{2} : \frac{3 ^{\frac{2}{3}} 3 2}{6} + i \frac{3 2 6 3}{2}

3 - \frac{2}{3} \frac{- 1 - 3 i}{2}

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

3 - \frac{2}{3}

Wende Radikal Regel an:

n - a = - n a,

wenn

n

ungerade ist

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

= - 3 \frac{2}{3}

= - \frac{- 1 - 3 i}{2} 3 \frac{2}{3}

Multipliziere Brüche:

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

= - \frac{( - 1 - 3 i ) 3 \frac{2}{3}}{2}

3 \frac{2}{3} = \frac{3 2}{3 3}

3 \frac{2}{3}

Wende Radikal Regel an:

n \frac{a}{b} = \frac{n a}{n b},

angenommen

a \geq 0, b \geq 0

= \frac{3 2}{3 3}

= - \frac{\frac{3 2}{3 3} ( - 1 - 3 i )}{2}

Multipliziere

(- 1 - 3 i) \frac{3 2}{3 3} : \frac{3 2 ( - 1 - 3 i )}{3 3}

(- 1 - 3 i) \frac{3 2}{3 3}

Multipliziere Brüche:

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

= \frac{3 2 ( - 1 - 3 i )}{3 3}

= - \frac{\frac{3 2 ( - 1 - 3 i )}{3 3}}{2}

Wende Bruchregel an:

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

= - \frac{3 2 ( - 1 - 3 i )}{2 3 3}

Wende Radikal Regel an:

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

32 = 2^{\frac{1}{3}}

= \frac{2 ^{\frac{1}{3}} ( - 1 - 3 i )}{2 3 3}

Wende Exponentenregel an:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{3}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{3}}}

= \frac{- 1 - 3 i}{3 3 \cdot 2 ^{- \frac{1}{3} + 1}}

Subtrahiere die Zahlen:

1 - \frac{1}{3} = \frac{2}{3}

= - \frac{- 1 - 3 i}{3 3 \cdot 2 ^{\frac{2}{3}}}

Rationalisiere

- \frac{- 1 - 3 i}{3 3 \cdot 2 ^{\frac{2}{3}}} : - \frac{3 ^{\frac{2}{3}} 3 2 ( - 1 - 3 i )}{6}

- \frac{- 1 - 3 i}{3 3 \cdot 2 ^{\frac{2}{3}}}

Multipliziere mit dem Konjugat

\frac{3 ^{\frac{2}{3}}}{3 ^{\frac{2}{3}}}

= - \frac{( - 1 - 3 i ) \cdot 3 ^{\frac{2}{3}}}{3 3 \cdot 2 ^{\frac{2}{3}} \cdot 3 ^{\frac{2}{3}}}

33 \cdot 2^{\frac{2}{3}} \cdot 3^{\frac{2}{3}} = 3 \cdot 2^{\frac{2}{3}}

33 \cdot 2^{\frac{2}{3}} \cdot 3^{\frac{2}{3}}

Wende Exponentenregel an:

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

3^{\frac{2}{3}} 33 = 3^{\frac{2}{3}} \cdot 3^{\frac{1}{3}} = 3^{\frac{2}{3} + \frac{1}{3}}

= 3^{\frac{2}{3} + \frac{1}{3}} \cdot 2^{\frac{2}{3}}

3^{\frac{2}{3} + \frac{1}{3}} = 3

3^{\frac{2}{3} + \frac{1}{3}}

Ziehe Brüche zusammen

\frac{2}{3} + \frac{1}{3} : 1

Wende Regel an

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

= \frac{2 + 1}{3}

Addiere die Zahlen:

2 + 1 = 3

= \frac{3}{3}

Wende Regel an

\frac{a}{a} = 1

= 1

= 3^{1}

Wende Regel an

a^{1} = a

= 3

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

= - \frac{3 ^{\frac{2}{3}} ( - 1 - 3 i )}{3 \cdot 2 ^{\frac{2}{3}}}

Multipliziere mit dem Konjugat

\frac{3 2}{3 2}

= - \frac{3 ^{\frac{2}{3}} ( - 1 - 3 i ) 3 2}{3 \cdot 2 ^{\frac{2}{3}} 3 2}

3 \cdot 2^{\frac{2}{3}} 32 = 6

3 \cdot 2^{\frac{2}{3}} 32

Wende Exponentenregel an:

