What is the general solution for sin^3(x)=-2/3 ?

The general solution for sin^3(x)=-2/3 is x=-1.06251…+2pin,x=pi+1.06251…+2pin

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sin^3(x)=-2/3

Solution

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

Solution

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

Degrees

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

Solution steps

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

Solve by substitution

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

Let:

sin (x) = u

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

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

For

x^{3} = f (a)

the solutions are

Apply radical rule:

n

is odd

Simplify

Apply radical rule:

n

is odd

Multiply fractions:

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

Apply radical rule:

assuming

a \geq 0, b \geq 0

Multiply

Multiply fractions:

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

Apply the fraction rule:

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

Apply radical rule:

Apply exponent rule:

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

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

Subtract the numbers:

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

Rationalize

Multiply by the conjugate

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

Apply exponent rule:

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

= 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}}

Combine the fractions

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

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 3^{1}

Apply rule

a^{1} = a

= 3

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

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

Multiply by the conjugate

Apply exponent rule:

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

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

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

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

Combine the fractions

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

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 2^{1}

Apply rule

a^{1} = a

= 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

Rewrite

in standard complex form:

Cancel

Factor

6 : 2 \cdot 3

Factor

6 = 2 \cdot 3

Cancel

Apply exponent rule:

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

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

Subtract the numbers:

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

Apply radical rule:

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

Apply exponent rule:

\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}}

Subtract the numbers:

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}}}

Apply radical rule:

Apply the fraction rule:

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

Remove parentheses:

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

Cancel

Apply radical rule:

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

Apply exponent rule:

\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}}}

Subtract the numbers:

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

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

Apply radical rule:

Multiply by the conjugate

Apply exponent rule:

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

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

Join

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

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

Since the denominators are equal, combine the fractions:

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 2^{1}

Apply rule

a^{1} = a

= 2

Multiply by the conjugate

Apply exponent rule:

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

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

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

Combine the fractions

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

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 2^{1}

Apply rule

a^{1} = a

= 2

Multiply by the conjugate

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

Apply exponent rule:

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

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

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

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

Combine the fractions

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

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 3^{1}

Apply rule

a^{1} = a

= 3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

Simplify

Apply radical rule:

n

is odd

Multiply fractions:

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

Apply radical rule:

assuming

a \geq 0, b \geq 0

Multiply

Multiply fractions:

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

Apply the fraction rule:

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

Apply radical rule:

Apply exponent rule:

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

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

Subtract the numbers:

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

Rationalize

Multiply by the conjugate

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

Apply exponent rule:

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

= 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}}

Combine the fractions

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

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 3^{1}

Apply rule

a^{1} = a

= 3

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

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

Multiply by the conjugate

Apply exponent rule:

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

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

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

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

Combine the fractions

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

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 2^{1}

Apply rule

a^{1} = a

= 2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

Rewrite

in standard complex form:

Cancel

Factor

6 : 2 \cdot 3

Factor

6 = 2 \cdot 3

Cancel

Apply exponent rule:

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

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

Subtract the numbers:

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

Apply radical rule:

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

Apply exponent rule:

\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}}

Subtract the numbers:

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}}}

Apply radical rule:

Apply the fraction rule:

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

Apply rule

- (- a) = a

Cancel

Apply radical rule:

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

Apply exponent rule:

\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}}}

Subtract the numbers:

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

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

Apply radical rule:

Multiply by the conjugate

Apply exponent rule:

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

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

Join

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

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

Since the denominators are equal, combine the fractions:

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 2^{1}

Apply rule

a^{1} = a

= 2

Multiply by the conjugate

Apply exponent rule:

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

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

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

Combine the fractions

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

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 2^{1}

Apply rule

a^{1} = a

= 2

Multiply by the conjugate

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

Apply exponent rule:

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

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

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

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

Combine the fractions

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

Apply rule

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

= \frac{2 + 1}{3}

Add the numbers:

2 + 1 = 3

= \frac{3}{3}

Apply rule

\frac{a}{a} = 1

= 1

= 3^{1}

Apply rule

a^{1} = a

= 3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

Substitute back

u = sin (x)

Apply trig inverse properties

General solutions for

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

No Solution

No Solution

No Solution

No Solution

Combine all the solutions

Show solutions in decimal form

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

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Frequently Asked Questions (FAQ)

What is the general solution for sin^3(x)=-2/3 ?
The general solution for sin^3(x)=-2/3 is x=-1.06251…+2pin,x=pi+1.06251…+2pin