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

2^{\frac{2}{3}} 32 = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{2}{3} + \frac{1}{3}}

= 3 \cdot 2^{\frac{2}{3} + \frac{1}{3}}

2^{\frac{2}{3} + \frac{1}{3}} = 2

2^{\frac{2}{3} + \frac{1}{3}}

Ziehe Brüche zusammen

\frac{2}{3} + \frac{1}{3} : 1

Wende Regel an

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

= \frac{2 + 1}{3}

Addiere die Zahlen:

2 + 1 = 3

= \frac{3}{3}

Wende Regel an

\frac{a}{a} = 1

= 1

= 2^{1}

Wende Regel an

a^{1} = a

= 2

= 3 \cdot 2

Multipliziere die Zahlen:

3 \cdot 2 = 6

= 6

= - \frac{3 ^{\frac{2}{3}} 3 2 ( - 1 - 3 i )}{6}

= - \frac{3 ^{\frac{2}{3}} 3 2 ( - 1 - 3 i )}{6}

Schreibe

- \frac{3 ^{\frac{2}{3}} 3 2 ( - 1 - 3 i )}{6}

in der Standard komplexen Form um:

\frac{3 2 \cdot 3 ^{\frac{2}{3}}}{6} + \frac{6 3 3 2}{2} i

- \frac{3 ^{\frac{2}{3}} 3 2 ( - 1 - 3 i )}{6}

Streiche

\frac{3 ^{\frac{2}{3}} 3 2 ( - 1 - 3 i )}{6} : \frac{- 1 - 3 i}{2 ^{\frac{2}{3}} 3 3}

\frac{3 ^{\frac{2}{3}} 3 2 ( - 1 - 3 i )}{6}

Faktorisiere

6 : 2 \cdot 3

Faktorisiere

6 = 2 \cdot 3

= \frac{3 ^{\frac{2}{3}} 3 2 ( - 1 - 3 i )}{2 \cdot 3}

Streiche

\frac{3 ^{\frac{2}{3}} 3 2 ( - 1 - 3 i )}{2 \cdot 3} : \frac{- 1 - 3 i}{2 ^{\frac{2}{3}} \cdot 3 ^{\frac{1}{3}}}

\frac{3 ^{\frac{2}{3}} 3 2 ( - 1 - 3 i )}{2 \cdot 3}

Wende Exponentenregel an:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{3 ^{\frac{2}{3}}}{3} = \frac{1}{3 ^{1 - \frac{2}{3}}}

= \frac{3 2 ( - 1 - 3 i )}{2 \cdot 3 ^{- \frac{2}{3} + 1}}

Subtrahiere die Zahlen:

1 - \frac{2}{3} = \frac{1}{3}

= \frac{3 2 ( - 1 - 3 i )}{2 \cdot 3 ^{\frac{1}{3}}}

Wende Radikal Regel an:

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

32 = 2^{\frac{1}{3}}

= \frac{2 ^{\frac{1}{3}} ( - 1 - 3 i )}{2 \cdot 3 ^{\frac{1}{3}}}

Wende Exponentenregel an:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{3}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{3}}}

= \frac{- 1 - 3 i}{3 ^{\frac{1}{3}} \cdot 2 ^{- \frac{1}{3} + 1}}

Subtrahiere die Zahlen:

1 - \frac{1}{3} = \frac{2}{3}

= \frac{- 1 - 3 i}{2 ^{\frac{2}{3}} \cdot 3 ^{\frac{1}{3}}}

= \frac{- 1 - 3 i}{2 ^{\frac{2}{3}} \cdot 3 ^{\frac{1}{3}}}

3^{\frac{1}{3}} = 33

Wende Radikal Regel an:

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

3^{\frac{1}{3}} = 33

= \frac{- 1 - 3 i}{2 ^{\frac{2}{3}} 3 3}

= - \frac{- 1 - 3 i}{2 ^{\frac{2}{3}} 3 3}

= - \frac{- 1 - 3 i}{2 ^{\frac{2}{3}} 3 3}

Wende Bruchregel an:

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

\frac{- 1 - 3 i}{2 ^{\frac{2}{3}} 3 3} = - (- \frac{1}{2 ^{\frac{2}{3}} 3 3}) - (- \frac{3 i}{2 ^{\frac{2}{3}} 3 3})

= - (- \frac{1}{2 ^{\frac{2}{3}} 3 3}) - (- \frac{3 i}{2 ^{\frac{2}{3}} 3 3})

Wende Regel an

- (- a) = a

= \frac{1}{2 ^{\frac{2}{3}} 3 3} + \frac{3 i}{2 ^{\frac{2}{3}} 3 3}

Streiche

\frac{3 i}{2 ^{\frac{2}{3}} 3 3} : \frac{6 3 i}{2 ^{\frac{2}{3}}}

\frac{3 i}{2 ^{\frac{2}{3}} 3 3}

Streiche

\frac{3 i}{2 ^{\frac{2}{3}} 3 3} : \frac{6 3 i}{2 ^{\frac{2}{3}}}

\frac{3 i}{2 ^{\frac{2}{3}} 3 3}

Wende Radikal Regel an:

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

33 = 3^{\frac{1}{3}}, 3 = 3^{\frac{1}{2}}

= \frac{3 ^{\frac{1}{2}} i}{2 ^{\frac{2}{3}} \cdot 3 ^{\frac{1}{3}}}

Wende Exponentenregel an:

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

\frac{3 ^{\frac{1}{2}}}{3 ^{\frac{1}{3}}} = 3^{\frac{1}{2} - \frac{1}{3}}

= \frac{3 ^{\frac{1}{2} - \frac{1}{3}} i}{2 ^{\frac{2}{3}}}

Subtrahiere die Zahlen:

\frac{1}{2} - \frac{1}{3} = \frac{1}{6}

= \frac{3 ^{\frac{1}{6}} i}{2 ^{\frac{2}{3}}}

Wende Radikal Regel an:

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

3^{\frac{1}{6}} = 63

= \frac{6 3 i}{2 ^{\frac{2}{3}}}

= \frac{6 3 i}{2 ^{\frac{2}{3}}}

= \frac{1}{2 ^{\frac{2}{3}} 3 3} + \frac{6 3 i}{2 ^{\frac{2}{3}}}

\frac{6 3}{2 ^{\frac{2}{3}}} = \frac{6 3 3 2}{2}

\frac{6 3}{2 ^{\frac{2}{3}}}

Multipliziere mit dem Konjugat

\frac{3 2}{3 2}

= \frac{6 3 3 2}{2 ^{\frac{2}{3}} 3 2}

2^{\frac{2}{3}} 32 = 2

2^{\frac{2}{3}} 32

Wende Exponentenregel an:

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

2^{\frac{2}{3}} 32 = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{2}{3} + \frac{1}{3}}

= 2^{\frac{2}{3} + \frac{1}{3}}

Füge

\frac{2}{3} + \frac{1}{3}

zusammen:

1

\frac{2}{3} + \frac{1}{3}

Da die Nenner gleich sind, fasse die Brüche zusammen.:

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

= \frac{2 + 1}{3}

Addiere die Zahlen:

2 + 1 = 3

= \frac{3}{3}

Wende Regel an

\frac{a}{a} = 1

= 1

= 2^{1}

Wende Regel an

a^{1} = a

= 2

= \frac{6 3 3 2}{2}

= \frac{1}{2 ^{\frac{2}{3}} 3 3} + \frac{6 3 3 2}{2} i

\frac{1}{2 ^{\frac{2}{3}} 3 3} = \frac{3 2 \cdot 3 ^{\frac{2}{3}}}{6}

\frac{1}{2 ^{\frac{2}{3}} 3 3}

Multipliziere mit dem Konjugat

\frac{3 2}{3 2}

= \frac{1 \cdot 3 2}{2 ^{\frac{2}{3}} 3 3 3 2}

1 \cdot 32 = 32

2^{\frac{2}{3}} 3332 = 233

2^{\frac{2}{3}} 3332

Wende Exponentenregel an:

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

2^{\frac{2}{3}} 32 = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{2}{3} + \frac{1}{3}}

= 33 \cdot 2^{\frac{2}{3} + \frac{1}{3}}

2^{\frac{2}{3} + \frac{1}{3}} = 2

2^{\frac{2}{3} + \frac{1}{3}}

Ziehe Brüche zusammen

\frac{2}{3} + \frac{1}{3} : 1

Wende Regel an

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

= \frac{2 + 1}{3}

Addiere die Zahlen:

2 + 1 = 3

= \frac{3}{3}

Wende Regel an

\frac{a}{a} = 1

= 1

= 2^{1}

Wende Regel an

a^{1} = a

= 2

= 233

= \frac{3 2}{2 3 3}

Multipliziere mit dem Konjugat

\frac{3 ^{\frac{2}{3}}}{3 ^{\frac{2}{3}}}

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

233 \cdot 3^{\frac{2}{3}} = 6

233 \cdot 3^{\frac{2}{3}}

Wende Exponentenregel an:

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

3^{\frac{2}{3}} 33 = 3^{\frac{2}{3}} \cdot 3^{\frac{1}{3}} = 3^{\frac{2}{3} + \frac{1}{3}}

= 2 \cdot 3^{\frac{2}{3} + \frac{1}{3}}

3^{\frac{2}{3} + \frac{1}{3}} = 3

3^{\frac{2}{3} + \frac{1}{3}}

Ziehe Brüche zusammen

\frac{2}{3} + \frac{1}{3} : 1

Wende Regel an

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

= \frac{2 + 1}{3}

Addiere die Zahlen:

2 + 1 = 3

= \frac{3}{3}

Wende Regel an

\frac{a}{a} = 1

= 1

= 3^{1}

Wende Regel an

a^{1} = a

= 3

= 2 \cdot 3

Multipliziere die Zahlen:

2 \cdot 3 = 6

= 6

= \frac{3 2 \cdot 3 ^{\frac{2}{3}}}{6}

= \frac{3 2 \cdot 3 ^{\frac{2}{3}}}{6} + \frac{6 3 3 2}{2} i

= \frac{3 2 \cdot 3 ^{\frac{2}{3}}}{6} + \frac{6 3 3 2}{2} i

u = - 3 \frac{2}{3}, u = \frac{3 ^{\frac{2}{3}} 3 2}{6} - i \frac{3 2 6 3}{2}, u = \frac{3 ^{\frac{2}{3}} 3 2}{6} + i \frac{3 2 6 3}{2}

Setze in

u = sin (x)

ein

sin (x) = - 3 \frac{2}{3}, sin (x) = \frac{3 ^{\frac{2}{3}} 3 2}{6} - i \frac{3 2 6 3}{2}, sin (x) = \frac{3 ^{\frac{2}{3}} 3 2}{6} + i \frac{3 2 6 3}{2}

sin (x) = - 3 \frac{2}{3}, sin (x) = \frac{3 ^{\frac{2}{3}} 3 2}{6} - i \frac{3 2 6 3}{2}, sin (x) = \frac{3 ^{\frac{2}{3}} 3 2}{6} + i \frac{3 2 6 3}{2}

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

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

Wende die Eigenschaften der Trigonometrie an

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

Allgemeine Lösung für

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

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

x = arcsin (- 3 \frac{2}{3}) + 2 πn, x = π + arcsin (3 \frac{2}{3}) + 2 πn

x = arcsin (- 3 \frac{2}{3}) + 2 πn, x = π + arcsin (3 \frac{2}{3}) + 2 πn

sin (x) = \frac{3 ^{\frac{2}{3}} 3 2}{6} - i \frac{3 2 6 3}{2} :

Keine Lösung

sin (x) = \frac{3 ^{\frac{2}{3}} 3 2}{6} - i \frac{3 2 6 3}{2}

Keine L \overset{o}{¨} sung

sin (x) = \frac{3 ^{\frac{2}{3}} 3 2}{6} + i \frac{3 2 6 3}{2} :

Keine Lösung

sin (x) = \frac{3 ^{\frac{2}{3}} 3 2}{6} + i \frac{3 2 6 3}{2}

Keine L \overset{o}{¨} sung

Kombiniere alle Lösungen

x = arcsin (- 3 \frac{2}{3}) + 2 πn, x = π + arcsin (3 \frac{2}{3}) + 2 πn

Zeige Lösungen in Dezimalform

x = - 1.06251 \dots + 2 πn, x = π + 1.06251 \dots + 2 πn

Graph

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